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Marginal deformations and open string field theory Longton, Matheson Edward 2015

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Marginal Deformations and OpenString Field TheorybyMatheson Edward LongtonB.Sc., The University of Victoria, 2004M.Sc., The University of British Columbia, 2007A THESIS SUBMITTED IN PARTIAL FULFILLMENT OFTHE REQUIREMENTS FOR THE DEGREE OFDOCTOR OF PHILOSOPHYinThe Faculty of Graduate and Postdoctoral Studies(Physics)THE UNIVERSITY OF BRITISH COLUMBIA(Vancouver)August 2015c© Matheson Edward Longton 2015AbstractThe study of solutions to open string field theory remains very much a work in progress, evenfor the bosonic string. In this dissertation I consider in detail two of these solutions involvingmarginal deformations of the original boundary conformal field theory. The first is a previouslyunknown solution in which two D-branes are translated before tachyon condensation occurs.This solution is studied in the level truncation scheme, in a sector which is larger than theuniversal subspace, but still less than the whole string Fock space due to several symmetriesof the theory which take on a different content in the presence of two D-branes. This solutionbrings us a step closer to a full understanding of the relationship between the magnitude ofa marginal deformation in BCFT and the strength of the corresponding marginal operator inOSFT. The other solution I study was first written down formally by Kiermaier and Okawa, andinvolves the renormalization of an exactly marginal operator. I consider the same solution witha more general renormalization scheme and find a set of sufficient restrictions for the solution’svalidity. While this proceeds much as in the original work on this solution, I find some freedomin the solution as well as additional algebraic structure for renormalization schemes. I alsopresent a collection of procedures written in Maple which define and manipulate wedge stateswith insertions, as well as computing correlation functions for such states provided that allinserted operators are sufficiently simple. Using this code I am able to calculate the tachyonprofile of this solution for the time-symmetric rolling tachyon at 6th order in λ and describe itsproperties in comparison to previously known rolling tachyon profiles. I find the same unwantedoscillations that were seen in previous work on the time-asymmetric rolling tachyon.iiPrefaceThe research and calculations in this dissertation were carried out by Matheson Longton underthe supervision of Dr. Joanna Karczmarek. While Dr. Karczmarek often provided directionand conceptual assistance, nearly all of the work and writing is my own. With the exception ofnumerical integration routines noted in section 5.3.7, all digital computations described wereperformed by programs I wrote, primarily in the Maple language.Versions of chapters 3 and 4 have been published in the Journal of High Energy Physics as[1] and [2] respectively. These papers were based on the work I have done for this dissertation,and were cowritten starting from drafts of these chapters rather than the other way around.The work of chapter 3 was also presented at the conference String Field Theory and RelatedAspects V, which took place in Jerusalem in 2012.A paper containing much of chapter 5 has appeared on the arXiv [3] and has now beenaccepted for publication by the Journal of High Energy Physics. Appendix A has also appearedon the arXiv as [4], but was never intended for peer reviewed publication.iiiTable of ContentsAbstract . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . iiPreface . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . iiiTable of Contents . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ivList of Tables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . viList of Figures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . viiList of Programs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . viiiGlossary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ix1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12 Background . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52.1 String Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52.2 String Field Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82.2.1 Level Truncation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 112.2.2 Analytic Solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 153 Separated D-branes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 223.1 Preliminaries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 223.1.1 Symmetries and the String Field . . . . . . . . . . . . . . . . . . . . . . . 233.1.2 The Potential . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 273.2 Solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 313.2.1 Level 0 Calculation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 323.2.2 Level 1 Calculation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 323.2.3 Level 3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 363.2.4 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 363.2.5 Comparison to Previous Solutions . . . . . . . . . . . . . . . . . . . . . . 413.2.6 Restoration of SU(2) Symmetry . . . . . . . . . . . . . . . . . . . . . . . 424 Renormalized Marginal Operators . . . . . . . . . . . . . . . . . . . . . . . . . . 454.1 Setup . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 464.2 Quadratic Operators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 504.2.1 The “Little g” Scheme . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52ivTable of Contents4.2.2 Small Integrated Operators . . . . . . . . . . . . . . . . . . . . . . . . . . 534.2.3 Linearity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 544.2.4 Assumptions (4.5d), (4.5e), and (4.5f) . . . . . . . . . . . . . . . . . . . . 614.2.5 Assumptions (4.5a) and (4.5b) . . . . . . . . . . . . . . . . . . . . . . . . 624.3 Renormalizing Higher Order Operators . . . . . . . . . . . . . . . . . . . . . . . 654.3.1 Third Order Operators . . . . . . . . . . . . . . . . . . . . . . . . . . . . 654.3.2 Extension to All Orders . . . . . . . . . . . . . . . . . . . . . . . . . . . . 674.3.3 Alternative Little g Schemes . . . . . . . . . . . . . . . . . . . . . . . . . 714.3.4 Assumptions (4.5c), (4.5d), (4.5e), and (4.5f) . . . . . . . . . . . . . . . . 724.3.5 Comparison to Kiermaier and Okawa . . . . . . . . . . . . . . . . . . . . 744.3.6 Proof of the First BRST Condition (4.5a) . . . . . . . . . . . . . . . . . . 784.3.7 Proof of the Second BRST Condition (4.5b) . . . . . . . . . . . . . . . . 854.3.8 Linear Renormalization . . . . . . . . . . . . . . . . . . . . . . . . . . . . 904.4 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 944.4.1 Uniqueness . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 944.4.2 Boundary Condition Changing Operators . . . . . . . . . . . . . . . . . . 955 Rolling Tachyon . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 985.1 Rolling Tachyon Introduction and Conclusions . . . . . . . . . . . . . . . . . . . 985.2 The Tachyon Profile . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1015.2.1 Small λ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1025.2.2 Large λ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1065.3 Technical Details . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1105.3.1 Conventions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1125.3.2 Basic Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1125.3.3 Known Wedge States . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1135.3.4 The BRST Operator . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1155.3.5 Expectation Values . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1175.3.6 Handling and Exporting Integrands . . . . . . . . . . . . . . . . . . . . . 1185.3.7 Numerical Integration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1196 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 130Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 132Appendices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .A Action For Separated D-branes . . . . . . . . . . . . . . . . . . . . . . . . . . . . 136A.1 Level (3,9) Potential . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 136B Rolling Tachyon Code . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 172B.1 Maple Code . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 172B.2 Sample C++ Program . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 208vList of Tables3.1 The first few Neumann coefficients appearing in (3.20b) . . . . . . . . . . . . . . 283.2 Values of maximum marginal parameters taken from four different cases . . . . . 414.1 How to relabel the integration variables in (4.123) . . . . . . . . . . . . . . . . . 795.1 Deterministic results for the non-zero coefficients of the tachyon profile . . . . . . 1035.2 Suave results for the non-zero coefficients of the tachyon profile . . . . . . . . . . 1045.3 Comparison of rolling tachyon profile for three previously calculated solutions . . 1055.4 Numerical results for consistency checks which require further analysis . . . . . . 1265.5 Three consistency checks shown with several different sample sizes . . . . . . . . 1275.6 Deterministic tests of the equation of motion for the rolling tachyon . . . . . . . 1285.7 Deterministic evaluation of the action for the rolling tachyon . . . . . . . . . . . 129A.1 Comparison of field definitions with two other works . . . . . . . . . . . . . . . . 138A.2 Quadratic coefficients in the action up to level 3 . . . . . . . . . . . . . . . . . . 138A.3 Cubic couplings in the action up to level 3 . . . . . . . . . . . . . . . . . . . . . . 139viList of Figures2.1 The potential at level 0 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 122.2 The Riemann surfaces used in two common conformal frames . . . . . . . . . . . 162.3 The marginal solution of [5] . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 193.1 The level 0 solution for separated D-branes . . . . . . . . . . . . . . . . . . . . . 333.2 The level 1 solution for separated D-branes . . . . . . . . . . . . . . . . . . . . . 343.3 Potential of the off-diagonal solution at level 3 . . . . . . . . . . . . . . . . . . . 373.4 Tachyon and marginal field of the off-diagonal solution at level 3 . . . . . . . . . 373.5 The remaining components of the off-diagonal solution at level 3 . . . . . . . . . 383.6 Slope of the field Xa for small separations . . . . . . . . . . . . . . . . . . . . . . 383.7 Masses of several particles in an effective theory expanded about a marginalstring field . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 444.1 Comparison of two choices of integration region used for the renormalization ofintegrated operators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 775.1 The tachyon profile with only j = 0 coefficients . . . . . . . . . . . . . . . . . . . 1065.2 The time of the first zero of the rolling tachyon profile . . . . . . . . . . . . . . . 1075.3 The falloff of the j = 0 coefficients of the rolling tachyon profile . . . . . . . . . . 1085.4 Several attempts to determine a trend for the j > 0 coefficients in the rollingtachyon profile . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1095.5 The tachyon profile of the time-symmetric rolling tachyon solution . . . . . . . . 1115.6 Plots showing the reliability of error estimates . . . . . . . . . . . . . . . . . . . . 1215.7 Several integrals from the tachyon profile calculated with different values of N . 1225.8 Several integrals from the consistency checks calculated with different value of N 123viiList of ProgramsB.1 Maple procedures for wedge states with insertions . . . . . . . . . . . . . . . . . 173B.2 Sample C++ command lines as output by Maple . . . . . . . . . . . . . . . . . . 208B.3 Sample C++ functions before they are edited for use . . . . . . . . . . . . . . . . 209B.4 Sample C++ header file with function definitions . . . . . . . . . . . . . . . . . . 210B.5 The C++ function to be integrated in order to find β(0)3 . . . . . . . . . . . . . . . 211B.6 Sample C++ program for the numerical evaluation of integrals . . . . . . . . . . 212viiiGlossaryAcronymsbcc : A “boundary condition changing” operator implements a change in conformal bound-ary conditions. A pair of bcc operators is commonly used to change the boundarycondition on a finite segment of the boundary.BCFT : Boundary Conformal Field Theory is a CFT on a surface with a boundary. Theboundary conditions imposed there must also be conformal.BRST : Named for Becchi, Rouet, Stora, and Tyutin, the BRST operator, QB, is used to definethe physical states of string theory.CFT : Conformal Field Theory.OPE : The Operator Product Expansion. A pair of operators can be expanded as φi(0)φj(z) =∑k Ckij(z)φk(0) provided there are no other operators closer to the pair than each other.OSFT : Open String Field Theory.SFT : String Field Theory. Since we will be focused on open strings this will be used inter-changeably with OSFT.Notation: : Normal ordering as defined in Polchinski’s book [6]. This sorts raising and lower-ing operators, or equivalently removes singular parts of expressions with operatorinsertions.◦◦◦◦g Normal ordering using a specified subtraction, g. Occasionally g will be omitted,in which case the subtraction is 1(s1−s2)2 . Defined in section 4.3.2.[x, y] , {x, y} The commutator and anti-commutator.[. . .]r Renormalization of operators by a generic renormalization scheme assumed tosatisfy certain assumptions.[. . .]G Renormalization of operators by a “big G” scheme which regulates distances be-tween operators and then subtracts counterterms which depend on the regulator.ixNotation[. . .]g Renormalization of operators by a “little g” scheme which subtracts off regulator-independent functions which are to be integrated over the same (possibly regu-lated) domain as the operators they renormalize.◦ Function composition operator.† Hermitian conjugation, also sometimes referred to as h.c..‡ This conjugation is used to define the reality condition for a string field Ψ = Ψ‡,which ensures that the action will be real-valued. It is defined by A‡ = bpz−1A†.Unlike bpz and hermitian conjugation, this conjugation does not exchange braand ket states.〈 , 〉 While this is treated like an inner product for string field theory, it is really asymplectic bilinear form. It is used in the action to get a real number from apair of string fields.〈. . .〉S This is the CFT expectation value defined on the Riemann surface S. Often thesurface S will be Wn, the wedge state of circumference n+ 1.∗ The star-product of string field theory glues two worldsheets together, allowingfor string interactions.b c Floor operator. The notation bn/2c will often appear in the limit of sums, whenthe sum is over the number of pairs of operators to contract.α′ The fundamental scale of the theory is determined by this constant with di-mension length squared. The target space length scale is√α′. We will use theconvention α′ = 1 except where it is helpful to include it explicitly.αn, bn, cn Raising and lowering operators for the matter and ghost sectors of the theory.αn are matter operators, bn are anti-ghost operators, and cn are ghost operators.The operator α0 is neither a creation or annihilation operator, as the vacuum isan eigenvector.β(j)n Coefficients of the rolling tachyon profile. The tachyon profile of the rollingtachyon is defined by the series T (t) = 2∑∞n=0∑bn/2cj=0 λnβ(j)n cosh ((n− 2j)t).bpz The bpz operator is defined by bpz(z) = −1z , and has the effect of exchangingthe “in” and “out” states of a string.c(t) The ghost operator, which is commonly inserted on the boundary of a worldsheet.The cn operators are its modes.dˆ The physical distance separating two D-branes in chapter 3. d = dˆpi√2α′definessimple units for this distance.Γa,b (x, y, . . .) The region (a, b) minus any places where two or more of the coordinates (x, y, . . .)are within  of each other. This regulates the region (a, b) so that divergencesdue to operator collisions do not occur.xNotationh.c. Hermitian conjugation, also referred to with †.Ln Virasoro raising and lowering operators. These are modes of the energy-momentumtensor in some sector, which is normally indicated by a superscript.λ Commonly used to parameterize the strength of a marginal deformation. Specif-ically, in chapters 4 and 5 the marginal deformation V will always appear in thecombination λV .Ω The twist operator which reverses the orientation of the string, σ → pi − σ.QB The BRST operator QB = 12pii∮ (dz jb + dz˜ j˜B), where jB = cTm+: bc∂c : +32∂2cand Tm is the matter part of the energy tensor.V (t) Commonly used to refer to a marginal operator which is inserted on the boundaryof a worldsheet. Throughout most of chapters 4 and 5 this will have the self-OPEV (0)V (t) ∼ 1t2 .V (a, b) The marginal operator V (t) integrated between a and b:V (a, b)n =∫ ba dnt∏ni=1 V (ti).Xa The coefficient of the marginal term representing translation for the solution ofchapter 3 on separated D-branes.xiChapter 1IntroductionThe most basic objects in string theory are strings. These can be closed strings (loops) or openstrings (with ends), but in either case the worldsheet interpretation of string theory forms aconformal field theory, or CFT. These are two dimensional CFT’s, as they describe dynamicswithin the two dimensions of a string: length and time. In the case of closed strings, the CFTcan be taken on the plane R2, but for open strings the endpoints create a boundary for the space.The boundary conditions placed on the string at its endpoints must be conformal, resulting ina boundary conformal field theory, or BCFT. While for some time these boundary conditionswere viewed as nothing more than that, in [7] it was recognized that they could also be thoughtof in terms of D-branes. D-branes are objects which the ends of strings are “attached” to, andthey can have varying size, shape, dimension, energy, and even other properties such as charge.In this picture, the tachyon in the spectrum of the bosonic open string can be explained ascorresponding to the instability of the D-brane it is attached to. This is where open string fieldtheory is useful: while a given string theory only studies the strings allowed by a given D-braneconfiguration, a single string field theory can describe many different D-brane configurations.Classical solutions of open string field theory, or OSFT, describe the different conformalboundary conditions available for a given conformal field theory. Since in the language ofstring theory these boundary conditions are the D-brane configurations, finding these classicalsolutions is an important tool for understanding D-branes. If we have such a solution we knowthat an associated D-brane configuration is allowed, and we can easily find its energy. Inprinciple the structure of OSFT can also tell us about the configuration’s stability and anymoduli. Time-dependent solutions can even teach us about D-brane dynamics. Unfortunately,the known solutions to OSFT only begin to scratch the surface of the entire space of D-braneconfigurations. While recent developments describe a procedure for constructing a solutionfrom a desired boundary condition, the solutions which have been explicitly written down andstudied are a very small subset of all allowed D-brane configurations.In the first part of this dissertation, we find a new solution using the level truncationapproximation scheme. One solution which is well known describes the decay of an existingspace-filling D-brane. When N of these D-branes are present, the solution for D-brane decaygains an SU(N) symmetry, so linear combinations of the D-branes can decay. The solutionrepresenting D-brane decay also exists when the initial D-brane is not space-filling, and in thiscase we can still gain an SU(N) symmetry by duplicating the initial D-brane. In the caseof D-branes with non-zero codimension, however, we can break this symmetry by placing theD-branes at different spatial locations. A non-trivial linear combination of D-branes cannotdecay if the D-branes are not in the same place, because the result — half a D-brane in oneplace, and half somewhere else — would not be a valid D-brane configuration. We will examinethe set of solutions to the approximate theory in the case where the initial configuration is twoD-branes separated by a known distance which will parameterize our set of solutions.1Chapter 1. IntroductionAnother solution of OSFT which should exist describes marginal deformations of the initialBCFT. While such solutions have been constructed analytically in a number of cases, they do notexist as solutions in the level truncated approximate theory. The truncation raises the marginaldirection in the string field space, and only points which are local extrema will be solutionsto the approximate theory. In practice this means that only the trivial solution survives fromthe marginal set. One such marginal deformation is the massless string mode perpendicularto a D-brane which is not space-filling. This marginal deformation causes translation of theD-brane, and it is the one we study in the first part of this document.We have seen that when two D-branes are initially separated either one can decay, but linearcombinations can only decay when they are coincident. In this case, the enhanced symmetry atthe point where the two D-branes coincide allows for the solution with a marginal deformationand a decay to survive level truncation. Although level truncation breaks a continuous setof solutions to a few discrete ones, by searching for an SU(2) of solutions to the full theorywe know that the level truncated action should have some extrema on that set, and solutionswill survive. We find such a solution and study its properties, parameterized by the initialseparation of the D-branes.We take the first step towards a map between the OSFT marginal parameter and thephysical impact of the marginal deformation without performing CFT calculations. Previousmaps have involved calculating a quantity in OSFT for a marginal solution and comparing thesame quantity to a marginally deformed CFT, but we attempt to do this directly. We havetaken the first steps but require a better understanding of the field redefinition which takes placewhen OSFT is reexpanded about one of its solutions. Even without the full mapping betweenthe OSFT marginal vev and the physical translation, however, we do find evidence that thereis a maximum physical translation that can be achieved through marginal solutions. While ithas long been known that there is a maximum value of the marginal vev in level truncatedOSFT, this maximum physical effect is unexpected and casts doubt on earlier explanationsof the maximum vev, in which the value of the marginal parameter was bounded but thecorresponding deformation of the BCFT was not.In the second part of this dissertation we examine a previously known formal solution fora class of marginal deformations. These marginal deformations are the more difficult case, inwhich the OPE of two copies of the associated operator is singular: V (0)V (t) ∼ 1t2 + O(1).The specific marginal deformation we are interested in is the so-called “rolling tachyon”, whichlocalizes a D-brane in the time direction. There are two different versions of the rolling tachyon.The simpler version has regular self-OPE and is not our focus. That rolling tachyon solutionrepresents an unstable D-brane which exists in the infinite past and then decays at a finitetime. Our focus will be on the time-symmetric rolling tachyon which exists only for an intervalof time, decaying in the past and future. The marginal operator associated with such a decayis V (t) =√2 cosh(X0(t)α′).First we examine the mathematical framework necessary for that solution in detail. Wesearch for the most general renormalization scheme we can apply to properly cancel the singu-larities arising from the OPE and satisfy the assumptions necessary for a formal solution. Weconsider two different approaches to constructing general renormalization schemes, which wecall the “big G” and “little g” schemes. We rigorously prove that the trivially exponentiatedversion of the little g scheme gives a two parameter family of finite operators satisfying allof the necessary assumptions. The big G scheme fails to give finite operators when naively2Chapter 1. Introductionexponentiated, but it offers an attractive framework for the construction of the most generalrenormalization scheme allowed at any finite order. We also briefly examine the renormalizationscheme originally proposed for this solution and show that it is nearly identical to our little gscheme. It is clear that there are at least two free parameters in the renormalization, and likelyinfinitely many, so we ask what this implies for uniqueness of the solution. The free parameterswe have identified correspond to rescalings of the renormalized operators. Because of the waythe renormalized operators appear in the solution, such a rescaling changes the solution and weexpect that it corresponds to gauge transformations, but this is not proven.Returning our focus from general marginal deformations with singular self-OPE to the caseof the rolling tachyon, I have written a computer program to explicitly construct this solutionat any order in the marginal deformation parameter λ. The program also computes arbitrarytachyon correlation functions in the conformal frame we are using. As a result, in addition tothe formal solution the program can also produce the tachyon profile in the form of unevaluatedintegrals. At moderate order in λ these integrals require numerical evaluation using 3rd partyintegration routines. In addition to the tachyon profile of the solution, the program can alsocompute the equation of motion and take its correlation function with any other string fieldbuilt from insertions of tachyon modes. Testing that such quantities vanish gives us evidencethat the formal solution does satisfy the equation of motion when explicitly constructed andrenormalized. The Maple procedures are designed to be flexible enough to manipulate manywedge states, and can be used to find correlation functions for other wedge states involvingonly the c ghosts and simple marginal operators. Adding more operators, however, should befairly straightforward provided the correlators with each other and with the operators alreadyconsidered can be written down in a closed form.Examining the tachyon profile, we find that for small λ the solution looks very much like twocopies of the simple exponential rolling tachyon. In particular, the coefficients which control thebehaviour for small λ show the same asymptotic behaviour as the exponential rolling tachyonsolutions. When λ is increased, however, the rest of the coefficients must be included and thefree parameters in the renormalization scheme come into play. The tachyon profile is not agauge independent quantity so the fact that it depends on the parameters tells us very little,aside from the fact that the solution really is affected by those rescalings. Whether they aregenuine gauge transformations is not known. It is known that the period of the tachyon field’soscillations is gauge dependent, so it is interesting that that period changes quite suddenlyas λ is increased to the point where all of the coefficients are important. This suggests thata change in marginal parameter λ for the time-symmetric rolling tachyon has a similar effectto a change of gauge in the time-asymmetric case. Finally, the presence of an approximatelyconstant period for small λ tells us that the coefficients responsible for that part of the tachyonprofile are a convergent series for any appropriate λ. Including additional coefficients is mostnaturally done by considering “rows” with constant momentum deficit. While there is not agreat deal of data, it suggests that each row converges, but there is no suggestion yet thatthe sum of rows is a convergent series. In the small λ limit, we show that only one row ofcoefficients contributes to the tachyon profile, so there is no sum of rows. If this sum does notconverge for large λ it raises the interesting possibility that marginal deformations really dohave a maximum strength, as has been found in level truncation studies including our work onseparated D-branes.This document is structured as follows. Chapter 2 discusses some of the work that has3Chapter 1. Introductionpreviously been done in string field theory, and describes the approaches that we will use.Chapter 3 finds the level truncated solution representing a combination of translation anddecay for separated D-branes, and discusses the properties of that solution and difficultiesinvolved in writing down a precise correspondence between the marginal parameter Xa and thetranslation distance dˆ. The work on formal solutions for marginal deformations with singularself-OPE is split over chapters 4 and 5. Chapter 4 expands on the work of Kiermaier andOkawa by describing in detail the construction and properties of a renormalization schemecompatible with a solution of OSFT, and chapter 5 deals with the numerical calculations usingthe renormalization scheme we find. Appendix A contains the level truncated action usedfor separated D-branes in chapter 3. Appendix B details the actual Maple program used toconstruct the rolling tachyon solution and its tachyon profile, and gives sample C++ code fornumerically evaluating the resulting integrals.4Chapter 2BackgroundAn obvious question in the study of boundary conformal field theories is exactly which boundaryconditions preserve the conformal symmetry. We refer to the collection of all CFT’s with allsuch boundary conditions for each as the BCFT landscape. We will only be interested in two-dimensional BCFT’s, since these are equivalent to string theories. In this context the boundaryconditions are identified with the D-brane configuration on which the open strings are allowed topropagate. A background independent study of the BCFT landscape is therefore an importanttool in the study of D-branes and their dynamics.The landscape of boundary conformal field theories contains all possible D-brane configu-rations of string theory, so we use string field theory to study it. Open string field theory is aninteracting theory of off-shell open strings, with closed strings appearing only in loop diagrams.It can be formulated using any BCFT as a starting point, and classical solutions of the equa-tion of motion describe new BCFTs which the theory can in principle be re-expanded about.The energy of a solution is the energy of the new D-brane configuration relative to the initialone, and the cohomology of the modified BRST operator for a solution describes the physicalexcitations of strings allowed on the configuration. While it is not at all clear that all BCFTsshould exist as possible solutions, every D-brane configuration for the starting string theoryshould. In this context, OSFT is background independent. It may also be that there are exoticsolutions corresponding to D-branes for string theories with different field content, for examplefinding an open superstring theory beginning with only bosonic strings. Such solutions, if theyexist, have not yet been found. The most well known solutions represent either the decay of aD-brane or a marginal deformation of the boundary conditions. We will focus primarily on thelatter.2.1 String TheoryWe begin by describing the first quantized bosonic string theory. Once we have done that wecan construct OSFT by replacing each string mode with a classical field. Quantizing thosefields will result in a quantum string field theory, but that subject is well beyond the scope ofthis dissertation.A common parameterization of the open string has endpoints at <w = 0, pi and time runningin the imaginary direction, where w is the worldsheet coordinate. A more useful parameteri-zation is z = e−iw which has the string endpoints on the real axis, with time running radiallyoutward from the origin. The field content of bosonic string theory is a matter field for eachdimension of the target space and two anticommuting Fadeev-Popov ghost fields which are in-troduced in gauge-fixing to the conformal gauge. All of these fields live in the two-dimensional52.1. String Theoryspace of the worldsheet. The action isS =12piα′∫d2z ∂Xµ∂¯Xµ +1pi∫d2z b∂¯c . (2.1)Our convention will be to set the constant α′ = 1, and we will only write it where it is useful forexplicitly dimensionful quantities. If we take Neumann boundary conditions at the endpointsof the string (=z = 0), the solution to this system has the following mode expansionXµ(z, z¯) = xµ0 − iα′pµ ln(zz¯) + i√α′2∑n6=0αµnn(z−n + z¯−n)(2.2)andb(z) =∑nbnz−n−2, c(z) =∑ncnz−n+1. (2.3)The modes αµ−n with n ≥ 1, b−n with n ≥ 2, and c−n with n ≥ −1 are raising operators,while αµ0 is a constant proportional to pµ. All of the other modes are lowering operators, andannihilate the vacuum. The commutation relations are[αµm, ανn] = mδm+n,0ηµ,ν , {bm, cn} = δm+n,0 , {bm, bn} = {cm, cn} = 0 . (2.4)In addition, the holomorphic energy-momentum tensor T (z) has a Laurent expansion in termsof the Virasoro operatorsT (z) =∞∑m=−∞Lmzm+2, (2.5)where the Virasoro operators Lm obey the Virasoro algebra[Ltotalm , Ltotaln]= (m− n)Ltotalm+n +c12(m3 −m)δm+n,0 . (2.6)The central charge of the theory is set to c = 0 by choosing the matter content appropriately.Specifically, for the bosonic string we can accomplish this by insisting on having D = 26 for ourtarget space. The anti-holomorphic part of the energy-momentum tensor has its own Virasoromodes with an identical algebra, but since we are working with open strings, our Riemannsurface will always have a boundary and the boundary conditions mean that T (z) and its anti-holomorphic partner T˜ (z¯) are not independent quantities. In terms of the fundamental matteroscillators, the matter Virasoro modes areLn ={ ∑∞m=0 αµ−mαµm, m = 012∑∞m=−∞ αµn−mαµm, m 6= 0 ,(2.7)and the Virasoro operators in the ghost sector areL(g)n =∞∑m=−∞(2n−m) : bmcn−m :−δn,0 , (2.8)where : : represents the standard normal ordering.62.1. String TheoryThe vacuum state |0; 0〉 with zero momentum is SL(2,R) invariant, meaning in the stateoperator correspondence it is mapped to the identity operator in the upper half plane. Thisvacuum is related to those in the notation of Polchinski [6] byc1 |0; kµ〉 = |kµ〉m ⊗ |↓〉 , (2.9)where |kµ〉m is the matter vacuum with momentum kµ and |↓〉 is one of the two ghost vacua.The number of states in this theory is too large as we have not yet done anything to removenull states and those with negative norm. This is done by identifying physical states as thecohomology of the BRST operator. The BRST operator has the formQB =∑ncnL−n +∑n,mm− n2: cmcnb−m−n :−c0 . (2.10)Physical states are those satisfying QB |φ〉 = 0, but the nilpotency of QB means that statesof the form QB |χ〉 are null (orthogonal to all physical states including itself) as well as beingphysical. The null states are identified with zero and removed from the spectrum. The lightestphysical state is the tachyon c1 |0; k〉 with mass given by m2 = − 1α′ , and the first excited stateαµ−1c1 |0; kν〉 is massless. An infinite tower of more massive states is created by acting on thetachyonic state with matter oscillators αµ−n and certain combinations of the ghost oscillatorsb−n and c−n.The state-operator correspondence allows us to view this theory in a different way. Insteadof operators such as αµn, cn, bn, and Ln acting on states such as the vacuum |0; kµ〉, we canview it as the local operators Xµ(z, z¯), c(z), b(z), and the energy-momentum tensor T (z)inserted in a Riemann surface. The canonical choice of Riemann surface is the infinite stripwith 0 ≤ <w ≤ pi for an open string, as the string’s spatial coordinate is defined to run from0 to pi while the worldsheet time coordinate is clearly unbounded. This is a conformal fieldtheory, however, and as such we have the conformal transformations available to us. Localconformal transformations are the holomorphic coordinate transformations z′ = f(z). Theconformal transformation z = e−iw maps the infinite strip to the upper half plane =z ≥ 0,which is in many cases the simplest Riemann surface to consider for open strings. Slices ofconstant worldsheet time are mapped to semicircular curves of constant radius, with the timeincreasing radially outward. The string endpoints lie on the boundary =z = 0, and the infinitepast is mapped to the point z = 0. When changing conformal frames any correlation functionsin the new frame are given by〈φ1(z′1, z¯′1) . . . φn(z′n, z¯′n)〉=n∏i=1(dz′dz)−hiz′=z′i(dz¯′dz¯)−h¯iz¯′=z¯′i〈φ1(z1, z¯1) . . . φn(zn, z¯n)〉 . (2.11)The exponents hi and h¯i here are called the conformal weights of the operators φi and definehow the operators transform under conformal mappings. The conformal weights can also bedefined in terms of an operator’s scaling dimension and spin [6][8]. An n-point function will becomputed by inserting operators at n distinct locations on the Riemann surface. Incoming oroutgoing open string states correspond to operators inserted on the boundary, and closed stringstates correspond to operators inserted in the bulk. For the open strings we will be studying, wemust specify the locations of exactly three operator insertions in order to fix the three moduliof the disc.72.2. String Field Theory2.2 String Field TheoryWhile there were several attempts to formulate an off-shell theory of strings, it was Witten’smodel [9] that succeeded in providing the foundations of OSFT. His theory is based on a Z2graded algebra satisfying several axioms.deg(a ∗ b) = deg(a) + deg(b), Q(a ∗ b) = (Q(a)) ∗ b+ (−1)aa ∗Q(b), Q2 = 0 (2.12)The nilpotent derivation Q turns out to be the BRST operator. The resulting theory is basedon a δ-function interaction of half-strings. It has a non-commutative star product that gluesthe right half (σ > pi2 ) of one string to the left half (σ <pi2 ) of the other, effectively joining twoopen-string worldsheets together. The ∗-product provides the interactions, but we still need amap from the space of string fields to the complex numbers in order to construct an action.The integration operation is introduced for this purpose. It glues the two halves of a string toeach other, also using a δ-function glue. Although actual calculations are not performed thisway, for the matter sector these operations are defined in [10] as∫Ψdef=∫ ∏0≤σ≤pidXµ(σ)∏0≤τ≤pi2δ [Xµ(τ)−Xµ(pi − τ)] Ψ [Xµ(τ)] , (2.13a)(Ψ ∗ Φ) [Zµ(τ)]def=∫ ∏0≤τ˜≤pi2dY µ(τ˜)dXµ(pi − τ˜)×∏pi2≤τ≤piδ [Xµ(τ)− Y µ(pi − τ)] Ψ [Xµ(τ)] Φ [Y µ(τ)] ,(2.13b)whereZµ(τ) = Xµ(τ) for 0 ≤ τ ≤pi2, and Zµ(τ) = Y µ(τ) forpi2≤ τ ≤ pi. (2.13c)Given this, the action isS = −12∫Ψ ∗QΨ−g3∫Ψ ∗Ψ ∗Ψ , (2.14)where g is a coupling constant and Q is the BRST operator. The first term is a kinetic termwhich reproduces the free string theory spectrum due to its use of the same BRST operator,while the second term introduces cubic interactions. Since a single string field is made up ofall possible string modes at all different momenta, the interaction mixes all of these infinitelymany fields. This action is valid even for string fields which are not on the mass shell.Because the action is cubic, it does have the problem that the action is unbounded bothabove and below. Quantum mechanically it is not a sound theory to study, but the classicaltheory still has perfectly valid solutions at the action’s critical points. Superstring field theorydoes not have this problem, so quantum superstring field theory should be a well defined thingto study, but at present classical solutions are not fully understood. The cubic bosonic theory istherefore a useful thing to study since many solutions have a similar structure, but are simpler inthe bosonic case. It is suggested in [11] that it may be possible to add higher order terms to theaction, but such terms would also appear as a violation of the associativity of the star-product.82.2. String Field TheoryThe ghost number of a string field can be defined as the number of c ghosts minus thenumber of b antighosts acting on the vacuum. With this convention, the∫operation vanishesunless the ghost number of its argument is 3. For string fields of definite ghost number, thismeans that the action is only non-zero for ghost number one. All physical string fields suchas the tachyon and massless vector have ghost number 1, while some toy models involve stringfields of ghost number 0 for simplicity. The action (2.14) has a large gauge symmetryδΨ = QΛ + Ψ ∗ Λ− Λ ∗Ψ , (2.15)where Λ is any string field of ghost number 0. Since the BRST operator Q increases a stringfield’s ghost number by 1, and ghost number is also additive under the ∗-product, the ghostnumber is preserved under such transformations provided that the string field Ψ has ghostnumber 1 and the gauge field Λ has ghost number 0.The simplest way to think of the string field is as a sum over all possible open string states.Ψ =∫d26p∏µ∑Iµ,J,KφIµ,J,K(p)αµI bJcK |0; p〉 (2.16)Here the upper case indices I, J,K represent multiple oscillator indices for products of operators,so that a random example of a term in the sum would beφ(−2,−1),(−2),(0,1)(p)α25−2α25−1b−2c0c1 |0; p〉 .Each state in the open string Fock space comes with a spacetime field as its coefficient, but inmost practical situations we are only interested in translationally invariant solutions. In thiscase the coefficient fields φ(p) are only non-zero for p = 0 and can be treated as simple constantcoefficients. The action (2.14) then has the form of a cubic polynomial in the infinitely manycoefficients.Let us now turn to some common notation and computational tools used when the stringfield is expressed in the open string Fock space. The integration operator and triple star productused in the cubic term of the action are written as∫A ∗B = 〈A,B〉 , 〈A,B ∗ C〉 = 〈A,B,C〉 . (2.17)Absorbing a factor of the coupling constant into the string field, Φ = gΨ, the action becomesS = −1g2(12〈Ψ, QΨ〉+13〈Ψ,Ψ,Ψ〉). (2.18)The ∗-product of two string fields is not at all simple to write down in the basis (2.16). For-tunately, Gross and Jevicky [12] found a state in the product of three SFT Hilbert spaces suchthat|Φ ∗Ψ〉3 = 〈Φ|1 〈Ψ|2 |V3〉 . (2.19)Specifically, the cubic term in the action is easily written in terms of this 3-vertex.〈Ψ,Ψ,Ψ〉 = 〈V3| |Ψ〉1 |Ψ〉2 |Ψ〉3 (2.20)92.2. String Field TheorySwitching from ket states to bra states is done using bpz conjugation: 〈Ψ| = bpz |Ψ〉. In theFock space basis this is similar to hermitian conjugation with additional sign factors. It isdescribed in more detail in section 3.1.1 and the bpz conjugates of simple states are given inequation (3.9). The action’s kinetic term is most easily calculated directly in terms of oscillators.〈Ψ|Q |Ψ〉 (2.21)The calculation of both terms in the action is explained in detail in section 3.1.Using the state-operator correspondence, we can also represent the string field withoutreferring to oscillator modes. In this form, a string field can be defined by its inner productwith an arbitrary test state. Any open string state can be expressed as an operator inserted atthe origin in the upper half plane. For example, we might define a string field A by〈φ,A〉 = 〈f1 ◦ φ(0)f2 ◦ OA(0)〉S (2.22)where f1 and f2 are conformal maps from the upper half disc to disjoint regions making upthe Riemann surface S, and OA is the operator corresponding to the state |A〉. We will oftenrelax the assumption made here that the string field is expressed as a single operator insertedat the origin of the upper half plane, and permit any number of operators inserted anywherein the portion of S associated with the image of the upper half disc under the function f2, orequivalently, the portion of S which is the image of |z| > 1 under f1. The region f1(|z| < 1) issometimes referred to as the conformal patch and is reserved for the test state representing theasymptotic future. We often think of the string field A as being the Riemann surface S withoperator insertions, but to be precise we should remember that it is really still a Fock space statedefined by its correlation function with an arbitrary test state |φ〉. Since the CFT correlationfunction must saturate the ghosts in order to produce non-zero results, a physical string fieldA requires a test state φ with ghost number 2. The test operator producing information aboutthe tachyon component of a string field is c∂c.Since OSFT can be formulated for any open string boundary conditions and solutions rep-resent D-brane configurations, there must be a new formulation of OSFT for every solution.Taking the original OSFT action and expanding it about the string field Ψ gives an action withthe same form except that the BRST operator is modified asQΨφ = Q0φ+ Ψ ∗ φ− (−1)(Ψ)(φ)φ ∗Ψ . (2.23)Q0 is the BRST operator for the initial reference BCFT (corresponding to the solution Ψ = 0)and φ is any string field. The sign factor in the last term uses the standard notation that whena string field appears in the exponent of −1 it is taken to mean the ghost number of that stringfield. This way the new terms in (2.23) combine to make a commutator for physical stringfields, but in general can be either a commutator or an anti-commutator. This is a significantamount of background independence, as only the field content is still defined in terms of the firstreference BCFT. Schnabl has referred to this as being “half-way” to background independence[13].A surge of interest in OSFT began in 1999 when Sen predicted an explanation for thetachyon in the spectrum of the bosonic open string and suggested that his conjectures could bebest tested in string field theory [14]. He suggested that any D-brane system with a tachyonin its spectrum can decay to the vacuum with no D-branes. He claimed that at this point in102.2. String Field Theorythe space of string fields, the negative value of the tachyon potential exactly cancels the initialtension of the D-branes, indicating the true vacuum with 0 energy. He also provided someevidence that there would be no dynamical degrees of freedom at that vacuum. This was anecessary result in order to explain the absence of the brane-antibrane system’s U(1) gaugefield from the true vacuum. The existence of solutions to OSFT representing certain otherD-brane configurations was also considered, with the discussion focusing on the fact that theseconfigurations should also be unstable to decaying to the same tachyon vacuum. In time, theseconjectures were rewritten and presented in their best known form [10].1. The tachyon potential has a locally stable minimum, whose energy density E , measuredwith respect to that of the unstable critical point, is equal to minus the tension of theD25-brane:E = −T25.2. Lower dimension D-branes are solitonic solutions of the string theory on the backgroundof a D25-brane.3. The locally stable vacuum of the system is the closed string vacuum. In this vacuum theD25-brane is absent and no conventional open string excitations exist.These conjectures imply that the open string with purely Neumann boundary conditions actu-ally lives on a space-filling D25-brane. The tachyon in the string spectrum is associated withthe instability of this D-brane.2.2.1 Level TruncationEarly attempts to find the tachyon vacuum solution of OSFT used the oscillator basis for thestring field. The first challenge in solving the equations of motion in this approach is the infinitenumber of fields. A suitable approximation scheme is needed to reduce the problem to a finitenumber of variables. The level truncation scheme sorts the states in the string field (2.16)according to their L0 eigenvalue [15]. It was initially used only for translationally independentsolutions so that the level essentially counted the indices on all of the oscillators acting on thevacuum, but the method was later extended to include the whole L0 eigenvalue so that solutionswith p 6= 0 could be considered [16]. The difference between a term’s eigenvalue and that of thezero momentum tachyon is called the level, and only terms in the string field with level up to afixed value are included. With a string field truncated to level L, it is sometimes helpful to alsotruncate the action (which has level 3L) to a lower level M . Of course M must still be at least2L in order to include the kinetic term for each field. Solutions to a level truncated system areapproximate solutions to the full OSFT equations of motion, with the approximation typicallyimproving as L is increased.A number of solutions are known in level truncated OSFT, but the first is the solutionrepresenting the tachyon vacuum. Sen and Zwiebach first found this solution for low levels [17],and that was followed by calculations at higher levels [18]. The string field Sen and Zwiebachused at level 2 had the form of (2.16) with constant coefficients:|T 〉 = t c1 |0〉+ u c−1 |0〉+ v1√13L−2c1 |0〉 . (2.24)112.2. String Field Theory0.20.2 2.1: The potential at level 0 as a function of the tachyon zero mode t.This makes several simplifying assumptions about which fields to include. First is the obviouschoice of a spatially uniform solution, so that the infinite tower of momentum modes for eachstring field can be dropped, keeping only the 0 momentum states based on |0〉def= |0; 0〉. Nextthey imposed Siegel gauge, which is b0 |T 〉 = 0. This gauge choice is imposed level by level, andit can be shown that this gauge choice is valid locally at |T 〉 = 0 for every level except level1, where the necessary gauge transformations become singular. If any level 1 terms are to bedropped from the string field, we must find a different justification. Fortunately, in this casethe twist symmetry of the theory allows us to consider only terms with even level. The twistsymmetry reverses the parameterization of the worldsheet space coordinate σ → pi − σ. Of thezero-momentum fields with Neumann boundary conditions that we are considering, all of theodd level fields change sign under this symmetry while the even level ones do not. The actionis invariant under twist, so any twist-odd fields can only appear in pairs, and their equations ofmotion will never be sourced by twist-even fields. It is consistent to set all twist-odd fields to0, reducing the space of solutions which will be found, but in no way invalidating the solutionswhich remain. Since the tachyon vacuum solution is in the twist-even subspace, imposing thissymmetry on the string field will greatly simplify calculations.The energy of any solution will depend on the entire string field, but it is often useful toconsider it as a function of only the tachyon vev t in tc1 |0〉. This is achieved by using theequations of motion to eliminate all of the fields except for the tachyon. This is not a uniqueprocedure since there are many different solutions, but for any given solution we can see a profilewhich will have critical points at values of t which correspond to solutions on the branch underconsideration. Often the energy is rescaled to units where the tension of the initial D-brane is1, and this is typically referred to as f(t). The simplest (level 0) tachyon potential V (t) has theform of a cubic polynomial shown in figure 2.1. For the tachyon vacuum solution this profileremains qualitatively the same when higher level fields are included. The trivial solution is at122.2. String Field Theoryt = 0 and the tachyon vacuum solution appears at t ≈ 0.456 in this approximation. We alsonotice that the potential is unbounded below for negative values of t. The physical interpretationof this region is unknown, but there are no classical solutions in that region, corresponding tono D-brane configurations. The energy of the tachyon vacuum solution was found to convergequite quickly to the correct value of negative the D-brane tension. At level 0 it reaches ≈ 68%of this energy, while by level 4 it is already ≈ 95%. It does pass the predicted asymptoticvalue of the energy at level 14, but extrapolations show it turning around and approachingthe correct energy from the other side [18]. The spectrum of physical open strings about thetachyon vacuum is expected to be empty, but verifying this with an approximate solution is adifficult task. The modified BRST operator for the theory expanded about a solution obviouslyhas to depend on that solution, as seen in (2.23). By using an approximate solution we willget an operator that is not the correct BRST operator. Additionally, by truncating the stringfield, we have broken gauge invariance, so QΨ will not be nilpotent. Such an approximateBRST operator will not have the correct cohomology. Attempts were made, however, such asby Giusto and Imbimbo [19][20], to show that the tachyon vacuum has no physical excitations.There the “modified kinetic operator” L˜0 = {QΨ, b0} is examined, with QΨ the BRST operatorfor the theory expanded about the tachyon vacuum solution. At each ghost number n theysearch for momenta where det L˜(n)0 = 0. Their assumption is that closely grouped zeros are theresult of a single degenerate zero being broken up by the level truncation approximation, andthey consider such a group as a single zero for the purposes of finding the cohomology of theBRST operator QΨ. They found that the cohomology is indeed empty at ghost number one, upto the limits imposed by level truncation. At other ghost numbers, however, the cohomologythey found was non-trivial. Since physical states all have ghost number 1, it is not at all clearwhat these states represent.While the vacuum solution confirms two of Sen’s conjectures, there is still the matter oflower dimensional D-branes appearing as solitons. In the framework of OSFT, lower dimensionD-branes are solutions. The so-called “lump solutions” were studied by Moeller, Sen andZwiebach [16] for codimension 1. They are approximately zero where the D-brane is localized,and take values approaching the tachyon vacuum far from the lump. This requires infinitelymany momentum modes, and is studied with a compactified direction so that the spectrum isdiscrete and can be level truncated to a finite number of modes at any given level. The stringfield analogous to (2.24) now has the form|T 〉 =12...∑n=0(tnc1 + unc−1 + vnc1LX−2 + wnL′−2 + znc1LX−1LX−1)(∣∣∣0;nR〉+∣∣∣∣0;−nR〉). (2.25)The vacuum state∣∣0; nR〉is the n-th momentum mode in the compact direction. In the non-compact directions the solution remains translationally invariant, so the momentum only hasone non-zero component. Here LXn are the Virasoro operators in the “lump” direction inwhich the D-brane is to be localized, while L′n are the matter Virasoro operators in all ofthe other directions. In principle the sum in (2.25) runs to ∞, but in the level truncatedcalculation performed, it is noted that the level (L0 eigenvalue) receives a contribution n2R2 fromthe momentum, so that at any fixed level the number of modes to consider is finite. In fact,the number of modes to consider decreases for terms which have a higher level at p = 0. Byexamining the solution for a D-(p − 1) brane at different levels and radii, they found that the132.2. String Field Theorysolution appears to converge quite quickly to a lump which is independent of the compactifiedradius, so long as that radius is large enough that the lowest non-zero momentum tachyon hasa negative “kinetic” energy (the quadratic term in the action). For smaller radii they did notfind any solutions. While such solutions may well exist, this is an example of the phenomenonthat string field theory has a great deal of trouble finding solutions with higher energy than theinitial CFT ground state.The third class of known solutions is those representing a marginal deformation of the initialBCFT. In level truncated SFT these solutions are found by turning on a marginal conformalprimary operator by hand and allowing it to excite other fields through the interaction term.Without including all of the infinitely many interactions, the resulting collection of fields willnot have the same energy as the exact string field, though it should be closer for higher levels.Since level truncation breaks the marginality of such operators at the non-linear level, the resultis not actually a solution at all. For example, a flat one-parameter family of solutions in thefull theory is broken to a collection of string fields with nearly-flat potential, and the equationsof motion drive solutions back to the extremum of the family. Sen and Zwiebach [21] studiedsuch solutions with a Wilson line marginal deformation by ignoring the equation of motion forthe marginal field and imposing the rest. This way a single marginal parameter results in aone-parameter family of string fields containing a single solution: the string field 0 at vanishingmarginal parameter. Since the action is close to flat, this family does represent string fieldswhich are almost solutions, and as the level of truncation was increased the action becomesflatter.Perhaps unexpectedly, Sen and Zwiebach found that these marginal string fields only re-main real for a finite range of the marginal parameter. In addition to becoming flatter whenthe level was increased, the action evaluated at the “solutions” of [21] remained real for pro-gressively larger ranges of the marginal parameter. The expectation from BCFT, however, isthat marginal deformations by a Wilson line should exist for any vev of the Wilson line. Sev-eral possibilities have been suggested. Since level truncation does give increasing values of themaximum marginal parameter with increased level, it is possible that the maximum increaseswithout bound at infinite level. The values of the maximum at finite level, however, do notincrease very fast and lead us to be skeptical of this possibility. In [21] it was suggested thatthe finite marginal parameter in SFT might correspond to an infinite marginal vev in the CFT.An elaboration of this idea [22] claims that the map from the SFT parameter to the CFT oneis double valued. In a toy model, once the SFT parameter reaches its maximum it begins todecrease on a different solution branch while the CFT parameter continues to increase. Inchapter 3 we will find evidence that the maximum SFT parameter does correspond to a finiteCFT parameter, and we will not see another solution branch corresponding to any higher valueof the CFT marginal parameter at the level which we do our calculations. Which explanationis correct, if any, presently remains unknown.More recently, a level truncated solution with positive energy was found [23] by constructinga string field theory from the BCFT of the Ising model. This model simplified calculations byreducing the number of conformal primaries, and that work was able to find solutions repre-senting all expected boundary conditions. What is unexpected is that the solution representingthe higher energy D-brane was found using the lower energy D-brane as the starting BCFT.Interestingly, that solution was complex until level 14 fields were considered, at which point itsuddenly became real and remained that way at higher levels. While it may be unlikely, this142.2. String Field Theorydoes present us with another possible explanation for the lack of marginal solutions beyond acritical value of the deformation strength. They could be there as complex solutions waitingfor high level calculations to reveal them.The issue of matching a marginal SFT parameter to the corresponding CFT parameterwas also examined in [24] for a different marginal deformation. While this is not directlyapplicable, it is a good representative of the type of comparison that is typically attempted. Aphysical quantity, in this case the energy momentum tensor, is calculated on both sides and thecorrespondence is found by setting the two to be equal. We will take an alternative approachto relating the marginal parameter to its physical effect. Rather than studying level truncatedstring fields which are close to being solutions of the equations of motion, we will examine oneswhich are solutions to all of the level truncated equations of motion, but involve a D-branedecay in addition to the marginal deformation. This is possible because we use a pair of D-branes. When two D-branes are coincident there is a continuous family of allowed decays, sincethe action gains an SU(2) symmetry. Separating the D-branes breaks the symmetry, and inprinciple a marginal deformation to restore the coincidence of the D-branes will also restorethe symmetry. In practice, however, level truncation prevents us from completely restoring thesymmetry (see section 3.2.6). This is why we want to have a continuous set of solutions. Despitethe hypothetical set of exact solutions being raised to have different values of the action, theremust still be some critical points in the set where the action has a maximum or a minimum, andthese will appear as solutions even in the level truncated model. We will try to then comparethe marginal parameter of the surviving solutions to the physical displacement which we canadjust as a parameter of the theory. We will not claim a definitive correspondence, as there is acontribution from the decay part of the solution, but the problem is reduced to understandingthis contribution.2.2.2 Analytic SolutionsGoing beyond level truncation, there are known analytical solutions to SFT. These are oftendefined using the test state method for representing the string field, as in (2.22). Alternatively,they are sometimes expressed as operators acting on known string fields such as the ∗-productidentity. This form of solution will simplify the theory provided the Riemann surface is chosenso that the star product is easy to implement. The simplest choice is to take the conformalmap z ∝ arctan ξ of the upper half plane. This gives a semi-infinite cylinder called a wedgestate, with the boundary remaining on the real axis. This conformal frame is shown in figure2.2b. A wedge state can be easily broken into any number of disjoint parts by changing thescaling of the conformal map so that the |z| < 1 portion of the upper half plane gets mappedto a vertical strip of any desired width, and then translating those strips along the real axis.What is important about this frame is that the star product acts by cutting open two cylindersand glueing them together to make a new cylinder with (typically) larger circumference. In theprocess, a part of each string field’s conformal patch is removed, so that only one conformalpatch, or “future” worth of Riemann surface remains. A wedge state with circumference n+ 1is denotedWn, and the star product acts on such states asWm ∗Wn =Wm+n. The wedge stateW0 with circumference 1, for example, is an identity string field, since it glues in to any stringfield exactly the width that is removed. Wedge states with insertions consist of such cylinderswith operators inserted outside the conformal patch, often on the boundary.152.2. String Field TheoryR Lξ-1 0 1(a)R L0 π2_π4_π4_-π2_-z(b)Figure 2.2: The Riemann surfaces used in the two common conformal frames of the upper halfplane (a) and the wedge (b). In each frame the infinite past is located at 0 and the τ = 0line on the open string worldsheet is drawn separating two shades of grey. The two conformalframes are related by z = tan−1 ξ. The coordinate ξ here is z in (2.2), since z is conventionallyused for the final coordinate being considered.The first analytical solution to SFT was found by Schnabl [25], and was constructed usingwedge states with insertions. The solution contains two pieces.Ψ = limN→∞(ψN −∞∑n=0∂nψn)(2.26)where the string field ψn is a Riemann surface with insertions which are defined for this con-formal frame using straightforward notation. The c ghost field in the conformal frame of thewedge is c˜, and the zero modes of the b ghost and virasoro operators in this conformal frameare B0 and L0 respectively.ψn =2pi2e−n2 (L0+L†0)[(B0 + B†0)c˜(−pi4n)c˜(pi4n)+pi2(c˜(−pi4n)+ c˜(pi4n))]|0〉 (2.27)The vacuum here is the Riemann surface for the conformal transformation z = arctan ξ: thewedge state of circumference pi with no insertions. Curiously, the first term does not contributeto any of the coefficients in the standard level truncated basis. In that basis it appears to be 0,but it is required in order for the string field to be a solution.Remarkably, this term is responsible for all of the physical content of the solution, as ex-plained by Erler and Maccaferri [26]. Given any two solutions Ψ and Φ of OSFT, they suggesteda method for splitting Ψ into two parts:Ψ = Φ1() + ψ12() . (2.28)This requires a “left gauge transformation” (Q+ Φ)U = UΨ which can always be constructedasU = Qb+ Φb+ bΨ , (2.29)162.2. String Field Theoryfor any string field b with ghost number −1. The first partΦ1() =1U + (Q+ Φ) (U + ) (2.30)is a gauge transformation of Φ with finite gauge parameter U + . The second termψ12() =U + (Ψ− Φ) (2.31)is simply the remainder in (2.28). In the limit as  → 0, two things can happen. If the stringfield U is invertible then it defines a gauge transformation relating Ψ and Φ, and the remainderterm ψ12() goes to zero. If the two are not gauge equivalent then the operator in the remainderis called the “boundary condition changing projector” and is denoted X∞ = lim→0 U+ . As faras I am aware, there is no proof that the gauge parameter defined by (2.29) will give a vanishingboundary condition changing projector if two solutions are gauge equivalent, but since Φ1()is a valid gauge transformation for all finite  if two solutions differ then X∞ will capture thephysical distinction between the two.In the case of Schnabl’s solution for the tachyon vacuum (2.26), the “phantom” termlimN→∞ ψN plays a role which is very similar to the boundary condition changing projec-tor. The way the solution is written, the second term is not a solution by itself, so it cannothave the form of ψ12 in (2.28). It was shown in [26] that the phantom term in Schnabl’s solutionis X∞Ψ rather than X∞(Ψ − Φ). The splitting of the solution is still into a part with all ofthe physical content and a part which is gauge equivalent to the perturbative vacuum. Whatis unusual in the case of Schnabl’s solution is that the phantom term is 0 when expanded inthe level truncation basis (2.16). While this seems to suggest that the phantom term is 0 andthat Schnabl’s solution is gauge equivalent to the perturbative vacuum, this is not the case,and limN→∞ ψN contains important non-perturbative information about the solution.The resolution to the seemingly vanishing phantom term most likely lies in the fact pointedout by Ellwood [27] that the vector space of the string field does not have a norm. The operation〈A,B〉 in OSFT is sometimes referred to as an inner product, but it is correctly identified as asymplectic bilinear form [11]. This is because the operation 〈A,B〉 lacks the positivity conditionof an inner product, that 〈x, x〉 ≥ 0 with the bound saturated only for x = 0. For the bilinearform itself this is trivial since two copies of a physical string field will not saturate the ghostsand we will always find 0. Even if we were to consider non-physical ghost numbers, by definitionthe bilinear form has the property that 〈x, y〉 = −〈y, x〉, so any string field at all will have zeronorm if we tried to use this as the inner product. A more reasonable choice to define a normwould be the kinetic term of the action 〈x,Qx〉, but for terms in the string field such as c−1 |0〉we find a kinetic term which is negative. The possibility 〈x, c0x〉 has similar problems. Withouta norm on the vector space, we cannot say if two string fields are equal or close to one another.In many cases the coefficients in the level truncation basis provide a good indication of a stringfield, but there is no guarantee that they give an accurate representation, especially since wecan never calculate all of the infinitely many coefficients appearing in that basis.Schnabl’s analytic solution was initially shown to have energy of −1 in units of the D-branetension, cancelling the energy of the D-brane that the starting CFT lived on, but the othertwo Sen conjectures were not immediately verified. The third conjecture, that the cohomologyat the tachyon vacuum is empty so that there are no physical open string degrees of freedom172.2. String Field Theorywithout a D-brane, was verified by Schnabl and Ellwood [28]. This is done by first proving thelemma that Q has no cohomology if and only if there exists a string field A such that QA = Iwith I the identity string field. The task of proving that QΨ has vanishing cohomology becomesa matter of finding such a string field A satisfying QΨA = I. This was accomplished with thestring field A = 1L0B0I.Once the first solution was written down, a few variants on it appeared. One in particulargeneralized the same approach as in [25] for conformal frames which are “special projectors” [29].A projector of the star product is a string field P such that P ∗P = P . As the circumference ofa wedge state with no insertions is taken to infinity we can see that it becomes a projector, sincethe star product with itself gives another cylinder with twice the already infinite circumference.This limit is known as the “sliver” frame. Wedge states can be constructed by acting on theidentity string field with exponentials of the Virasoro zero-mode together with its hermitianconjugate, L0 + L†0. The sliver state is called a “special projector” because the commutatorof L0 with its bpz conjugate bpzL0 satisfies [L0, bpzL0] = s (L0 + bpzL0) as well as someregularity conditions. Using instead the zero-modes from the conformal frame of a differentspecial projector yields alternative sets of states with algebraic properties similar to the wedgestates, and Schnabl’s solution will still exist with all operators from the sliver frame replacedby their counterparts in the new frame, because all of the necessary algebraic structure isreproduced in the new frame.A simpler form of the solution was found several years later [30]. This uses the so-calledKBc subalgebra generated by the three string fields K, B, and c which are defined by actingon the identity wedge state with simple operators.K =pi2(L1 + L−1)L |I〉 , B =pi2(b1 + b−1)L |I〉 , c =1pic(1) |I〉 (2.32)The subscript L indicates the part of the operator acting on the left half of the string. Thiscan be obtained by integrating the charges over the right half of the unit circle, from −i to i.These string fields satisfy[K,B] = 0, {B, c} = 1, B2 = c2 = 0, (2.33)QBK = 0, QBB = K, QBc = cKc. (2.34)Further details of the KBc subalgebra can be found in a number of places, for example [31] or[32]. Two different solutions, both gauge equivalent to each other and to the original tachyonvacuum solution, can be written down.Ψ = c (1 +K)Bc11 +K(2.35)Ψ =1√1 +Kc (1 +K)Bc1√1 +K(2.36)Both forms satisfy the equations of motion, and the second form satisfies the reality conditionon the string field which guarantees that the action will be real valued.Since the tachyon vacuum, another class of analytic solution has been found, correspondingto marginal deformations. These solutions are generally written as a taylor series in the marginalparameter λ, but the string field can be written down explicitly at each order. If the marginal182.2. String Field TheorycV V V V cVt1 tn-1t2121021 l +1nBf◦øFigure 2.3: The λn term for the marginal solution of [5] for a marginal operator V (t) with regularself-OPE. The ghost part drawn has fewer insertions than in (2.37) because the anticommutator{B, c} = 1 can be used to simplify it.deformation is regular, that is if its OPE with itself is regular, then solutions are quite wellknown [33][5][34]. These solutions take the form of wedge states with insertions of the marginaloperator along the boundary, along with a few ghost insertions. The positions of the marginalinsertions are mostly integrated over regions which depend on the specific solution, since thethree moduli of the disc tell us that only one marginal operator should be inserted at a fixedlocation, as the other two moduli are fixed by the ghost number 2 test state in (2.22). Thesolution of [5] is given as Ψ =∑∞n=1 λnΨ(n) where〈φ,Ψ(n)〉=∫ 10dt1 . . .∫ 10dtn−1〈f ◦ φ(0)cV (1)n−1∏i=1BcV (1 +i∑j=1tj)〉W1+∑n−1j=1 tj. (2.37)This solution can be seen in figure 2.3. The operator B =∫dz2piib(z) here does not have a fixedinsertion location, as it only matters which ghost operators it is between.One example of a marginal deformation with regular self-OPE is the exponential or time-asymmetric rolling tachyon. Solutions involving this deformation represent placing a D-branein the infinite past and letting it decay at a finite time. This is intended to show the dynamicsof D-brane decay. The tachyon profile T (t) is the coefficient field of the tachyon in the leveltruncation expansion, and for uniform decay it is only a function of time. The tachyon profilewas calculated for two analytical solutions in [5] and [34], as well as for level truncation in[35]. In each case the tachyon profile shows oscillations with exponentially growing amplitudebeginning when the D-brane decays and continuing without bound. This is in contrast to theexpectation that the late time limit of the tachyon profile should be the tachyon vev of thetachyon vacuum solution. A few possible explanations have been suggested. In [35] it was192.2. String Field Theoryargued that a time-dependent field redefinition could be used to make the time dependence ofrolling tachyon solution a simple exponential, matching the BCFT result for the correspondingmarginal operator. In [27] this idea was reinterpreted as meaning that a time-dependent gaugetransformation of the rolling tachyon solution could be used to give a solution with the expectedtime dependence. That same work claimed that the late time limit of the rolling tachyon solutionis in fact Schnabl’s solution, despite the numerical evidence of the tachyon profile.If the marginal deformation has a singular self-OPE, then a detailed renormalization schemeis required in order to prevent divergences from the places where integrated insertions collidewith fixed insertions or with each other. Such a renormalization scheme is constructed in [36].This will be covered in detail and expanded on in chapter 4. An alternate approach was laid outin [37], in which solutions are considered which are formally gauge equivalent to the perturbativevacuum, but with a gauge parameter outside the Hilbert space. In [38], the regular-OPE solutionof [34] was extended to operators with singular OPE. This approach used integrals away fromthe boundary of the worldsheet to find finite solutions which are equivalent to several previouslyfound solutions in different limits. Recently, another new approach was suggested in which aWilson line in the time direction is used to cancel the divergence of singular operators in thematter direction [39]. This approach does not apply to the case of the rolling tachyon becauseit requires that the time direction is left unused and free to soften the divergences. It also usesboundary condition changing operators, rather than the bare marginal operators which thesolution of [36] was constructed from. However, given a boundary condition changing operatorwhich is trivial in the time direction, this approach apparently produces solutions for any newboundary condition, not only marginally deformed ones. We will not be investigating solutionsof this type. We will further investigate the singular marginal deformations by finding explicitnumerical results for the tachyon profile in the renormalized approach. This will involve carefulexamination of the renormalization scheme used, and a re-evaluation of the freedom involvedin doing so. We find that the renormalization must be chosen very carefully to guarantee thatthe sufficient conditions for a solution are met, but that at least two free parameters remain.The renormalization scheme cannot be further restricted by structural properties of the stringfield, or by numerical evidence, and we believe the remaining freedom may represent residualgauge transformations of the solution.Simply constructing string fields which satisfy the equations of motion does not mean verymuch if nothing is known about what the solutions represent, so gauge invariant quantities areof particular interest. The simplest gauge invariant quantity is the energy of a solution, whichis easily calculated. For a second test, however, the most clearcut quantity to consider is theboundary state. This is a state in the closed string Fock space which describes the D-brane inthe sense that all amplitudes for closed strings interacting with the D-brane are given by overlapswith the boundary state. The task of calculating the boundary state was first accomplished in[40], and more recently a different approach was taken in [41]. The first method of computingthe boundary state gives an exact closed string state as the result of inserting multiple copies ofthe OSFT solution around the boundary of a disc. In practice this is transformed to the sliverframe so that glueing in multiple copies of the solution is only a matter of considering largercylinders with more insertions on the boundary. The second method writes the boundary statein a basis of Ishibashi states. For each bulk conformal primary operator Vα, the associated202.2. String Field TheoryIshibashi state is the closed string state ‖Vα〉〉 satisfying(Ln − L¯−n) ‖Vα〉〉 = 0 . (2.38)In this basis it is shown that the coefficient of each basis state can be simply calculated fromthe string field in terms of BCFT amplitudes. Using this approach, the overlap of the boundarystate |BΨ〉 corresponding to a string field solution Ψ with any closed string insertion Vcl is givenby〈Vcl| c−0 |BΨ〉 = −4pii 〈I| Vcl(i) |Ψ−ΨTV〉 . (2.39)This method has the advantage that it can be applied to level truncated solutions as well asanalytic solutions. The disadvantage is that there are in principle infinitely many Ishibashicoefficients to compute in order to find the whole boundary state. In practice there may beonly a finite number which are non-zero or are of interest, so the approach is still effective.While the first approach, [40], was used to calculate the boundary state for Schnabl’s tachyonvacuum solution and a few regular marginal deformation solutions, the other approach, [41],has been applied to many recent solutions. Originally, it was applied to an analytical tachyonvacuum solution and the numerical lump solution of [16]. After that, it was used to suggestthat for level truncated marginal solutions with a periodic marginal direction the marginalparameter covers one fundamental domain [42]. Another level truncated solution representingchange of boundary conditions for the Ising model also included a discussion of boundary states[23]. The recent analytical solutions of [38, 39] also include calculations of the boundary statefor marginal deformation and tachyon vacuum solutions respectively. I am not familiar withany cases where the boundary state calculated from an OSFT solution does not agree withthe predictions of BCFT, so it appears to be a remarkably robust tool for the study of thephysical interpretation and properties of solutions to string field theory. In the case of thiswork the boundary state could be helpful for identifying the physical meaning of new solutionsand determining which related solutions are gauge-equivalent, but calculation of the boundarystate for the solutions we study here will be left for future work.21Chapter 3Separated D-branesAlthough string field theory is in principle a background independent theory, it still requiresan initial choice of BCFT to determine its field content. In the majority of studies, the initialBCFT is chosen to be a single D-brane, often a space-filling one so that the full rotationalsymmetry is preserved. In the event that multiple D-branes are considered, the initial BCFTis almost always just the product of the single D-brane theory with a suitable gauge group,so that the theory describes a collection of identical D-branes. In this chapter we consider asituation where the starting locations of the D-branes are different so that the initial BCFT ismore complicated, since the off-diagonal sector consists of strings that are stretched betweenthe two D-branes and have a different spectrum than the diagonal sector.One of the outstanding problems in OSFT is regarding solutions for marginal deformations.As a simple example, consider a single D-brane which is not space-filling. One marginal de-formation is the translation of this D-brane, and we would expect that we could position thisD-brane anywhere leading to an unbounded deformation. In string field theory, such configu-rations would correspond to a continuous set of solutions with a marginal parameter. In leveltruncation studies, however, this marginal parameter always has an upper bound which doesnot appear to grow fast enough as the level is increased, leading to speculation that only a finiterange of marginal deformations are allowed. While several attempts have been made to explainthis [21][22][24], here we will use the separated D-brane system to gain a new perspective onthe problem by using the initial separation as a reference distance.We will see that we are genuinely unable to find solutions corresponding to large D-branetranslations, and that this seems unlikely to change with the inclusion of more terms in thestring field. We will also use this reference distance to move one step closer to an alternativecalculation of the relationship between the marginal parameter in OSFT and the marginaldeformation of the BCFT. Previous methods have all compared quantities calculated in stringfield theory to those calculated in the BCFT, but we are able to rephrase the problem entirelyin terms of OSFT calculations. The solution described here is the first step in this comparison,and I will leave the analysis of necessary field redefinitions about a marginal solution to futurework.The work presented in this chapter has been published in the Journal of High Energy Physics[1] and presented at the conference String Field Theory and Related Aspects V, which tookplace in 2012.3.1 PreliminariesFollowing a procedure similar to that of [17], we begin by defining the states in the expansionof the string field|Φ〉 = (tijc1 + xijα−1c1 + hijc0 + . . .) |ij〉 . (3.1)223.1. PreliminariesWe are considering N parallel D-branes, so the state |ij〉 is the unexcited string beginning onbrane i and ending on j. The sum over brane indices is implied. The matter oscillator α−1belongs to the CFT transverse to the branes. In principle we should have xµαµ−1, but we assumethat all of the oscillators parallel to the brane are unexcited as we will discuss later. Of coursethere is an infinite set of higher level states such as α−2 |ij〉, but to begin we will truncate theexpansion here, at level (1,3). This means that the string field is truncated at level 1 and theaction is truncated at level 3. Level is determined by the L0 eigenvalue, counting from thetachyon at level 0.The embedding function for the coordinates tangent to the branes are the standard stringexpansion with Neumann boundary conditions in (2.2). The transverse embedding functionshave Dirichlet boundary conditions and areXµD = dˆµi + idˆµj − dˆµi2pilnzz¯+ i√α′2∑n 6=0αµnn(1zn−1z¯n)(3.2)and we find the positions of the branes are dˆi. For the special case of codimension one branesthis is only the mode expansion for X25, so we can drop the target space index. The zero-modesareα250 |ij〉 = −dˆj − dˆi√2α′pi|ij〉def= (di − dj) |ij〉 . (3.3)Ignoring momentum parallel to the branes, the matter virasoro zero-mode isL0 |ij〉 =12αµ0α0µ |ij〉 =12(dj − di)2 |ij〉def=12d2ij |ij〉 (3.4)By definition dii = 0, and for two branes we can always set d1 = 0 and d2def= d.We will only need to focus on terms in the string field with zero-momentum. For thesemodes, the potential of the string field is proportional to the action (2.18). If we divide out themass of the D-brane, essentially a choice of units, the potential isV = 2pi2(12〈Φ|QB |Φ〉+13〈Φ,Φ,Φ〉). (3.5)Technically, since we are working in a non-compact space, the D-brane mass is infinite, butwe can still think of this as the limit of the potential as the spacetime volume goes to infinity.Alternatively, we could calculate the potential energy density and work in units where theD-brane energy density is 1, and we would find the same form.3.1.1 Symmetries and the String FieldThe string field is constructed by acting on the vacuum c1 |ij〉 with both matter and ghostcreation operators. We can group these operators into CFTX , CFT′, and CFTg representingthe X25 matter sector, the Xµ6=25 sector, and the ghost sector respectively. For our purposeswe only need the subspace with ghost number one. Of course there are as many ways to acton a state with the composite operators LX−n, L′−n, and (Lg)−n as with the simple oscillatorsα−n, b−n, and c−n while preserving ghost number, so the Virasoro basis is equally valid as long233.1. Preliminariesas all states are linearly independent. As discussed in [16][8], this is the case provided thereare no null states. When we find a null state we must replace it, and its descendants, with theconformal family descended from another primary state.At level 1 we find that L′−1c1 |ij〉 is a null state, so we must instead include the primarystates αµ−1c1 |ij〉. While LX−1c1 |ij〉 is not technically a null state for a stretched string vacuum,LX−1c1 |ii〉 is, and in any case it is a simple rescaling of α25−1c1 |ij〉 so we will use that state in bothcases to avoid having to consider different states in the different sectors. There are no othernull states at the levels we will consider. The states αµ−1 with µ 6= 25 can be dropped due torotational invariance. All matter oscillators in the brane-parallel directions must come in pairswith their spacetime indices contracted. Any term with an odd number of matter oscillatorsin these directions can be ruled out, which means we can drop the extra conformal primaryα′−1c1 |0〉 and its descendants in these directions.The String Field Theory action is invariant under the twist operator Ω as discussed in[10]. Twist is defined by reversing the parameterization of the worldsheet space coordinate,σ → pi − σ. Typically Ω acts on individual states as (−1)L0+1, with every odd-level statealso being twist odd. Since the action is twist-even we can never have a single twist-odd stateappearing coupled to twist-even states, so those states’ equations of motion are trivially satisfiedby setting all twist-odd states to zero. This is the justification used to drop the odd-level statesfrom the string field in analyses of D-brane decay. With Dirichlet boundary conditions, however,the twist eigenvalue of the operator Xµ also changes sign so that every matter oscillator αncontributes an extra sign to the twist eigenvalue. In addition we have the added complicationthat the vacuum for stretched strings is not a twist eigenvalue. For a single D-brane the vacuum|0〉 is twist-odd, but with multiple branes Ω(|ij〉) = − |ji〉, so we can construct twist-even andtwist-odd vacua as|ij〉e =12(|ij〉 − |ji〉), |ij〉o =12(|ij〉+ |ji〉). (3.6)Instead of simply acting on the twist-odd vacuum with a twist-odd collection of operators,we must now also consider the possibility of acting on the twist-even vacuum with twist-evenoperators.We can drop any terms from the string field that are odd under (−1)L0+nαΩvac where nα isthe number of α25n matter oscillators with dirichlet boundary conditions. While any operatorcontent can create a twist-even state by applying it to the correct choice of vacuum, this willrestrict each state to have either a symmetric or anti-symmetric coefficient matrix.The string field can now be written down, and at level 3 it is|Φ〉 =(tijc1 + hijc0 + uijc−1 + vijL′−2c1 + wijLX−2c1 + oij(b−2c−1c1 − 2c−2)+ o˜ij(b−2c−1c1 + 2c−2) + pijL′−3c1 + qijLX−3c1 + . . .)|ij〉+(xijc1 + fijLX−1c1 + rijc−1 + sijL′−2c1 + yijLX−2c1 + zijLX−1LX−1c1 + . . .)αX−1 |ij〉 . (3.7)In order to be twist-even the fields t, x, u, v, w, r, s, y, and z are symmetric matrices, while h,f , o, o˜, p, and q are anti-symmetric.When dealing with the twist-even subspace on a single D-brane the coefficient of each stateis a single real number (or real-valued function). Little discussion is given to this point, as realnumbers seem a natural choice. Now that the coefficients are matrix-valued we must decide howto correctly generalize this. In [11] it was shown that in order to guarantee the reality of the243.1. Preliminariesaction we must impose the condition that the string field be invariant under the compositionof hermitian conjugation and inverse bpz conjugation.|Φ〉 = bpz−1 ◦h.c. |Φ〉 = |Φ〉‡ (3.8)As in [36], we denote this combined conjugation by ( )‡. For a generic single state constructedfrom the ghost number one vacuum by applying operators α−n and L−n, we know that hermitianconjugation simply switches the order of operators and the sign of each index. Bpz conjugationhas a similar effect, but introduces a sign factor.bpz−1 〈0| = |0〉 , bpz−1 Ln = (−1)nL−n, bpz−1 φn = (−1)n+hφ−n (3.9)for any primary field φ with conformal dimension h. For composite operators, we multiply theirbpz conjugates in the reverse order, but with an additional sign factor when there are multipleghosts. This additional factor depends on the number of fermionic operators as (−1)nf (nf−1)/2.Since we are looking at ghost number one, this is equivalent to including an extra (−1) withevery b ghost. When we work out the action of bpz−1 ◦h.c. on any term, we can find the samesign as the twist eigenvalue, ignoring the possibility of the twist-even vacuum. If the twist-evenvacuum is used then the sign factor from this conjugation is opposite the twist eigenvalue. Wecan now say that for any state built on the twist-odd vacuumaij |φ; ij〉 = (aij |φ; ij〉)‡ = bpz−1(a†ij 〈φ; ij|) = a†ijΩφ |φ; ij〉 (3.10)will guarantee the reality of the action, where Ωφ represents the twist eigenvalue of the state|φ; ij〉. This gives the resultaij = Ωφa†ij . (3.11)For states built on the twist-even vacuum we similarly findaij = −Ωφa†ij . (3.12)Since all the states we are considering are twist-even, these conditions reduce to requiring thecoefficient matrices, which are already either symmetric or anti-symmetric, to be real-valued.If we were to examine states with odd twist eigenvalues then we would need to assign themimaginary coefficient matrices. However, since those terms must always appear quadraticallyin the action, making their coefficient matrices real would introduce a factor of −1 multiplyingall such terms, and the action would remain real.The term hijc0 |ij〉 is a special case. Level truncation calculations are typically done inSiegel gauge, which is defined by setting b0 |Φ〉 = 0. This removes a number of terms includinghij , but the standard proof that Siegel gauge states are complete and distinct fails for level 1states because the eigenvalue of L0 is 0. Since Siegel gauge can be imposed level by level, this isnot a problem for studies of tachyon condensation, where level 1 states are dropped completely.While the standard proof fails this does not mean that Siegel gauge is necessarily invalid. Inany case, the matrix hij has already been reduced to a real anti-symmetric matrix by the realitycondition and twist symmetry. With the diagonal part of the matrix removed we can once againimpose Siegel gauge on the off-diagonal parts as long as the initial D-brane separation is non-zero. Separating the D-branes gives L0 a non-zero contribution and rescues the proof of Siegel253.1. Preliminariesgauge’s validity. Even if we take d = 0 we can still use the exchange symmetry described nextto ensure that hij is consistently 0, but if we were to study solutions involving the twist-oddsector then we should include h and Siegel gauge would be violated at level 1.Once we restrict our attention to the case of two D-branes we will find another simplesymmetry related to exchanging the two D-branes. Consider any single cubic term in theaction and define the function f byf(dij , djk, dki)AijBjkCkidef= AijBjkCki 〈iAj , jBk, kCi〉 . (3.13)Because all brane indices are summed, terms will appear in pairs likef(0, d12, d21)A11B12C21 + f(0, d21, d12)A22B21C12. (3.14)It is straightforward to see that for operators A, B, and C with a definite number of matteroscillators in the dirichlet direction, the function f will transform under d → −d in the sameway as the product of the three operators does under X25 → −X25, picking up a factor of(−1)nα . Now (3.14) will take the formf(0, d12, d21)A11B12C21 + f(0,−d12,−d21)A22B21C12= f(0, d12, d21) (A11B12C21 + (−1)nαA22B21C12) (3.15)This part of the action is invariant under taking X25 → −X25 and swapping the D-braneindices. Terms in the action which take place entirely on a single D-brane trivially satisfy thesame symmetry, as does the quadratic part of the action. We can then consistently restrictourselves to string fields which are even under this operation. There are solutions which arenot, such as the solution where one D-brane decays and the other does not, but it happens thatthe solution we are interested in is exchange-even so we can focus on that part of the stringfield. Choosing a parameterization of the matricesAij =(As −Aa as + aaas − aa As +Aa)(3.16)we see that if the operator A has an even number of dirichlet matter oscillators then we canconsistently set Aa = αa = 0. If, on the other hand, nα is odd then we can set As = αs = 0.Because we will begin our solution by exciting only twist-even and exchange-even fields at level0 we can restrict all higher levels to the same subspace. The most dramatic consequence ofthis is to completely drop the fields h, o, o˜, p, and q because they have an even nα but arealready restricted to have real anti-symmetric matrices through the twist symmetry and realitycondition.Alternate basisWe have chosen to write the string field up to level 3 as the two primary fields c1 |ij〉 andα−1c1 |ij〉 and their virasoro descendants, but we could have chosen an alternate form. It canbe useful to see the string field written in terms of the basic matter and ghost oscillators. In263.1. Preliminariesthis basis we have|Φ〉 =(tijc1 + hijc0 + xijα250 c1 + uijc−1 + v˜ijα−1 · α−1c1 + w˜ijα25−1α25−1c1 + f˜ijα25−2c1+ oij(b−2c−1c1 − 2c−2) + o˜ij(b−2c−1c1 + 2c−2) + pijα−1 · α−2c1 + q˜ijα25−1α25−2c1+rijα25−1c−1 + s˜ijα−1 · α−1α25−1c1 + y˜ijα25−1α25−1α25−1c1 + z˜ijα25−3c1 + . . .)|ij〉 . (3.17)The terms were defined this way so that the new coefficients denoted with tildes have the sameproperties under twist and exchange symmetries as their partners in (3.7). The fields andtheir coefficients are related to the ones in (3.7). Defining states such as tijc1 |ij〉 = tij |t〉 fornotational convenience, we can state the relationships between the states themselves as|v〉 =12|v˜〉 , |s〉 =12|s˜〉 , |w〉 =12|w˜〉+ α0|f˜〉, |f〉 = |f˜〉+ α0 |w˜〉|q〉 = |q˜〉+ α0 |z˜〉 , |y〉 =12|y˜〉+ |z˜〉+ α0 |q˜〉 , |z〉 = 2 |z˜〉+ 3α0 |q˜〉+ α0α0 |y˜〉 .(3.18)With this transformation known, it is simple to go from solutions in one basis to another usingthe relationships between the coefficients. Let φI |φI〉 be an arbitrary state and its coefficient(with brane indices suppressed), then for every change of states |φI〉 = CIJ∣∣∣φ˜J〉the coefficientsare related by φ˜J = φICIJ .This change of variables becomes singular at specific separations. While we can still formallywork in the virasoro basis, we must bear in mind that it is only complete almost everywhere.Off-diagonal components of the fields w and f are linearly dependent when the matter zero-mode has eigenvalue α0 = 1√2 , and the fields q, y, and z contain a linearly dependent part atboth α0 = 1√2 and α0 =√ The PotentialWith the initial brane configuration and string field defined, we can work out the string fieldtheory potential. The solutions we are looking for will be the critical points. For calculations,the potential of (3.5) can be written using the 3-string vertex defined by〈Ψ1,Ψ2,Ψ3〉 = 〈Ψ1|Ψ2 ∗Ψ3〉 = 〈V3|∣∣∣Ψ(1)1〉 ∣∣∣Ψ(2)2〉 ∣∣∣Ψ(3)3〉. (3.19)This vertex can be written as〈V3| =34√326∑i,j,k〈ij| 〈jk| 〈ki| c(1)−1c(2)−1c(3)−1c(1)0 c(2)0 c(3)0 eΞ , (3.20a)Ξ =∑r,s∞∑m,n=0(12α(r)µm Nrsmnα(s)n,µ + c(r)m Xrsmnb(s)n), (3.20b)where the coefficients N rsmn and Xrsmn are known. The first few coefficients are given by table3.1 and it will be useful to define N˜ rsmn = Nrsmn +Nsrnm. The creation and annihilation operatorshere have an extra upper index indicating which oscillator vacuum they act on. The Neumanncoefficients N rsmn were derived in [12] with a minor correction to the matter coefficients appearing273.1. Preliminariesm n N rrmn Nr(r+1)mn Nr(r−1)mn0 0 ln 4√3/9 0 00 1 0 −2√3/9 2√3/91 0 0 2√3/9 −2√3/90 2 −2/27 1/27 1/271 1 −5/27 16/27 16/272 0 −2/27 1/27 1/270 3 0 22√3/729 −22√3/7291 2 0 32√3/243 −32√3/2432 1 0 −32√3/243 32√3/2433 0 0 −22√3/729 22√3/729m n Xrrmn Xr(r+1)mn Xr(r−1)mn0 0 0 0 00 1 0 0 01 0 0 4√3/9 −4√3/90 2 0 0 01 1 11/27 8/27 8/272 0 16/27 −8/27 −8/270 3 0 0 01 2 0 40√3/243 −40√3/2432 1 0 −80√3/243 80√3/2433 0 0 −68√3/243 68√3/243Table 3.1: The first few Neumann coefficients appearing in (3.20b).in [43]. The conventions introduced in (3.20b) are the same ones appearing in [44], though Ihave included the zero-modes which did not appear there. The ghost coefficients Xrs0n = 0 form = 0 are included here only for clarity, as those terms are typically omitted from the definitionof Ξ.In order to calculate the cubic terms in (3.5) we use the Baker-Haussdorf formulaeXY =(Y + [X,Y ] +12[X, [X,Y ]] + . . .)eX (3.21)which allows us to commute the exponential eΞ past each of the operators in our string fieldand get a new set of operators in its place. The triple product can then be computed usingstraight-forward commutator algebra. A number of useful commutation relations are given in(3.22).[α(r),µm , α(s),νn]= mδm+nδr,sδµ,ν (3.22a){c(r)m , b(s)n}= δm+nδr,s (3.22b)[α(r),µm , L(s)n]= mα(r),µm+nδr,s (3.22c)[(Lg)(r)m , b(s)n]= (m− n)bm+nδr,s (3.22d)[(Lg)(r)m , c(s)n]= −(2m+ n)cm+nδr,s (3.22e)[Ξ, α(q),µ−k]=k2∞∑m=0N˜ rqmkα(r),µm (3.22f)[Ξ, L(q)−k]=∞∑m,n=0m4N˜ qrmn{α(r)n,µ, α(q),µm−k}(3.22g)[Ξ,[Ξ, L(q)−k]]=14∞∑i,j,m,n=0inN˜ rqmnN˜qsij α(r),µm α(s)j,µδn+i−k (3.22h)283.1. Preliminaries[Ξ, c(q)−k]=∞∑m=0Xrqmkc(r)m (3.22i)[Ξ, b(q)−k]= −∞∑n=0Xqsknb(s)n (3.22j)[Ξ, (Lg)(q)−k]=∞∑m,n=0(2k +m)Xqsmnc(q)m−kb(s)n −∞∑m,n=0(n+ k)Xrqmnc(r)m b(q)n−k (3.22k)[Ξ,[Ξ, (Lg)(q)−k]]= −∞∑i,j,m,n=0(j + k)XqsmnXtqij c(t)i b(s)n δm+j−k−∞∑i,j,m,n=0(2k + i)XrqmnXqtij c(r)m b(t)j δn+i−k(3.22l)[Ξ,[Ξ,[Ξ, L(q)−k]]]=[Ξ,[Ξ,[Ξ, (Lg)(q)−k]]]= 0 (3.22m)The operator L(r) refers to the matter part of the Virasoro operator on the r-th string vacuum,and the ghost part of the Virasoro operator is (Lg)(r). The results shown include the entirematter Virasoro operators L(r)n = (L′)(r)n + (LX)(r)n , so for commutators involving only a partof that, the spacetime indices behave as expected. For example,[α(r),25m , (L′)(s)n]= 0 and[Ξ, (LX)(q)−k]=∑∞m,n=0m4 N˜qsmn{α(s),25n , α(q),25m−k}. It is useful to note that since the right handside of[Ξ,[Ξ, L(q)−k]]is summed over non-negative integers we can use the fact thatmnδm+n−2 =δm−1δn−1 to simplify the case where k = 2. As an example, we can now show thateΞL(q)−2 =(L(q)−2 +m4N˜ qrmn(α(r)µn α(q)m−2,µ + α(q)µm−2α(r)n,µ) +18N˜ rqm1N˜qs1jα(r)µm α(s)j,µ)eΞ. (3.23)Once Ξ is all the way to the right, all of its operators except for the zero-modes will annihilatethe vacuum. Using the zero-mode Neumann coefficients we findeΞ |0〉(1) |0〉(2) |0〉(3) =(43√3) 12 ((α(1)0 )2+(α(2)0 )2+(α(3)0 )2)|0〉(1) |0〉(2) |0〉(3) . (3.24)Throughout this we have assumed the rule (3.22b) despite the fact that for r 6= s theoperators act in different sectors. It could be argued that since c(r)m acts only on the r-sector, itis bosonic in the s-sector and the commutator should be zero rather than the anti-commutator.This, however, would only amount to some extra sign factors, which can be checked againstknown results found by other methods. The use of anti-commutators with our form of the 3-vertex gives a result for the coefficient of u3 which matches the coefficient in [17], which appearsto have been calculated directly from the disc amplitude. We chose the u3 term to considerbecause if we were to change the sign on Xrsmn with r 6= s we would not get the same result.If one of the three fields is different than the others then there are three ways to order them.Since the triple product 〈A,B, C〉 is invariant under cyclic permutations of fields with ghostnumber one, though, we only need to calculate it in one order and then multiply the end resultby 3. If all three are different then there are six ways to order the fields. Again we can use293.1. Preliminariescyclicity of the triple product to reduce the problem to two distinct orders, and then multiplyeach by 3. The twist operator will tell us how the two distinct orderings are related to eachother. We know from [10] that〈A,B〉 = 〈Ω(A),Ω(B)〉 (3.25a)Ω(A ∗ B) = Ω(B) ∗ Ω(A) (3.25b)for string fields with ghost number one. From this we can easily see that the reverse orderingis〈C,B,A〉 = 〈Ω(A),Ω(B),Ω(C)〉 . (3.26)For the twist-even fields we are considering this results in 6 times the coefficient for one orderingof the fields. In more generality, we can still use this result as long as we remember to includethe necessary signs and exchange the D-brane indices as Ω acts on each string field.In order to check the vertex, we can use it to compute several couplings that are alreadyknown. Since most work neglects the zero-modes we have to use [16], which contains a detailedaction for states with non-zero momentum. Because they use states which are eigenvalues of |p|rather than the signed momentum, we must take care to include all the necessary combinatorics.We should then be able to match their couplings. As an example, we will examine their couplingfor t0t1v1.1 We begin by using our 3-vertex to find the cubic term for two copies of the fieldt and one field that we call w. We then choose α(1)0 = 0 and α(2)0 = −α(3)0 =√2R to matchthe zero-modes to the momenta |p| = 1R . To avoid complications due to conventions, we usethe value of α0 for which the L0 eigenvalue matches the contribution of the momentum tothe level in that work. There are two ways to pick which state has positive momentum; wecould have swapped α(2)0 and α(3)0 , so we pick up a combinatoric factor of 2, but each of thecosines appearing with t1 and v1 has a factor of 12 when expressed in terms of the necessaryexponentials, contributing an additional 14 . The three fields in question are all distinct, so thereis a factor of 6 for the number of ways to order them. The cubic term in the action has a factorof 13 in front which we must also remember to include. We then find exactly the same result asin [16]:(−3√38R2−15√3128)(43√3) 2R2.The couplings for t0t1v1, t1t2v1, t0v1v1, t1v0v1, and t1w0v1 have all been checked this way.Finally, we need to be concerned with the boundary conditions. The Neumann coefficients,N rsmn and Xrsmn were derived for Neumann boundary conditions, and since we are starting with D-branes that have codimension 1, we will also need to consider the vertex for Dirichlet boundaryconditions. Fortunately, T-duality guarantees that the vertex is identical. With Neumannboundary conditions, the radius of a compact direction only appears through the allowed valuesof α0. We can contract the radius to 0 while holding α0 = −d constant by giving the gauge fieldthe value A25;2,2 = −dˆ2piα′ =−d√2α′, A25;ij = 0 otherwise, equivalent to θ2 = Rdˆα′ . When we thenT-dualize to get a non-compact Dirichlet theory we will have the same value of α0 we wanted,and the same mass spectrum. Because the 3-vertex can be defined by the 3-point functionin (3.19) and the amplitude is the same with Neumann or Dirichlet boundary conditions byT-duality, the 3-vertex will have the same form and the same coefficients in either case.1 While several of the states we will use are also in [16], only t and u will be labeled the same.303.2. SolutionsThe quadratic term in (3.5) is much easier to compute. The only sums in this term appeardue to the BRST operatorQB = cnL−n +m− n2: cmcnb−m−n : −c0. (3.27)The other thing we need in order to calculate 〈Φ|QB |Φ〉 is the bra state. Conjugation is doneusing the bpz conjugate which we already defined in (3.9).The quadratic terms are verified similarly to the cubic ones. For the coefficient of v1v1 wehave a factor of 24 from the two cosines and the number of ways to make the momentum vanish.Letting α0 =√2R and including the12 from the action we find that the coefficient is18+98R2+1R4in perfect agreement again. The coefficients of t0t0, t1t1, t2t2, u0u0, u1u1, v0v0, v1v1, w0w0,w1w1, and v1z1 have all been checked. The coefficient for z1z1 requires an additional detail.While v1z1 can be easily matched by multiplying an extra α0 into the quadratic term to matchthe normalization difference between our f and their z, when we calculate quadratic terms withz on the left we must remember to multiply our coefficient of f by bpz(α0) = −α0. With thisadditional overall sign, we agree with [16].The potential of (3.5) can be evaluated, and at level (1, 3) it isV = 2pi2[12((−1 +∆ij2)tijtji +∆ij2xijxji − 2hijhji)+33√326(tijtjktki +32N˜ r301αr0tijtjkxki +32(N˜3211 +12N˜ r201 N˜s301αr0αs0)tijxjkxki+14(N˜1211 N˜3r10αr0 + N˜2311 N˜1r10αr0 + N˜3111 N˜2r10αr0 +12N˜ r101 N˜s201 N˜t301αr0αs0αt0)xijxjkxki+169tijhjkhki +89N˜ r301αr0hijhjkxki)(43√3) 12 (∆ij+∆jk+∆ki)](3.28)Here the states have already been removed by taking the inner product with the vertex, so theαr0 should be taken to mean the eigenvalue of α0 against the r-th vacuum in the 3-string state|ij〉 |jk〉 |ki〉. For example, α(2)0 = dj − dk. A full listing of the higher level couplings used inthis work is given in appendix A.3.2 SolutionsNow once again restricting our attention to N = 2 branes, we can use the twist and exchangesymmetries to simplify the potential. First we will define the coefficient matrices such as tijappearing in the string field expansion (3.1).tij =(Ts − Ta τeiθtτe−iθt Ts + Ta)=(Ts ττ Ts), xij =(Xs −Xa χeiθxχe−iθx Xs +Xa)=(−Xa 00 Xa)(3.29)313.2. SolutionsTs =t11+t222 and Ta =t22−t112 are the symmetric and anti-symmetric parts of the diagonal.A complete list of terms which are even under both twist and exchange symmetries is nowrelatively short at this level.tij =(Ts ττ Ts), xij =(−Xa 00 Xa), uij =(Us υυ Us)vij =(Vs νν Vs), wij =(Ws ωω Ws), fij =(0 φ−φ 0)rij =(−Ra 00 Ra), sij =(−Sa 00 Sa), yij =(−Ya 00 Ya), zij =(−Za 00 Za)(3.30)3.2.1 Level 0 CalculationAs a first approximation, we will include only the lowest level states in the string field expansion.Setting xij and all higher level terms to zero and keeping only tij we can write down thetruncated potential.V = 2pi2[−T 2s − T2a + (−1 +d22)τ2 +27√332(T 3s + 3TsT2a + 3(43√3)d2Tsτ2)](3.31)This polynomial has 5 critical points. The trivial point Ts = Ta = τ = 0 corresponds tothe perturbative vacuum where neither D-brane has decayed. There are three other diagonalsolutions with Ts = ±Ta =T02 and Ta = 0, Ts = T0, where T0 is the tachyon value decaying asingle D-brane at level 0. These represent solutions where one or both D-branes have decayedseparately. These four solutions are all expected, since the presence of another brane does notaffect a brane’s ability to decay. This is the diagonal subset of the solutions found in [28].There is a fifth solution, shown in figure 3.1, in which there is an off-diagonal tachyon τ 6= 0.In the zero separation limit, it is the solution where a symmetric linear combination of the twobranes has decayed; an SU(2) rotation of the solution where one of the branes decays. As theD-brane separation is increased the solution goes to zero at d =√2. This solution is only validfor d ≤√2, as for larger separations it becomes complex and violates the condition that thematrix tij must be hermitian. We can explore the physical interpretation of this solution athigher levels.3.2.2 Level 1 CalculationNow that we have an approximation to the tachyon sector of the solutions we are looking for,we can begin including more terms in the string field (3.7). The transverse scalars xij will beof key interest, which is why we have explicitly included them in section 3.1. With the explicitparameterization of the matrices tij and xij given in (3.29) as well as the Neumann coefficients323.2. Solutions(a) (b)Figure 3.1: The level 0 solution for separated D-branes which decays a non-trivial linear com-bination of two D-branes. The energy is shown in (a) and the non-zero components Ts (solidline) and τ (dashed line) of tij are in (b).N rsmn, we can rewrite the full level 1 potential of (3.28) asV = 2pi2[−T 2s − T2a + (−1 +d22)τ2 +d22χ2 + 2γ2+27√332T 3s +81√332TsT2a +3√32Ts(X2s +X2a) + 3√3TaXsXa+(43√3)d2(81√332Tsτ2 +278dXaτ2 −278dTaτχ−3√32Tsγ2+3√3(1−d22)Xsτχ+3√32(1 +d24)Tsχ2 +d32Xaχ2 − 2dXaγ2)](3.32)Here we find the first example of an unexpected quadratic term. The tachyon’s mass isclearly −1 with an additional d22 if it’s stretched, and the level 1 vector has a mass of 0 +d2ij2which makes perfect sense. The ghost term γ corresponding to the off-diagonal part of hij ,however, has mass 2 and does not change at all when stretched. We will see at level 2 that thesecond ghost term, c−1 |0〉, is actually tachyonic, despite being at level 2. It even becomes moretachyonic as the string is stretched. Fortunately, this is a ghost mode, so in all likelihood it isone of the non-normalizable modes that are not present in the cohomology.As discussed in section 3.1.1 we do not need to consider the entire set of 6 fields. Usingthe twist and exchange symmetries, only Ts, τ , and Xa are excited with the off-diagonal level333.2. Solutions(a) (b) (c)Figure 3.2: The level 1 solution for separated D-branes which decays a non-trivial linear com-bination of two D-branes. The energy (a) with the level 0 energy dotted for comparison, thetachyon matrix (b) Ts (solid) and τ (dashed), and the transverse scalar Xa (c).0 solution as our starting point, so we can remove all of the other variables, and get a modelfor the effective potential of this solution.V = 2pi2[−T 2s + (d22− 1)τ2 +27√332T 3s +3√32TsX2a +(43√3)d2(81√332Tsτ2 +278dXaτ2)](3.33)This system is now simple enough that it can be solved exactly. Apart from the obvious caseswhere either nothing happens or both branes decay, one solution is found. The physical branchesof this solution are shown in figure 3.2.Since the field responsible for transverse fluctuations of a brane is α−1, the Xa field gives aconstant positive shift to the position of brane 2, and a negative shift to brane 1, representingthe first term in their separation. The negative value of Xa on this solution means that thetwo branes have moved closer together. The correct sign can be easily verified by noting thatan increase to the value of Xa in (3.33) contributes a positive τ2 term. An increased effectivetachyon mass corresponds to increasing the brane separation felt by the tachyon.There is every reason to expect that there are analytical solutions to string field theoryconsisting of a marginal deformation followed by a Schnabl type decay. In fact an analyticsolution of this form was found for a single D-brane shortly after this work was published [45].Since we know from [28] that any appropriate linear combination of two branes can decay aslong as they are coincident, any such linear combination of separated branes should be able todecay as long as they first move to become coincident. We believe that the solution we havefound is one of these. That only one specific linear combination of the branes has been foundto decay like this is no surprise, since the SU(2) symmetry of two D-branes was broken byseparating them. In principle we expect this symmetry to be restored after the translation, butsince level-truncated solutions are approximate the symmetry has not been restored here.In the limit as d → 0 we see that Xa goes to 0 as well, since the branes do not need tomove when they are already coincident. In fact, for small d the solution behaves as Xa =− 427d + O(d3), which is linear in the physical starting separation of the branes. To see if343.2. Solutionsthe magnitude is reasonable, we can find the value of Xa for which the SU(2) symmetryof the tachyon is restored. We accomplish this by equating the eigenvalues of the Hessianmatrix for the three tachyon fields and solving. What we find is that χ = 0 and, remarkably,Xa = − 427d+O(d3). Because of the severe level truncation approximation, we only expect thisto hold for small separations. xij is the leading term contributing to a translation, as shown in[21]. Infinitely many other fields will contribute at higher orders in separation, and indeed theO(d3) terms in the two calculations of Xa above do not even have the same sign.At d >√2 this off-diagonal solution becomes double-valued, and the two branches meet andcease to exist for d & 1.61. It is easy to see that d =√2 is the point where the stretched tachyonbecomes massless. While there is no proof that the level truncation scheme is a consistentapproximation, we can understand that the more massive modes tend to be less importantcontributions to a solution. This was refined in [16] where they pointed out that the level-truncation scheme should use the entire L0 eigenvalue, including the zero-modes. What thismeans for us is that for d ≥√2 we should be including the diagonal fields before off-diagonalfields from one level previous. While at level 1 this would not make a difference to our solutionsince χ is already 0, we should bear in mind that our solutions are not necessarily valid forlarge d. For larger distances we should still be able to translate D-branes, but we don’t knowif such solutions can be found from a level-truncated system.We should note that the reality condition on the string field is a sufficient condition toguarantee the reality of the action, but not a necessary condition. In fact, when the level 0solution becomes complex the potential remains real for all separations. At level 1, however,the potential has an imaginary part that grows polynomially for separations above the criticaldistance dmax ≈ 1.61. As a result we will consider the reality condition on the string field ratherthan the reality of the action when discussing the range of separations for which our solutionremains valid.Direction of translation by T-dual methodI have claimed that the negative sign of xij indicates that the branes move towards each otherbecause of the effect it has on the mass of the stretched tachyon. We can also check thatthe negative sign of Xa opposes the initial separation by considering the T-dual picture. Theposition dˆi is dual to −2piα′Aii25 where Aii25 is the gauge flux on the i-th D-brane. The actionwith this charge isS =12piα′∫Mdzdz¯∂Xµ∂¯Xµ + i∫∂MAµdXµ. (3.34)A shift in the position of a D-brane by  is a shift Aii25 → Aii25−2piα′ , and under this perturbationthe action changes ase−S = e−S0(1 +i2piα′∫∂M∂X25dz +O(2))(3.35)On the other hand, the state xijα25−1c1 |ij〉 corresponds to inserting the operators√2α′ ixij∂X25(0)c(0) at the origin on the upper half plane. For any operator we can write|A〉 =∫DXie−S[Xi]A(0) (3.36)353.2. Solutionsso we can see that for a translation of a single D-brane, we getxα25−1c1 |0〉 = ix√2α′∫DXie−S[Xi]∂X25(0). (3.37)Splitting the solution into a classical piece Xcl and a perturbation X ′ so that DXi = DX ′,(3.37) becomes= ix√2α′e−S[Xcl](∂X25cl (0)∫DX ′e−S[X′] +∫DX ′e−S[X′]∂X ′25(0))(3.38)= ix√2α′e−S[Xcl]∂X25cl (0)∫DX ′e−S[X′] (3.39)= ix√2α′∂X25cl (0)∫DXie−S[Xi] (3.40)where the second term in (3.38) vanished because the integrand is odd. The sign of the operatorinsertion associated with a positive constant times the state α25−1c1 |0〉 is clearly the same as thesign associated with the operator insertions from a gauge field which induces a positive shift inD-brane position in the T-dual picture. This is confirmation that when we find a solution withXa < 0 the D-branes have moved towards each other.3.2.3 Level 3We will now include all of the fields listed in (3.30). The full potential used can be found inappendix A, and must then be simplified by dropping all fields which are not even under bothtwist and exchange symmetries. Beyond level 1 we can only get numerical solutions for thecritical points of the potential, but we can do this for any initial separation d. We have plottedeach field in figures 3.3-3.5.The tij and xij fields look similar to what we found at level 1. In fact, as seen in figure 3.6,the Xa field has very nearly the same slope at d = 0 as it did before. The additional terms allgo to 0 at d =√2 just as the string field has at all lower levels. This appears to be related tothe fact that d =√2 is a special point where the tachyon is exactly massless, and that is notaffected by level. The maximum separation before solutions become complex is now d ≈ 1.92,which is another increase beyond previous levels, but not a very large change. In physical units,this separation is dˆ ≈ 8.55√α′.3.2.4 DiscussionThe level 3 slope of Xa at small separations is now Xa ≈ −0.1745d, which is slightly steeperthan the −0.1481 that we found at level 1. It is tempting to relate this to the marginal vevrequired to translate a D-brane a distance d2 . The translation induced by small values of themarginal field would then bexα−1 |0〉 → d ≈ 2.866x, dˆ ≈ 12.73√α′x, (3.41)but this is not the whole story. We will consider a hypothetical exact solution Ψ correspondingto the combination of translation and off-diagonal decay that we are studying. The solution363.2. Solutions(a) (b)Figure 3.3: The potential of the off-diagonal solution at (a) level 3 and (b) levels 0 through 3with the second branch omitted.(a) (b)Figure 3.4: The tachyon and marginal fields of the off-diagonal solution plotted at level 3 overthe allowed range of separations, d. In (a) we see the tachyons Ts (solid) and τ (dashed), whilein (b) we have the field Xa.373.2. Solutions(a) (b) (c) (d)(e) (f) (g) (h)Figure 3.5: The remaining components of the off-diagonal solution at level (3,9) plotted overthe allowed range of separations, d. When two fields are plotted together, the solid line is thediagonal component, while the dashed line is off-diagonal. (a) Us and υ, (b) Vs and ν, (c) W˜sand ω˜, (d) φ˜, (e) Ra, (f) Sa, (g) Ya, and (h) Za as defined in (3.30). The matrices wij andfij were replaced by their alternate forms defined in (3.18) to avoid singularities. All othercomponent fields are 0 at level 3.Figure 3.6: The slope of the field Xa for small separations. Level 3 is the solid line, with lowerlevels having shallower slopes.383.2. Solutionsshould then be split into a part Ψ¯ which is a solution for the translation leaving the D-branescoincident, and a part Ψ′ which represents the decay, but is not a solution about the perturbativevacuum by itself. Each component of the string field can then be split in the same way, so thatwe will write the marginal component of the string field as Xa = X¯a + X ′a. We will label allother fields as φI = φ¯I + φ′I with I an index. Since Ψ¯ is a solution to the full OSFT, we canexpand the theory about that point and must find the same action we would expect for twocoincident D-branes, up to a field redefinition. We will write this field redefinition asX˜a = cXXX′a +∑IcXIφ′I , (3.42)φ˜I = cIXX′a +∑JcIJφ′j , (3.43)where X˜a is the marginal field belonging to the new theory expanded about Ψ¯. The action willhave the form of the action on coincident D-branes, but in terms of the redefined fields:Sd(X¯a +X′a, φ¯I + φ′I) = Sd=0(X˜a, φ˜I) . (3.44)Since this off-diagonal decay of the coincident D-branes is an SU(2) rotation of a single branedecay, it will not involve the marginal field, so X˜a = 0.It is clear that the action Sd has terms like X¯aX ′aφ′I arising, for example, from the TsXaXaterm in the action of (3.33). Since for small separations we know X¯a ∝ d, the action has termslike dX ′aφ′I . We then infer that for small separationscXX → 1, cXI ∼ d→ 0 (3.45)cIX → 0, cIJ → δIJ . (3.46)The redefined marginal field X˜a = 0 together with our ansatz for the field redefinition thengives us that for small separationsX ′a ≈ −∑IcXIφ′I ≈ −∑IcXI φ˜I . (3.47)But for D-brane decay many of the fields φ˜I such as the tachyon are non-zero, so X ′a ∝ d.Unless there is a cancellation, which we have no reason to expect, this correction to X¯a is thesame order of magnitude as X¯a itself. What we have accomplished is to remove any BCFTcalculations from the question of relating the two parameters. Using this method, the problemhas been reduced to one entirely within OSFT, to find the precise field redefinition. Findingthe redefinition, however, is beyond the scope of this work.We have interpreted the solution as an off-diagonal D-brane decay after a translation makesthe two branes coincident, so we expect that as the level of approximation increases the inter-pretation should be approximately valid for a larger range of separation d. From figure 3.3bwe can see that as each even level lowers the energy exactly as happens with a single D-branedecay, both odd levels flatten out the preceding level somewhat. While it is surprising thatthe level 1 curve is so flat, we can understand that it is flatter than the higher level curvesshown. In [21], they studied the behaviour of the potential near the single D-brane vacuum asa marginal parameter was turned on, and they found that the leading quadratic term did not393.2. Solutionsdecrease monotonically as the level was increased. Instead the largest single increase in thatquadratic term came from level 1 to level 2, which may explain why we saw a similar increasein our solution.For d = 0 we see that the entire string field takes the value 12 (1 11 1 ) Φ0. It remains the samesymmetric SU(2) rotation of the string field Φ0 which describes the decay of a single D-brane.All of the fields which are descended from the primary state α25−1c1 |0〉 go to 0 as d→ 0, since notranslation is needed when the D-branes are initially coincident. We might also think that thetwo tachyon curves in figure 3.4a should remain together longer as the level increases, but this isnot true. The approximate string field responsible for translating a D-brane without any decaydoes include a tachyonic component. For our situation where we move both D-branes equallyin opposite directions, exciting Xa, the tachyon in question is Ts and the off-diagonal term isnot excited at all. This explains the separation between the two curves in each of the plotsfor the tachyon tij , as well as uij , vij , and wij as the separation is increased and a translationbecomes necessary.There is one new feature at level 3 which did not appear at level 1. The change of variablesdescribed in (3.18) is singular at d = 1√2, and so we find a pole in the fields ω and φ at thatpoint. In the alternate basis, however, the fields ω˜ and φ˜ defined in (3.17) remain finite as dapproaches the singularity. We do not see any singularity in the level 3 fields rij , sij , yij , and zijbecause their off-diagonal components are protected by exchange symmetry, and the diagonalparts have α0 = 0 so that the transformation has no singularities there.While the correspondence between the marginal parameter and the physical translationremains unknown, the physical translation of the solution is known exactly in this case. Thatthe maximum translation remains finite is troubling. There is no reason why the translation of aD-brane should be limited, but this and all previous marginal solutions have found a maximumvalue of the deformation parameter. In previous cases there was still the possibility that thephysical translation was unbounded, either by a singular correspondence between the marginalparameter and its physical effect, or through a second branch of the solution with decreasingmarginal parameter but increasing effect as was found in a toy model [22]. In our case, however,we know exactly what the physical translation distance is, and we already see a second branchwhich does not have a larger displacement. It seems unlikely that either of these hypothesescould apply based on our solution. If D-brane translations are to be unbounded then we musthope that the small increases in the maximum with each level do not shrink too quickly andform a divergent series. One last possibility comes from recent work [23] where a level truncatedsolution underwent a sudden change from complex to real at level 14. Perhaps certain levelscould have similar behaviour in our system and cause the solution to exist for a much largerrange of D-brane separations. It is not known how this problem is resolved.We have not done anything with the second branch of the solution, which begins at d =√2.The relatively flat energy of this branch suggests that it represents a truly marginal deformation,and if it is a pure translation then we could use the slope of that branch to directly determinea relationship. Unfortunately we do not know the physical interpretation of this branch, so wecannot be sure that it is only a translation, or if it is how far the D-branes move.It is worth mentioning that the boundary state was calculated for a level-truncated marginalsolution in [42], and that method should be applicable here too. We know that the first branch ofour solution represents a single D24-brane at the origin, so we would expect that the boundarystate constructed by the method of [41] should reflect this. It would be interesting to see if the403.2. SolutionsLevel X¯a X(0)a a¯s t¯1/√2 t(0)1 /√21 -0.1798 -0.1557 0.2963 0.296 0.2862 -0.2464 -0.2431 0.3214 0.321 0.3093 -0.2579 -0.2579 0.3301 0.330 0.316Table 3.2: Values of maximum marginal parameters taken from four different cases and variouslevels. X¯a is taken from this work, a¯s from [21], and t¯1 from [24]. For the situations whereit was considered, we have also included the value of the marginal parameter which has thegreatest physical effect, X(0)a and t(0)1 . The values from [24] are estimated from plots.boundary state for the second branch corresponds to two D-branes at the origin, as we wouldexpect from the energy of the solution and the sign of Xa. We also do not know preciselywhat happens near the maximum separation, but as we expect higher level terms to play alarger role for such separations a boundary state based on an approximate solution is not likelyto give accurate results. While a calculation of the boundary state would not give resultsindependent of CFT (the boundary state must be compared to known CFT results), it is apowerful gauge-independent tool for determining physical properties of a SFT solution.3.2.5 Comparison to Previous SolutionsWe can also examine the extremum of the Xa field at the three levels where it was computed.At levels 1 and 2 we can see that the largest Xa is actually reached before the two branchesmeet. At level 3 we see what appears to be a cusp, though it may actually be the beginningsof a loop, based on one data point for the branch starting at d =√2 having an Xa value whichis slightly below the value at the other branch at that point. The extremum values themselvesare comparable to the maximum value of the marginal parameter found in [21] before their twobranches also meet and become complex. The difference between our methods is that they areturning on a gauge field by itself, T-dual to translating a single D-brane without any decay,while in our system the translation will combine with the off-diagonal decay which allows us toget a true level truncated solution without dropping one equation of motion. While our valuesof Xa are smaller than their a¯s, they are on the same order of magnitude, and are growing fasteras the level increases, going from 60% at level 1 to nearly 78% at level 3. Numerical values arelisted in table 3.2.Another marginal deformation that has been studied numerically is the lump solution atmarginal radius [24]. Sen examined a very similar question to ours, attempting to match thestrength of the marginal parameter to the physical effect of that parameter on the CFT. In thatwork, the energy-momentum tensor was used to link the two. Despite the fact that the marginaldeformation studied was different, the two should be dual to each other with the rescalingλ = t1 ↔√2as (3.48)where as is defined in [21] and is equivalent to our x on a single D-brane, and the parameterson the left hand side of the duality are defined in [24]. In this way we were able to extend table3.2 to include the value of the marginal parameter with the greatest physical effect. The rate ofgrowth with level is similar in both cases, and since the t1 marginal deformation is understood413.2. Solutionsto be periodic with period 1, this suggests that our marginal deformation remains valid onlyfor a finite range of the marginal parameter even at arbitrarily high level. We also notice thatthe value of the marginal parameter corresponding to the peak physical effect on the CFT sideis quite close to the maximum value of the parameter in all cases. In [24] it was speculated thatas the level increased the greatest effect t(0)1 might coincide with the maximum of the parametert¯1, which is actually occurring at a much lower level in our solution than in that work. However,there is some suggestion that in our case the two points may begin moving apart again.We can also use this relationship between the two marginal deformations to estimate avalue for the slope (3.41). For this we will assume the exact result proposed in [24], that asthe truncation level becomes large the lump marginal deformation causes the dimension of theinitial D-brane to reduce by one at precisely t1 = 12 . From [46] we see that this marginaldeformation is also dual to turning on a periodic gauge field on a circle of radius 1, and thatthe value of the marginal parameter which reduces the D-brane dimension is exactly half theperiod. This corresponds to a translation half way around the circumference of the circle, adistance pi. We also assume that the translation is a linear function of the marginal parameter.It is this assumption of linearity that we cannot justify, and which means our comparison willbe rough at best. With all of this in mind, we can now say thatdˆ = pi ←→ t1 =12←→ as =12√2(3.49)asα−1 |0〉 → dˆ = 2√2pias ≈ 8.89as. (3.50)This slope is the same order of magnitude as the one we found from our off-diagonal solutioncombining a translation and a decay. In fact as we increased the level we saw our slope headingin that direction, though we certainly do not expect perfect agreement at infinite level becausewe had to assume a linear relationship between translation and marginal parameter in orderto extrapolate this slope from only two known positions on the circle, and because we haveseen that the presence of a D-brane decay will alter the slope of the marginal parameter in thecombined solution studied here.3.2.6 Restoration of SU(2) SymmetryHere we will attempt an alternate method of relating the marginal parameter to the physicaldisplacement of the D-branes. Without a D-brane decay, the marginal parameter is directlyrelated to the displacement, so we can consider other properties of the theory besides off-diagonal decays which are special for coincident D-branes. An obvious choice is to look for theSU(2) symmetry of the theory, which is only present for coincident branes.We consider the SU(2) symmetry of a system of D-branes to be restored if the effectivequadratic terms of the potential for our fields can be written as CIφIijφIji with some mass CI .Taking the tachyon as an example, the mass would be −pi2 and the quadratic term in the actionwould be −pi2tijtji. This means that the eigenvalues of the second derivative matrix over thecoordinates Ts, Ta, and τ must all be equal. Allowing for field redefinitions, we search for themore general condition that the second derivative eigenvalues come in degenerate sets of 3.We hope that near such a degenerate point the masses of the fields would also show the samesplitting for stretched strings as we find with initially separated D-branes.423.2. SolutionsIn [21] Sen and Zwiebach found marginal “solutions” by dropping the one equation ofmotion for the field that we call x. This gave them a one parameter family of string fieldswhich represent the marginal deformation of a D-brane. Due to level truncation, their stringfield was not exactly marginal, but was a good approximation for small values of the parameter.Taking the same approach, we can adjust the parameter to try to restore the SU(2) symmetrywhich was broken by beginning with separated D-branes. With only one parameter we find itimpossible to restore the degeneracy for more than one cluster of fields.We will plot the eigenvalues of the second derivative matrix expanded about such stringfields which are approximately marginal and approximately solutions to the level truncatedtheory. In this case we have two parameters to adjust. We want to pick some value of theinitial separation and see if there is some value of the marginal parameter which restores thedegeneracy of the eigenvalue triplets. Focusing on the sextuplet of figure 3.7a formed by themixing of the tachyon and the level 2 ghost uij , which have the same mass on coincident D-branes, we see three eigenvalues which are mostly unaffected by the marginal parameter, aswell as three eigenvalues which clearly are affected. The parabola opening upwards makes goodphysical sense, as it has a minimum at Xa ≈ −0.022, which we can interpret as the symmetry-restored point where the D-branes are once again coincident. As the marginal parameter movesaway from this point the stretched string becomes heavier. This point would correspond toXa ≈ −0.44d which is significantly steeper than the relationship at level 1, X(1)a = − 427d.Since this is only the minimum of one eigenvalue it does not necessarily represent a restoredSU(2) symmetry. The stretched u ghost does become lighter as the D-branes separate, so adownward-opening parabola is not unexpected, but this is a double eigenvalue with its maximumat Xa = 0, which does not agree with either our expectation that a non-zero marginal parameteris required to restore the symmetry and coincidence of the D-branes, or our expectation thatonly the stretched u mode should become more tachyonic as the D-branes are separated.We can also examine the other sets of eigenvalues which we hope to be degenerate. Forthese fields the masses on a single D-brane tend not to be degenerate, so when we considertwo D-branes we need to look at sets of three eigenvalues unlike the tachyon case where theuij field’s degeneracy with tij meant that we had to consider a set of six eigenvalues. Whathappens in most cases is that two of the eigenvalues, corresponding to the diagonal fields, willbe degenerate and relatively flat, while the off-diagonal eigenvalue is quadratic in the marginalparameter. We would hope to see an extremum that coincides with the diagonal values, alllocated near the same value of the marginal parameter. Instead we see extrema beyond thediagonal, so that the two sets of eigenvalues cross twice as in figure 3.7b, and while most of theextrema are located at Xa = 0, some are not. Some sets of fields have no flat eigenvalue, eithersimilar to figure 3.7a without the constant modes, or having two parabolas opening in the samedirection.This suggests that the level-truncated approach of [21] does not give accurate results evenfor small separations. Note that while we focus on the matrix of second derivatives, in leveltruncation, the first derivatives of the potential do not all vanish, since the equation of motion forXa is not satisfied. This effect decreases with increased truncation level. The computation usedtwist-even fields only. The approximate solution is also exchange-even, and we included all twist-even (both exchange-even and exchange-odd) fields in the computation of the second derivativematrix. Unfortunately, the features we were looking for do not seem to be unambiguouslyvisible at level (3,9). Apparently, the cubic couplings to higher level fields with non-zero vev433.2. Solutions(a) (b)Figure 3.7: Masses of several particles in an effective theory expanded about a marginal stringfield. Several eigenvalues of ∂φ(i)∂φ(j)V are plotted over the marginal parameter Xa and with aninitial separation of d = 0.05. The six eigenvalues in (a) have eigenvectors consisting primarilyof the components of tij and uij , while the three in (b) are primarily associated with sij . Singleeigenvalues are dashed lines, while degenerate pairs are solid.contribute nontrivially to the masses of the lower modes when the D-branes are translated. Itwould be interesting to see whether this can be improved at higher levels.44Chapter 4Renormalized Marginal OperatorsIn this chapter we will study the solution of [36] which represents a marginal deformation ofthe conformal boundary condition in the initial BCFT. What sets this solution apart fromearlier marginal solutions in OSFT is that it still holds when the marginal operator V whichgenerates the deformation has a singular self-OPE, V (0)V (t) ∼ 1t2 . This is done by using arenormalization scheme for integrated operators, which must satisfy six assumptions in order forthe solution to satisfy the equation of motion. When this particular solution was first presentedin [36], a model renormalization scheme was included to demonstrate the process. Our purposeis to examine the space of possible renormalization schemes compatible with the conditionsrequired for a real SFT solution.Recently, an alternative method was given for constructing finite solutions to OSFT fromoperators with singular OPE [39]. This solution, however, does not apply to our main focus, therolling tachyon, because it requires that the boundary condition changing operator σ acts as theidentity in the time direction. For the rolling tachyon, the time direction is the only matter fieldwhich is affected by the boundary condition changing operator. Even in cases where the timedirection is untouched, the solution of [39] may not always be useful. It is based on the boundarycondition changing operator relating the desired solution to some known reference solution, andthis may not always be available. It may still be possible to use without knowing what thatboundary condition changing operator is, as long as some of their properties and three-pointfunctions are known, but the method we will study does not require the bcc operators at all.The simplest renormalization scheme replaces every pair of marginal operators which cancollide with a counterterm which properly cancels the divergence, and then sums over all possiblepairwise replacements. The OPE completely determines the singular part of the counterterm,but we can also ask whether the counterterm should have a finite part and if so what it should be.For a pairwise counterterm, the six assumptions will restrict the finite part to an arbitrary linearfunction of the length of boundary over which the operators are integrated. Considering moregeneral schemes, we allow for higher order counterterms beyond simple pairwise subtractions.At each order in the marginal operator we will find new counterterms which can be added. Thesingular part of each counterterm is once again determined by the OPE and the countertermsat lower order, but the finite parts will each have some amount of freedom. This gives us aninfinite dimensional space of renormalization schemes which satisfy four of the six assumptions.If we restrict this space to renormalization schemes which are linear, it may be a useful startingpoint for future studies of renormalized marginal boundary operators.The two assumptions which are not immediately satisfied by the space we have outlinedare also the most physically meaningful. The BRST conditions (4.5a) and (4.5b) representthe conformality of the new boundary conditions corresponding to the OSFT solution. If theydo not hold then deforming the initial BCFT with a marginal operator renormalized in thisway will no longer give conformal boundary conditions. Unfortunately, the sheer size of the454.1. Setupspace of renormalization schemes makes a thorough examination of which schemes preserveconformality of the boundary beyond our reach. Instead we consider the “little g” schemewhich closely resembles the model used in [36]. This choice is made because the integrandswhich appear in the little g scheme are more clearly finite than what we see in more generalapproaches. As a result, we will actually prove that the BRST conditions hold for this schemeat all orders in the marginal parameter.This examination of the space of renormalization schemes is really just a starting point,as there are many unanswered questions which remain. Most obviously, is it possible to provefiniteness and the BRST conditions for some larger subset of general renormalization schemes.There is no reason to expect that only one renormalization scheme will satisfy the BRSTconditions, so there is a question of uniqueness. If we have two equally valid renormalizationschemes which are supposed to represent the same marginal boundary deformation, are theyrelated to each other by a nontrivial rescaling of the marginal parameter, or do they somehowrepresent different deformations? Since the little g scheme is actually a two-parameter familyof valid renormalization schemes, we have infinitely many such schemes. We expect that theyare related by gauge transformations, but this will not be proven.Another unanswered question pertains to the conformal properties of boundary conditionchanging operators. For many boundary conditions, the boundary condition changing operatoris a conformal primary operator, and so it must behave in a given way when acted upon bythe BRST operator, but when we explicitly construct this for the little g scheme, there is anextra term which behaves differently. It may be that this gives an additional constraint onthe counterterm, requiring such extra terms to vanish. Since solutions to OSFT correspond toBCFT’s, however, we would expect such a constraint to appear without needing to considerthe BCFT side, for example as a violation of the equation of motion, but we have not seenany evidence that the OSFT solution fails with the counterterm considered. Perhaps suchrenormalization schemes correspond to non-primary boundary condition changing operators.This chapter is structured as follows. In section 4.1 we begin by briefly reviewing the solutionof [36]. We then construct the most general renormalization scheme for two marginal operatorsin section 4.2. Because the problem is simpler at this order, we will thoroughly examine theproperties of this scheme. Moving to higher orders in section 4.3, we use a third order exampleto see how the addition of extra counterterms is allowed at each order. We then have to restrictourselves to the little g scheme in order to get useful results which hold at all orders. In section4.3.8 we describe a simple method for defining general linear renormalization schemes, and thengive some evidence that the little g scheme can be described this way. Finally, in section 4.4we will describe a few unanswered questions regarding the structure of general renormalizationschemes.A version of the work presented in this chapter has been published in the Journal of HighEnergy Physics [2].4.1 SetupMany analytical solutions to string field theory are constructed as wedge states with inser-tions, as in (2.22). For the case of solutions representing marginal deformations of the startingboundary CFT, the operators inserted on the boundary are generally the marginal operatorin question as well as some ghosts. While several such marginal solutions have been studied,464.1. Setup[5][33][34], they have been limited to marginal deformations with regular self-OPE. The caseof singular self-OPE was examined in [36] in a more formal setting. Here we will review thesolution of [36]. More recently, an alternative solution has been proposed [38]. In that solutionthe singular OPE of the marginal operators is handled by performing extra integrals into thebulk of the worldsheet to soften the divergences until a finite result is achieved. The solution of[37] also deals with singular marginal deformations, but its focus was on the photon marginaldeformation. It is likely that the same approach could be used for the rolling tachyon, but wewill not be examining this more closely. It has also been suggested [32] that that solution ismost likely equivalent to the one we study.A few pieces of notation must be taken care of. The marginal operator is V (t), and is takento have self-OPEV (t)V (0) ∼1t2+O(1) (4.1)with no 1t term. In CFT a deformed boundary condition on the interval (a, b) is achieved byinserting an exponential of the marginal operator integrated between a and b, defined in termsof a Taylor series in the deformation parameter λ:eλV (a,b) =∞∑n=0λnn!V (a, b)n , (4.2)whereV (a, b)n =(∫ badtV (t))n=∫(a,b)ndnt V (t1) . . . V (tn) . (4.3)Since V (t) has a singular self-OPE, the above expressions need to be regulated. We will denotethe regulated (or renormalized) operators by enclosing them with [ ]r. The string field isdefined in terms of its inner product with an arbitrary test state φ. Such a definition wouldtypically appear as〈φ,Λ〉 = 〈f ◦ φ(0) . . .〉Wn . (4.4)In this example the ellipsis represents the operator content to be inserted in order to define thestring field Λ, and Wn is the wedge state (of circumference n+ 1) on which the inner productis to be taken. This example simplifies the typical case in which a string field theory solution isthe superposition of such states defined on wedges with many different circumferences. For ourpurposes the function f will always be f(ξ) = 2pi arctan ξ which maps the upper half plane tothe wedge state W1. f ◦φ(0) is the conformal transformation of the operator φ(t) correspondingto the state φ, inserted at 0.The formal solution of [36] depends on six assumptions which must be satisfied by the renor-malization procedure. If these conditions are satisfied, the formal solution constructed in [36]will satisfy the SFT equations of motion and be real; however, different renormalization schemescan possibly lead to different SFT solutions. These conditions are basically physical conditionswhich ensure that when [eλV (a,b)]r is inserted on the boundary, the effect is a conformal change474.1. Setupof boundary conditions on the interval (a, b), and nothing else. The conditions areQB[eλV (a,b)]r=[eλV (a,b)OR(b)]r−[OL(a)eλV (a,b)]r, (4.5a)QB[OL(a)eλV (a,b)]r= −[OL(a)eλV (a,b)OR(b)]r, (4.5b)[. . . eλV (a,c) . . .]r=[. . . eλV (a,b)eλV (b,c) . . .]r, (4.5c)[. . . eλ1V (a,b)eλ2V (c,d) . . .]r=[. . . eλ1V (a,b)]r[eλ2V (c,d) . . .]r, b < c , (4.5d)[eλV (a,b)]rand[OL(a)eλV (a,b)]rdo not depend on the circumference of the wedge , (4.5e)[eλ∫ ba dtV (t)]r=[eλ∫ ba dtV (a+b−t)]r. (4.5f)The first two conditions ensure that the resulting boundary condition is conformal. Thefirst condition defines two local (unintegrated) operators OL and OR, which play an importantrole in the solution. The first condition, (4.5a) requires the existence and finiteness of therenormalized operators[OL(a)eλV (a,b)]r and[eλV (a,b)OR(b)]r, implying that the OPE of themarginal operator V with OL,R is not so singular that it cannot be renormalized within thescheme we choose. The second of these two assumptions, (4.5b) expresses the fact that QBis anti-commuting. The third condition, (4.5c) ensures that changing the boundary conditionon the interval (a, b) and (b, c) using the same deformation parameter should be the same aschanging the boundary condition on the interval (a, c). In other words, renormalization shouldnot spoil factorization of exponentials. This condition was called the “replacement condition”in [36] to differentiate it from the factorization condition (4.5d). We will continue to use thisterm. The factorization condition guarantees that the insertion [eλV (a,b)]r does not modifythe boundary conditions away from the interval (a, b). In particular, it requires that whenproducts of operators that are inserted away from each other are renormalized, it is sufficientto renormalize each term separately. In other words, the renormalized operator factorizes foroperators with disjoint support. Next, (4.5e) is a kind of locality condition: the assumption thatthe subtractions involved in renormalizing operators depend only on the operators in question,and not on the size of the wedge state on which they are inserted. Finally, in order to obtain areal solution it is important not to violate the reflection symmetry of the operators, (4.5f). Inaddition to these explicitly stated conditions, a very natural condition of translation invariancewas also implied in [36]. Counterterms may depend on the operator being renormalized as wellas its properties such as the size and shape of the region the operator is integrated over, butshould not depend on the location of the operator.At this point, it is relevant to ask what classes of operators we need to provide a renormal-ization scheme for. Clearly, we need to be able to renormalize exponentials and their products.This is done order by order, so operators such as V (a, b)n must be renormalizable. Further, theaction of the BRST operator QB on V (a, b) (QBV (t) = ∂∂t(cV (t))) immediately implies thatOL(a) = λcV (a) +O(λ2) and OR(b) = λcV (b) +O(λ2). Thus, we must be able to at least writedown such operators as[V (a)eλV (a,b)]r. In fact, we will see that this is sufficient: we need torenormalize products of exponentials of integrated operators with possible insertions of a singleunintegrated V on either the left, or the right, or both. These operators also arise naturallywhen derivatives are taken, for example: ∂∂a[eλV (a,b)]r.484.1. SetupOnce we have decided on the renormalization scheme for[eλV (a,b)]r, derivative operatorssuch as QB[eλV (a,b)]r and∂∂a[eλV (a,b)]r will be fixed. The choice of renormalization schemefor such operators as[V (a)eλV (a,b)]r can influence the explicit form of operators OR,L and theexistence of natural properties such as∂∂a[eλV (a,b)]r?= −[V (a)eλV (a,b)]r, (4.6)but it does not change QB[eλV (a,b)]r or∂∂a[eλV (a,b)]r themselves. In other words, our choiceof renormalization scheme for operators with unintegrated insertions will not affect the SFTsolution. However, it does affect the linearity of the renormalization scheme (for example,property (4.6)). The implications of both this and the replacement condition for linearity will bediscussed in section 4.2.3, and a general procedure for constructing a fully linear renormalizationscheme order by order will be discussed in section 4.3.8.With these assumptions on the renormalization scheme, a solution for regular marginaloperators can be generalized to operators with singular self-OPE. The solution isΨ =1√UAL1√U+1√UQB√U, (4.7)where the wedge states U and AL need to be defined.U ≡ 1 +∞∑n=1λnU (n), AL ≡∞∑n=1λnA(n)L (4.8a)where 1 is the ∗-product identity and the wedge states in the λ-expansion are given by〈φ,U (n)〉=〈f ◦ φ[V (n)(1, n)]r〉Wn, (4.8b)〈φ,A(n)L〉=n∑l=0〈f ◦ φ[O(l)L (1)V(n−l)(1, n)]r〉Wn, (4.8c)〈φ,A(n)R〉=n∑l=0〈f ◦ φ[V (n−l)(1, n)O(l)R (n)]r〉Wn. (4.8d)We should notice that with this definition U (1) = 0. The operator O(l)L/R is the l-th coefficientwhen OL/R is expanded in powers of λ. In practice, we will show thatOL(t) = λcV (t)−12λ2∂c(t) + λ2C1c(t), OR(t) = λcV (t) +12λ2∂c(t) + λ2C1c(t) (4.9)so only O(1)L and O(2)L are non-zero. This is explained in section 4.2.5 and generalizes theargument of [36] which found the same form but with C1 = 0. Since everything has beendefined in terms of series in λ, we define powers of string fields using the appropriate powerseries, with the star product implied whenever string fields are multiplied. For exampleU−1 = 1−∞∑n=1λnU (n) +(∞∑n=1λnU (n))∗(∞∑n=1λnU (n))− . . . (4.10)= 1− λU (1) + λ2(U (1) ∗ U (1) − U (2))− . . . . (4.11)494.2. Quadratic OperatorsThe solution Ψ is a gauge transformation of the simpler solutionΨL = ALU−1, (4.12)which is a solution that does not satisfy the reality condition already mentioned in chapter 3guaranteeing reality of the string field theory action: Ψ = bpz ◦h.c.Ψ = Ψ‡. ΨL will not be thefocus of our study, but it is much simpler to show that this form is a solution to the equationof motion.Using (4.8b), (4.5a), and (4.8c) it is straightforward to show that QBU = AR−AL. We canthen computeQBΨL = QB(AL ∗ U−1) (4.13a)= (QBAL) ∗ U−1 +AL ∗ U−1 ∗ (QBU) ∗ U−1 (4.13b)= (QBAL +AL ∗ U−1 ∗AR) ∗ U−1 −ΨL ∗ΨL (4.13c)so the problem of showing that the equation of motion is satisfied is reduced to the problem ofshowing that QBAL +AL ∗U−1 ∗AR = 0. This is accomplished by using (4.5b) to write QBALin terms of operators inserted on a wedge state, then showing the structural equivalence of thetwo terms by applying (4.5c) and (4.5d). This is the topic of appendix A of [36], and we willnot repeat the details here.While the purpose of this chapter is to give a detailed construction of renormalized operatorssuitable for use in the solution (4.7), we will also study the general structure of allowed oper-ators along the way. The operators we use will generalize the example renormalization schemeprovided by Kiermaier and Okawa in [36]. The formal approach they used for describing thesingularity structure of well regulated operator collisions (for example in equation (4.35)) canbe used to explicitly define a general renormalization scheme for quadratic operators, as we willdemonstrate in section 4.2. When renormalizing products of more than two operators, however,this approach breaks down and we must resort to a less general approach in order to prove theassumptions (4.5). This is the primary concern of section Quadratic OperatorsThe renormalization of operators discussed above is necessary whenever two operators withsingular OPE become arbitrarily close to one another. As we mentioned previously, the OPEis assumed to be V (0)V (t) ∼ 1t2 , but the finite part of the OPE can contain operators otherthan the identity. In order for a relatively simple renormalization scheme to be possible, werequire that any divergences which appear when a marginal operator V approaches this OPEare not too bad. More precisely, we require the finiteness condition which is equation (4.10) of[36]. This condition can be restated as requiring that◦◦∏iV (ti)◦◦ = exp(−12∫ds1ds21(s1 − s2)2δδV (s1)δδV (s2))∏iV (ti) (4.14)remains finite even when more than two of the coordinates ti collide simultaneously. This is acondition on the marginal operator V and will not hold for all marginal operators.504.2. Quadratic OperatorsSince the solution (4.7) is given order by order in λ, the renormalized exponentiated inte-grated operators such as[eλV (a,b)]r should be interpreted as a series, and the renormalizationof powers of integrated operators is what we will need to define, rather than the exponentials.To start with, we will see what counterterms are compatible with renormalizing two operators.Even at this order the structure of renormalized operators differs from the regular case in anumber of ways.We have seen that there are two ways for operators to collide in this solution: they can bothbe integrated with overlapping regions, or one can be fixed at the end of an integration region.The latter case is simpler so we will start there. Setting any other operator insertions aside,the singularity we are studying appears in∫ ba+dt 〈V (a)V (t)〉Wn =1−pin+ 1cot(pin+ 1(b− a))+O () , (4.15)where n + 1 is the circumference of the wedge state’s boundary. The renormalized operator[V (a)V (a, b)]r can now be written as[V (a)V (a, b)]G = lim→0[∫ ba+dt V (a)V (t)−GLab], (4.16)where the counterterm GLab is implicitly a function of . The subscript G on the renormalizationbracket indicates this approach to subtracting off counterterms which are given as functions Gof epsilon. Requiring finiteness means that the singular part must be the integral of the OPE,but the finite part is not constrained by this, soGLab =1+O(0) . (4.17)This cancels the divergence in (4.15). Similarly, we haveGRab =1+O(0) . (4.18)Next we will define the renormalization of the doubly integrated operator∫ b−adt1∫ bt1+dt2 〈V (t1)V (t2)〉Wn =b− a+ln −1+lnpi(n+ 1) sin(b−an+1pi)+O(). (4.19)For this operator we have the form[V (a, b)2]G = 2 lim→0[∫ b−adt1∫ bt1+dt2V (t1)V (t2)−GDab], (4.20)where the counterterm isGDab =b− a+ ln +O(0) . (4.21)The factor of 2 in (4.20) is due to the fact that the operators have been written in a specific order.Instead of including integrals with t2 < t1, we used the indistinguishability of the operators anddoubled the result.514.2. Quadratic OperatorsAlthough it never appears in the solution of the form (4.7), we will need an additionalrenormalized operator in order to prove the assumptions. When two integrated operatorscollide only at a shared edge, we get the operator[V (a, b)V (b, c)]G = lim→0[∫ badt1∫ cb∨(t1+)dt2V (t1)V (t2)−GEabc]. (4.22)Here the notation b ∨ (t1 + ) represents the minimum of b and t1 + . In this case the onlydivergence is logarithmic, and we getGEabc = − ln +O(0) . (4.23)4.2.1 The “Little g” SchemeWhile the renormalization scheme we have just described is extremely useful at this order, whenmore than two operators need to be renormalized, it is often useful to take a different approach.While the “little g” scheme is not as useful at quadratic order, it is perfectly valid, so we willintroduce it here.The critical problem which the “big G” scheme, [. . .]G, has is that the operators are inte-grated and the counterterm is not. At higher orders this will mean that a renormalized operatoris the sum of terms with different numbers of integrals over different regions, which makes someproofs intractable, and causes many other issues including for the finiteness of the scheme. Evensetting aside the analytical proofs, when performing numerical calculations the renormalizationscheme described so far is extremely cumbersome. The integrals over various regulated regionsmust be evaluated repeatedly as the regulator  approaches 0. Since each integral can generallynot be performed analytically and requires a numerical algorithm, this would be an extremelytime consuming process. In addition, since the limit only exists once all of the countertermsare included, each integral itself will diverge as  → 0, and the results will most likely not betrustworthy due to large roundoff errors.The solution is to combine the counterterms into the integrand. We want functions gD(t1, t2),gL(a, t), and gE(t1, t2) which once integrated over the correct -regulated regions will give thecounterterms GDab, GLab, and GEabc respectively. This can be accomplished bygDab(t1, t2) =1(t1 − t2)2+2(b− a)2(1 + ln(b− a) + . . .) , (4.24a)gLab(t1, t2) =1(t1 − t2)2+1(b− a)2+. . .b− a, (4.24b)gEabc(t1, t2) =1(t1 − t2)2+1(c− b)(b− a)(1 + ln((c− b)(b− a)c− a)+ . . .), (4.24c)where the ellipsis in each right hand side is the finite part of the corresponding countertermG. These functions have no more dependence on ; it has been shifted to the region overwhich they are integrated. Additionally, we did not have to choose the functions we did. Anyfunction with the same integral over the appropriate regulated regions will produce an equivalentrenormalization scheme. In particular the constant terms appearing in each of these functionscould have been any function of t1 and t2 with the same integral, but we have chosen constant524.2. Quadratic Operatorsvalues purely for simplicity. This will be examined in more detail in section 4.3.3. The leftcounterterm gLab(t1, t2) is nearly always used with one insertion at a since that is the endpointof the interval. In this case we should feel free to use the shorthandgLab(t)def= gLab(a, t) . (4.25)Now if we define the renormalization by[V (a, b)2]g = 2 lim→0∫ b−adt1∫ bt1+dt2(V (t1)V (t2)− gDab(t1, t2)), (4.26a)[V (a)V (a, b)]g = lim→0∫ ba+dt(V (a)V (t)− gLab(t)), (4.26b)[V (a, b)V (b, c)]g = lim→0∫ badt1∫ cb∨(t1+)dt2(V (t1)V (t2)− gEabc(t1, t2)), (4.26c)then this renormalization scheme is identical to the “big G” scheme when renormalizing thesequadratic operators. The finiteness of these integrals stems from the fact that the integrandsare now completely finite. Whenever two operators approach each other, the counterterm hasa singular term which cancels the divergence and gives a finite quantity. This is apparent fromthe finiteness condition (4.14) which we impose on the marginal operator V .4.2.2 Small Integrated OperatorsHere we will take a moment to briefly discuss one unusual feature of our renormalization scheme,demonstrating that it may not be viable for all wedge states. It naively seems reasonable toexpect that a fully renormalized integrated operator over a set of measure 0 would be 0 itself.Specifically, we will look at the equationlimb→a[V (a, b)2]r = 0 (4.27)and see if it can be satisfied.The OPE for exponential operators on the boundary gives us the form we will be considering.:V (t1) : :V (t2) : =1(t1 − t2)2+O((t1 − t2)0) (4.28)The operator (4.27) is then12limb→a[V (a, b)2]G = limb→alim→0[∫ b−adt1∫ bt1+dt2(1(t1 − t2)2+O((t1 − t2)0))−b− a− ln + . . .] (4.29a)= −1 + limb→a[− ln(b− a) +∫ b−adt1∫ bt1+dt2 O((t1 − t2)0)+ . . .](4.29b)= −1− limb→aln(b− a) . (4.29c)534.2. Quadratic OperatorsWe see that the operator has a finite piece which could be cancelled by making a specific choiceof the finite piece of GDab similar to [36]. But there is also a divergent part which cannot becancelled without the finite piece having ln(b− a) and this would conflict with the assumptions(4.5). Integrating over a smaller region has caused this operator to diverge, and we cannotconsistently set it to 0 despite our intuition. We will also see the necessity of this divergencewhen we compute the derivative of a renormalized operator in section 4.2.3. The logarithmicdivergence of[V (a−∆, a)2]G provides the counterterm that makes V (a)V (a, b) finite. Thestructure of the renormalization scheme satisfying the replacement condition (4.5c) forces us toaccept a divergence for some integrated operators in order for others to remain finite.What this has shown us is that some operators must be handled with care despite beingrenormalized, if we wish to see finite results. Fortunately, the solution (4.7) is built entirelyout of operators with integer domains, so it is safe from this kind of divergence. This doesremain a concern for the possible future construction of renormalized solutions on wedge statesof continuous circumference, as in [34].4.2.3 LinearityThe replacement condition (4.5c) as stated is trivially satisfied because eλV (a,c) = eλV (a,b)eλV (b,c)holds for the bare operators. The implied condition, however, is for us to take the assumptionat each order in λ and bring the combinatoric sum outside of the renormalization.λnn![V (a, c)n]r =n∑j=0λnj!(n− j)![V (a, b)jV (b, c)n−j]r (4.30)Repeated application of this means that operators of the formV (a1)Vn1(a1, b1) . . . Vnk(ak, bk)V (bk) (4.31)satisfy linearity provided a1 < b1 < a2 < . . . < bk so that the intervals are of finite length anddo not overlap. The replacement condition also does not provide linearity if the fixed operatorsV (a1) and V (bk) are not inserted at the same place in each term of the sum. A strongercondition is needed if we want full linearity, including the ability to perform derivatives andintegrals either before or after renormalization with equal results.2 We will return to this oncewe have examined the consequences of the replacement condition alone.Beginning with (4.5c) we will examine its consequences. The singular parts of GLab, GDab andGEabc are already known, so we will focus primarily on the finite parts. Imposing the obviousrequirement of translation invariance we will take the ansatzGLab =1+ CLb−a , (4.32a)GDab =b− a+ ln + CDb−a , (4.32b)GEabc = − ln + CEc−b,b−a , (4.32c)2Integration only commutes with renormalization if both sides are well defined and finite. For an examplewhere this does not work see the discussion of (4.73).544.2. Quadratic Operatorswhere the functions CL/D/E have no more -dependence. This means that our renormalizationscheme is[V (a)V (a, b)]G = lim→0[∫ ba+dt V (a)V (t)−1− CLb−a], (4.33a)[V (a, b)2]G = 2 lim→0[∫ b−adt1∫ bt1+dt2 V (t1)V (t2)−b− a− ln − CDb−a], (4.33b)[V (a, b)V (b, c)]G = lim→0[∫ badt1∫ cb∨(t1+)dt2 V (t1)V (t2) + ln − CEc−b,b−a]. (4.33c)We now take (4.5c) and (4.5d) and explore some restrictions which we can impose on thecounterterms. The first quantity we see is[V (a, c)]r = [V (a, b)]r + [V (b, c)]r , (4.34)which is trivial as there is no renormalization involved. If we insert a fixed operator at a,however, we find our first condition.[V (a)V (a, c)]G = [V (a)V (a, b)]G + [V (a)]G [V (b, c)]G (4.35a)∫ ca+dtV (a)V (t)−1− CLc−a =∫ ba+dtV (a)V (t)−1− CLb−a+∫ cbdtV (a)V (t)(4.35b)CLc−a = CLb−adef= CL (4.35c)The finite part of the counterterm for collision of an integrated operator with a fixed operatoris in fact constant, and does not depend on the size of the integration region. In addition tolinearity, this result involved the assumption that there are no counterterms for the renormal-ization of operators which do not meet. If we had allowed a finite part in the renormalizationof [V (a)V (b, c)]G even though it is not necessary, we could not say anything about the constantCL. Fortunately, neglecting counterterms for any two operators which do not collide is clearlyequivalent to the assumption (4.5d), so this is not only permitted, but required.This property is more general than just the case [V (a)V (a, b)]r. If we insert an arbitrarylocal operator OA to the left of an integrated operator we get the same result.[OA(a)∫ cadtOB(t)]r=[OA(a)∫ badtOB(t)]r+[OA(a)∫ cbdtOB(t)]r(4.36a)=[OA(a)∫ badtOB(t)]r+OA(a)[∫ cbdtOB(t)]r(4.36b)OA(a)∫ ca+dtOB(t)−G(AB)ac = OA(a)∫ ba+dtOB(t)−G(AB)ab +OA(a)∫ cbdtOB(t) (4.36c)G(AB)ac = G(AB)abdef= G(AB) (4.36d)Using only the linearity of the renormalization and a generalization of assumption (4.5d) tothe operators being considered, we have shown that the counterterm for one fixed and one554.2. Quadratic Operatorsintegrated operator must always be independent of the limits of integration. Using exactlythe same reasoning as the above two arguments, we apply linearity and factorization to therenormalized operator[V (a, b)V (b, d)]r = [V (a, b)V (b, c)]r + V (a, b)V (c, d) . (4.37)This tells us that the counterterm for integrated operators colliding at an edge is also indepen-dent of the integration ranges:CEc−b,b−a = CE . (4.38)We will rarely discuss the right handed counterterm GR because it is normally identicalto GL. The same argument applies that we used for CL, and we know that CRb−a = CR is aconstant. Strictly speaking, the condition (4.5c) does not imply any relationship between CLand CR, but stronger linearity conditions such as (4.6) do. Even without the strong linearitycondition, a natural extension of the reflection condition (4.5f) would imply equality of the twoconstants. If we require that[∫ ba dt2V (t1)V (t2)V (t3)|t1=a,t3=b]ris invariant under the reflectionoperation ti → b+ a− ti, then we getV (a)V (a+, b−)V (b)−GRV (a)−GLV (b) = V (a)V (a+, b−)V (b)−GRV (b)−GLV (a) (4.39)and immediately have GR = GL which implies CR = CL. This equation is not required,however. In fact, because the solution is built out of the wedge state U and AL/R (which canbe obtained by acting on U with QB), it cannot have any dependence on the constants CL/Rwhich don’t appear in U . We will see an explicit CL dependence later on, but this is onlyintroduced to cancel the implicit dependence from the renormalization of the operators in AL.Since CL and CR play no part in the solution, we should not expect any constraints on them.The constraints imposed by (4.6) come about because that condition gives the renormalizationscheme a linear structure that goes beyond what is necessary for a solution.Next we will examine the replacement condition (4.5c) for two integrated operators:[V (a, c)2]r =[V (a, b)2]r +[V (b, c)2]r + 2 [V (a, b)V (b, c)]r . (4.40)Our restriction on CDb−a can now be calculated.[V (a, b)V (b, c)]G =12[V (a, c)2]G −12[V (a, b)2]G −12[V (b, c)2]G (4.41a)∫ badt1∫ cb∨(t1+)dt2 (V (t1)V (t2)) + ln − CE=(∫ c−adt1∫ ct1+dt2 −∫ b−adt1∫ bt1+dt2 −∫ c−bdt1∫ ct1+dt2)(V (t1)V (t2))−c− a+ b− c+ a− b+ ln − CDc−a + CDb−a + CDc−b (4.41b)CDc−a − CDc−b − CDb−a − CE = 0 (4.41c)564.2. Quadratic OperatorsIn order to satisfy this, the finite renormalization terms must be CD(∆t) = C0 + C1∆t, andCE = −C0. Together with CL and CR this leaves us with four free parameters which are notfixed. In chapter 5 we will reduce the number by one by fixing CR = CL and consider thethree remaining parameters to be free when we numerically check the validity of the solutionand examine the tachyon profile.A stronger linearity conditionThis is all that the weak version of linearity can tell us about the counterterms. Strong linearity,however, gives us one additional restriction. We would naturally expect that the derivative ofa renormalized operator might behave in the same way as an unrenormalized operator. By thisI mean that we could choose to require that∂a[V (a, b)2]r?= −2 [V (a)V (a, b)]r (4.42)and consequently the related condition∫ badt [V (t)V (t, c)]r?=[∫ badt V (t)V (t, c)]r. (4.43)This will also be relevant when we examine the boundary condition changing operator in section4.4.2. The integral condition is simpler to study, so we write[∫ badt1∫ ct1dt2 V (t1)V (t2)]G=∫ badt1[∫ ct1dt2 V (t1)V (t2)]G(4.44a)= lim→0∫ badt1(∫ ct1+dt2 V (t1)V (t2)−1− CL)(4.44b)= lim→0((∫ b−adt1∫ bt1+dt2 +∫ badt1∫ cb∨(t1+)dt2)V (t1)V (t2)−b− a− (b− a)CL).(4.44c)But we can also split the integral before renormalizing it, and write[∫ badt1∫ ct1dt2 V (t1)V (t2)]G=[12V (a, b)2 + V (a, b)V (b, c)]G(4.44d)= lim→0(∫ b−adt1∫ bt1+dt2V (t1)V (t2)−b− a− ln − CDb−a+∫ badt1∫ cb∨(t1+)dt2V (t1)V (t2) + ln − CE).(4.44e)The singularities and -regulated operators cancel – as they must for a consistent theory – andwe findCDb−a = −CE + (b− a)CL . (4.45)574.2. Quadratic OperatorsThis has the same effect as the replacement condition but with an added constraint on CL:CD∆ = C0 + ∆C1 , CE = −C0 , CL = C1 . (4.46)Because a similar calculation gives CR = C1, the strong linearity condition also enforces theextra reflection condition CR = CL which is convenient.While the derivative must give the same condition, we will need the result later, so we deriveit here. Consider the expression ∂a[V (a, b)2]r using the definition of a derivative.∂a[V (a, b)2]G = lim∆→0[V (a, b)2]G − [V (a−∆, b)2]G∆(4.47a)= − lim∆→0[V (a−∆, a)2]G + 2[V (a−∆, a)V (a, b)]G∆(4.47b)We saw in section 4.2.2 that when the renormalized product of two marginal operators isintegrated over a small region the result diverges aslimb→a12[V (a, b)2]G = −1− C0 − (b− a)C1 − limb→aln(b− a) . (4.48)We have included the C1 term which vanishes because the overall 1∆ will allow us to keep it.The next corrections, however, are O((b − a)2) and can be safely dropped. We will use thisresult to get∂a[V (a, b)2]G = 2 lim∆→0lim→0− (V (a−∆, a)V (a, b)) − ln + ln ∆ + 1 + C1∆∆. (4.49)Here we make the observation that(V (a−∆, a)V (a, b)) + ln − ln ∆− 1=∫ 0dz(∫ ba−z+dt V (a− z)V (t)−1)+∫ ∆dz(∫ badt V (a− z)V (t)−1z). (4.50)The first term is small:∫ 0dz(∫ ba−z+dt V (a− z)V (t)−1)= (∫ ba+dt V (a)V (t)−1)+O(2) . (4.51)Without the factor of  in front, the  → 0 limit would give the renormalized operator[V (a)V (a, b)]r, so this quantity must go to zero in that limit despite the1 term. Since the-dependence came from the renormalized operators inside the ∆ → 0 limit, the  limit mustbe taken first, so this operator vanishes despite the 1∆ factor appearing in (4.49). The otherterm is∫ ∆dz(∫ badt V (a− z)V (t)−1z)=∫ ∆0dz(∫ badt V (a− z)V (t)−1z)+O() (4.52a)= ∆ limz→0(∫ badt V (a− z)V (t)−1z)+O() +O(∆2) (4.52b)= ∆ [V (a)V (a, b)]G + ∆CL . (4.52c)584.2. Quadratic OperatorsSo by applying (4.50), we get that∂a[V (a, b)2]G = −2 lim∆→0(1∆(∆ [V (a)V (a, b)]G + ∆(CL − C1) +O(∆2)))(4.53a)= −2 [V (a)V (a, b)]G + 2(C1 − CL) . (4.53b)Of course from (4.47a), if we are to require linearity to the extent that the limit ∆ → 0commutes with renormalization along with the sum of two terms, then we would have∂a[V (a, b)2]r =[lim∆→0V (a, b)2 − V (a−∆, b)2∆]r(4.54a)=[∂aV (a, b)2]r (4.54b)= −2 [V (a)V (a, b)]r . (4.54c)So we once again see that strong linearity requires CL = C1.We have found a condition on the counterterms that gives linearity for any operators wehave considered so far at quadratic order, but we would still like to prove that the full linearitycondition holds for all operators, given the condition CL = C1. To prove linearity, we can con-struct a linear operator which acts on the singular operators to produce the renormalized ones.This will be done as the composition of two linear operators: one to produce the counterterms,and the other to properly regulate the integrals with  separations. The two linear operatorswill not commute with each other, so we must apply the one to produce counterterms first.The operatorL =∫dxdy δ(x− y)GLδδV (x)δδV (y)+12lim∆→0∫dxdy(δ′(x− y + ∆)− δ′(x− y −∆))GEδδV (x)δδV (y)(4.55)replaces a pair of operators which can collide with the appropriate counterterm. We will referto the first term as L1 and the second as L2. Taking the δ function to be the limit of symmetricpeaks at zero so that∫∞0 δ(x)dx =12 , the following are straightforward to show.L1 (V (a)V (a, b)) = 2∫ badx GLδ(x− a) = GL (4.56a)L2 (V (a)V (a, b)) =∫ badx GE(δ′(x− a+ ∆)− δ′(x− a−∆))= 0 (4.56b)L1 (V (a, b)V (b, c)) = 2∫ badx∫ cbdy δ(x− y)GL = 0 (4.56c)L2 (V (a, b)V (b, c)) =∫ badx∫ cbdy(δ′(x− y + ∆)− δ′(x− y −∆))GE = GE (4.56d)L1(V (a, b)2)= 2∫ badx GL = 2(b− a)GL (4.56e)L2(V (a, b)2)=∫ badx∫ bady(δ′(x− y + ∆)− δ′(x− y −∆))GE = −2GE (4.56f)594.2. Quadratic OperatorsFrom the definitions of the counterterms, we know that GDab = −GE+(b−a)GL, so the operatorL correctly produces the counterterms for each operator being considered. In the event thatthe two operators never meet, the δ functions will ensure that no counterterms are added, andthe factorization property (4.5d) will hold. To add in the appropriate counterterms for anyoperator consisting of two (or less) marginal operators, we apply 1 − L. This proves that theinclusion of these counterterms is a linear operation.The full renormalization procedure must -regulate the operator insertions as well as in-cluding the counterterms. We will define this order by order for all orders, so that we can usethe notation throughout the rest of this chapter. We apply different linear operators dependingon the number of V operators that are inserted. Let A be the most general operator withn insertions of V , A =∫M V (t1) . . . V (tn) where M is some measure on Rn. For example,V (a)V (a, b)2 is associated with a uniform measure on {a}× (a, b)× (a, b) (where {a} is a pointand (a, b) is an interval). When adding two such operators, we simply add the correspondingmeasures. The map A → (A) acts on the measure M(A) associated with A by setting itto zero for any point (t1, . . . , tn) such that |ti − tj | <  and leaving it unchanged otherwise.Denote this map by U so that U (M(A)) = M ((A)). Since the action of the map U onany given point within a measure depends only on the coordinates of that point, we have thatU(M(A) + M(A˜)) = U(M(A)) + U(M(A˜)). A → (A) is a linear map for any operator,which together with the existence of the linear operator L shows that renormalization definedby[A]r = ((1− L)A) (4.57)is a linear map.Now let us consider the operator[V (a, b)2]G again. We already know from the definitionthat this is(V (a, b)2) − 2GDab, but we can also write it as[V (a, b)2]G =[∫ badt(∫ tads+∫ btds)V (t)V (s)]G. (4.58a)If we are to assume full linearity then we get=∫ badt ([V (a, t)V (t)]G + [V (t)V (t, b)]G) (4.58b)= lim→0∫ badt((V (a, t)V (t)) + (V (t)V (t, b)) −GR −GL)(4.58c)= lim→0[(V (a, b)2) − (b− a)(GR +GL)]. (4.58d)This has only produced part of the necessary counterterm, as ln  and C0 are both absent fromGL/R. This suggests that a fully linear renormalization scheme is impossible, but the existenceof the linear operators L and U tells us otherwise. The answer is that when L acts on the604.2. Quadratic Operatorsoperators there is an extra term.LV (t)V (t, b) =∫ btdx(2GLδ(x− t) + lim∆→0+GE(δ′(x− t+ ∆)− δ′(x− t−∆)))V (t)V (t, b)(4.59a)=(GL +GE lim∆→0+(δ(b− t+ ∆)− δ(b− t−∆)))V (t)V (t, b) (4.59b)=(GL −GE lim∆→0+δ(b− t−∆))V (t)V (t, b) (4.59c)The last term vanishes when t is fixed (as in V (a)V (a, b)), but not when t is allowed to approachb continuously. Infinite sums and infinitesimal regions are some of the differences between thereplacement condition and the stronger linearity we have been discussing, and this is an exampleof that. To get a fully linear renormalization scheme the left counterterm should really beGLab =1+ C1 + (ln + C0) lim∆→0+δ(b− a−∆) . (4.60)4.2.4 Assumptions (4.5d), (4.5e), and (4.5f)We did not discuss the factorization condition (4.5d) in much detail in the last section, but wedid effectively prove that it holds for the renormalization scheme discussed. The δ functionsappearing in every term of the operator L in (4.55) mean that there are no counterterms foroperators which do not meet at a point.The assumption (4.5e) is trivial in our construction, since at no point in the renormalizationof the integrated operators have we considered the wedge state on which they are embedded.By constructing the counterterms using the local OPE rather than the two-point functions, wehave avoided any difficulties that this assumption may have caused.For the sixth assumption we will ask ourselves the same question we asked about linearity:what does this condition really mean and what do we really require? The construction of thesimple solution (4.12) does not require a reflection condition at all. It is only when we wishto impose the reality condition that we need the renormalized operators to preserve the bareoperators’ twist symmetry in a particular way. Looking at page 29 of [36], we see that theprecise condition required isU ‡ = U, A‡L = AR, A‡R = AL . (4.61)As before, the operator ‡ represents the composition of inverse bpz conjugation and hermitianconjugation, which has the effect of reversing the orientation of all operators inserted in theworldsheet.For the string field U this is quite simple, as only the fully integrated operator [V (a, b)n]rappears. By the definition, (4.33), the counterterms are constant in terms of the integrationvariables. We must also consider the region of integration used for (V (a, b)n), to ensure that itis invariant under the reflection. This amounts to the understanding that we can parameterizethe region in a number of equivalent ways. For example,∫ b−(n−1)adt1 . . .∫ btn−1+dtnn∏i=1V (ti) =∫ ba+(n−1)dtn . . .∫ t2−adt1n∏i=1V (ti) . (4.62)614.2. Quadratic OperatorsThe two regions are identical, as are many other similar parameterizations. If we perform thechange of variables ti → a+ b− tn−i+1 on the right hand side we get∫ b−(n−1)adt1 . . .∫ btn−1+dtnn∏i=1V (a+ b− ti) , (4.63)which is precisely the effect of (4.5f) acting on the original integrated operator. In the last stepwe changed the order in which the operators are written in order to make our equation look likethat of (4.5f), but since thy are bosonic operators we are free to do so. That the countertermsare also invariant under taking ti → a+b− ti and then being rearranged is trivial since they arenot functions of the coordinates ti at all. The reparameterization of the integrals in (4.62) wasthe important step, but it is an identity for the subsets of Rn in question, and would actuallyhold for any integrand.When we consider the other condition, A‡L = AR, we must now consider what happens whenfixed operators are inserted at the endpoints. With the argument that U is symmetric in mind,the only thing left to show is that the operators inserted at the endpoints, OL/R are mappedto each other by twist. This is guaranteed by the BRST operator having the property(QBU)‡ = −(−1)UQBU‡ , (4.64)which since U is even means that AR −AL is odd under this conjugation. NowA‡L −A‡R = AR −AL (4.65a)naturally leads us to the conclusionA‡L = AR , A‡R = AL . (4.65b)This can also be viewed as a statement about the transformation property of OL/R, but thatapproach would require consideration of each term separately.4.2.5 Assumptions (4.5a) and (4.5b)These two assumptions are easily proven at second order in λ. Here we will review the proofof each at the lowest nontrivial order and discuss problems which can occur at higher orders asthey arise. We will also see what happens when we attempt an alternate form of the proof, andresolve the apparent inconsistencies that arise. Throughout this section we will omit writingthe limit → 0, and it should be inferred.Recall that the first assumption isQB[eλV (a,b)]r=[eλV (a,b)OR(b)]r−[OL(a)eλV (a,b)]r. (4.5a)At second order this statement reads as12QB[V (a, b)2]G =2∑n=01(2− n)!([V (a, b)2−nO(n)R (b)]G−[O(n)L (a)V (a, b)(2−n)]G)(4.66)624.2. Quadratic Operatorswhere OL/R =∑n λnO(n)L/R is a local operator to be determined. Since the counterterm is notan operator, it vanishes when acted on by QB. The behaviour of the primitive operators whenacted on by the BRST charge is not difficult to determine by integrating the BRST current ona contour about the operator in question. The results we will need areQBV (t) = ∂t(cV (t)) , QB(cV (t)) = 0 ,QBc(t) = c∂c(t) , QB∂c(t) = c∂2c(t) .(4.67)Using the first of these and the definition of the renormalization scheme, we can start workingout the left hand side of (4.66) explicitly.12QB[V (a, b)2]G =12QB(V (a, b)2) (4.68a)= QB∫ b−adt1∫ bt1+dt2V (t1)V (t2) (4.68b)=∫ b−adt1∫ bt1+dt2 (∂t1cV (t1)V (t2) + V (t1)∂t2cV (t2)) (4.68c)The next step is to integrate by parts:12QB[V (a, b)2]G =∫ b−adt V (t) (cV (b)− cV (t+ )) +∫ ba+dt (cV (t− )− cV (a))V (t)(4.69a)= V (a, b− )cV (b)− cV (a)V (a+ , b) +∫ ba+dt (cV (t− )V (t)− V (t− )cV (t))(4.69b)= V (a, b− )cV (b)− cV (a)V (a+ , b) +∫ ba+dt V (t− )V (t) (c(t− )− c(t)) . (4.69c)In the remaining integral we notice that the ghost factor is O() and will suppress an finitecontributions from the matter part. Now when we rewrite the matter factor using the OPE,we only need to keep the relatively simple divergent term.∫ ba+dt V (t− )V (t) (c(t− )− c(t)) =∫ ba+dt12(−∂c(t) +22∂2c(t))(4.70a)= −1(c(b)− c(a+ )) +12(∂c(b)− ∂c(a)) (4.70b)= −1(c(b)− c(a)) +∂c(a)+12(∂c(b)− ∂c(a)) (4.70c)12QB[V (a, b)2]G = V (a, b− )cV (b)−c(b)+12∂c(b)− cV (a)V (a+ , b) +c(a)+12∂c(a)(4.71a)= [V (a, b)cV (b)]G + CRc(b) +12∂c(b)− [cV (a)V (a, b)]G − CLc(a) +12∂c(a)(4.71b)634.2. Quadratic OperatorsThis has the form of (4.66) whereOR(b) = λcV (b) +λ22∂c(b) + λ2CRc(b) , OL(a) = λcV (a)−λ22∂c(a) + λ2CLc(a) . (4.72)The operators OL/R have an explicit dependence on CL/R, but this only serves to cancelthe dependence of the renormalization scheme so that the full operators[eλV (a,b)OR(b)]G and[OL(a)eλV (a,b)]G are independent of CL/R. This has to be the case because in the left handside of (4.5a) neither of QB or[eλV (a,b)]G has or introduces any dependence on those parame-ters. It is because of this cancellation that the reflection condition (4.5f) does not impose anyrestrictions on CL/R.There is another approach which looks very simple but has a subtle difficulty. This approachinvolves a less well-regulated expression which produces a part of the correct result without anycounterterms or higher order pieces of OL/R. To demonstrate, we will again consider the secondorder calculation. We proceed as before to12QB[V (a, b)2]G =∫ badt (V (a, t− )∂(cV (t)) + ∂(cV (t))V (t+ , b)) (4.73a)=∫ badt (V (a, b)∂(cV (t))) . (4.73b)Now the -bracket is a linear operator so we would normally expect it to commute with theintegral, and we would get12QB[V (a, b)2]G =(V (a, b)∫ badt ∂(cV (t)))(4.73c)= (V (a, b)(cV (b)− cV (a))) . (4.73d)This has the correct first term, but no counterterms or ∂c terms. Where has this approachfailed? While the linearity of the -bracket implies that we should be able to bring the integralover t inside, in practice this does not work. We have assumed that by thinking of the integralas the limit of Riemann sums, the linearity of the -bracket justifies bringing the integral inside,but this is not always the case. An integral such as∫dtt3 is not finite itself and integrating byparts ignores the singularity by using a principal value prescription in order to avoid gettingan undefined result. More accurately, this means introducing a small regulator to protect thesingularity. When the -bracket is taken first, there is no singularity in the integrand because serves the role of regulator. When the integral is performed first, we must have two regulators,and the direction of the limit in this two-dimensional plane has changed. It is this direction ofthe limit which prevents principal value integrals from commuting with the -bracket.Taking the form of OL/R found from the calculation above, the final assumption to proveat quadratic order, (4.5b), isQB([cV (a)V (a, b)]G −12∂c(a) + CLc(a))= −cV (a)cV (b) . (4.74)This is relatively straightforward to show. We begin by expanding the left hand side, and then644.3. Renormalizing Higher Order Operatorsapply (4.67).QB([cV (a)V (a, b)]G −12∂c(a) + CLc(a))= QB(cV (a)∫ ba+dtV (t)− c(a)GLab −12∂c(a) + CLc(a))= −cV (a) (cV (b)− cV (a+ ))−c∂c(a)− CLc∂c(a)−12c∂2c(a) + CLc∂c(a)(4.75)As we did for the first BRST condition, we write V (a)V (a+ ) = 12 + O(0) and we now alsowrite c(a)c(a+ ) = c∂c(a) + 122c∂2c. Several terms cancel and we end up withQB([cV (a)V (a, b)]G −12∂c(a) + CLc(a))= −cV (a)cV (b) (4.76)Thus, the second BRST condition is satisfied at this order.4.3 Renormalizing Higher Order OperatorsWhen only two operators needed to be renormalized, the big G and little g schemes providedidentical results. At higher orders, however, the two approaches naturally extend in differentways. We will carefully examine and compare these two schemes at third and fourth order, andfind that the big G scheme is incorrect at higher orders. Focusing on the little g scheme, wewill prove that it satisfies all of the assumptions (4.5) at any order. This proceeds similarly to[36], but I will include all of the technical details necessary for more rigorous proofs.While we do not know whether the BRST conditions can be proven for other renormalizationschemes, it seems likely that there is still some freedom to choose differing schemes. We willbriefly consider this question beginning with full linearity as a starting point. Consideringlinearity first gives a relatively straightforward path to writing down a general renormalizationscheme order by order which is compatible with all of the assumptions except for the BRSTconditions. Those two assumptions, however, look extremely difficult with this approach, so wewill not pursue it any farther here.4.3.1 Third Order OperatorsBefore we plunge into a computation at all orders, we will consider an extension of the big Gscheme to third order, i.e. the renormalization of a product of three V s.At this order, we define a regularized operator involving a single cubic integrated operatorby following the same regularization pattern as we did for the quadratic operator:[V (a, b)3]G = lim→0[(V (a, b)3) − 6G(3),Dab V (a, b)], (4.77)whereG(3),Dab =b− a+ ln + C(3),Db,a . (4.78)654.3. Renormalizing Higher Order OperatorsThe extra superscript (3) indicates that these are the counterterms at third order. We alsodefine a regularized operator involving two integrated operators:[V (a, b)2V (b, c)]G = lim→0[(V (a, b)2V (b, c)) − 2G(3),DEabc V (b, c)− 2G(3),Eabc V (a, b)], (4.79)and three operators:[V (a, b)V (b, c)V (c, d)]G = lim→0[(V (a, b)V (b, c)V (c, d)) −G(3),EEabcd V (a, b)−G(3),EEabcd V (c, d)],(4.80)whereG(3),Eabc = − ln + C(3),Eabc , (4.81a)G(3),DEabc =b− a+ ln + C(3),DEabc , (4.81b)G(3),EEabcd = − ln + C(3),EEabcd . (4.81c)Notice that we have four new and potentially different finite constants. Using translationinvariance together with the factorization and replacement conditions in a way similar to thatpresented in the quadratic case, we can show thatC(3),Eabc = −C0 (4.82a)C(3),EEabce = −C0 (4.82b)C(3),Dab = (b− a)C1 + C(3)0 (4.82c)C(3),DEabc = (b− a)C1 + C0 , (4.82d)where the constants C0 and C1 are necessarily the same as the ones used at quadratic orderbut C(3)0 is a new independent constant. One can check, by examining all combinations, thatthe replacement condition at third order is satisfied for any value of this constant.We also need to define renormalized operators involving fixed insertions at the endpoints.Using factorization and replacement conditions, these can be constrained to[V (a)V (a, b)2]G = lim→0[(V (a)V (a, b)2) − 2V (a)G(3),DLab − 2V (a, b)G(3),Lab](4.83a)[V (a)V (a, b)V (b)]G = lim→0[(V (a)V (a, b)V (b)) − V (a)G(3),RLab − V (b)G(3),LRab](4.83b)[V (a)V (a, b)V (b, c)]G = lim→0[(V (a)V (a, b)V (b, c)) − V (a)G(3),ELabc − V (b, c)G(3),LEabc], (4.83c)664.3. Renormalizing Higher Order OperatorswhereG(3),DLab =b− a+ ln + C(3),DL0 + (b− a)C1 , (4.84a)G(3),Lab =1+ CL , (4.84b)G(3),RLab =1+ CR , (4.84c)G(3),LRab =1+ CL , (4.84d)G(3),ELabc = − ln − C0 , (4.84e)G(3),LEabc =1+ CL . (4.84f)There are two new constants: C(3),DL0 and its partner, C(3),DR0 , along with C(3)0 from (4.82).Just like CL and CR, however, the constants C(3),DL0 and C(3),DR0 cannot change the SFTsolution and can only affect the form of the BRST insertions OL and OR. Only C(3)0 and thequadratic order constants C0 and C1 have a physical effect on the solution.It is clear that if we were to continue our order-by-order approach to renormalization, wewould find new free parameters. However, at quartic and higher orders, this approach in un-wieldy: it is hard to write down the most general renormalized operator that is demonstrativelyfinite. To study renormalization to all orders, we will no longer try to study the space of allrenormalizations and instead focus on a particular renormalization scheme. The scheme wechose will have C0 and C1 as free parameters, however we will not add new constants at everyorder. We will return to the question of classifying all renormalization schemes in sections 4.3.8and Extension to All OrdersThe big G scheme with strong linearity was defined in (4.57), and we would naturally want toextend this by exponentiating it. We would define the scheme by[A]Gdef= lim→0(e−LA) . (4.85)The simplest exponentiation using the same L as in (4.55) would correspond at third order toC(3)0 = C(3),DL0 = C(3),DR0 = C0 and CL = CR = C1 as before. In practice this would mean[V (a, b)n]G = lim→0bn/2c∑k=0(−1)kn!k!(n− 2k)!(GDab)k(V (a, b)n−2k). (4.86)The little g scheme, on the other hand, does not have as simple an exponential form, but theidea is the same. Since we were not focused on the little g scheme at quadratic order, we should674.3. Renormalizing Higher Order Operatorsrestate the exact form of the counterterms we use:gDab(t1, t2) =1(t1 − t2)2+2(b− a)2(1 + ln(b− a) + C0 + (b− a)C1) , (4.87a)gLab(t1, t2) =1(t1 − t2)2+1(b− a)2+CLb− a, (4.87b)gEabc(t1, t2) =1(t1 − t2)2−1(c− b)(b− a)(1 + ln((c− b)(b− a)c− a)+ C0). (4.87c)We require that for every pair of marginal operators which can meet, we subtract off a coun-terterm defined with the appropriate range. For example,[(V (a, b))n]g = lim→0∫Γa,bdnt∑σ∈Snbn/2c∑k=0(−1)k2kk!(n− 2k)!k∏i=1gDab(tσ(i), tσ(i+k))n∏j=2k+1V (tσ(j)) .(4.88)We define Γa,b (t1, . . . , tn) to be the region (a, b)n minus the places where any two coordinatesare within  of each other. This is the same region used in the definition of (V (a, b)n) but nowthe counterterms are included in the integral. We will often suppress the coordinate list of Γabwhen it is unambiguously implied by the coordinates being integrated over. From this form of[V (a, b)n]g there are two directions we can go. Since the region we integrate over is symmetricwe can remove the sum over the symmetric group to get the slightly more compact form[(V (a, b))n]g = lim→0∫Γa,bdnt∑0≤k≤n2(−1)kn!2kk!(n− 2k)!k∏i=1gDab(ti, ti+k)n∏j=2k+1V (tj) . (4.89)An alternative viewpoint is to write the integrand of (4.88) as a “normal ordered” operatorwhich subtracts a counterterm for every pair of operators:◦◦∏iV (ti)◦◦gdef= exp(−12∫ds1ds2 g(s1, s2)δδV (s1)δδV (s2))∏iV (ti) (4.90)The subscript g on the normal ordering is our notation to indicate what counterterms to sub-tract. When we normal order more complicated products including things like fixed marginaloperators at an endpoint, the normal ordering ◦◦ . . .◦◦g should be interpreted as meaning to sub-tract whichever g counterterm, gD, gL/R, or gE is appropriate to the region the operators willeventually be integrated over. So, for example, in the context of[V (a)V (a, b)2]g =∫ badt1dt2◦◦V (a)V (t1)V (t2)◦◦g, (4.91a)our notation◦◦◦◦gindicates◦◦V (a)V (t1)V (t2)◦◦g= V (a)V (t1)V (t2)−V (a)gDab(t1, t2)− gLab(t1)V (t2)− gLab(t2)V (t1) . (4.91b)684.3. Renormalizing Higher Order OperatorsIt is useful to notice that all of the little g counterterms have the same divergent part,1(t1−t2)2. This is the same function used in the finiteness condition (4.14), so we can representthe little g normal ordering as◦◦∏iV (ti)◦◦g=◦◦exp(∫dr1dr2 (finite terms)δδV (r1)δδV (r1))∏iV (ti)◦◦ 1(s1−s2)2(4.92)which is clearly still finite as long as (4.14) holds. This demonstrates how using one countertermor another, or even different counterterms within a single normal ordered operator as is the casewhen both gD and gL are needed, will always result in a finite operator as long as all of thecounterterms have the same singular term 1(s1−s2)2 . Since the normal ordering gives finiteoperators for any choice of coordinates, we can now remove the holes in the integral and write(4.88) as[V (a, b)n]g =∫ badnt◦◦n∏i=1V (ti)◦◦g. (4.93)It is possible to write the renormalization scheme in an exponential form as well:[eλV (a,b)]g=∞∑n=0λnn!∫ badnt◦◦n∏i=1V (ti)◦◦g(4.94a)=∞∑n=0∫ badnt ◦◦ eλV ◦◦g . (4.94b)The notation in the second line is similar to that commonly used, for example, for the Chern-Simons action on a D-brane: under an n-dimensional integral, we include all the terms fromthe Taylor expansion of the integrand that have the right number of variables to saturate theintegral. It is easy to see that this is the same definition as that in equation (4.88).As long as the integrand is fully normal ordered, the limit  → 0 can be used to simplifythe integration region to (a, b)n and the little g scheme seems to exponentiate quite cleanly.Once we start separating terms and examining the structure in detail, however, the regulatedregion Γa,b (t1, . . . , tn) is highly nontrivial instead of being the product of lower dimensionalregions. One drawback of this is that linearity is not so clear for this scheme. By subtractingoff counterterms which depend on the region of integration in a non-trivial way, and specifyingthat we only subtract off counterterms if the operators can meet, we make it unfeasible to writedown a linear operator like L for this scheme. We might ask whether the two schemes are stillidentical, as they were at quadratic order, but it can be shown that they are not. The criticaldifference between the big G and little g schemes is illustrated by considering(2GDab)k∫Γa,bdt1 . . . dtii∏j=1V (tj) (4.95a)=∫Γa,b (t1,...,ti)×Γa,b (s1,s2)×...×Γa,b (s2k−1,s2k)dt1 . . . dti ds1 . . . ds2ki∏j=1V (tj)k∏j=1gDab(s2j−1, s2j)(4.95b)694.3. Renormalizing Higher Order Operators6=∫Γa,bdt1 . . . dti ds1 . . . ds2ki∏j=1V (tj)k∏j=1gDab(s2j−1, s2j) . (4.95c)If n = 2k + i then the first line is a term in [V (a, b)n]G and the third line is the correspondingterm in [V (a, b)n]g. We might try to argue that since the integrand has no singularity whereone of the sj approaches a tj or an sj belonging to another counterterm, the difference vanishesas → 0 and the difference between the integration regions shrinks. The flaw in this reasoningis that when, for example, s1 is within 2 of some t, then the integrand does become large for|s2 − t| < , which is one of the small regions of difference between the two integrals. As aconcrete example, it can be shown that for any finite function f ,lim→0(∫ badt∫Γa,bds1ds2 −∫Γa,bdt ds1ds2)f(t)g(s1, s2) = (6 + 2 ln 2)∫ badt f(t) . (4.96)This leads to a more severe issue. By comparing the big G and little g schemes at fourthorder, we can show that they are not both finite. From the form (4.89) we know that124[V (a, b)4]g = lim→0∫Γabd4t(1244∏i=1V (ti)−14V (t1)V (t2)gDab(t3, t4) +18gDab(t1, t2)gDab(t3, t4)).(4.97a)For the last two terms we can add and subtract the same thing integrated over the factorizedregions used by the big G scheme:= lim→0[124(V (a, b)4) −12GDab(V (a, b)2) +12(GDab)2+14(∫Γabd2t∫Γabd2s−∫Γabd2t d2s)V (t1)V (t2)gDab(s1, s2)−18(∫Γabd2t∫Γabd2s−∫Γabd2t d2s)gDab(t1, t2)gDab(s1, s2)](4.97b)=124[V (a, b)4]G+14lim→0(∫Γabd2t∫Γabd2s−∫Γabd2t d2s)(V (t1)V (t2)− gDab(t1, t2))gDab(s1, s2)+18lim→0(∫Γabd2t∫Γabd2s−∫Γabd2t d2s)gDab(t1, t2)gDab(s1, s2) . (4.97c)We know that the two schemes are not the same beyond second order, so the appearance ofterms representing the difference is expected. The term with the integral of (V (t1)V (t2) −gDab(t1, t2))gDab(s1, s2) is the equivalent of (4.96) at fourth order. It is the integral over a smallregion of a divergent quantity giving a finite result, times the integral of the finite function704.3. Renormalizing Higher Order OperatorsV (t1)V (t2) − gDab(t1, t2). The last term, however, integrates two divergent quantities over thisregion, and is itself infinite.(∫Γabd2t∫Γabd2s−∫Γabd2td2s)gDab(t1, t2)gDab(s1, s2) = 4(43+ 4 ln 2 +53ln 3)(b− a)+O(ln )(4.98)This proves that the two schemes are not only different, but cannot both be finite at fourthorder. We have already shown that the little g scheme is finite by demonstrating that theintegrand is finite. This confirms that the big G scheme must not properly renormalize alloperators, so the trivial exponentiation of the quadratic renormalization scheme is not correct.To see exactly where the big G scheme has failed, consider the operator V (a, b)n. Eachterm with k counterterms in the corresponding renormalized operator [V (a, b)n]G represents thecollision of 2k marginal operators, and subtracts k powers of the divergence for two collidingoperators. If each pair of operators which are colliding were integrated over the full range(a, b)2 the counterterm would be correct, but the small differences due to avoiding the otheroperators will give rise to subleading divergences. For k = 1 the worst divergence is O(−1) sothe subleading terms are finite, but for k ≥ 2 these subleading divergences need to be cancelled.This is what the big G scheme fails to do. By integrating counterterms over the same regions asthe colliding operators, the little g scheme naturally produces the correct subleading divergencesat each k without having to explicitly write them down, or even know what they are.4.3.3 Alternative Little g SchemesNow we will consider a renormalization scheme with a different counterterm, g˜Dab(t1, t2) =gDab(t1, t2) + ∆ab(t1, t2). The difference ∆ab is assumed to be a finite function of t1 and t2.If the divergent parts of g and g˜ do not agree then g˜ will not properly renormalize simplequadratic operators unless the extra divergences in g˜ integrate to a finite contribution, but wewill not consider this case.The new renormalized operator is[V (a, b)n]g˜n!def=∫ badnt∑σ∈Sn∑0≤k≤n2(−1)k2kk!(n− 2k)!k∏i=1g˜Dab(tσ(2i−1), tσ(2i))n∏j=2k+1V (tσ(j)) (4.99a)=∫ badnt∑σ∈Sn∑0≤k≤n2(−1)k2kk!(n− 2k)!×k∑m=0(km) m∏l=1∆Dab(tσ(2l−1), tσ(2l))k∏i=m+1gDab(tσ(2i−1), tσ(2i))n∏j=2k+1V (tσ(j))(4.99b)=∫ badnt∑σ∈Sn∑0≤m≤n2m∏l=1(−1)m2mm!∆Dab(tσ(2l−1), tσ(2l))×∑m≤k≤n2(−1)k−m2k−m(k −m)!(n− 2k)!k∏i=m+1gDab(tσ(2i−1), tσ(2i))n∏j=2k+1V (tσ(j))(4.99c)714.3. Renormalizing Higher Order Operators=∫ badnt∑0≤m≤n2(−1)m2mm!m∏l=1∆Dab(t2l−1, t2l)×∑σ∈Sn−m∑m≤k≤n2(−1)k−m2k−m(k −m)!(n− 2k)!k∏i=m+1gDab(tσ(2i−1), tσ(2i))n∏j=2k+1V (tσ(j))(4.99d)=∑0≤m≤n2(−1)m2mm!m∏l=1∫ badt2l−1dt2l∆Dab(t2l−1, t2l)∫ badtm+1 . . . dtn∑σ∈Sn−m∑m≤k≤n2(−1)k−m2k−m(k −m)!(n− 2k)!k∏i=m+1gDab(tσ(2i−1), tσ(2i))n∏j=2k+1V (tσ(j))(4.99e)=∑0≤m≤n21m!(−12∫ bad2s∆Dab(s1, s2))m×∫ badn−2mt∑σ∈Sn−m∑0≤k≤n2−m(−1)k2k(k)!(n− 2m− 2k)!×k∏i=1gDab(tσ(2i−1), tσ(2i))n−2m∏j=2k+1V (tσ(j))(4.99f)=∑0≤m≤n21m!(n− 2m)!(−12∫ bad2s∆Dab(s1, s2))m[V (a, b)n−2m]g . (4.99g)While the exponential form would automatically make the combinatorial factors ‘work out’,using this form makes it easier to ensure that the integrand stays finite at every step, a cru-cial part of the proof. This result implies, in particular, that if∫ ba d2s∆Dab(s1, s2) = 0, thenthe operator renormalized using g˜Dab is the same as that renormalized using gDab. However, if∫ ba d2s∆Dab(s1, s2) 6= 0, then the new operator is different, but the difference exponentiates as[eλV (a,b)]g˜= e−12λ2∫ ba d2s∆Dab(s1,s2)[eλV (a,b)]g. (4.100)Extending this idea to include a fixed operator at the endpoint is not too difficult. It isbest done as a two step process, where only one counterterm is altered in each step. We defineg˜Lab(s1, s2) = gLab(s1, s2) + ∆Lab(s1, s2). After a small calculation, we find the expected result[V (a)eλV (a,b)]g˜= e−12λ2∫ ba d2s∆Dab(s1,s2)[(V (a)− λ∫ bads∆Lab(a, s))eλV (a,b)]g. (4.101)We will not have any need for an alternative edge counterterm g˜Eabc, so it is not included here.4.3.4 Assumptions (4.5c), (4.5d), (4.5e), and (4.5f)We have mentioned that linearity is more difficult to show for the little g scheme than forthe trivial (but divergent) exponentiation of the big G scheme. While an argument for the724.3. Renormalizing Higher Order Operatorslikelihood of strong linearity will be given in section 4.3.8, it remains beyond proof for now. Wecan, however, prove that weak linearity (4.5c) holds at all orders.Using the finiteness of the “normal ordered” operators, we extend the definition of the littleg scheme to multiple integrated operators:[ p∏i=1eλiV (ai,ai+1)]g=∞∑k1=0. . .∞∑kp=0p∏i=1∫ ai+1aidkit◦◦p∏i=1eλiVai,ai+1◦◦g(4.102a)=∞∑k1=0. . .∞∑kp=0p∏i=1∫ ai+1aidkit( p∏i=1e−12λ2i gDai,ai+1p−1∏i=1e−12λiλi+1gEai,ai+1,ai+2p∏i=1eλiVai,ai+1).(4.102b)Here the operators Vai,ai+1 are interpreted as vanishing outside of their natural domain (ai, ai+1).The normal ordering here will insert both gDai,ai+1 and gEai,ai+1,ai+2 counterterms depending on theoperators being replaced, and those functions should also be defined to vanish outside of theirnatural domains. Instead of taking the operators and functions to vanish when appropriate, wecould have said that when we taylor expand the exponentials each operator can only be locatedat one of the coordinates integrated between the correct endpoints. These two approaches areobviously equivalent. The factor of 12 appearing with gEai,ai+1,ai+2 is due to the fact that we mustconsider ~t in the region (ai+1, ai+2)× (ai, ai+1) in addition to the region (ai, ai+1)× (ai+1, ai+2)which was used to relate gEai,ai+1,ai+2 to GE , just as the 12 appearing with gDai,ai+1 must also makeup for the fact that we consider t2 < t1.For our study of the replacement condition we will only need to use two integrated operators,and the ellipses representing operators inserted at the endpoints are not important for the proof.We will focus on[eλV (a,b)eλV (b,c)]g=∞∑i=0∞∑j=0∫ badit∫ cbdjt ◦◦ eλV ◦◦g . (4.103)Now we will define an alternative little g scheme, as in the previous section. We chooseg˜Dac(t1, t2) =gDab(t1, t2) , a < t1, t2 < bgDbc(t1, t2) , b < t1, t2 < cgEabc(t1, t2) , a < t1 < b < t2 < c, or a < t2 < b < t1 < c .(4.104)This is chosen so that the statement[eλV (a,b)eλV (b,c)]g =[eλV (a,c)]g˜ is trivial. The replacementcondition holds as long as g˜ and g are equivalent renormalization schemes, and from the previoussection, we know that two little g schemes are equivalent as long as the difference ∆Dac = g˜Dac−gDacvanishes when integrated. Since this is only two-dimensional integration, up to corrections whichvanish as → 0, we have12∫Γa,cd2t(g˜Dac(t1, t2)− gDac(t1, t2))= GDab +GDbc +GE −GDac . (4.105)The divergent parts have to and do cancel, because both the g and g˜ schemes use the samedivergent part of the counterterms. This is why we are justified in freely swapping the domain of734.3. Renormalizing Higher Order Operatorsintegration between (a, c)2 used in (4.100) and the Γa,c shown here. We included the divergentpart of the counterterms simply to demonstrate that this is the same condition we found atquadratic order. As a result the replacement condition (4.5c) holds for the little g scheme atall orders with the same conditions that we found at quadratic order.The factorization property (4.5d) is imposed by the fact that we defined the little g schemeto include counterterms only for pairs of operators which are integrated over domains whichintersect at at least one point. By defining it this way, factorization is trivial. The locality andreflection properties (4.5e) and (4.5f) can be shown using exactly the same arguments as weused at quadratic order. The g counterterms are still independent of any global properties ofthe Riemann surface in which they are inserted, and while not completely independent of theinsertion coordinates, they depend only on (t2 − t1)2. The fact that this dependence is evenand translationally invariant means that reflection will still go through without any problems.4.3.5 Comparison to Kiermaier and OkawaThe renormalization scheme used by Kiermaier and Okawa [36] is nearly equivalent to the littleg scheme. We will construct an alternative little g scheme which is equal to ours up to O()corrections, and which is also equal to the renormalization scheme of [36] for all operators whichappear explicitly in the solution Ψ.The renormalization of the operator[V (a, b)2]r is performed using pair-wise subtractions,as we did, but consists of two stages. The first stage gives a finite operator which does not obeyall of the assumptions (4.5). This operator is denoted by normal ordering.◦◦ V (t1)V (t2)◦◦G = V (t1)V (t2)− Gm(t1, t2) (4.106)whereGm(t1, t2) = 〈V (t1)V (t2)〉Wm =pi2(m+ 1)2 sin2(pi(t2−t1)m+1) (4.107)is the two-point function in the matter BCFT. The two point function depends on the globalproperties of the Riemann surface, and in this case is defined to use the semi-infinite cylinderWm, which has circumference m + 1. The wedge index m is only given a fixed value once thefull wedge state has been built and the correlation function is taken – it is not a fixed value forthe renormalization of a given wedge state. This presents the first obvious problem for such arenormalization scheme: the counterterms are not local and violate (4.5e). This will be solvedby the second step in the renormalization process, but first we must define the normal orderingof integrated operators.Unlike the normal ordering we have defined, when acting on integrated operators they definethis operation to include an -regularization as well. In practice we can define this by applyingthe -regularization and a limit in all cases, and similarly to equation (4.9) of [36] define 3◦◦∏iV (ti)◦◦Gdef= lim→0exp(−12∫dt1dt2 Gm(t1, t2)δδV (t1)δδV (t2))(∏iV (ti)). (4.108)3This is only valid for integrated operators and fixed operators with finite separation. In the case of ◦◦ V (t)n ◦◦Gas in equation (4.10) of [36], this definition fails, but we will not need to consider this case.744.3. Renormalizing Higher Order OperatorsApplying this to operators which actually appear in the solution, we have◦◦ V (a, b)2 ◦◦G = lim→0∫ b−adt1∫ bt1+dt2 (V (t1)V (t2)− Gm(t1, t2))+ lim→0∫ ba+dt1∫ t1−adt2 (V (t1)V (t2)− Gm(t1, t2))(4.109a)and◦◦ V (a)V (a, b)◦◦G = lim→0∫ ba+dt (V (a)V (t)− Gm(a, t)) . (4.109b)The renormalization of higher powers of operators as defined by (4.108) looks exactly as ex-pected, subtracting off the two-point function for every pair of operators.◦◦(V (a, b))n ◦◦G = lim→0∫Γa,bdnt∑0≤k≤n2(−1)kn!2kk!(n− 2k)!k∏i=1Gm(ti, ti+k)n∏j=2k+1V (tj) (4.110)Now that we have defined the normal ordering which gives a finite version of each inte-grated operator in question, we need to repair the locality condition (4.5e). Kiermaier andOkawa accomplished this by adding back in the finite part of the two-point function for eachcounterterm. This is designed to cancel the m-dependence of the propagator. The finite pieceswhich are used are〈V (a, b)2〉r = lnpi2(m+ 1)2 sin2(pi(b−a)m+1) = lnGm(a, b) , (4.111a)〈V (a)V (a, b)〉r = −pim+ 1cot(pi(b− a)m+ 1). (4.111b)At quadratic order, the finished renormalized operators are[(V (a, b))2]G =◦◦(V (a, b))2 ◦◦G +〈V (a, b)2〉r , (4.112a)[V (a)V (a, b)]G =◦◦ V (a)V (a, b)◦◦G + 〈V (a)V (a, b)〉r . (4.112b)More useful for our purposes is the extension of this to all orders.[eλV (a,b)]G= e12λ2〈V (,b)2〉r ◦◦ eλV (a,b) ◦◦G , (4.113a)[V (a)eλV (a,b)]G= e12λ2〈V (,b)2〉r ◦◦ (V (a) + λ 〈V (a)V (a, b)〉r) eλV (a,b) ◦◦G . (4.113b)At first glance, this may look quite different from the little g scheme, but by noticing the re-sult of section 4.3.3 we can combine the finite term〈V (a, b)2〉r, which acts like∫d2s∆Dab(s1, s2),with the counterterm containing the divergence Gm to get an alternate form of this renormal-ization scheme which involves only one step. We need to decide what function we should use to754.3. Renormalizing Higher Order Operatorsget〈V (a, b)2〉r once we integrate it. Fortunately, we notice that in [36] this function is definedby equation (4.38):〈V (a, b)2〉rdef= 2 lim→0(∫ b−adt1∫ bt1+dt2 Gm(t1, t2)−b− a− − ln ). (4.114)Half of the needed function is already supplied by Gm. The rest can be easily found by inte-grating our function gDab with the appropriate choice of constants, C0 = −1 and C1 = 0. Thelittle g scheme which is equivalent to the renormalization scheme of [36] uses the countertermgˆDab(t1, t2) = Gm(t1, t2)−(Gm(t1, t2)− gDab(t1, t2)∣∣C0=−1,C1=0)(4.115a)= gDab(t1, t2)∣∣C0=−1,C1=0. (4.115b)Of course for the renormalization scheme of [36] to be completely equivalent to this gˆscheme, they must be equivalent for more operators than just[eλV (a,b)]gˆ. The calculation for[V (a)eλV (a,b)]gˆ goes through exactly like the doubly integrated case and givesgˆLab(a, t) = gLab(a, t)∣∣CL=0 . (4.116)Together with the always-similar right handed operator, this shows equivalence for all operatorswhich appear explicitly in the solution. The remaining operator we are interested in is actuallytreated slightly differently in [36]. The -bracket used in [36] for the edge collision operator[V (a, b)V (b, c)]G is not the linear one we have been using. Instead they used(V (a, b)V (b, c))KO =∫ b− 2adt1∫ cb+ 2dt2 V (t1)V (t2) . (4.117)The difference between this and our -bracket is illustrated in figure 4.1. The finite termassociated with this operator is〈V (a, b)V (b, c)〉rdef= lim→0(∫ b− 2adt1∫ cb+ 2dt2 G(t1, t2) + ln ), (4.118)which makes the total counterterm subtracted for that operator equivalent togˆEabc(t1, t2) = gEabc(t1, t2)∣∣C0=−1. (4.119)The choice of C0 exactly replaces the missing contribution from integrating 1(t2−t1)2 over thesubregion where one of the coordinates is within 2 of b.Whether or not this renormalization scheme satisfies replacement or even linearity at higherorders is not a simple question. The -bracket is not linear because it sets the measure to zerofor coordinates like ( b+a2 , b) when the operator being renormalized is V (a, b)V (b, c) but leavesthe measure intact at the same location for V (a, b)V (b). This lack of linearity for this operatorsuggests that the renormalization scheme built on it should fail to be linear as well, and withthe replacement condition being closely related to linearity it must be re-evaluated as well. Ifwe were to treat this as a big G scheme, subtracting off fixed counterterms from integrated764.3. Renormalizing Higher Order Operatorsa b cabc c-ϵa+ϵ(a)a b cabc c-ϵa+ϵb-ϵ/2(b)Figure 4.1: Comparison of the two choices of integration region used for the renormalization ofthe integrated operators(V (a, b)2) and(V (b, c)2) (diagonal hatching), and (V (a, b)V (b, c))(cross-hatched): (a) using our prescription for the renormalizations. (b) using the prescriptionof [36]. The difference is the grey strips, which are not covered using the latter choice. Thedashed line indicates the singularity due to colliding operators.operators, then in addition to finiteness failing as we know it must, the replacement conditionwould fail as well. The non-linear -bracket would cause operators to exponentiate differentlyfrom counterterms and the form of CDab would have to be different at each order to make up for it.On the other hand, as long as we insist on using a little gˆ scheme to define the renormalization,the fully symmetrized form of the renormalized integrand remains finite as the limit  → 0 istaken, so this combination of operators and counterterms does not care about alterations ofthe region of integration which are O(). This means that the renormalization scheme of [36]most likely satisfies replacement, and perhaps even strong linearity (when CL = CR = C1) asa result of this insensitivity to holes. We will not dwell on proving this, however, as we havealready proven the replacement condition for our little g scheme with the linear -bracket.Because our scheme is so similar to the Kiermaier and Okawa scheme, it is not surprisingthat in the next sections when we prove the BRST assumptions (4.5a) and (4.5b) we willtake an approach very similar to the one they used. None of the steps they used to provethose two conditions are wrong, but a few will need more justification than was originallygiven. Specifically, the first BRST condition requires the lemma (4.132), and the second BRSTcondition requires that we take great care to ensure that integrands are finite whenever smallchanges are made to the integration region. We must also be clear about why the first BRSTcondition can be safely applied without corrections even when multiplied by divergent factorsor other operators close to the integration region. These technical details, while they do notchange the structure of how we approach the proof, are certainly not trivial and will each beexplained as they are encountered.774.3. Renormalizing Higher Order Operators4.3.6 Proof of the First BRST Condition (4.5a)Because the little g scheme is very similar to the example renormalization scheme from Kier-maier and Okawa in [36], we will follow the proof of (4.5a) there quite closely. The renormalizedoperator we start with this time is, as in (4.89),1n![(V (a, b))n]g =∫Γa,bdt1 . . . dtnbn/2c∑k=0(−1)k2kk!(n− 2k)!k∏i=1gDab(ti, ti+k)n∏j=2k+1V (tj) . (4.120)As in section 4.2.5, we will omit the limit  → 0 throughout this and the next section (wherewe prove the second BRST condition). Because the counterterm gDab appears very frequently,as long as there is no ambiguity we will also drop the indices and simply refer to it as g for thissection and the next only.We wish to show that1n!QB [V (a, b)n]g =2∑l=11(n− l)!([V (a, b)n−lO(l)R (b)]g−[O(l)L (a)V (a, b)n−l]g). (4.121)To begin with we use the well known action of the BRST operator on the marginal deformation(4.67).1n!QB [V (a, b)n]g = QBbn/2c∑k=0(−1)k2kk!(n− 2k)!∫Γa,bdnt(k∏i=1g(ti, ti+k))n∏j=2k+1V (tj)(4.122a)=bn−12 c∑k=0(−1)k2kk!(n− 2k − 1)!∫Γa,bdnt(k∏i=1g(ti, ti+k))n−1∏j=2k+1V (tj) ∂tn (cV (tn)) (4.122b)=bn/2c∑k=0(−1)k2kk!(n− 2k)!∫Γa,bdnt ∂tn(n− 2k)k∏i=1g(ti, ti+k)n∏j=2k+1V (tj)c(tn) (4.122c)In the last line we have simply noticed that a term with k = n/2 vanishes, so we can include itin the sum if n is even. We can now add and subtract the following quantity:bn/2c∑k=1(−1)k2kk!(n− 2k)!∫Γa,bdnt ∂tn2kn−2k∏j=1V (tj)n−k∏i=n−2k+1g(ti, ti+k)c(tn) (4.123a)=bn/2c−1∑k=0(−1)k+12k+1(k + 1)!(n− 2k − 2)!×∫Γa,bdnt ∂tn2(k + 1)n−2k−2∏j=1V (tj)n−k−1∏i=n−2k−1g(ti, ti+k+1)c(tn)(4.123b)= −bn/2c−1∑k=0(−1)k2kk!(n− 2k − 2)!∫Γa,bdnt ∂tnk∏i=1g(ti, ti+k)n−2∏j=2k+1V (tj)g(tn−1, tn)c(tn)(4.123c)784.3. Renormalizing Higher Order OperatorsIn the first two lines, (4.123a) and (4.123b), the V (tj) insertions are listed before the g(ti, ti+k)factors, but since the integration region is symmetric we can relabel any index except for tn.As long as the number of factors of each type is the same and the combinatorics match up, theexpressions are equal. In this instance, we can go from (4.123b) to (4.123c) by relabelling theindices as in table 4.1. Now we take (4.122c) and we add (4.123a) and subtract (4.123c). ThisV gt1 t2 tn−2k−2 tn−2k−1 tn−k−2 tn−k−1 tn−k tn−1 tn↓ ↓ · · · ↓ ↓ · · · ↓ ↓ ↓ · · · ↓ ↓t2k+1 t2k+2 tn−2 t1 tk tn−1 tk+1 t2k tnTable 4.1: How to relabel the integration variables in going from (4.123b) to (4.123c).gives us1n!QB [V (a, b)n]g =bn/2c∑k=0(−1)k2kk!(n− 2k)!∫Γa,bdnt ∂tn(n− 2k)k∏i=1g(ti, ti+k)n∏j=2k+1V (tj)c(tn) + 2kn−2k∏j=1V (tj)n−k∏i=n−2k+1g(ti, ti+k)c(tn)+bn/2c−1∑k=0(−1)k2kk!(n− 2k − 2)!∫Γa,bdnt ∂tnk∏i=1g(ti, ti+k)n−2∏j=2k+1V (tj)g(tn−1, tn)c(tn) .(4.124)We will consider the two integrals separately, defining A and B for convenience.Adef=∫Γa,bdntbn/2c∑k=0(−1)k2kk!(n− 2k)!∂tn(n− 2k)k∏i=1g(ti, ti+k)n∏j=2k+1V (tj)c(tn) + 2kn−2k∏j=1V (tj)n−k∏i=n−2k+1g(ti, ti+k)c(tn)(4.125a)Bdef=bn/2c−1∑k=0(−1)k2kk!(n− 2k − 2)!∫Γa,bdnt ∂tnk∏i=1g(ti, ti+k)n−2∏j=2k+1V (tj)g(tn−1, tn)c(tn)(4.125b)By splitting the expression into A and B, we have given a precise implementation of the idea[V (a, b)n]r ↔[V (a, b)n−1]r V (a, b)− (n− 1)[V (a, b)n−2]rGDab . (4.126)We have taken a quantity with n−1 renormalized insertions and one insertion not renormalized(inserted at tn), and broken it into A with n renormalized operators, and B with n− 2 of themand an extra counterterm. Put another way, the BRST operator acts like a derivative hitting794.3. Renormalizing Higher Order Operatorsonly the marginal operators and not the counterterms. By using (4.126) we can write this as aderivative hitting the entire expression regardless of whether the coordinate has an operator ora counterterm, plus a term where the derivative only hits the counterterm. It is this trick thatallows us to proceed.While this has given us a longer expression, it is advantageous because we can make theintegrand of A finite and do the integral over tn. If we symmetrize all of the dummy coordinatesother than tn, we get an integrand∂tnc(tn)1(n− 1)!∑σ∈Snbn/2c∑k=0(−1)k2kk!(n− 2k)!k∏i=1g(tσ(i), tσ(i+k))n∏j=2k+1V (tσ(j))=1(n− 1)!∂tn(c(tn)◦◦n∏i=1V (ti)◦◦g)(4.127)which is manifestly finite. In this form, it is safe to change the integration region to (a, b)n andperform the trivial integral over tn using the fundamental theorem of calculus. We can thenchange the region of integration to Γa+,b− (t1, . . . , tn−1) and use the still-symmetric region ofintegration to reorder the coordinate labels again and get the simpler expressionA =∫Γa+,b−dn−1tbn/2c∑k=0(−1)k2kk!(n− 2k)!(n− 2k)k∏i=1g(ti, ti+k)n−1∏j=2k+1V (tj)(cV (b)− cV (a))+2kn−2k∏j=1V (tj)n−k−1∏i=2−2k+1g(ti, ti+k)(gDab(tn−k, b)c(b)− gDab(tn−k, a)c(a)) . (4.128)This has localized tn at the boundary, turning[V (a, b)n−1∫ ba dtn∂tncV (tn)]rinto somethingsimilar to[V (a, b)n−1(cV (b)− cV (a))]r. The reason that the derivative does not do exactlythis is that when tn appears in a counterterm it gives c(a)gDab(a, ti), whereas in order to getthe correctly renormalized operator with a fixed insertion we need c(a)gLab(a, ti). Correcting forthis, we can write thatA =1(n− 1)![V (a, b)n−1(cV (b)− cV (a))]g+1(n− 2)![V (a, b)n−2]g(c(b)∫ badt(gRab(t, b)− gDab(t, b))− c(a)∫ badt(gLab(a, t)− gDab(a, t))).(4.129)While it is not normally correct to write a fully renormalized operator times a counterterm,as in the second line here, in this case it is allowed because the counterterms’ divergent partscancel so that everything is finite and independent of . The limit does not prevent us from804.3. Renormalizing Higher Order Operatorstreating the two factors independently. From the definitions in (4.87) we know thatgLab(x, y)− gDab(x, y) =1(b− a)2+CLb− a+2(b− a)2(1 + ln(b− a) + C0 + (b− a)C1) (4.130a)=1(b− a)2+CLb− a+ fDab . (4.130b)where fDabdef= 2(b−a)2 (1 + ln(b− a) + C0 + (b− a)C1) is the constant part of gDab. ThenA =1(n− 1)![V (a, b)n−1(cV (b)− cV (a))]g+1(n− 2)![V (a, b)n−2]g(c(b)− c(a)b− a+ c(b)CR − c(a)CL − (c(b)− c(a))(b− a)fDab).(4.131)Returning now to the other integral, B in (4.125), we notice that the integrand divergeswhenever tn−1 and tn approach each other, but not when these two variables approach any ofthe others. This alone is not enough to factorize the region of integration, but with (4.126)in mind we notice that the rest of the integrand (including the sum and combinatoric factors)is what we would see for[V (a, b)n−2]g, so there are no divergences due to ti approaching anypoint for i < n− 1. We can show, as a lemma, that∫Γa,bdnt∫Γa,bd2s f(~t) ∂s2(gDab(s1, s2)c(s2))=∫Γa,bdnt d2s f(~t) ∂s2(gDab(s1, s2)c(s2))(4.132)for any function f(~t) which is finite on (a, b)n. The difference of the two regions can be writtenin terms of three other regions.(∫Γa,bdnt∫Γa,bd2s−∫Γa,bdnt d2s)f(~t) ∂s2(gDab(s1, s2)c(s2))=n∑i=1∫Γa,bd2s(∫Γa,b ∩|ti−s1|<dnt+∫Γa,b ∩|ti−s2|<dnt)f(~t) ∂s2(gDab(s1, s2)c(s2))−n∑i=1∫Γa,bd2s∫Γa,b ∩|ti−s1|<∩|ti−s2|<dnt f(~t) ∂s2(gDab(s1, s2)c(s2))(4.133)The first and second lines of the right hand side both vanish independently, so we will computethem separately, starting with the first line.Because the function f is finite and is integrated over a region with area of order , wenotice that each of those integrals over ~t is  times a finite function of one of the two remainingcoordinates. Specifically, by definingF (s) =1n∑i=1∫Γa,b ∩|ti−s|<dnt f(~t) , (4.134)814.3. Renormalizing Higher Order Operatorsthe first line of (4.133) is∫Γa,bd2s ∂s2(gDab(s1, s2)c(s2))(F (s1) + F (s2)) . (4.135a)We will not need to know the precise form of F (s) so long as it and its derivative are finite.With the full expression having an  factor out front from the small area of the ti integral, weknow that the finite part of gDab will not play any role, and we only need to consider the singularterm. Integrating by parts, we have∫ b−ads(c(b)F (s) + c(b)F (b)(b− s)2−c(s+ )F (s)− c(s+ )F (s+ )2)+ ∫ ba+(c(s− )F (s) + c(s− )F (s− )2−c(a)F (s) + c(a)F (a)(s− a)2)− (∫ b−ads1∫ bs1+ds2 +∫ ba+ds1∫ s1−ads2)c(s2)F ′(s2)(s2 − s1)2. (4.135b)The integrals with 1(b−s)2 and1(s−a)2 can be done explicitly by taylor expanding F (s) about theappropriate endpoint. The integrals with 12 can be put over a common region by shifting thecoordinate s in one of them. For the double integrals, we will taylor expand the numeratorabout s1 in order to perform the s2 integral.2cF (b)− 2cF (a) +∫ ba+ds(c(s− )− c(s)) (F (s− ) + F (s))− (∫ b−ads1∫ bs1+ds2 +∫ ba+ds1∫ s1−ads2)(cF ′(s1)(s2 − s1)2+∂(cF ′)(s1)s2 − s1+ . . .)(4.135c)Evaluating this further, we get2cF (b)− 2cF (a)− 2∫ bads ∂c(s)F (s)− ∫ b−ads(cF ′(s)−cF ′(s)b− s)−∫ ba+ds(cF ′(s)−cF ′(s)s− a) (4.135d)= 2cF (b)− 2cF (a)− 2∫ bads ∂c(s)F (s)− 2∫ bads cF ′(s) +O( ln ) (4.135e)= 2cF (b)− 2cF (a)− 2∫ bads ∂s (cF (s)) +O( ln ) (4.135f)which goes to zero in the → 0 limit.Turning now to the last line in (4.133), where ti is close to both s1 and s2, we defineF2(s1, s2) =1n∑j=1∫Γa,b ∩|ti−s1|<∩|ti−s2|<dnt f(~t), F3(s1, s2) = F2(s1, s2)c(s2) . (4.136)824.3. Renormalizing Higher Order OperatorsBoth of these functions are finite for the same reasons as F (s) above: they are finite operatorsintegrated over a region with area proportional to , and then divided by . The term we wishto evaluate is(∫ ba+2ds1∫ s1−s1−2ds2 +∫ b−2ads1∫ s1+2s1+ds2 +∫ a+2a+ds1∫ s1−ads2 +∫ b−b−2ds1∫ bs1+ds2)F2(s1, s2)∂s2(gDab(s1, s2)c(s2)). (4.137a)As with the other term, we will integrate this by parts.∫ ba+2ds(F3(s, s− )2−F3(s, s− 2)42)+ ∫ b−2ads(F3(s, s+ 2)42−F3(s, s+ )2)+ ∫ a+2a+ds(F3(s, s− )2−F3(s, a)(s− a)2)+ ∫ b−b−2ds(F3(s, b)(b− s)2−F3(s, s+ )2)− (∫ ba+2ds1∫ s1−s1−2ds2 +∫ b−2ads1∫ s1+2s1++∫ a+2a+ds1∫ s1−ads2 +∫ b−b−2ds1∫ bs1+ds2)∂s2(F2(s1, s2))c(s2)(s2 − s1)2(4.137b)For the terms with a 12 we will gather like denominators, shifting the integration variable whennecessary to match intervals. For the other single integrals, the functions F3(s, a) and F3(s, b)can be taylor expanded about the endpoints a and b and only the first term will contribute, withthe rest of the taylor series giving at most terms of order O( ln ). For the double integrals, wewill also taylor expand ∂s2F2(s1, s2)c(s2) in s2 about s2 = s1 and again only the first term willcontribute. In addition, the last two double integrals will not contribute at all because the s1integrals there provide extra suppression.∫ ba+2dsF3(s− 2, s)− F3(s, s− 2)4+∫ ba+dsF3(s, s− )− F3(s− , s)+ F3(b, b)∫ b−b−2ds(b− s)2+ F3(a, a)∫ a+2a+ds(s− a)2− (∫ ba+2ds1∫ s1−s1−2ds2 +∫ b−2ads1∫ s1+2s1+ds2)∂2(F2(s1, s1))c(s1)(s1 − s2)2(4.137c)Here ∂2F2 is the derivative with respect to the second parameter, and ∂1 will be with respectto the first. Now we taylor expand the numerators on the first line and evaluate an integral foreverything else.12∫ bads (∂1 − ∂2)F3(s, s) +F3(b, b)− F3(a, a)2−(∫ ba+2ds+∫ b−2ads)∂2F2(s, s)c(s)2(4.137d)In order to remove the middle term, we would like to change (∂1 − ∂2) to − (∂1 + ∂2) = −∂s in834.3. Renormalizing Higher Order Operatorsthe first term, which we can do by adding an extra ∂1 piece.−12∫ bads ∂sF3(s, s) +F3(b, b)− F3(a, a)2+∫ bads ∂1F3(s, s)−∫ bads ∂2F2(s, s)c(s) (4.137e)=∫ bads (∂1F2(s, s)− ∂2F2(s, s)) c(s) (4.137f)Now we look back at the definition of F2(s1, s2) and see that it is a symmetric function of itstwo parameters, so that the two derivatives are equal when acting on the line s1 = s2. We thushave zero for all of (4.133).We can now finally evaluate B. This lemma tells us that the domain of integration isequivalent to Γa,b (t1, . . . , tn−2) × Γa,b (tn−1, tn) and we evaluate the integrals with respect totn−1 and tn. The expression in question wasB =bn/2c−1∑k=0(−1)k2kk!(n− 2k − 2)!∫Γa,bdnt ∂tnk∏i=1g(ti, ti+k)n−2∏j=2k+1V (tj)g(tn−1, tn)c(tn)(4.138a)=bn/2c−1∑k=0(−1)k2kk!(n− 2k − 2)!∫Γa,bdn−2tk∏i=0g(ti, ti+k)n−2∏j=2k+1V (tj)×(∫ b−adt1∫ bt1+dt2 +∫ ba+dt1∫ t1−adt2)∂t2 (g(t1, t2)c(t2))(4.138b)=[V (a, b)n−2]g(n− 2)!(∫ b−adt(c(b)(t− b)2−c(t+ )2)+∫ ba+(c(t− )2−c(a)(t− a)2)+∫ badt (c(b)− c(t+ ) + c(t− )− c(a)) fDab) (4.138c)=[V (a, b)n−2]g(n− 2)!(c(b)− c(a)+c(a)− c(b)b− a+∫ b−adtc(t)− c(t+ )2+ (c(b)− c(a))(b− a)fDab) (4.138d)=[V (a, b)n−2]g(n− 2)!(c(b)− c(a)+c(a)− c(b)b− a−c(b− )−∂c(b)2+c(a)+∂c(a)2+ (c(b)− c(a))(b− a)fDab) (4.138e)=[V (a, b)n−2]g(n− 2)!(c(a)− c(b)b− a+∂c(b)2+∂c(a)2+ (c(b)− c(a))(b− a)fDab). (4.138f)844.3. Renormalizing Higher Order OperatorsPutting the pieces back together, we getA+ B =1(n− 1)![V (a, b)n−1(cV (b)− cV (a))]g+[V (a, b)n−2]g(n− 2)!(c(b)− c(a)b− a+ c(b)CR − c(a)CL − (c(b)− c(a))(b− a)fDab)+[V (a, b)n−2]g(n− 2)!(c(a)− c(b)b− a+∂c(b)2+∂c(a)2+ (c(b)− c(a))(b− a)fDab)(4.139a)=[V (a, b)n−1cV (b)]g(n− 1)!+[V (a, b)n−2]g(n− 2)!(∂c(b)2+ CRc(b))−[cV (a)V (a, b)n−1]g(n− 1)!+(∂c(a)2− CLc(a)) [V (a, b)n−2]g(n− 2)!.(4.139b)If we take (4.139b) and multiply it by λn and then sum over n, we arrive at the precise formwe wanted.QB[eλV (a,b)]g=[eλV (a,b)OR(b)]g−[OL(a)eλV (a,b)]g(4.140)whereOL(a) = λcV (a) + λ2CLc(a)−λ22∂c(a), OR(b) = λcV (b) + λ2CRc(b) +λ22∂c(b) . (4.141)The first BRST condition is satisfied for the little g scheme without any higher order correctionsto OL/R.4.3.7 Proof of the Second BRST Condition (4.5b)To prove the second BRST condition, we want to evaluateQB(n− 1)![cV (a)V (a, b)n−1]g −QB(n− 2)!12∂c(a)[V (a, b)n−2]g . (4.142)We have left off the CL term in OL because we know that the constants CL and CR are puregauge in that they never appear in the solution. We are free to make the choice to set them allto zero, which we will do throughout this section in order to simplify the calculation.Let’s take a moment to discuss some notation we will be using to simplify this section. Wehave previously defined the little g scheme by[V (a, b)n]g =∫ badnt◦◦n∏i=1V (ti)◦◦g, (4.143)but for a given interval (a, b) we could have chosen to write it as[V (a, b)n]gDab =∫ badnt◦◦n∏i=1V (ti)◦◦gDab. (4.144)854.3. Renormalizing Higher Order OperatorsThis notation will allow us to use a counterterm gDab whose parameters do not match the regionof integration of V exactly. For example[V (a+ , b)n]gDab =∫ ba+dnt◦◦n∏i=1V (ti)◦◦gDab. (4.145)Further, since the counterterms gD and gL/R will need to be modified independently, we willuse a notation [ · ]gDab,gL/Rabto list the appropriate counterterms and, when necessary, theirparameters.Using the first BRST condition, is it straightforward to show that the second term in (4.142)is−QB(n− 2)!12∂c(a)[V (a, b)n−2]g = −1(n− 2)!c∂2c(a)2[V (a, b)n−2]g+1(n− 3)!∂c(a)2[V (a, b)n−3cV (b)]g +1(n− 3)!c∂c(a)2[V (a)V (a, b)n−3)]g+1(n− 4)!∂c(a)2[V (a, b)n−4]g∂c(b)2. (4.146)In order to compute the other term, however, we will need to know how the first BRST conditionworks for alternative little g schemes. Recalling the general rules for alternative schemes, (4.100)and (4.101), we writeQB[eλV (a,b)]g˜= e−λ22∫ ba d2s∆Dab(s1,s2)QB[eλV (a,b)]g(4.147a)= e−λ22∫ ba d2s∆Dab(s1,s2)([eλV (a,b)OR(b)]gD,gR−[OL(a)eλV (a,b)]gD,gL)(4.147b)=[eλV (a,b)OR(b)]g˜D,gR−[OL(a)eλV (a,b)]g˜D,gL. (4.147c)Since OL/R are defined for the standard little g scheme, they have the simple form of (4.141).To shift the counterterms from gL/Rab to g˜L/Rab = gL/Rab + ∆R/Lab we must use equation (4.101):=[eλV (a,b)(OR(b) + λ2∫ badt∆Rab(t, b))]g˜D,g˜R−[(OL(a) + λ2∫ badt∆Lab(a, t))eλV (a,b)]g˜D,g˜L(4.147d)=[eλV (a,b)OR(b)]g˜−[OL(a)eλV (a,b)]g˜+ λ2∫ badt(∆Rab(t, b)−∆Lab(a, t)) [eλV (a,b)]g˜.(4.147e)The BRST condition has the same form and the same operators OL/R provided that the left andright differences ∆L/Rab are equal, or at least have the same integral. The specific alternative864.3. Renormalizing Higher Order Operatorsscheme we are interested in is the one where the counterterms are too big for the region ofintegration by a small constant amount, . Since we are taking CL/R = 0 in this section, inthis case ∆Rab(t1, t2) = ∆Lab(t1, t2) =1(b−a+)2 −1(b−a)2 so the first BRST condition goes throughwithout any additional terms.With the preliminaries out of the way, the main part of the proof of (4.5b) consists ofcalculating the first term in (4.142).QB(n− 1)![cV (a)V (a, b)n−1]g =QB(n− 1)!∫ badn−1t◦◦cV (a)n−1∏i=1V (ti)◦◦g(4.148a)At this point, we introduce a small parameter  which is implicitly taken to zero. Since theintegrand is finite, we can modify the integration region. We make an -sized modification tothe integration region at a to examine the divergence there and write=QB(n− 1)!∫ ba+dn−1t(cV (a)◦◦n−1∏i=1V (ti)◦◦gDab−(n− 1)c(a)gLab(a, t1)◦◦n−1∏i=2V (ti)◦◦gDab),(4.148b)where the counterterms used in the normal ordering no longer match the integration regiondue to the  regulator. The lack of holes for the bulk of the integrated operators means wecan rewrite those as renormalized integrated operators, where the implicit regulator should betaken to zero before . Also using the fact that QB(cV ) = 0, we find= −cV (a)(n− 1)!QB[V (a+ , b)n−1]gab+c(a)(n− 2)!∫ ba+dt gLab(a, t)QB[V (a+ , b)n−2]gab−c∂c(a)(n− 2)!∫ ba+dt gLab(a, t)[V (a+ , b)n−2]gab.(4.148c)874.3. Renormalizing Higher Order OperatorsThe BRST operator can now act on these renormalized operators using (4.147e) since the-regulator is holding the unintegrated insertion ‘away’, resulting in= −cV (a)(n− 2)!([V (a+ , b)n−2cV (b)]gab−[cV (a+ )V (a+ , b)n−2]gab)−cV (a)(n− 3)![V (a+ , b)n−3]gab(12∂c(b) +12∂c(a))+c(a)(n− 3)!∫ ba+dt gLab(a, t)([V (a+ , b)n−3cV (b)]gab−[cV (a+ )V (a+ , b)n−3]gab)+c(a)(n− 4)!∫ ba+dt gLab(a, t)[V (a+ , b)n−4]gab(12∂c(b) +12∂c(a))−c∂c(a)(n− 2)!∫ ba+dt gLab(a, t)[V (a+ , b)n−2]gab.(4.148d)Rearranging and recombining some of the integrands into finite combinations, and in one placeusing the fact that∫ ba+ dt gLab(t) = −1 +O(), we get= −1(n− 2)!∫ ba+dn−2t◦◦cV (a)n−2∏i=1V (ti)cV (b)◦◦gab−1(n− 3)!∫ ba+dn−3t◦◦cV (a)n−3∏i=1V (ti)◦◦gab12∂c(b)−1(n− 3)!∫ ba+dn−3t12c∂c(a)◦◦V (a)n−3∏i=1V (ti)◦◦gab+1(n− 2)!∫ ba+dn−2t(cV (a) ◦◦ cV (a+ )n−2∏i=1V (ti)◦◦gab −c∂c(a)◦◦n−2∏i=1V (ti)◦◦gab−(n− 2)c(a)gLab(a, t1)◦◦cV (a+ )n−2∏i=2V (ti)◦◦gab).(4.148e)Now that we have some finite integrands, we can once again heal the hole at the left endpoint.= −1(n− 2)![cV (a)V (a, b)n−2cV (b)]g −1(n− 3)![cV (a)V (a, b)n−3]g12∂c(b)−c∂c(a)2(n− 3)![V (a)V (a, b)n−3]g +c∂2c(a)2(n− 2)![V (a, b)n−2]g(4.148f)884.3. Renormalizing Higher Order OperatorsIn the last term of (4.148e) we have used the expansion◦◦ cV (a)cV (a+ )V (a+ , b)n−2 ◦◦gab = O() (4.149a)= cV (a) ◦◦ cV (a+ )V (a+ , b)n−2 ◦◦gab −c(a)c(a+ )2◦◦ V (a+ , b)n−2 ◦◦gab− (n− 2)c(a)∫ ba+dt gLab(a, t)◦◦ cV (a+ , b)V (a+ , b)n−3 ◦◦gab(4.149b)= cV (a) ◦◦ cV (a+ )V (a+ , b)n−2 ◦◦gab −c∂c(a)◦◦ V (a+ , b)n−2 ◦◦gab−12c∂2c(a) ◦◦ V (a+ , b)n−2 ◦◦gab− (n− 2)c(a)∫ ba+dt gLab(a, t)◦◦ cV (a+ , b)V (a+ , b)n−3 ◦◦gab(4.149c)to replace the parentheses in (4.148e) with c∂2c(a)2◦◦∏n−2i=1 V (ti)◦◦gab . This requires defining◦◦ V (a)V (a + ) . . .◦◦gab , which should technically not have any counterterm for the two fixedoperators since they do not meet, but it is clear that including a 1(t1−t2)2 counterterm for thosetwo operators gives a normal ordered operator which is finite as  → 0. It is also clear thatwhich finite part we choose for that counterterm is irrelevant since the ghost factor will suppressit anyways.It is also worth mentioning that we can use explicit third order calculations to show that anyof these steps is correct at the order where the number of operators is manageable. Specifically,I have checked that at third order (4.148c) and (4.148d) both match the expected result12QB[cV (a)V (a, b)2]g = − [cV (a)V (a, b)cV (b)]g−12cV (a)∂c(b) +12c∂2c(a)V (a, b)−12c∂cV (a) .(4.150)Finally we add the two pieces (4.148f) and (4.146) together to seeQB(n− 1)![cV (a)V (a, b)n−1]g −QB(n− 2)!12∂c(a)[V (a, b)n−2]g (4.151a)= −1(n− 2)![cV (a)V (a, b)n−2cV (b)]g −1(n− 3)![cV (a)V (a, b)n−3]g12∂c(b)−c∂c(a)2(n− 3)![V (a)V (a, b)n−3]g +c∂2c(a)2(n− 2)![V (a, b)n−2]g−c∂2c(a)2(n− 2)![V (a, b)n−2]g +1(n− 3)!12∂c(a)[V (a, b)n−3cV (b)]g+c∂c(a)2(n− 3)![V (a)V (a, b)n−3]g +1(n− 4)!12∂c(a)[V (a, b)n−4]g12∂c(b)(4.151b)= −1(n− 2)![cV (a)V (a, b)n−2cV (b)]g −1(n− 3)![cV (a)V (a, b)n−3]g12∂c(b)+1(n− 3)!12∂c(a)[V (a, b)n−3cV (b)]g +1(n− 4)!12∂c(a)[V (a, b)n−4]g12∂c(b) .(4.151c)894.3. Renormalizing Higher Order OperatorsMultiplying by λn and summing this over n givesQB[(λcV (a)−λ22∂c(a))eλV (a,b)]g= −[(λcV (a)−λ22∂c(a))eλV (a,b)(λcV (b) +λ22∂c(b))]g. (4.152)This proves that the assumption (4.5b) holds for the little g scheme at all orders.4.3.8 Linear RenormalizationFor quadratic operators we were able to define a fully linear renormalization scheme by makinga specific choice for the constants CL = CR = C1. Since CL and CR do not affect the solutionin any way, this was like a gauge choice. Unfortunately, we saw that this big G scheme wasinsufficient to get finiteness at fourth order and higher. The method we used to construct alinear renormalization scheme is not particularly well suited to products of many operators, butwe can still sketch how it works and show that at least for third order the little g scheme withthe same choice of constants CL = CR = C1 can be defined in terms of linear operators.We begin with a straightforward and general extension of the quadratic result to higherorders, and define a renormalization scheme by[O(t1, . . . , tn)]rdef= lim→0(e−Lr (O(t1, . . . , tn))) . (4.153)We know from section 4.2.3 that the simplest (but not finite) big G scheme can be found bychoosingLG =∫dxdy δ(x− y)GLδδV (x)δδV (y)+12lim∆→0∫dxdy(δ′(x− y + ∆)− δ′(x− y −∆))GEδδV (x)δδV (y). (4.154)Since we have already thoroughly examined the allowed renormalization schemes at quadraticorder, we know that the action of Lr on any two marginal operators V (t1)V (t2) must matchLG. An appropriate choice of ansatz is thenLr =∞∑n=2∫dnx Lrn(x1, . . . , xn)n∏i=1δδV (xi), (4.155)withLr2(x, y) = LG2 (x, y) = GLδ(x− y) +12lim∆→0GE(δ′(x− y + ∆)− δ′(x− y −∆)). (4.156)The higher order counterterms Lrn with n > 2 are not determined, giving us a huge space ofpossible counterterms to consider. This is not unexpected, and is how this approach generatesnew constants such as the C(3)0 and C(3),DLo that we saw in section 4.3.1. Obviously the spaceof all functions Lrn is much larger than the free parameters which are allowed at each order, but904.3. Renormalizing Higher Order Operatorsthis can be reduced somewhat by the assumptions (4.5). The assumptions give restrictions onthe functions, but at arbitrary order the space will remain too large for us to fully examine.The first restriction we can find, which follows trivially from the factorization condition(4.5d), is the need for δ-functions to put all of the operators being replaced at a single point.A coefficient, L, which does not include some sort of δ-function would be non-zero even foroperators which never meet, and violate the factorization condition.Since operators are renormalized pairwise, we expect only counterterms in which the numberof surviving operators is the same as the number we started with mod 2. At third order, theobvious candidates areLr3(x, y, z) = A(3)V (x)δ(x− y)δ(x− z) +B(3)V (x)δ(x− y)δ(y − z) . (4.157)Other permutations of the coordinates are redundant because the integrals in (4.155) sym-metrize the result. At this point we might ask why we are not considering slightly more generalcounterterms, such as the second line in (4.55), which contains derivatives of the δ function.We know from the explicit third order calculations of section 4.3.1 that the only free param-eters we expect at third order are C(3)0 , C(3),DL0 , and C(3),DR0 , and just as linearity enforcedCL = CR at quadratic order, we expect C(3),DL0 = C(3),DR0 , so there should not be more thanthe two counterterms we already have in A(3) and B(3). A simple example of a counterterm wehave not included is L′3 = V (x)δ(x− y) (δ′(y − z + ∆)− δ′(y − z −∆)) which when applied tothe operator V (a, b)3 adds a fixed marginal operator at each endpoint, V (a) + V (b). This isunwanted because it will not satisfy the BRST conditions.Using the calculations of section 4.3.8 and comparing those results to the generic third orderrenormalization scheme of section 4.3.1, we can show that the cubic linear operator of (4.157)gives rise toC(3)0 = C0 +A(3) +B(3), C(3),DL0 = C(3),DR0 = C0 +A(3) +78B(3) . (4.158)What we have discussed so far at third order is only a shift in the finite part of the big Gscheme’s counterterms at third order. We saw from (4.97) that there are subleading divergencesat fourth order which are not cancelled by the quadratic counterterms of the big G scheme. Asa result, we will require an additional counterterm at fourth order. While explicitly performingfourth order calculations can be quite difficult, the form of the difference between the big Gand little g schemes suggests that we need a counterterm likeLr4(x1, x2, x3, x4) ∼1δ(x1 − x2)δ(x2 − x3)δ(x3 − x4) +O(ln ) . (4.159)With this approach, we could in principle write down the most general finite scheme at quarticorder that satisfies conditions (4.5c) and (4.5d). Then, we would need to check that the BRSTconditions do not impose any extra restriction on the free parameters. This would allow us todiscover whether there are any free parameters at quartic order that affect the SFT solution ina nontrivial way, without analyzing all possible restrictions due to the replacement conditionat this order. It is, of course, extremely likely that such free parameters do exist.914.3. Renormalizing Higher Order OperatorsLinearity of the little g schemeThe little g scheme was not defined as in (4.153), so we would like to show that it is in that spaceof linear renormalization schemes. Although we do not have a proof that this is the case, we willfind the exact form of L up to third order. Because we have exhaustively studied renormalizationschemes at quadratic order, we know that linearity is only possible when CL = CR = C1, andwe must impose this condition on the constants of the little g scheme as well. Taking (4.157)as our starting point, we define a new third order renormalization scheme usingLg˜3(x, y, z) = A(3)V (x)δ(x− y)δ(x− z) +B(3)V (x)δ(x− y)δ(y − z) . (4.160)We then find constants A(3) and B(3) such that [. . .]g˜ = [. . .]g at third order. We do this byevaluating the distinct operators[V (a, b)3]r,[V (a)V (a, b)2]r, and[V (a, b)V (b, c)2]r using bothrenormalization schemes. For the first of these operators, it is obvious that∫dxdydz Lg˜3(x, y, z)δδV (x)δδV (y)δδV (z)V (a, b)3 = 6(A(3) +B(3))∫ badt V (t) , (4.161)and then[V (a, b)3]g˜ =lim→0[(V (a, b)3) − 3(∫ badt∫Γa,b (s1,s2)d2s V (t)gDab(s1, s2) + 2(A(3) +B(3))∫ badt V (t))].(4.162)By comparing this to (4.96), we must choose A(3) + B(3) = −3 − ln 2 in order to get the littleg scheme. A full derivation of (4.96), as well as similar formulae for other regions, is not veryinstructive. The important step is simply to write down an explicit form of the region tointegrate over, corresponding to the difference of the two integrals on the left hand side. Themost common case is that of (4.96), so here we will include that region:(∫ badt∫Γa,bds1ds2 −∫Γa,bdtds1ds2)f(t, s1, s2)=[∫ b−a+2dt∫ t+t−ds1∫ s1−ads2 +∫ b−2a+dt∫ t+t−ds1∫ bs1+ds2 −∫ b−2a+2dt∫ tt−ds1∫ t+s1+ds2+∫ a+2a+dt∫ t−ads1∫ t+s1+ds2 +∫ b−b−2dt∫ bt+ds1∫ s1−t−ds2+∫ a+adt∫ t+ads1∫ bs1+ds2 +∫ bb−dt∫ bt−ds1∫ s1−ads2 + s1 ↔ s2]f(t, s1, s2) . (4.163)This holds for any function f(t, s1, s2), but we often have a symmetric function of s1 and s2,in which case we can trade the exchanged indices s1 ↔ s2 for an overall factor of 2. (4.96) isderived by explicitly performing all of the integrals over s1 and s2 in this formula, and thenseries expanding in .924.3. Renormalizing Higher Order OperatorsFor the second operator we wish to check,[V (a)V (a, b)2]g˜, it is important that we distin-guish the two terms in (4.160) because∫ badxdz V (x)δ(x− a)δ(x− z) =38V (a) , (4.164a)∫ badxdz V (x)δ(x− a)δ(a− z) =14V (a) . (4.164b)With this in mind, the extra counterterms for this operator are given by∫dxdydz Lg˜3(x, y, z)δδV (x)δδV (y)δδV (z)(V (a)V (a, b)2)= A(3)(2∫ badydz V (a)δ(a− y)δ(a− z) + 4∫ badxdz V (x)δ(x− a)δ(x− z))+B(3)(2∫ badydz V (a)δ(a− y)δ(y − z) + 4∫ badxdz V (x)δ(x− a)δ(a− z))(4.165a)=(2A(3) +74B(3))V (a) . (4.165b)This represents the difference between the g˜ scheme and the big G scheme. We compare thisto the little g scheme by explicitly finding the difference between the g and G schemes for thisoperator. It islim→0[2(∫Γa+,bdtds−∫ badt∫ ba+ds)gLab(a, s)V (t)+(∫Γa+,bd2s−∫Γabd2s)gDab(s1, s2)V (a)]= −2 (3 + ln 2)V (a) . (4.166)This tells us that 2A(3) + 74B(3) = −2(3 + ln 2), which together with A(3) +B(3) = −(3 + ln 2)means that in order for the g˜ and g schemes to match we must chooseA(3) = −(3 + ln 2), B(3) = 0 . (4.167)The final condition to check, that[V (a, b)V (b, c)2]g˜ matches the corresponding operatorfrom the little g scheme, proceeds similarly. This time the delta functions are on regions whichcan only touch at a point, so the correction to the pairwise renormalization is∫ badxdydz Lg˜3(x, y, z)δδV (x)δδV (y)δδV (z)(V (a, b)V (b, c)2)= 0 . (4.168)For the schemes to be equivalent we need to show that the corresponding little g calculation islim→0[(∫ badt∫Γbc (s1,s2)d2s−∫ badt∫Γb∨(t+),c (s1,s2)d2s)V (t)gDbc(s1, s2)+2(∫ bads1∫ cbdt∫ cb∨(s1+)ds2 −∫ bads1∫Γb∨(s1+),c (t,s2)ds2dt)V (t)gEabc(s1, s2)]= 0 . (4.169)934.4. DiscussionFor the first term, we can see that the region in the s1, s2 plane can be non-zero only whenb−  < t < b. This further suppresses what would otherwise be a finite correction, and we findzero. The second term requires some calculation to verify, but the largest surviving correctionthere is O( ln ).At third order we needed only the one termLg3(x, y, z) = −(3 + ln 2)V (x)δ(x− y)δ(x− z) (4.170)to correct the big G scheme and make it equivalent to the little g scheme. In the notation ofsection 4.3.1 the little g scheme hasC(3)0 = C(3),DL0 = C(3),DR0 = C0 − (3 + ln 2) . (4.171)At higher orders we know that the big G scheme is not finite so there will be additionalcounterterms carrying the subleading divergences, and we expect additional finite contributionsat each order. At any given order, however, there are many possible counterterms to add, so Iexpect that there will always be enough degrees of freedom to represent the little g scheme asa linear operator. We will leave any attempts to prove this for future work.4.4 DiscussionThere are a number of open questions regarding the choice of renormalization schemes and theirproperties. Here we briefly look at what we know and do not know about them.4.4.1 UniquenessThe little g scheme alone has two free parameters which alter the way operators are renor-malized. The space of allowed linear renormalization schemes, while still not fully understood,looks much larger, and the space of renormalization schemes satisfying the replacement con-dition (4.5c) instead of full linearity is larger still. In contrast, the solutions built from theserenormalized operators represent BCFT’s in which the conformal boundary condition has beendeformed in the same way. The BCFT is parameterized by only one parameter, λ, so we areleft to wonder what the other renormalization parameters do.We have already noticed that some of the extra parameters do not appear in the solution.This is the case for CL and CR, as well as C(3),DL0 , C(3),DR0 and any other left- or right-constants. The solution is built from the wedge states U and AL which only contain the fullyintegrated operator [V (a, b)n]r and the operator [OL(a)V (a, b)n]r which is derived from theBRST transformation of the fully integrated one. Only constants which appear in the fullyintegrated operator will be in the solution. Constants such as CL which naively appear inOL must cancel against the same constant in [cV (a)V (a, b)n]r. This does not deal with theconstants appearing in the fully integrated operator, however. By third order we already haveC0, C1, and C(3)0 which must be understood. Using the little g scheme as an example, we willstart with a renormalization scheme that chooses C0 = C1 = 0 and call it [. . .]g0 . Then theresult (4.100) tells us that these parameters are simply a rescaling of the renormalized operator[eλV (a,b)]g= e−λ2(C0+(b−a)C1)[eλV (a,b)]g0. (4.172)944.4. DiscussionTo see whether this rescaling changes the solution, consider the definition (4.8b) of the wedgestate U which the solution is primarily constructed from. While it may appear that C0 gives asimple rescaling of U , the interval on which V is integrated is different at each order: b−a = n−1.Including a λ-dependent rescaling factor, the width of the integration interval will no longermatch the number of marginal insertions, and the final expression for U will be changed. Weleave the question of whether SFT solutions given by different values of C1 and C0 are relatedby a gauge transformations to future work, and offer only one more observation: introducing anonzero C1 is the same as replacing V (t) with V (t)− λC1. It is worth mentioning that the C1dependence of the derivative in (4.53b) can be found from this rescaling. Clearly∂a(e−λ2(C0+(b−a)C1)[eλV (a,b)]g0)= e−λ2(C0+(b−a)C1)∂a[eλV (a,b)]g0+ λ2C1[eλV (a,b)]g(4.173)produces the C1 term in (4.53b), but the CL dependence must still be found separately in orderto make the result independent of that constant.Leaving now the confines of the renormalization scheme defined in section 4.3, we can askwhether changing C(3)0 changes the SFT solution. It is easy to see that generalizing the rescalingin equation (4.172) to include higher order terms, as in[eλV (a,b)]g˜= e−(C0λ2+C(4)0 λ4+...)−(C1λ2+C(4)1 λ4+...)(b−a)[eλV (a,b)]g0, (4.174)does not result in a change of C(3)0 from the value it has in the scheme of section 4.3, C(3)0 =−(3 + ln 2) + C0. There is, however, another simple change in the renormalization schemeswhich does affect C(3)0 : a renormalization of the perturbation parameter λ. Specifically, we cantake [eλV (a,b)]g˜=[e(λ+6∆C(3)0 λ3+...)V (a,b)]g, (4.175)where ∆C(3)0 = C(3)0 + (3 + ln 2) − C0. The conclusion is then that changing C(3)0 away from−(3 + ln 2) + C0 does affect the SFT solution, but in a benign and easy to understand way:by reparameterizing the deformation flow. This observation also explains why there is noindependent parameter C(3)1 .4.4.2 Boundary Condition Changing OperatorsWe might assume (as has been the focus of recent work [39]) that the point where the boundarycondition is changed behaves as if a boundary condition changing operator σ was inserted there.Specifically, a BCFT with σL(a)σR(b) inserted on the boundary has a new boundary conditionbetween a and b and the original boundary condition elsewhere. In our case this is implementedusingσL(a)σR(b) =[eλV (a,b)]r(4.176)without having to know an explicit form of the local operators σL/R. As a result we know thatQB (σL(a)σR(b)) =[eλV (a,b)OR(b)]r−[OL(a)eλV (a,b)]r, (4.177)954.4. Discussionbut for any conformal primary operator φh with conformal weight h, we would expect that [47]QBφh(t) = c(t)∂tφh(t) + h (∂c(t))φh(t) . (4.178)These two results suggest that[OL(a)eλV (a,b)]r= −c(a)∂a[eλV (a,b)]r− h(λ)∂c(a)[eλV (a,b)]r. (4.179)We saw in (4.53b) how the derivative acts on quadratic operators, so expanding (4.179) at orderλ2 we see[cV (a)V (a, b)]r −12∂c(a) + CLc(a)= [cV (a)V (a, b)]r − (C1 − CL)c(a)−2∑l=01(2− l)!h(l)∂c(a)[V (a, b)2−l]r, (4.180)where h(l) are the taylor coefficients of the conformal weight h(λ). This equation is only satisfiedif h(λ) = λ22 and C1 = 0. C1 was a free parameter in the construction of a SFT solution,and can still take any value, but this suggests that the boundary condition changing operatorcorresponding to the new boundary condition is only primary if the operators are renormalizedwith C1 = 0.As usual, at higher orders there is still more to consider, but in this case there may not beany further restrictions. At third order, using the ansatzes of section 4.3.1 it is straightforwardto show that∂a[V (a, b)3]r = −3[V (a)V (a, b)2]r + 6V (a, b)(C1 − CL)+ 6V (a)(C(3)0 − C(3),DL0). (4.181)The first BRST condition, (4.5a), can also be worked out at third order while including theextra constants C(3)0 , C(3),DL0 , and C(3),DR0 . This results in slightly altered operators at theendpoints.OL(a) = λcV (a)−12λ2∂c(a) + λ2CLc(a) + λ3(C(3),DL0 − C(3)0)cV (a) +O(λ4) (4.182a)OR(b) = λcV (b) +12λ2∂c(b) + λ2CRc(b) + λ3(C(3),DR0 − C(3)0)cV (b) +O(λ4) (4.182b)This has the same extra terms as the derivative, so the condition C1 = 0 is still enough to givea primary boundary condition changing operator with conformal weight h(λ) = λ22 .In many works, when arbitrary boundary condition changing operators are considered, it isassumed for simplicity that they are primary. Here we see a case where the boundary conditionsrelated to generic renormalization require non-primary bcc operators. It is surprising that achange to the renormalization scheme, which we expect to correspond to a gauge transformationor reparameterization of λ, can have such an impact on the associated bcc operator. We sawin section 4.4.1 that the C1 dependence of the derivative comes from rescaling the exponentialoperator. In this context it is natural that C(3)0 should not affect the primarity of the bccoperator since that rescaling is simply a reparameterization of λ, while the other constants area more complicated rescaling. This represents a possible topic of further investigation.964.4. DiscussionWe know that the BRST operator is similar to a derivative and a c ghost when acting onthe marginal operator V (a, b), so it should not be surprising that the additional terms at higherorders are the same as those appearing in the derivative. If this relationship holds at all orders itcould greatly simplify the calculation of OL/R at arbitrary order, but any proof of such a claimwould likely entail the calculation of corrections to OL/R for arbitrary counterterms anyways.Alternatively, similarities between QB and the derivative might lead to a better understandingof how to prove the BRST conditions. This remains an open question.97Chapter 5Rolling TachyonNow that we have examined the algebraic structure of renormalized integrated marginal oper-ators, we will apply this knowledge to a specific case: the rolling tachyon. This is particularlyinstructive due to the existence of previously studied and related solutions with regular self-OPE. We can use our renormalization scheme to examine how the presence of renormalizedoperators affects the solution.We will begin this chapter by summarizing our results and their relation to the literature.This is followed by the details of the tachyon profile and a discussion of the results. We thenproceed to discuss how the calculations were performed. This begins in sections 5.3.1-5.3.6with an explanation of the computer program written to construct the solution algebraically.In section 5.3.7 we discuss how numerical integration was performed, and how floating pointroundoff errors due to the counterterms were handled. In this section we also examine theequation of motion and the action in order to check the validity of our results and see howaccurate the numerical process is and whether the results converge as the precision of thenumerical process is increased.Most of this chapter has been posted on the arXiv [3] and has recently been accepted forpublication by the Journal of High Energy Physics.5.1 Rolling Tachyon Introduction and ConclusionsAs we have seen, in a boundary CFT the boundary condition can be deformed on any sectionof the boundary by exponentiating a marginal operator integrated along it, as ineλ∫dt V (t) . (5.1)The marginal parameter λ controls the strength of the deformation. In Open String FieldTheory, allowed D-brane configurations are in one to one correspondence with classical solutions.The rolling tachyon is the time-dependent solution which corresponds to a decaying D-brane.There are two rolling tachyon solutions obtained by different marginal deformations of the D-brane CFT. The simpler case, the exponential rolling tachyon, involves the marginal operatorV (t) = eX0(t) and represents a D-brane which exists in the infinite past and then decays ata finite time. This case has been studied using level truncation methods [35, 48] as well asanalytically [5, 33, 34], and is relatively simple because the OPE of the marginal operatorwith itself is finite. The more difficult case uses the time-symmetric marginal operator V =√2 cosh(X0), which has the singular self-OPE V (0)V (x) ∼ 1x2 . This rolling tachyon correspondsto placing a D-brane at t = 0 and letting it decay at both t = −∞ and t = +∞. Notice thatthis t is time and not the worldsheet boundary coordinate frequently referred to in chapter 4.In SFT, the tachyon profile is the tachyon component of the string field as a function oftime. Higher level modes are, of course, part of the solution Ψ, but are not calculated. In the985.1. Rolling Tachyon Introduction and Conclusionssymmetric case, the tachyon profile has the formT (t) = 2∞∑n=1bn/2c∑j=0λnβ(j)n cosh((n− 2j)t) , (5.2)where β(j)n are coefficients which can be calculated numerically and bn/2c is the greatest integerless than or equal to n/2, so that n − 2j ≥ 0. In this notation the deformation strength λ istaken to be negative for physical solutions [5]. In the exponential case, λ controls the time atwhich the D-brane decays, while for the time-symmetric case it determines the lifetime of theD-brane, with longer lifetimes corresponding to λ closer to zero.In BCFT studies of D-brane decay such as [49, 50] it was noticed that the point λBCFT = 12√2should exhibit some kind of special behaviour with this normalization for V . The recent workof [51] tells us that the parameter λ here is equivalent to the BCFT parameter for small λ(up to the sign, which is a matter of convention), but that the precise relationship for strongerdeformations is in fact gauge-dependent. They also found that λSFT in string field theory has amaximum which occurs close to, but not in general at, the critical value of the BCFT parameter.This does not represent any limitation on λBCFT, as the relationship is not one-to-one, and itdoes not necessarily limit our λ parameter either, as the relationship between it and λSFT of[51] will also have higher order corrections. In any event, since the solution we study is writtenas a taylor series about λ = 0, we should not expect such features at large λ to be evident.In the regular OPE case, instead of the double sum and time-symmetric cosh functions,energy conservation tells us that there is only a single sum of exponentials involving coefficientswith β(0)n :Treg(t) =∞∑n=1λnβ(0)n ent . (5.3)While the β(j)n are in general gauge-dependent, for one choice of gauge it was proven that thesum in the regular OPE case converges for all λ, with the asymptotic behaviour β(0)n ∼ e−γn2[34]. Numerical data suggests that this is true in other gauges as well, as shown in figure5.3b. The trouble with this is that the tachyon profile itself exhibits wild oscillations whichgrow exponentially in magnitude, while the vacuum without any D-branes is a well definedand finite point in string field space. How these two very different looking string fields arereconciled has been the source of much speculation (see, for example, [27]). Both that workand the calculation of the boundary state in [40, 41] indicate that these wild oscillations are notphysical. The boundary state appears to asymptotically approach the tachyon vacuum despitethe component fields taking values which are very different. This may imply that there is atime-dependent gauge transformation relating the rolling tachyon solution to one where thestring field smoothly interpolates the perturbative and tachyon vacuum states. It has also beensuggested that the energy of the D-brane should be radiated away in the form of closed strings,and the wild oscillations come from attempting to describe closed string physics using only openstrings. Our results confirm that the tachyon profile has the same growing oscillatory behaviourin the time-symmetric case, and do not appear to exclude any of the current hypotheses.When studying the marginal operator V =√2 cosh(X0) leading to the time-symmetricrolling tachyon, we must be careful to avoid singularities arising from the operator’s OPE.995.1. Rolling Tachyon Introduction and ConclusionsAnalytic solutions for marginal deformations require the insertion of many copies of the marginaloperator with separations that are integrated over, and there will be divergences when twooperators approach each other. Fortunately there are several solutions which are intended tohandle this issue [36, 37, 38, 39]. The most recent work, by Erler and Maccaferri, does not applyto solutions which have a non-trivial time direction, so we cannot use it for the rolling tachyon.Fuchs, Kroyter and Potting’s solution was designed with the photon marginal deformation inmind, but it is possible that it could describe the rolling tachyon as well. The solution of [38] isa generalization of [34] to operators with singular OPE, and it could be applied to the rollingtachyon. In fact it has been suggested that this solution could give the tachyon profile in theform (5.9), which would help settle the convergence issue.Our focus, however, will be on the work of Kiermaier and Okawa. They proposed the generalconstruction dependent on the existence of a suitable renormalization scheme [36], which weinvestigated and refined in chapter 4. We found a general renormalization scheme satisfyingthe necessary conditions and showed that it has at least two free parameters, suggesting thatthe tachyon profile could have free parameters as well. Here we will perform the first explicitnumerical calculations for the time-symmetric rolling tachyon solution, and we will show thatthe tachyon profile is a finite function which does in fact depend on the free parameters.When we implement the solution Ψ of [36] with our renormalization scheme we can findthe tachyon profile for the symmetric rolling tachyon. This involves algebraically constructingthe wedge states with insertions corresponding to the solution, taking expectation values, andthen performing the required integrals numerically. Here this is done up to 6th order in λ. Itwill have the form (5.2), where now the function is symmetric in t and all the β(j)n are non-zero. The marginal operator√2 cosh(X0) contains the operators e±X0with both signs, andthe coefficients β(j)n correspond to terms with n − j factors of one of the two operators and jfactors of the other. Since renormalization has the effect of adding counterterms for collisionsof operators with opposite sign, the j = 0 coefficients involve no counterterms and behave verysimilarly to exponential solutions. These show the same β(0)n ∼ e−γn2asymptotic behaviour,implying that the sum∑∞n=1 λnβ(0)n cosh(nt) converges absolutely for all λ. For |λ|  1, as isthe case when the D-brane survives for a long time, the j > 0 coefficients are suppressed dueto extra factors of λ. This results in a decay process which looks very much like the regularcase, as the decay is well separated from the “anti-decay” by the D-brane’s lifetime. Once thislifetime is long enough, further shrinking λ even has the same effect on the decay time as itwould with the exponential rolling tachyon, simply shifting the time of the decay.For the β(j)n coefficients with j > 0, each coefficient is calculated using a number of coun-terterms determined by j. The counterterms in turn are functions of the two parameters of therenormalization scheme, C0 and C1. The bulk coefficients are therefore polynomial functionsof C0 and C1. Because these coefficients are not constants, patterns such as the asymptoticbehaviour for j = 0 could depend on the choice of C0 and C1. Considering only the j = 1coefficients, with C0 = C1 = 0 they are quite a good fit to β(1)n ∼ e−γ1(n−2)3. In fact there isno choice of those constants for which the exponential quadratic behaviour β(1)n ∼ e−γ1(n−2)2fits as closely. This suggests that the sum∑∞n=2 λnβ(1)n cosh((n− 2)t) also converges, but theremay still be some choices of C0 and C1 for which this is not the case, or for which the radiusof convergence in λ is finite. For example, choosing the constants so that β(1)n are a best fitto the exponential cubic behaviour results in β(2)n which are increasing, at least for the three1005.2. The Tachyon Profilecoefficients we can calculate with j = 2.So how does the inclusion of all the β(j)n coefficients affect the shape of the tachyon profile?We show that for |λ|  1 these coefficients are negligible, but as the D-brane lifetime is decreasedthere comes a point where more coefficients must be considered. Some terms cease to dominatefor any range of time, and the number of oscillations actually decreases. The missing oscillationmeans that the effective “period” is significantly decreased. What this means physically is notclear, since the period is a gauge dependent quantity related to the coefficient γ in the exponentof the asymptotic behaviour. The tachyon profile for small λ is very similar to that of [5], whilefor larger λ it has some features similar to the tachyon profile of [34], so perhaps the solutionis interpolating between regular-OPE solutions in different gauges as the marginal deformationstrength is changed. Understanding this phenomenon is left for future work.While here we will only calculate the tachyon profile, it would be interesting in the future tostudy the boundary state using the approach of [41]. Previous work on the boundary state fortime-asymmetric rolling tachyon solutions suggests that it is the same as predicted by BCFT,and our results suggest that we can expect that that would hold true for the time-symmetricrolling tachyon at weak deformation parameter as well. For larger λ, however, it would bevery interesting to see if the renormalization parameters really are gauge, or if they affect theboundary state. In particular, C1 controls whether the boundary condition changing operatoris a conformal primary, and it is not at all clear what physical effect that will have.5.2 The Tachyon ProfileThe solution of [36] presents a promising framework for construction of a time-symmetric rollingtachyon solution, but it was not applied to any specific marginal deformation. Taking thatapproach and inserting the marginal deformation V =√2 cosh(X0) we are able to numericallycompute the tachyon profile up to 6th order in the deformation parameter λ. Since the tachyonprofile has previously been calculated for several exponential rolling tachyon solutions, we cancompare our results in order to determine what qualitative differences appear in the time-symmetric case. It is also useful to have explicit numerical evidence that the renormalizationscheme we found in chapter 4 is effective and the solution remains finite despite the singularOPE that the marginal operator has with itself.The solution takes the form of a wedge state with insertions on the boundary. Whileone insertion will always be at a fixed location, the rest are integrated. The renormalizationprocedure replaces pairs of operators with appropriate counterterms under the integral. Eachoperator V contains two terms carrying ±1 unit of “momentum” in the time direction, but thecounterterms are functions and carry no momentum. In (5.2) the coefficient β(j)n clearly containsthe part of the tachyon profile with n factors of λ and k = n− 2j units of this momentum, so itfollows that the coefficients β(0)n contain no counterterms. This is as it should be since operatorse±X0with the same sign have a regular OPE; the singular OPE of the cosh(X0) marginaloperator comes entirely from the collision of exponentials with opposite sign. The index j,which counts the momentum deficit, also has the effect of counting the maximum number ofcounterterm factors. For the coefficients β(j)n , table 5.1 shows their values as calculated bythe Cuhre algorithm, and with the exception of two terms we will use those coefficients. Fortechnical reasons explained in section 5.3.7, the two terms marked with asterisks will use values1015.2. The Tachyon Profilefound by the Suave algorithm instead, and those are shown in table 5.2. Occasionally we willwant to think of the tachyon profile in terms of these timelike momentum modes, and writeT (t) =∞∑k=02 cosh(kt)∞∑n=kλnβ(n−k2 )n . (5.4)This form is equivalent to (5.2) as long as we define β(j)n to vanish for non-integer j, as well asfor n = j = 0.The tachyon profile for several different solutions with regular OPE has been calculatedbefore. It has the simpler form of T (t) =∑∞n=0 λnβn√2nent where the coefficients βndef= β(0)nare only non-zero for maximal momenta. In table 5.3 we compare the coefficients for thosesolutions to the ones we have found. We have changed the normalization of their coefficientsby 2−n2 for better comparison with our coefficients, due to the relative normalizations of themarginal operators eX0and√2 cosh(X0). Our coefficients show very similar falloff to [5] as nis increased, though we do not expect exact agreement between any of the sets of coefficientsbecause the tachyon profile is a gauge dependent quantity. We believe that each of these listsis related to the others by such gauge transformations, but constructing them is beyond thescope of this work.As in [5], we take λ to be negative in order to study physical solutions. With this assumption,the tachyon profile (5.2) can be rewritten asT (t) =∞∑n=1bn/2c∑j=0(−1)nβ(j)n(en(ln |λ|+t)−2jt + en(ln |λ|−t)+2jt), (5.5)where in practice the sum over n only runs up to some cutoff N where the coefficients can becomputed. When only the j = 0 coefficients and the first term in parentheses are considered,as in the regular OPE case, we can clearly see that a change of ln |λ| will only shift the timeof the D-brane decay. For the singular case, however, the tachyon profile will have a differentshape depending on the strength of the marginal deformation, controlled by ln |λ|. The renor-malization scheme also contains the constants C0 and C1, which will appear nontrivially in β(j)nwith j > Small λWe begin our analysis with the case |λ|  1, where only the coefficients β(0)n need to beconsidered. Following the notation of [5, 34], we will refer to these coefficients as βndef= β(0)n .We will focus on (5.5), which receives significant contributions from the first term in parentheseswhen t > 0 and from the second term when t < 0. Knowing that T (t) is an even function,we will assume t > 0 and not need to consider the second term. Since we are considering−1 λ < 0, each term in the sum of (5.5) will be suppressed by the exponential until t is largecompared to − ln |λ|. For a large fixed t, terms with j > 0 will be small relative to others, soonly the j = 0 coefficients need to be considered. Since this is the case, the tachyon profile doesnot depend on the renormalization constants C0 and C1 at all. This had to be the case sincethere is no renormalization when all of the marginal operators have momentum in the samedirection. We can then unambiguously plot the tachyon profile for small |λ|. In figure 5.1 we1025.2.TheTachyonProfilen j β(j)n1 0 1√22 1 (−1.29904 . . .± 3 · 10−11) + (0± 1 · 10−14)CL2 0 (0.0760297 . . .± 8 · 10−16)3 1 (−1.30572± 4.3 · 10−5)− (0.707107 . . .± 3 · 10−14)C1 − (0± 2 · 10−4)CL3 0 (9.150± 0.019) · 10−44 2 (0.655579± 6 · 10−6) + (0± 3 · 10−4)CL + (3.2858± 0.0021)C1 + (4.9± 7.8) · 10−15CLC1+(0± 1 · 10−3)C0 + (0± 1 · 10−14)CLC04 1 −(0.4488± 0.0031) + (0± 8 · 10−4)CL − (0.2349± 0.0023)C1 + (1.4± 0.7) · 10−7C04 0 (1.17222± 0.00013) · 10−65 2 (0.723± 0.011) + (0± 1 · 10−3)CL + (4.387± 0.041)C1 + (0± 0.02)CLC1 + (3.53553 . . .± 7 · 10−15)C21+(0± 6 · 10−3)C0 + (0± 0.01)CLC05 1 (−0.01221± 1.2 · 10−4) + (0± 2 · 10−5)CL − (5.86± 0.34) · 10−3C1 − (1.27± 0.61) · 10−4C05 0 (1.598± 0.007) · 10−106 3 (−0.3± 0.4)∗ + (0± 3 · 10−3)CL − (2.572± 0.026)C1 + (0.3± 1.2) · 10−3CLC1 − (23.9401± 0.0013)C21+(1.5± 3.1) · 10−14CLC21 + (0.0955± 0.0030)C0 + (0± 5 · 10−3)CLC0 − (0.135± 0.015)C0C1−(1.2± 3.9) · 10−14CLC0C1 + (5.8± 1.5) · 10−6C206 2 (0.4991± 0.0050) + (0.4± 1.3) · 10−5CL + (1.912± 0.019)C1 + (0.8± 4.1) · 10−3CLC1 + (1.715± 0.024)C21+(1.879± 0.025) · 10−2C0 + (0± 2 · 10−3)CLC0 + (4.77± 0.38) · 10−2C0C1 − (2.37± 0.09) · 10−7C206 1 (−2.686± 0.027) · 10−5 − (1.6± 5.9) · 10−8CL − (9.1± 2.2) · 10−6C1 − (7.3± 0.7) · 10−7C06 0 (2.18± 0.04) · 10−15 ∗Table 5.1: Deterministic results for the non-zero coefficients β(j)n of the tachyon profile for the cosh rolling tachyon with singularself-OPE. Results are calculated with the Cuhre and QAG algorithms. The constant CL is part of the renormalization scheme,but it can not influence the solution, so we safely set it to zero in our analysis. CL was included in these numerical results onlyto demonstrate that it does not contribute to the solution at all.∗ These two coefficients found using the Cuhre algorithm appear to be unreliable, so the corresponding Suave results in table 5.2will be used for analysis instead.1035.2.TheTachyonProfilen j β(j)n1 0 1√22 1 −(1.2985± 0.0003)− (4.134± 0.007) · 10−6CL2 0 (7.61± 0.07) · 10−23 1 −(1.301± 0.005)− (0.001± 0.010)CL − (0.707107 . . .± 7 · 10−18)C13 0 (8.99± 0.09) · 10−44 2 (0.659± 0.007) + (0.2± 2.6) · 10−3CL + (3.288± 0.003)C1 + (0± 1 · 10−17)C1CL + (0.5± 3.4) · 10−3C0+(0± 5 · 10−9)C0CL4 1 −(0.449± 0.004)− (0.1± 1.4) · 10−3CL − (0.235± 0.002)C1 + (1.39± 0.07) · 10−4C04 0 (1.163± 0.002) · 10−65 2 (0.722± 0.012) + (1.3± 1.3) · 10−3CL + (4.38± 0.04)C1 + (0.014± 0.034)C1CL + (3.53553 . . .± 3 · 10−8)C21+(0.1± 6.5) · 10−3C0 − (0.3± 1.3) · 10−2C0CL5 1 −(1.21± 0.01) · 10−2 + (2.2± 1.1) · 10−5CL − (5.81± 0.06) · 10−3C1 − (1.17± 0.01) · 10−4C05 0 (1.27± 0.01) · 10−106 3 −(0.307± 0.004)− (1.6± 2.9) · 10−3CL − (2.55± 0.05)C1 + (0.5± 3.6) · 10−2C1CL − (23.943± 0.008)C21+(0± 1 · 10−17)C21CL + (9.5± 0.5) · 10−2C0 − (8.0± 6.6) · 10−3C0CL − (0.12± 0.02)C0C1+(0± 1 · 10−17)C0C1CL + (5.0± 4.4) · 10−5C206 2 (0.497± 0.005)− (2.1± 1.1) · 10−4CL + (1.92± 0.02)C1 − (1.7± 2.6) · 10−3C1CL + (1.718± 0.013)C21+(1.774± 0.014) · 10−2C0 − (1.2± 2.3) · 10−3C0CL + (4.71± 0.05) · 10−2C0C1 − (1.156± 0.016) · 10−4C206 1 −(2.54± 0.02) · 10−5 − (3.5± 0.3) · 10−8CL − (1.222± 0.012) · 10−5C1 − (7.618± 0.014) · 10−7C06 0 (2.3± 0.3) · 10−15Table 5.2: Suave results for the non-zero coefficients β(j)n of the tachyon profile for the cosh rolling tachyon with singular self-OPE.These Monte Carlo results are shown for comparison with the deterministic results of table 5.1.1045.2. The Tachyon Profile[5] [34] [35] Ψ heren βn1 1√21√21√21√22 0.0760295 0.290 0.0760297 0.07602973 7.59312 · 10−4 0.0506 7.732 · 10−4 9.149 · 10−44 6.54812 · 10−7 4.18 · 10−3 9.8145 · 10−7 1.173 · 10−65 4.93424 · 10−11 1.54 · 10−4 8.734 · 10−11 1.275 · 10−106 3.50136 · 10−16 2.45 · 10−6 7.903 · 10−13 2.26 · 10−157 2.41180 · 10−22 1.64 · 10−8Table 5.3: Comparison of rolling tachyon profile for three previously calculated solutions withregular self-OPE. Calculated from the solutions of [5], [34], and [35]. The n = k coefficients forour calculations based on [36] are included for comparison.see ln |T (t)| for ln |λ| = −4. Each “peak” represents the range of t for which T (t) is dominatedby a specific exponential in the sum. For different values of |λ| the shape of the oscillating partof the tachyon profile remains unchanged, and the whole half-plot shifts horizontally, with thesize of the plateau in the middle changing as expected.The size of the plateau which describes the time when the D-brane exists can be estimatedby the time of the first zero of the tachyon profile. This time is plotted in figure 5.2a, andis linear for the region where |λ|  1 is valid. The slope is −1 as it had to be from (5.5)when t is significantly larger than 0. We can also examine the “period” of the oscillations.The oscillations result from each exponential overtaking the one before, so we can calculate anestimate of their spacing by setting adjacent terms to be equal.βnen(ln |λ|+tn) = βn+1e(n+1)(ln |λ|+tn) (5.6a)tn = − ln |λ|+ ln(βnβn+1)(5.6b)∆tn = ln((βn)2βn−1βn+1)(5.6c)While there is no reason to expect this a priori, let us suppose that ∆t is a constant. In thiscase we haveβn+1βn= e−∆tβnβn−1, (5.7a)which is a recursion relation with the solutionβn ∝ ρne−n22 ∆t . (5.7b)The factor ρn can always be removed by taking β(j)n →β(j)nρn and simultaneously λ → λρ,which does not alter the tachyon profile. In one particular solution for the rolling tachyon withregular OPE, Kiermaier, Okawa, and Soler [34] found that their solution’s coefficients had theasymptotic behaviourβn ∼ e−γn2+O(n lnn) , (5.8)and in [33] it was shown that a solution equivalent to the one in [5] has coefficients which closelyfit bn ∼ e−γn2without significant corrections. We have just shown that this same recurring1055.2. The Tachyon Profile(a) (b)Figure 5.1: The tachyon profile T (t) with only the j = 0 coefficients considered. This is plottedfor ln |λ| = −4, where the approximation is valid. a) Black is for positive values of T (t) and redis for negative values. b) The tachyon profile with all Suave coefficients from table 5.2 is alsoshown in orange and grey, and where the Cuhre-only results deviate is indicated with a blueline. We can see that the Suave results are qualitatively equivalent to the ones we use.pattern can be derived from the assumption of exponentially growing oscillations with constantperiod. In figure 5.3 we see the best fit lines for our j = 0 coefficients, as well as those of severalother known solutions, to the form βn ∼ e−γn2. This was only predicted to be a fit for onesolution at large n, but we see good agreement in all cases, even with n never rising past 6 or 7for any of the solutions considered. In figure 5.3a the fit is to the deterministic results of table5.1 for n ≤ 5 and the Suave result for n = 6, but the Suave results with smaller n are shownas the red points for reference. It is curious that the coefficients fall so close to the e−γn2lineswithout any correction, even such as choosing ρ 6= 1 in (5.7b). While this trend was derived inour case from a constant period of oscillation, if it holds at higher n it guarantees that the edgecoefficients are a convergent series. The fact that all of the solutions appear to behave similarlysuggests that they are also all convergent.5.2.2 Large λOnce we loosen the |λ|  1 restriction, we must consider all of the coefficients β(j)n and searchfor patterns there. Due to the small number of coefficients, and particularly the small numberof rows with constant j, it is not possible to get a good understanding of any patterns orasymptotics for these coefficients, but we can speculate as to possible trends. The first thingwe notice from table 5.1 is that the sign of the coefficients appears to alternate as (−1)j2 . Thisis not strictly true even for the coefficients we have calculated, however, as choosing non-zeroC0 and especially C1 will alter many of the coefficients and can affect their sign. With only asmall number of the coefficients known, we do not know whether the large n asymptotics are1065.2. The Tachyon Profile(a) (b)Figure 5.2: The time of the first zero of the rolling tachyon profile as a function of ln |λ|.An approximation to the asymptotic behaviour is shown as a dashed line. a) Only the β(0)ncoefficients are considered, so the plot is not valid for large |λ|. b) The whole tachyon profile isconsidered with coefficients from the fit of figure 5.4a.fixed or can be changed by a choice of the two free parameters. We can, however, attempt toforce a few patterns and see which appear more naturally.As a first choice we pick C0 = C1 = 0 and notice that the j = 1 coefficients appear to be agood fit to β(1)n ∼ e−γ1k3with k = n−2j, which is shown in figure 5.4a. While this can be madeto fit even better by a choice of renormalization constants, this would lead to some of β(2)n beingless than zero or to that row having increasing magnitudes. On the other hand, if we attemptto pick renormalization constants which are a fit to β(1)n ∼ eγ1k2, as shown in figure 5.4c, we donot find as good a fit. The same is true of β(1)n ∼ eγ1n2using n instead of k in the exponent. Itappears that the j = 1 coefficients have a tendency towards the cubic exponential decay, whilefor j = 2 we lack enough points to reach any conclusions. The red points in figure 5.4 againrepresent the Suave coefficients, and we see that β(2)n have significantly different values oncethe renormalization constants are changed, but a look at table 5.2 suggests that this is mainlydue to large errors in the Suave coefficients, so it is unlikely that the deterministic plots wouldchange significantly if more sample points were used.While five points is not a lot of data, the β(1)n coefficients suggest that each row with constantj may eventually be a convergent series for at least some choice of renormalization constants.Showing that the full tachyon profile converges when these rows are added together, however,remains impossible until much higher order calculations can be performed. In particular, thelarge dependence of coefficients such as β(3)6 on C1 is troubling since it suggests that if that con-stant is of order 1 then the sequence β(n−k2 )n with constant k could have increasing magnitudes.1075.2. The Tachyon Profile(a) (b)Figure 5.3: The falloff of the j = 0 coefficients of the rolling tachyon profile shown as − lnβ(0)nversus n2. A linear graph indicates that βn ∝ e−γn2holds, with γ given by the slope. The bestlinear fit is also shown. a) The solution shown here, with slope 0.9599. Red points are Suavevalues. b) Our solution as well as the other three presented in table 5.3. Square points are for[5], crosses are for [34], and circles for [35].Thinking of (5.4) asT (t) = 2∞∑k=0β(k)eff (λ) cosh(kt) , (5.9)the effective coefficients β(k)eff (λ) would then be defined by series which do not converge fornon-zero renormalization constants. Looking at table 5.1, even with vanishing renormalizationconstants, the magnitudes of the terms βj2j do not drop off very fast. We know that for smallλ the series is convergent, but this suggests that the radius of convergence in λ may be finite.This could support either the claim of [22, 51] that there is a second branch with decreasingmarginal parameter, or our result from chapter 3 that the marginal deformation simply has anunexpectedly finite maximum. Either case would be worth further investigation.Now that we have seen what the bulk coefficients look like, we can begin examining thetachyon profile for larger values of |λ|. Of course as we do this we must be aware that weare missing all coefficients β(j)n with n ≥ 7, and as we increase the strength of the marginaldeformation those coefficients will begin to play a larger role, but we can still get a qualitativeidea of the impact of the bulk coefficients on the tachyon profile. Since none of the optimizedfits in figure 5.4 were significantly better than simply setting the renormalization constants tozero, we will choose that from now on. Other reasonable choices will not give results that arequalitatively different. For large negative λ we see a tachyon profile in figure 5.5 with fewer1085.2. The Tachyon Profile(a) (b)(c) (d)Figure 5.4: Several attempts to determine a trend for the j > 0 coefficients in the rollingtachyon profile. ln |β(j)n | is plotted vs. functions of k = n− 2j with several different choices forC0 and C1. Black points are Cuhre values while red points are from the Suave algorithm. Ina) we set C0 = C1 = 0 and plot j = 1 coefficients on the left and j = 2 on the right. The j = 1coefficients with C0 and C1 optimized for the best linear fit appear in b), and c) attempts thesame linear fit assuming a k2 horizontal axis rather than k3. d) attempts a linear fit to boththe j = 1 and j = 2 sets of coefficients assuming a k2 horizontal axis, with j = 1 coefficients onthe left and j = 2 on the right.1095.3. Technical Detailsoscillations than we had with just the edge coefficients from figure 5.1. As |λ| decreases, theadditional oscillation appears at ln |λ| ≈ −1.948. Once ln |λ| . −2.5 the profile has stabilizedand the plateau continues growing as |λ| shrinks, just as we know it should from our discussion ofthe tachyon profile for small |λ|. The disappearance of this oscillation for large |λ| is because thej > 0 coefficients cannot be neglected in this region, and they change the effective coefficientsin (5.9).The behaviour we see for these large values of |λ| is not unprecedented; in [34] the tachyonprofile had coefficients (seen in table 5.3) which did not decrease as quickly as other time-asymmetric solutions. Taking the asymptotic ansatz β ∼ e−γn2and decreasing γ beyond acritical value causes some oscillations to vanish, which is what happens in [34] where some ofthe exponentials did not dominate for any range of time. As λ is increased for our solution,the effective coefficients in (5.9) change in a way that causes an oscillation to disappear. If wekeep increasing λ beyond this point, some of the coefficients will even change sign, but once thetachyon profile has stabilized with the missing oscillation it does not change significantly. Forλ this large, however, we should not trust our results since coefficients with higher n will havelarger contributions.Aside from the obvious, that the singular OPE case is a symmetric function where theD-brane exists for a limited time while with regular OPE it exists until it decays at a finitetime, the qualitative difference between the tachyon profiles seems to be that the period andnumber of oscillations can change this way. Because the strong deformation tachyon profilewe have found is similar to the profile of [34], it suggests that changing the strength of themarginal deformation in the time-symmetric case is much like changing gauge in the time-asymmetric case. If the late time behaviour is equivalent to the tachyon vacuum under a time-dependent gauge transformation, as has been hypothesized [35], then in this case the gaugetransformation should depend on both time and the marginal parameter in a non-trivial way.That our solution appears qualitatively like time-asymmetric ones for both weak and strongdeformation parameter suggests that such gauge transformations remain a valid explanationof the oscillations in the time-symmetric case, although we are not aware of any examples ofgauges where coefficients have negative sign, so there may be a limit to the range of λ wherethis approach is valid.5.3 Technical DetailsIt takes a surprising amount of Maple and C++ code to fully construct the solution Ψ for amarginal deformation with singular self-OPE and then evaluate the integrals corresponding toits tachyon profile. This section describes what steps are necessary and why, as well as someof the limitations that remain. Despite the limitations, this ends up being general enoughto calculate more than just the tachyon profile of the solution, so the action and equation ofmotion are also considered. This section gives a rough description of the program used, witha focus on what it is and is not able to do. To see precisely how these steps are accomplished,appendix B contains the full commented code used. We then study a few tests of the validityof the program by using it to calculate known quantities.1105.3. Technical Details(a) (b)(c) (d)Figure 5.5: The tachyon profile of the time-symmetric rolling tachyon solution. We plot thelog of the tachyon profile at a) ln |λ| = −0.5, b) ln |λ| = −1.92, c) ln |λ| = −1.98, and d)ln |λ| = −2.5. Positive values represented by black, negative values by red. The renormalizationconstants C0 and C1 are both set to zero.1115.3. Technical Details5.3.1 ConventionsThe first step in writing a program to perform calculations on wedge states with operatorinsertions is to decide how the mathematical objects will be represented. Maple has datatypesfor mathematical expressions, but in our case we need more. A wedge state is a semi-infinitecylinder with a given circumference, and operators are inserted at varying locations whichfor our purposes will always be on the boundary. We define the wedge datatype as a three-part list containing the width of the wedge, a set of variables with their (possibly integrated)positions in the wedge, and finally the list of operators to insert, along with any constants.When we add wedges, we do not want a wedge with circumference equal to the sum of the twowedges’ circumferences any more than we would want one with the arithmetic sum of two setsattempting to describe the insertion locations. In order to avoid this, all wedge sums should bewritten using the inert operator &+ to avoid Maple’s default behaviour of adding lists elementby element. &+ is treated as a binary operator and can be used infix, but has no evaluationrules and will be left alone by Maple’s simplification routines.When we eventually perform calculations, we will have to truncate to a finite number ofmarginal operator insertions, or equivalently a finite order in the marginal parameter λ. Asthe order increases, the number of terms will increase very quickly, and if we are not carefulthe execution time for simple operations such as the star product will become a problem. Tohelp with this, we will define an alternative form of the wedge state datatype. This is calledwpoly and has the same first two items as the wedge type, but the third term representing theoperators to insert is replaced with a list representing the taylor series of those operators inλ. When only a given order in λ is desired, parts of this list which give results at too high anorder can be skipped. We then define a few more labels for determining the type of an objectfor conditional processing: wsing evaluates to either of wedge or wpoly, wsum is a sum of wsingtypes added using &+, and wtype evaluates to either of wsing or wsum and so covers all wedgestate types.Multiplication in Maple is naturally commutative, so keeping track of ghosts requires someextra thought. I have chosen to represent the ghost operators using the functions c(t) anddc(t) which are undefined and will never be evaluated. For most situations the order of ghostinsertions is defined by their ordering along the boundary. c(1)*c(2) means c(1)c(2), andso does c(2)*c(1), while to get c(2)c(1) we would need to write -c(1)*c(2). A problemarises, however, when two ghosts are coincident, as in c∂c(t). In this case I chose to break thetie lexicographically using the name of the insertion location. This means that c∂c(0) mustbe represented by something like c(t[1])*dc(t[2]) where the insertions will only be set to0 once the expectation value has been taken and there are no more ghost operators in theexpression. Because of this, two ghosts should never be inserted using the same coordinate,and care must be taken when creating wedge states to guarantee that the sign is correct for theghost insertions given.5.3.2 Basic FunctionsWith our datatypes defined, we can now move on to begin writing routines to manipulate them.The most obvious operation we will perform on wedge states is the star product, which we willdo with a function called star. This takes two or more wedge states or sums of wedge statesand multiplies them from left to right. The danger in this is apparent when we consider an1125.3. Technical Detailsexample like A ∗ A, where both copies of the wedge state A would have their insertions at thesame locations. In addition to shifting all insertions in the second wedge state to the right bythe circumference of the first wedge state, we must ensure that the name of the coordinate usedto label those locations is unique. For this, I wrote a subroutine called newvar, which renamesvariables to prevent conflicts. The star function must be careful to call this on the coordinatesin the second wedge state in lexicographic order so that the ordering is preserved and the ghostfactors will not inadvertently change sign. Our star product can take an additional namedparameter, LAMBDA MAX, which will truncate the resulting string field to the given power of λ.This is especially efficient when all wedge states to be multiplied are given in the wpoly form.At times it can also be very useful to be able to simplify expressions involving inert sums ofwedge states. For example, in a long sum many terms can become zero when acted on by theBRST operator or truncated to a given order of λ, so the function plus0 will identify vanishingwedge states and remove them. This function will also flatten sums, effectively removing extraparentheses in what is an associative operation. When the operator content of a wedge statecan be expanded, there is wexpand to do this and make each term its own wedge state, and thefunction wcombine attempts to perform the inverse operation. As with addition, multiplicationof a wedge state by a scalar should not be done with the standard Maple multiplication operatorbecause that would multiply each item of the list that represents the wedge state. Instead, thereis ctimesw to multiply the operator content of a wedge state by any expression and leave therest of the wedge unchanged. Finally, when we want to examine only one component of thetaylor series in λ, there is pickoff to return the taylor coefficient wedge state of its input at agiven order. Of course pickoff is extremely simple for wpoly types.All of the basic operations we have seen in this section apply equally to all wedge datatypes.There are alternative cases for the wedge and wpoly types, and they properly distribute overthe inert addition &+.5.3.3 Known Wedge StatesWe now turn our attention away from the basic operations defined on wedge states, and towardsthe computation of the specific solution found in [36] and reviewed in chapter 4. The first wedgestates which are defined are zero and the star product identity, each coming in both wedge andwpoly versions. The solution is built from the string fields AL and U , as well as the powers U12and U−12 . Each of these string fields is in turn defined as a sum of wedge states with insertions.So far we have talked about manipulating wedge states with insertions, but we have notdiscussed what those insertions are. The renormalized integrated operators we found in chap-ter 4 are complicated constructions which will each need to be created before they can beused in numerical calculations. The operators we will need are [V (a, b)n]r,[V (a)V (a, b)n−1]r,[V (a, b)n−1V (b)]r, and[V (a)V (a, b)n−2V (b)]r. Because the action of the BRST operator on allbut the last of these is well understood from the assumptions (4.5a) and (4.5b), expanding themin terms of bare marginal operators and counterterms too early would be a mistake. Insteadwe will make use of inert functions, as we did with addition, and name these renormalizedoperators V ren, V L ren, V R ren, and V LR ren respectively. Each of these should be givenfive parameters, representing the total number of marginal insertions, n, the left and right end-points, a and b, a list of the variable names where the marginal operators are to be inserted, anda name used for the total circumference of the wedge state the operator is inserted on. The last1135.3. Technical Detailsparameter is never used when the operators are defined by our little g renormalization scheme,but the sample scheme of [36] which is discussed in section 4.3.5 requires it, so we will includethat parameter for backward compatibility. Once all operations which are simpler on the to-tal renormalized operators have been performed, the inert functions representing them can bereplaced with active ones that use bare insertions of the marginal operator and counterterms.The active functions are named V r, V Left, V Right, and V LR.The active functions representing renormalized operators insert counterterms using the lit-tle g scheme, which depend on the insertion coordinates. They are inserted for every pair ofcoordinates, as in ◦◦∏V (ti) ◦◦g, with the fully symmetrized integrand being finite at all coordi-nates, as described in (4.88) of section 4.3.2. The counterterms to be inserted are the functionsG r, G Left, and G Right. The capitalization of the counterterms in the code goes back to theuse of the two-point function in [36], but the functions they represent are the correct little gscheme terms. Finiteness of the integrand is important for numerical calculations. If we were totry something like a big G scheme where the integrals of the marginal operators are regulatedby  and the counterterms are explicit functions of , actual integration would be extremelydifficult. First we would have to implement the regulated integration regions, which are notsimple at all, and then we would have to perform the integrals many times so that the limit in could be taken numerically. This is not practical, so instead we guarantee that the integrandsare always finite and then integrate over unregulated regions once. The minor disadvantage ofthis approach is that we cannot integrate terms in the result separately, since they may havecancelling singularities. The more serious related problem occurs when the divergent termsbeing added to give a finite integrand have differing rounding errors. This will be consideredin more detail in section 5.3.7.Once renormalized integrated operators are defined, we can begin using them to makestring fields. The string field U is given order by order as a wedge state with n fully integratedoperators inserted between 1 and n. The 0th order of U is defined to be the star product identity,and this means that functions of U can be defined in terms of Taylor series of the function about1. The first order term in U vanishes because it consists of a single (non-renormalized) operatorintegrated over a vanishing region. The function Utail returns the string field U − 1 for usein these Taylor expansions. Specifically, we need Uinv, Usqrt, and Uinvsqrt, which return thestring fields U−1, U12 , and U−12 in the wpoly type. Finally, we also define the functions A Land A R to give the corresponding string fields in wpoly form. It is worth mentioning thatsince the wedge states used for both U and AL/R have different circumferences at each orderin λ, each order must be its own wedge state and the lists representing the Taylor series of theoperator insertions will each have only one non-zero element.The solutions Psi L real, Psi R real, Psi L, and Psi R all have simple definitions in termsof the string fields defined so far. As in [36], they areΨ =1√UAL1√U+1√UQB√U , (5.10a)=1√UAR1√U+√UQB1√U, (5.10b)ΨL = AL1U, (5.10c)ΨR =1UAR . (5.10d)1145.3. Technical DetailsThe real solutions Ψ in the first two lines are equal, while ΨL and ΨR are related to Ψ bygauge transformations. Having several forms of the solution is useful for debugging, since wecan compare calculations. The number of terms in each of these forms of the solution growsvery rapidly as the order in λ is increased, but it is still possible to do 6th order calculations ina very reasonable amount of time.The last string field we will need is the conformal patch. When we define string fields interms of wedge states with insertions, what we are really saying is that the string field Φ issuch that its overlap with an arbitrary test state is the same as the expectation value of theoperator corresponding to the test state φ inserted on the corresponding wedge state. See alsothe discussion of (2.22). The test state we will be using is quite simple, having ghost numbertwo and no matter content except for an amount of momentum in the time direction. Theoperator corresponding to this test state is, up to the conformal factor going from the upperhalf plane to Schnabl frame,φ(0) = ∂c(0)c(0)ekX0(0) . (5.11)We implement this by defining a wedge state with width 0 and the appropriate operator content,so that when it is multiplied by a wedge state the result has all of the operators needed for theexpectation value in question, all inserted in the correct positions. The zero width is so thatthis conformal patch does not alter the circumference of the resulting wedge state.5.3.4 The BRST OperatorIn the construction of the solution Ψ, the BRST operator is not terribly complicated. This isbecause it only acts on U±12 which has ghost number zero and no fixed operators. In this caseonly an implementation of the first BRST condition (4.5a) is needed. Acting separately on eachterm in a sum of wedge states, Q B goes through any product of renormalized operators andreplaces each one with the sum of terms representing the right hand side of1n!QB [V (a, b)n]g =1(n− 1)![V (a, b)n−1cV (b)]g −1(n− 1)![cV (a)V (a, b)n−1]g+1(n− 2)!(O(2)R (b)−O(2)L (a)) [V (a, b)n−2]g (5.12a)whereO(2)R (b) =12∂c(b) + CLc(b) , O(2)L (a) = −12∂c(a) + CLc(a) . (5.12b)As we have already mentioned, in this chapter we assume that CR = CL, but do not neglectthat constant.While this is all we need in order to construct the solution, if we wish to test the equationof motion we will need to implement the second BRST condition (4.5b). This raises a numberof additional issues for a clear definition of QB. One obvious difficulty is in the fact that QB isa graded derivation satisfying QB(A ∗B) = (QBA) ∗B + (−1)GAA ∗ (QBB). For wedge stateswith insertions of operators such as [cV (a)V (a, b)n1 ]g [V (c, d)n2 ]g we want to have QB act onthe first renormalized operator as a whole without picking up a sign change from acting onoperators to the right of a ghost until it gets to the second renormalized operator. This is anissue whenever the fixed operator does not appear in the last renormalized group. To do this,for the purposes of determining sign we view integrated operators as being inserted at their left1155.3. Technical Detailsendpoint, and ghosts as being inserted slightly to the right of where they actually are. Thenby considering the action of QB on all operators from left to right and changing sign wheneverwe pass a ghost, the signs will be correct.The other problem with applying the BRST operator to operators of ghost number onecomes from the fact that we are treating the matter and ghost parts separately, whereas insection 4.3.6 we had to consider them together so that the ghosts could properly soften some ofthe divergences. What we will do is assume that QB only acts on renormalized operators withone fixed insertion in the form[cV (a)V (a, b)n−1]g or the right handed version; there is alwaysa c ghost included with the fixed insertion. We then take the resultQB([cV (a)V (n−1)(a, b)]g−12[∂c(a)V (n−2)(a, b)]g)= −[cV (a)V (n−2)(a, b)cV (b)]g−12[cV (a)V (n−3)∂c(b)]g+12[∂c(a)V (n−3)(a, b)cV (b)]g+14[∂c(a)V (n−4)(a, b)∂c(b)]g(5.13)and rearrange it to getQB[V (a)V (n−1)(a, b)]reff= −[V (a)V (n−2)(a, b)V (b)]rc(b)−[V (a)V (n−3)(a, b)]rO(2)R (b)−1c(a)O(2)L (a)[V (n−3)(a, b)V (b)]rc(b)−1c(a)O(2)L (a)[V (n−4)(a, b)]rO(2)R (b)−1c(a)O(2)L (a)[V (n−3)(a, b)V (b)]rc(b)−1c(a)O(2)L (a)[V (n−4)(a, b)]rO(2)R (b)−O(2)L (a)[V (a)V (n−3)(a, b)]r−1c(a)(O2L(a))2[V (n−4)(a, b)]r. (5.14)The notation used here is[V (n)(a, b)]rdef= 1n! [V (a, b)n]r, which is standard in [36] and is con-venient for writing renormalized operators which have an exponential form. The right handedversion of this isQB[V (n−1)(a, b)V (b)]g= −c(a)[V (a)V (n−2)(a, b)V (b)]g−O(2)L (a)[V (n−2)(a, b)V (b)]g−[V (n−4)(a, b)]gO(2)R (b)O(2)R (b)1c(b)−[V (n−3)(a, b)V (b)]gO(2)R (b) . (5.15)Obviously these cannot be used if they are not multiplied by c(a). We have also assumed thatQB does not act on the ghosts. In reality this is not true, but as long as ghosts appear only inthe specific combinations of OL/R produced by (4.5a) we can pretend that QB only acts on thematter sector and combine the ghost part into those results. That is the approach taken here,and while it is definitely a hack from the point of view of all allowed renormalized operators, itis enough to produce correct results for the string fields we are interested in, namely those usedin the solution Ψ, as well as the equation of motion and the action evaluated on the solutionΨ.The BRST code which replaces one operator with another is quite dense so I will explainone term in detail. The term − 1c(a)O(2)L (a)[V (n−3)(a, b)V (b)]c(b) coming from the third andfifth terms in (5.14) will be our example.1165.3. Technical Details877 grandterm[i]:= grassign*subsop (1=op(1,grand[i]) -2,4=[op(3..op(1,grand[i]) -1,op(4,grand[i])),op(4,grand[i])[2]], V_R_ren(op(grand[i]))):878 outw:=outw &+ [expr[1], remove(has ,remainv ,op(4,grand[i])[op(1,grand[i])]) union map((x->lhs(x)=op(2,rhs(x))),removev),‘*‘(-2,OL2(op(4,grand[i])[1]),c(op(4,grand[i])[2]),op(remove(has ,grandterm ,c(op(4,grand[i])[1]))))]:The first line alters the call to the inert function representing a renormalized integrated operator.The maple function subsop performs substitutions which replace the operands of its argument.In this case the first operand of V L ren is reduced by 2 since there are two fewer insertions ofthe marginal operator, and the fourth operand is changed from [t1, . . . , tn] to [t3, . . . , tn−1, t2].The operator itself is then changed from V L ren to V R ren without any further changes to itsarguments. In the second line we see the construction of the wedge state to be added to theoutput. In the coordinate selection part, the integrated t2 coordinate is replaced with t2 = bby removing it and then taking the union with the removed coordinate evaluated at its rightendpoint. Finally, the product ‘*‘(...) gives the operators inserted along the boundary. Theinitial 2 comes from the sum of the third and fifth terms in (5.14), and then the ghost operatorsO(2)L and c are inserted at the left and right hand endpoints respectively. The remainder of theoperators are inserted at the end, with the exception of c(t[1]), which is removed from theproduct.5.3.5 Expectation ValuesThe string fields we are interested in are defined by having the same overlap with an arbitrarytest state as a given collection of wedge states with insertions. We have now described everythingneeded to produce those wedge states with insertions, and include an insertion for a useful classof test states. This leaves us with the question of how to find those CFT expectation values.This will be done in two steps, with the algebraic work to produce integrands done in Maple,and the numerical integration done in C++.The algebraic portion of the correlation functions is found with the function corr. The firststep is to replace all of the inert renormalized operators with active ones and then wexpand theresult so that each wedge state contains a single product of operators and functions. Next wego through the operator content and collect each type of operator. Obviously, it is not possibleto handle every operator which exists in the CFT, but the only ones we need for our purposesare ekX0(t), c(t), and ∂c(t). If we want to see how other marginal deformations compare, thedeformation ∂X has the same self-OPE as the rolling tachyon√2 coshX0, so we also allow thisoperator in addition to the previous three. In the matter sector this is implemented by goingto the upper half plane where we have〈n∏i=1eiki·X(ti)m∏j=1∂sjXµj (sj)〉= limα→0〈(n∏i=1eiki·X(ti))m∏j=1∂sjαeαXµj (sj) e−α∑mj=1Xµj (u)〉= limα→0m∏j=1∂sjα∏i<i′(ti − ti′)2α′ki·ki′∏i,j(ti − sj)−2iα′αkµji∏j<j′(sj − sj′)−2α′α2δµj′µj . (5.16)1175.3. Technical DetailsThe ghost sector is easily represented in Schnabl frame, where we can use the result fromequation (D.11) of [25],〈c(x)c(y)c(z)〉Wpi−1 = sin(x− y) sin(x− z) sin(y − z) . (5.17)The matter sector result is only valid when∑ni=1 ki = 0 so that momentum is conserved, andthe ghost factor will be zero unless the ghost number is three. Both of these restrictions arechecked for, and zero is returned if either one is not satisfied.So far, the correlators have been done on the operator content without using the secondelement of the wedge datatype which says where the insertions really are. Now the list ofcoordinates is separated into those which are integrated and those which are fixed. For eachintegrated coordinate, the expectation value is wrapped in Maple’s inert integration function,Int, so that excessive execution time is not wasted trying to evaluate the integrals at this stage.For each fixed insertion, an attempt is made to substitute the insertion location for the variablename. This may fail if, for example, diametrically opposed insertions on the wedge state causea divergence which is cancelled by the conformal factors. If direct substitution fails, a limit isused instead, and if that still cannot be evaluated for whatever reason an inert limit functionis used. Fortunately, the solution is well enough behaved that the inert limit should never beneeded.5.3.6 Handling and Exporting IntegrandsAs a result of the corr function, we now have a long sum of integrals, many of them multi-integrals. Each integral represents one product of matter exponential and ghost operatorsappearing in the quantity being evaluated. As we have mentioned, these integrals are unreg-ulated because we are using the little g scheme which has finite integrands. At this point,however, many of the integrands are still divergent because a single product of (unrenormal-ized) operators is naturally divergent in this setup. Only when all of the terms associated witha renormalized operator are added together do we have a finite integrand. Since the integralsare unregulated we have to add the integrands before we can integrate. Some numerical inte-gration routines require that the region of integration is the unit hypercube, (0, 1)n. Of courseit is a simple matter to rescale functions at the time of integration, but by making the changeof variables at this point, the process of adding integrands is simplified. Instead of searchingthrough all of the terms to identify which ones have matching integration domains, the functionto cube rescales every integrand to be integrated from 0 to 1, and then simply adds all termswhich are integrated over the same dimension. This gives sums with even more terms than weneeded for finiteness, and would only be an issue if we were interested in finding the values ofspecific terms separately.In order to perform the integrals using C++ routines, we need to output the integrandsthat we have produced so far to a file. A single file containing a hundred integrands as C++functions would require a great deal of work to incorporate into a working program, so weinclude code to keep track of what should be done to each of those functions, and even whatoutput to print with descriptions of each term. The function tacconst is set up to do all of this.First it constructs the solution at a given order in λ, and optionally the action and equationof motion. Next it finds the tachyon profile of the solution, and if the equation of motion wascomputed the overlap of that with several ghost number one test states, as unevaluated integrals.1185.3. Technical DetailsNow each of those quantities is still a function of the renormalization constants C0, C1, andCL, but because of the structure of the renormalization scheme, they are actually polynomialfunctions, so by differentiating we can separate the integrals into Taylor coefficients. Now bytreating each of these Taylor coefficients as a separate integral to be computed, the integrandsare entirely floating point functions which can be evaluated and integrated numerically. A lineis then written to a file to call a C++ function called CubeInt on each of these integrands,since that is what we will call the function to integrate over the unit hypercube. After that,printf statements are written to the file which will output a few characters describing whatquantity is being given, followed by the reconstructed Taylor series for that quantity with theTaylor coefficient integrals replaced by their floating point results. Finally, each of the integrandfunctions is written to the file. This is done at the end because they can often make up manymegabytes of text, and it is easier to separate them into a header file if they are all together.5.3.7 Numerical IntegrationThe integration is handled by off-the-shelf C++ routines. The CUBA library appears to bea good choice in most cases [52].4 It is a collection of four algorithms for multi-dimensionalnumerical integration, three of which use pseudo-random sampling while the fourth is a deter-ministic algorithm. Since we are working at sixth order in λ and the solution has ghost numberone (corresponding to the number of fixed moduli), there are never more than five integratedcoordinates in a given integral. While Monte Carlo algorithms do scale better as the dimensionrises, in five or less dimensions it appears that the deterministic algorithm, Cuhre, is slightlymore efficient. As we will see, Cuhre is also more reliable in most cases. Unfortunately, itonly integrates functions of more than one variable, so in the one-dimensional case we use theQAG routine from the GNU scientific library.5 Each of the routines in the CUBA and GNU li-braries provides its own error estimate, and the CUBA library routines also provide a chi-squareestimate of the probability that the error is sufficient.A single quantity to be calculated numerically, such as an individual term in the tachyonprofile of table 5.1 or in one of the consistency checks of tables 5.6 and 5.7, generally consistsof a small number of integrals. Each integral contains all of the terms in the solution which areintegrated over a given number of coordinates, or equivalently all of the terms with the samenumber of integrated operators. Most quantities are the sum of two integrals, but a few areonly a single integral, and quantities in the action can consist of more than two.ConvergenceIn order to get as much data as possible, the collection of integrals we look at here will include allof the ones used in calculating the tachyon profile as well as the action and several componentsof the equation of motion. The action and equation of motion will be discussed as checks ofconsistence later in this section.It is difficult to study convergence of the one-dimensional integrals due to the fact that theQAG algorithm does not report the number of samples used. We can, however, take several fullsets of data for these integrals and compare the different calculations of the same integrals. We4The CUBA library is distributed from http://www.feynarts.de/cuba/.5The GNU scientific library is found at https://www.gnu.org/software/gsl/.1195.3. Technical Detailsfind that they all agree with each other well within the error estimates. The only troublingone-dimensional case is that of a constant integrand. This can be seen in the kinetic energy ofthe solution at fourth order in λ, which is the particularly simple quantity∫ 10 dt32 −32 . Theintegration algorithm performs operations on the constant causing tiny roundoff errors which,when the constant value is subtracted, causes the result to differ from zero. This would notbe a problem except that error analysis in numerical integration is based on variation of theintegrand, and as such gives an estimated error which is extremely small. While in principleerror estimates should account for the roundoffs inherent in their algorithm, in practice this doesnot seem to be the case. The error estimate becomes small enough that the reported result canactually be incorrect, and even unstable with respect to changing the desired accuracy. Mostintegrands worth using a numerical algorithm to integrate undoubtedly vary enough that thisis not normally an issue. This is the only instance of such a problem that we encounter, butsince the integral is trivial we do not need to rely on numerical integration for its value.We now turn our attention to the Cuhre algorithm, so we will only be considering integralsover two or more dimensions. The first question we will ask here is whether the error barsreported by the integration algorithms are sufficient. Because we do not know the correct resultsfor most of the integrals, we evaluate each integral with at least three different choices for thesample size, N . Making the assumption that the calculation with the largest N is “correct”, wecan compare the difference between each computed integral and the most accurate one to theerror estimate reported by that integral. This is shown in figure 5.6a, where blue points havesufficient error bars and green points are within twice the error bars. The few red points are theoutliers which differ from their largest N partners by more than twice their error estimates. Ina moment we will compare every computed integral with other calculations of the same integral,so the red points here will also have greater than 2σ difference in that comparison. Many ofthese points come from the same integrals, so there are actually not very many integrals whichwill need to be examined in detail. Our choice to prefer the Cuhre algorithm over the adaptiveMonte Carlo algorithm Suave is justified by figure 5.6b, where we see that the Suave algorithmis as likely as not to underestimate the error. In its defence, Suave frequently reports a 100%χ2 estimate that the reported error is insufficient, but this is not particularly helpful in findingaccurate values.The Suave algorithm can still be useful for comparison, however. Its error bars are nothelpful, but if the same quantity computed in Cuhre and Suave differs by more than a fewpercent it is worth closer examination. This is how the two terms marked with asterisks intable 5.1 were identified. The integrals responsible for the troublesome behaviour of these twoquantities are plotted in figures 5.7a and 5.7b. There were some other terms which were flaggedby this test, but they converged reasonably well once the sample size was increased sufficiently.With many integrals each calculated for several different values of N , we have a sizeablecollection of data to examine. Among the 848 Cuhre integrals, there is only one instance ofthe error estimate increasing as the sample size was increased, so we can safely say that thereported error bars decrease monotonically as N → ∞. We can then examine the quantity|xi−xj |√∆x2i+∆x2jfor every pair of calculations of the same integral. We find that 88% of pairs arewithin σ of each other, while 94% are within 2σ. Those which disagree by more than 2σ canbe studied individually, since they correspond to only 15 different integrals. Of those, four onlydisagree due to a single computation each with very low N (about 500 samples) that has a 50%1205.3. Technical Details(a) (b)Figure 5.6: Plots showing the reliability of error estimates. The vertical axis is the difference oftwo calculations relative to the more precise of the two, and the horizontal axis is the relativeerror reported by the numerical integration. The Cuhre algorithm results are shown in a) andSuave results are in b). Points with differences greater than twice the reported errors are red,those with differences between one and two times the error are green, and those with errorestimates large enough to cover their difference from the “best” value are blue.χ2 chance of being incorrect. One of the remaining 11 integrals is also the one responsible forthe tachyon profile coefficient β(0)6 , so adding in the integral responsible for the other flaggedterm in table 5.1 we have 12 integrals to examine. These are plotted with various values of Nin figures 5.7 and 5.8. The values and their error bars are shown in blue, and when appropriateto the scale of the plot the corresponding Suave results are also shown in green.The first two plots represent the parts of the tachyon profile for which we used the Suaveresults. In figure 5.7a the problem is not that the results are inconsistent, but that the errors areso large that the results are meaningless. It looks likely that as N is increased the results willcontinue to converge to something quite close to the Suave result. For the other integral, figure5.7b shows that as N is made extremely large we finally find something like the much moreconsistent Suave results. The slope, however, does not yet appear to be significantly slowingdown, so we cannot be sure that it is convergent. The rest of the plots in figures 5.7c-5.7e showthe other integrals that contribute to the tachyon profile and have more than a 2σ variationbetween points. They all show signs that once N is sufficiently large they converge quite well.Only for smaller N do the error estimates appear to be insufficient.Moving on to integrals which contribute to consistency checks, in figure 5.8 we see that themajority are fine. Only 5.8c does not appear to converge. As with the examples shown in figure5.7, this may well be linear until some critical N where it begins to converge. In addition, whileonly the quantities composed of small sums of these integrals are supposed to vanish, in manycases each of the integrals will vanish independently. These seven examples all become closer1215.3. Technical Details(a) (b)(c) (d) (e)Figure 5.7: Several integrals from the tachyon profile calculated with different values of N . Theerror bars are those reported by the Cuhre algorithm. When the Suave algorithm gives resultswhich fit on the same scale they are included as the green data.1225.3. Technical Details(a) (b) (c)(d) (e) (f)(g)Figure 5.8: Several integrals from the consistency checks calculated with different value of N .The error bars are those reported by the Cuhre algorithm. When the Suave algorithm givesresults which fit on the same scale they are included as the green data.1235.3. Technical Detailsto vanishing as N is increased, and only 5.8f is not getting very close to zero. Convergence ofthese results does not appear to be very much of an issue, and I expect that if we continued toincrease N by another factor of ten they would all continue to approach zero.Roundoff errorsEach integrand may contain a number of terms which are divergent either on the boundaryof the region, ti ∈ {a, b}, or on a diagonal, ti = tj . While the renormalization is designedspecifically so that these divergences will cancel, individual terms evaluated near these regionscan be very large. Because we are limited to the double precision floating point datatype, eachterm has a relative precision of approximately 10−16. Any time an individual term is morethan 1016 times the theoretical value of the integrand evaluated at the same point, the machineuncertainty coming from that term can dominate the result. We would hope that since this onlyhappens for a small subset of the points sampled the effect will be negligible as the numberof points increases, but this is not the case. If we take a random sample of N points, as isdone for Monte Carlo integration, we would expect the closest point to a given boundary (orother codimension 1 subspace) to be ∼ 1N away. Individual terms, however, often have a1t2divergence from the OPE of the marginal operator, which would lead to ∼ N2 divergence forthe closest point. This grows faster than the denominator, N , so the roundoff error in theresulting integral should increase linearly with the number of points. The deterministic case isactually worse because some points are intentionally chosen near or even right on the boundary.To combat this, whenever a sample point is close to a boundary or a diagonal, we can replaceit with a nearby point giving a decent approximation to the integrand. The integrand functionis effectively replaced by one where the value is held constant on small strips. While thismeans that a perfect integration with no uncertainty would give an incorrect result, the errorsintroduced this way are less problematic than the roundoff errors when we sample many pointswithout any regulation.When we discussed the differences between the big G and little g renormalization schemes,I mentioned that the lack of a regulator was an advantage of the little g scheme. Here we haveintroduced another regulator, so we naturally ask why this is not a problem. The regulator inthe big G scheme was required by the theory in that scheme, and we wanted the limit as itapproached zero. This regulator is to prevent roundoff error, which is the unavoidable resultof using a floating point datatype. Since we are regulating the integration region anyway, wemight ask why (aside from issues regarding finiteness at higher orders) we did not use thebig G scheme. By using the little g scheme, the integrand is independent of the regulator,which simplifies the integration process. There is not a different integrand for each value of theregulator, and instead of a limit, we only use a small value of the regulator, namely 3 ·10−4, forwhich the integrands always evaluate with negligible errors. The choice of regulator for theseroundoff errors is arbitrary, but we can estimate the error we have introduced by using thesame regulator to replace the value of the integrand with zero near boundaries and diagonals.Fortunately, the differences are minor compared with the statistical errors which are accountedfor by the algorithms’ reported uncertainties.1245.3. Technical DetailsConsistency checksSince the programs to construct wedge states with insertions and produce and evaluate integralscorresponding to the tachyon profile are quite complicated, it is worth using them to evaluatesome known quantities. We will see that the numerical integration process gives results whichare consistent with expectations the majority of the time, despite the presence of countertermsand the uncertain nature of numerical integration. An obvious choice for a quantity which weknow is the equation of motion, which should vanish. The equation of motion, however, hasghost number two, which means that its expectation value by itself will trivially vanish becausethe ghosts are not saturated. In order to test that QBΨ+Ψ∗Ψ vanishes we test that its overlapswith various other string fields all vanish. Because the equation of motion is stronger than justrequiring that the equation of motion annihilates all states and actually tells us that it shouldvanish exactly, as long as the string field is constructed properly these correlation functionsshould work out to zero whether they themselves were computed correctly or not. In orderto test that a non-trivial result also gives the correct answer, we look to the action. Becausethis is an exactly marginal solution, we expect the energy to vanish, and because the energy isproportional to the action, the action should vanish as well. The action has ghost number threeand does not need any additional test states inserted. That this also vanishes is our first strongtest that non-trivial expectation values are computed successfully. A summary of these testcalculations and their results using the deterministic algorithms is found in tables 5.6 and 5.7 atthe end of this section, and all of them are expected to vanish. The majority of the results areconsistent with zero, but a few exceptions require detailed examination. These six examples arein table 5.4, which restates their values using the deterministic algorithms and then includes thecorresponding results with Monte Carlo calculations and with deterministic calculations usinga vanishing integrand near borders and diagonals where cancelling singularities may occur.The values in tables 5.6 and 5.7 all use a border with width  = 3 ·10−4. When the integrandis sampled within  of a boundary or a diagonal, the closest point on the edge of this strip isused instead. In the last column of table 5.4, when the integrand was sampled at points withinthese strips, zero was returned instead. The difference between these results gives an estimateof how important the regulated region is to the final result of the integral, and we can see thatin most cases it is small compared to the error estimates.Looking at table 5.4, the first two quantities, 〈ceX(0),EOM(3)〉 and 〈Ψ(1),EOM(3)〉, havevery similar behaviours because the second integrand is√2 times the first. They are onedimensional integrals, so we can do them analytically and find that the results are exactlyzero. The integrands for these two are increasing as they approach each of the boundaries,which would suggest that the discrepancy comes from the regulated region, but the resultswith 0 inserted near the boundaries are identical. In fact, the QAG algorithm always seemsto give identical results with either choice of regulation near the boundaries, suggesting thatthat it does not pick points too close to the limits of integration. If we redo these integralsrequesting much higher accuracy, however, we find results which are less precise but consistentwith zero. Perhaps requesting a higher accuracy causes the algorithm to notice the cusp wherethe regulated strips at the boundary are, and that increases the error estimate. The kineticenergy at fourth order is a peculiar case of rounding errors, and it was already discussed inthe context of convergence. None of the integration algorithms is correct all of the time, andwe should not expect them to be, but QAG is problematic because it gives less control over the1255.3. Technical DetailsQuantity Cuhre/QAG Suave Cuhre/QAG with 0〈ceX(0),EOM(3)〉 (−6.1± 0.6) · 10−11 (4.1± 0.4) · 10−4 (−6.1± 0.6) · 10−11〈Ψ(1),EOM(3)〉 (−8.5± 0.8) · 10−11 (6.1± 0.7) · 10−4 (−8.5± 0.8) · 10−11〈Ψ, QBΨ〉(4) (−8.7± 1.2) · 10−10 (−1.5± 1.3) · 10−2 (−8.7± 1.2) · 10−10∂∂CL 〈ce3X0 ,EOM(5)〉 (6.2± 2.1) · 10−5 (−0.3± 1.1) · 10−6 (−0.4± 1.3) · 10−4∂∂C0〈c,EOM(6)〉 (−2.2± 0.5) · 10−4 (8.7± 7.8) · 10−4 (0.1± 2.1) · 10−2∂∂C1〈Ψ(3),EOM(3)〉 (−3.0± 0.2) · 10−8 (−5.1± 0.7) · 10−4 (0.8± 1.7) · 10−3Table 5.4: Numerical results for consistency checks which require further analysis. Among theresults which are expected to vanish, these six have deterministic results which do not. Theyare given using the standard deterministic algorithms Cuhre and QAG, using the adaptive MonteCarlo algorithm Suave, and using the deterministic algorithms with the integrand replaced with0 on the regulated strips near potential singularities instead of using the value of the integrandat a safe nearby point.sample size and does not report the total number of points it uses. We cannot show theseresults with different sample sizes, as we do for other quantities of interest. Because these areone-dimensional integrals, however, we can expect that the majority of them will be computedvery accurately.For the other three quantities, the integrals are multidimensional so we can evaluate themusing the Cuhre algorithm with several different sample sizes and see if they become closerto zero. This is shown in table 5.5. The first two of these both have error estimates thatdecrease as the sample size is increased, and values that decrease even faster. For the first,we see agreement once the sample size is large enough, and for the second we can suspectthat the result will continue to tend towards zero. The most important integrals in these twoquantities were shown in figures 5.8a and 5.8d respectively, where we can see the convergenceto 0 as N is increased. In the case of the final quantity, however, the result clearly does notvanish. If we change the thickness of the regulating border, however, it causes a significantfluctuation, and for extremely thin borders both the result and the uncertainty become muchlarger. This suggests that it is a genuine case of the border region having a significant impact.Unfortunately, the only way to resolve this issue would be to use higher precision floating pointdatatypes.1265.3. Technical DetailsAlgorithm Quantity Result Dimensions and Sample SizesCuhre ∂∂CL 〈ce3X0 ,EOM(5)〉(6.2± 2.1) · 10−5216055332131(3.7± 1.6) · 10−5 18005 54229(2.3± 1.3) · 10−5 25545 76581(0.2± 1.7) · 10−6 81055 243205Cuhre ∂∂C0 〈c,EOM(6)〉(−2.2± 0.5) · 10−4332131464107(−7.1± 2.6) · 10−5 54229 162027(−4.0± 1.7) · 10−5 76581 229347(−9.1± 5.2) · 10−6 243205 729045Cuhre ∂∂C1 〈Ψ(3),EOM(3)〉(−2.96± 0.19) · 10−8332131(−2.99± 0.13) · 10−8 54229(−2.99± 0.11) · 10−8 76581(−3.01± 0.06) · 10−8 243205Table 5.5: Three of the consistency checks in table 5.4 are shown with different sample sizes.The first two are computed as the sum of two integrals with different dimensions, while thethird is a single integral. We expect to see results which tend towards zero as the sample sizeis increased.1275.3. Technical Details〈c,EOM(2)〉 0〈ce2X0,EOM(2)〉 0〈ceX0,EOM(3)〉 −(6.1± 0.6) · 10−11〈ce3X0,EOM(3)〉 0± 1.1 · 10−16〈c,EOM(4)〉 −(0.6± 1.9) · 10−6 − (0.8± 1.3) · 10−6CL〈ce2X0,EOM(4)〉 (1.7± 2.5) · 10−6 + (9.2± 9.7) · 10−7CL〈ce4X0,EOM(4)〉 (2.8± 8.7) · 10−11〈ceX0,EOM(5)〉 (1.8 ± 2.5) · 10−5 − (1.3 ± 1.8) · 10−5CL − (0.2 ± 3.2) · 10−4(CL)2 + (2.1 ± 2.0) ·10−5C1 + (0.6± 2.3) · 10−5C0〈ce3X0,EOM(5)〉 (0.2± 2.3) · 10−5 + (6.2± 2.1) · 10−5CL− (0.3± 2.7) · 10−5C1− (0.7± 6.6) · 10−6C0〈ce5X0,EOM(5)〉 (0.2± 3.8) · 10−11〈c,EOM(6)〉 −(0.8± 2.0) · 10−3 + (0.7± 1.9) · 10−4CL − (1.6± 5.4) · 10−5(CL)2 − (1.8± 2.3) ·10−4C1 + (0.2± 7.1) · 10−3CLC1 − (2.2± 0.5) · 10−4C0 + (1.9± 6.5) · 10−5C0CL〈ce2X0,EOM(6)〉 −(0.1± 1.3) · 10−3 + (0.7± 3.7) · 10−4CL + (0.2± 1.9) · 10−3(CL)2 + (0.1± 2.1) ·10−3C1 + (0± 1.5 · 10−2)CLC1 + (0.9± 5.1) · 10−4C0 − (0.1± 2.9) · 10−3CLC0〈ce4X0,EOM(6)〉 0± 1.1 · 10−5 + (1.4± 1.0) · 10−6CL − (0.2± 5.9) · 10−6C1 + (0.4± 8.9) · 10−7C0〈ce6X0,EOM(6)〉 −(0.6± 1.3) · 10−14〈Ψ(1),EOM(2)〉 0〈Ψ(2),EOM(2)〉 0〈Ψ(3),EOM(2)〉 0〈Ψ(4),EOM(2)〉 0〈Ψ(5),EOM(2)〉 0〈Ψ(1),EOM(3)〉 −(8.5± 0.8) · 10−11〈Ψ(2),EOM(3)〉 0〈Ψ(3),EOM(3)〉 −(0.5± 1.3) · 10−5 − (0.1± 3.4) · 10−6CL − (3.0± 0.2) · 10−8C1〈Ψ(4),EOM(3)〉 0〈Ψ(1),EOM(4)〉 0〈Ψ(2),EOM(4)〉 −(0.3± 2.4) · 10−3 + (2.3± 5.2) · 10−5CL − (0.2± 1.8) · 10−5(CL)2〈Ψ(3),EOM(4)〉 0〈Ψ(1),EOM(5)〉 (2.4 ± 3.9) · 10−5 − (1.9 ± 2.5) · 10−5CL − (0.3 ± 4.5) · 10−4(CL)2 + (3.0 ± 2.8) ·10−5C1 + (0.8± 3.2) · 10−5C0〈Ψ(2),EOM(5)〉 0〈Ψ(1),EOM(6)〉 0Table 5.6: Deterministic tests of the equation of motion for the rolling tachyon. Superscriptsrepresent the order in λ of each quantity. Cuhre/QAG results shown.1285.3. Technical Details〈Ψ, QBΨ〉(2) 0〈Ψ,Ψ ∗Ψ〉(2) 0〈Ψ, QBΨ〉(3) 0〈Ψ,Ψ ∗Ψ〉(3) 0〈Ψ, QBΨ〉(4) −(8.7± 1.2) · 10−10 + (0± 4.2 · 10−14)CL〈Ψ,Ψ ∗Ψ〉(4) 0〈Ψ, QBΨ〉(5) 0〈Ψ,Ψ ∗Ψ〉(5) 0〈Ψ, QBΨ〉(6) (0.02± 0.30) + (0± 3.5 · 10−2)CL + (0± 0.11)(CL)2 + (0.5± 9.6) · 10−13(CL)3 + (0±6.7 · 10−2)C1 − (0.1± 2.3) · 10−13CLC1 + (0± 1.1 · 10−2)C0 + (0.1± 1.5) · 10−14CLC0〈Ψ,Ψ ∗Ψ〉(6) 0Table 5.7: Deterministic evaluation of the action for the rolling tachyon. Kinetic and cubic termsare found separately as a consistency check. Superscripts represent the order in λ. Cuhre/QAGresults shown.129Chapter 6SummaryThis thesis consisted of three main projects. All three involved solutions to string field theorywith marginal deformations of the initial D-brane configuration, and two also had connectionsto the tachyon condensation that represents D-brane decay. We saw both the level truncationand analytical approaches to string field theory. While there are still unanswered questions,progress has been made on each of these topics.In chapter 3 we saw a new solution to level truncated string field theory. Building on knownsolutions involving tachyon condensation and marginal deformations, this solution combinedthose two phenomenon. The result is a solution where a pair of separated D-branes move tobecome coincident and then a linear combination of the two decays. A combination of tachyondecay after a marginal deformation has since been studied analytically in [45], finding that theenergy is independent of the marginal deformation without examining any limitations on thepresumably unbounded analytical marginal deformation, but at the time the system examinedhere had only been seen in the context of separated brane-antibrane decay [53]. That systemlacked the enhanced symmetry when the D-branes are coincident, but otherwise had very similarresults.We saw that the part of the string field primarily responsible for D-brane translation isapproximately a linear function of the separation as long as that separation is small. Unfor-tunately, we are also able to show that a field redefinition mixes the terms, so that this linearrelationship is not by itself the correspondence between the marginal parameter and its physicaleffect. We also saw that both the parameter and the physical deformation have finite maximumvalues. This is in conflict with the natural expectation that translation of a D-brane shouldbe unbounded, but in agreement with all previous level-truncated results on marginal deforma-tions. In fact, we found evidence that contradicts early explanations for this maximum whichinvolved either a singular correspondence or a second branch of the solution, since in our caseit is the physical translation distance that is known and bounded.In chapter 4 we focused on the work of Kiermaier and Okawa [36]. For a class of marginaldeformations with the OPE V (0)V (t) ∼ 1t2 they showed that it is possible to define a solutionusing renormalized marginal operators. That work also provides an example of a renormaliza-tion scheme which is compatible with the sufficient conditions for their solution. Here I havecarefully studied those conditions and proven that they are all satisfied for a slightly more gen-eral family of renormalization schemes with two parameters. This also filled some gaps in theoriginal proofs. Considering the most general case, we saw that there are likely to be infinitelymany free parameters, but for our proofs we only included two. We hope that this freedom isconnected to gauge transformations, but that remains unproven. After studying two differentapproaches to renormalizing the marginal operators, we saw that they are not equivalent be-yond quadratic order, and that the more natural approach at quadratic order needs subleadingcorrections in order to remain finite beyond cubic order.130Chapter 6. SummaryOne well known example of a marginal deformation is the rolling tachyon. Physically itcorresponds to the time-dependent process of D-brane decay. The time-asymmetric rollingtachyon has been studied extensively, and no further calculations were necessary here. Thetime-symmetric case, however, was not previously studied due to the fact that it has a singularself-OPE of precisely the form that Kiermaier and Okawa’s solution was intended to renormalize.In practice, the solution is difficult to construct, but we found in chapter 5 that it can be doneup to 6th order in λ by combining symbolic algebra and numerical integration programs. Sucha calculation to explicitly construct component fields of a renormalized marginal solution hadnot previously been attempted. In this way we found that the tachyon profile for small λ isqualitatively no different from what we would find if we took the tachyon profile of the time-asymmetric case and symmetrized it. 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That work was intended as supplementary materialto [1], and is not intended for publication in a peer reviewed journal.A.1 Level (3,9) PotentialIn level-truncated studies of string field theory, the string field is expanded in a basis of con-formal primaries and their descendants and then truncated at some eigenvalue of the virasorozero-mode operator. In this note we present the potential, proportional to the action, for allterms in the string field up to level 3. Our intended theory is a collection of D-24 branes whosetransverse directions are all aligned, but which are not necessarily coincident. The choice ofD24-branes means that there should be one direction which is singled out, and this is the mainassumption we have made on our string field. Any system of bosonic D-branes with a rotationalsymmetry in 25 of the 26 dimensions should be described at level 3 by this potential. The stringfield we use will then be equivalent to (3.7) studied in chapter 3:|Φ〉 =∑d(tdc1 + hdc0 + udc−1 + vdL′−2c1 + wdL25−2c1 + od(b−2c−1c1 − 2c−2)+ o˜d(b−2c−1c1 + 2c−2) + pdL′−3c1 + qdL25−3c1 + . . .)|0; d〉+∑d(xdc1 + fdL25−1c1 + rdc−1 + sdL′−2c1 + ydL25−2c1 + zdL25−1L25−1c1 + . . .)α25−1 |0; d〉 . (A.1)where LXn = L25n are the virasoro operators in the transverse direction, and L′n contain the sumover the other 25 directions of matter oscillators. The ghost CFT is handled explicitly in termsof bn and cn operators.Unlike in (3.7), in this case the vacuum states and the coefficients are not labeled withD-brane indices. They are labeled only by d, the eigenvalue of α250 . The vacuum is defined byα250 |0; d〉 = d |0; d〉, and αµ6=250 |0; d〉 = 0. This does not uniquely determine the vacuum state,as there are many different ways to achieve a given zero-mode, but once a physical situation ischosen the appropriate values can be inserted for d. In our case this meant replacing the sumover d with a sum over brane indices and the eigenvalue with the separation. Since the actionis more general than the case we have considered the coefficients given here are in as generala form as possible. While the case of non-zero α0 eigenvalues in the rotationally symmetricdirections has not been considered, the twist-odd states have been. The action in this appendixdoes contain all of the necessary couplings to include twist-odd states in the string field.With the string field defined, we can move on to writing down the action. The quadraticterm is a function of the zero-mode eigenvalue d, assuming that the vacuum states have been136A.1. Level (3,9) Potentialput into a basis which is orthogonal. For every set of three fields there is a cubic term which is afunction of the three zero-modes of the strings. For the study of separated D-branes or non-zeromomentum states these zero-modes must sum to 0, but we will leave them unconstrained. Theaction can be written asS = −12∑l,m∑dAlm(d)φ(l)†d φ(m)d −∑l,m,n∑d1,d2,d3Blmn(d1, d2, d3)(43√3) 12 (d21+d22+d23)φ(l)d1φ(m)d2φ(n)d3 ,(A.2)where l, m, and n run over the set of all fields we are considering. The cubic interaction termis normally written with a factor of 13 , but I will define the coefficients Blmn to include it. Thisaction is equivalent to the potential of (3.5) after rescaling to the appropriate units V = −2pi2S.The purpose of this appendix is then to explicitly list the coefficient functions Alm and Blmnfor the first few levels. The quadratic coefficients Alm(d) are presented in table A.2, and thecubic coefficients Blmn(d1, d2, d3) are in table A.3.For our special case of separated D-branes, where the vacuum states are represented byChan-Paton factors, the cubic terms use d1 = di − dj , d2 = dj − dk, d3 = dk − di, and the fieldassociated with such a state has the appropriate indices such as φ(t)d1 = tij . For example, thepotential for only the fields tij and xij would contain the termsS = −12(Atttijtji +Axxxijxji)−(Bttttijtjktki + 3Bttxtijtjkxki + 3Btxxtijxjkxki +Bxxxxijxjkxki)(A.3)where the separation parameters of the coefficients were omitted for brevity. The other orderingsof fields, such as Btxttijxjktki, are equivalent to the ones shown once the D-brane indices aresummed over, which is why the factors of 3 appear.There are in principle as many as six orderings for the cubic interaction of each set of threefields, but we have shown in section 3.1.2 using the twist operator that we only need to calculateone of them. Similarly we know from the properties of the inner product and the hermicity ofQB that Alm = ±Aml. Specifically, we know thatAlm(d)φ(l)†d φ(m)d = Aml(d)φ(m)†d φ(l)d , (A.4a)Blmn(d1, d2, d3) = Bmnl(d2, d3, d1) = Bnlm(d3, d1, d2) , (A.4b)Blmn(d1, d2, d3) = −ΩlΩmΩnBnml(−d3,−d2,−d1) , (A.4c)where Ωl is the twist eigenvalue of the operator associated with the state∣∣φ(l)〉. Since thequadratic reordering rule involves the behaviour of the coefficient field under hermitian con-jugation and not just the masses Alm, table A.2 includes both orderings for the few non-zerooff-diagonal quadratic terms.The separation parameters d1, d2, and d3 in (A.2) are the eigenvalues of the operatorsα250 acting on the vacua of the three string fields, so this allows our coefficients Blmn to beuseful in describing other situations, such as lump solutions. The same coefficients can be usedwith a different choice of parameters. This is also true of the quadratic terms Alm. The firstterms Alm(0) and Blmn(0, 0, 0) were given in [21] and an action including terms with non-zeromomentum was given in [16], so for comparison a brief dictionary of fields is given in table A.1.The couplings Alm and Blmn found here can then be used to reproduce the action in either ofthose works up to level (3,9). An example of such a calculation was given in section Level (3,9) PotentialHere t x h u v w f o p q r s y z[21] t as NA u w v NA NA NA NA s r r¯ y[16] t NA NA u w v z∗ NA NA NA NA NA NA NATable A.1: A comparison of our field definitions to those used in two other works on leveltruncated solutions.*There is a d-dependent difference in normalization for this field.l m Alm(d)t t 1/2 d2 − 1x x 1/2 d2h h −2u u −1− 1/2 d2v v 252 +254 d2ww 1/4 (4 d2 + 1)(d2 + 2)wf 3/2 d(d2 + 2)f w −3/2 d(d2 + 2)f f −(d2 + 2)(d2 + 1)o o 8 + 2 d2o˜ o˜ −8− 2 d2p p −100− 25 d2q q −1/2 (3 d2 + 2)(4 + d2)q y −5/2 d(4 + d2)y q 5/2 d(4 + d2)q z −6 d(4 + d2)z q 6 d(4 + d2)r r −2− 1/2 d2s s 254 d2 + 25y y 1/4 (4 + d2)(4 d2 + 9)y z 3/2 (3 d2 + 2)(4 + d2)z y 3/2 (3 d2 + 2)(4 + d2)z z 3 (4 + d2)(d2 + 2)(d2 + 1)Table A.2: The quadratic coefficients for the fields in the action up to level 3. Off-diagonalterms which are not shown are all 0.138A.1. Level (3,9) Potentiall mn Blmn(d1, d2, d3)t t t 2764√3t t x 932 d1 −932 d2t t h 0t t u 1164√3t t v − 125128√3t t w 1128√3(4 d12 − 8 d1d2 + 4 d22 + 4 d3d1 + 4 d3d2 − 8 d32 − 5)t t f 164√3(2 d2 + 4 d3d12 − 8 d3d2d1 + 4 d3d22 − 9 d3 + 2 d1)t t o 0t t o˜ 0t t p 0t t q 196 (d1 − d2)(2 d1 − 15 d3 + 2 d2)t t r 1196 d1 −1196 d2t t s − 125192 d1 +125192 d2t t y 1192 (d1 − d2)(4 d12 + 4 d3d1 − 8 d1d2 − 37− 8 d32 + 4 d22 + 4 d3d2)t t z 196 (d1 − d2)(4 d32d12 − 8 d32d2d1 + 6 d3d1 − 22− 27 d32 + 4 d32d22 + 6 d3d2)t x x −1/16√3(−4− d1d2 + d12 − d3d1 + d3d2)t x h 0t x u − 1196 d1 +1196 d3t x v 125192 d1 −125192 d3t xw −1/48 d13 − 1/6 d2 − 1/16 d3d2d1 − 1/24 d33 + 1/48 d32d2 + 1/48 d3d22 + 1/24 d12d2 + 1/16 d32d1 +37192 d1 +59192 d3 − 1/48 d1d22t x f −1/24 d3d13 − 1/12 d32d2d1 − 1/24 d3d1d22 − 1/48 d1d2 + 1/24 d32d22 − 516 d3d2 + 1/24 d32d12 −1/48 d12 − 332 d32 + 4396 d3d1 + 1/12 d3d2d12 + 1/3t x o 0t x o˜ 0t x p 0t x q − 1432√3(−17 d3d12+15 d3d2d1−15 d32d2+28 d3+2 d13+15 d32d1+2 d3d22−2 d1d22+24 d2−40 d1)t x r − 11432√3(−4− d1d2 + d12 − d3d1 + d3d2)t x s 125864√3(−4− d1d2 + d12 − d3d1 + d3d2)t