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The AdS/CFT correspondence and string theory on the pp-wave Ramadanovic, Bojan 2008

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The AdS/CFT Correspondence and String Theory on the pp-wave by Bojan Ramadanovic B.Sc., Simon Fraser University, 2001 M.Sc., University of British Columbia, 2003 A THESIS SUBMITTED IN PARTIAL FULFILMENT OF THE REQUIREMENTS FOR THE DEGREE OF DOCTOR OF PHILOSOPHY in The Faculty of Graduate Studies (Physics and Astronomy)  THE UNIVERSITY OF BRITISH COLUMBIA (Vancouver) October 2008  ©  Bojan Ramadanovic, 2008  Abstract  ii  Abstract Aspects of the AdS/CFT correspondence are studied in the pp.-wave/BMN limit. We use the light cone string field theory to investigate energy shifts of the one and two impurity states. In the case of two impurity states we find that logarithmic divergences, in the sums of intermediate states, actually cancel out between the Hamiltonian and a Q-dependent contact term”. We show how non-perturbative terms, that have previously plagued this theory, vanish as a consequence of this cancelation. We argue from this that every order of internal impurities contributes to the overall energy shift and attempt to give a systematic way of calculating such sums for the case of the simplest 3-string vertex (one proposed by diVecchia). We extend our analysis of the mass shift to the case of the most advanced 3-string vertex (proposed by Dobashi and Yoneya). We find agreement between our string field theory calculations and the leading order CFT result in the BMN limit. We also find strong similarities between our result and higher orders in the field theory, including, on the string side, the disappearance of the half-integer powers which generically do not exist in the field theory calculations. We also study the orbifolding of the pp-wave background which results in the discrete quantization of the light-cone momentum. We present the string field theory calculation for such a discreet momentum case. We also observe how a particular choice of the orbifold, results in the string theory corresponding to the quantization of the finite size giant magnon on the CFT side. We study this theory in detail with particular emphasis on its super algebra.  Contents  iii  Contents Abstract  ii  Contents  iii  Acknowledgements  vi  Foreword  vii  1  Introduction 1.1 Perturbative string theory 1.1.1 Classical bosonic string 1.1.2 Supersymmetry 1.1.3 Closed string spectrum arId the background fields 1.2 T- Duality and Dirichiet branes 1.2.1 T-duality 1.2.2 Properties of D-branes 1.3 AdS/CFT correspondence 1.3.1 Origin of the AdS/CFT 1.3.2 Black holes and stacks of branes 1.3.3 Early equivalences Entropy and absorption cross-section 1.3.4 Maldacena Conjecture 1.3.5 Evidence of the AdS/CFT and holography 1.3.6 Summary  1 1 1 6 11 13 13 15 17 17 18 20 23 23 27  AIS/CFT and the interacting strings 2.1 Strings on the plane-wave and the BMN limit 2.1.1 Penrose limit of the AdS 5 x S 5 2.1.2 Strings on the pp-wave limit 2.1.3 The Supersymmetry group of the pp-wave 2.1.4 BMN limit 2.1.5 Interactions in the BMN limit 2.2 String field theory on the pp-wave background 2.2.1 Bosonic particle in the plane wave background 2.2.2 Bosonic string field theory 2.2.3 Fermionic contribution to the vertex 2.2.4 Interaction super-algebra and the pre-factors 2.2.5 The contact interaction term  28 28 28 29 31 32 34 35 36 38 40 40 44  -  2  Contents  2.3  2.4  2.5 3  iv  2.2.6 Impurity conserving channel Higher impurity channels 2.3.1 Trace state 2.3.2 Four impurity channel 2.3.3 Generalizing to arbitrary impurities 2.3.4 Conclusion Alternate vertices 2.4.1 DVPPRT vertex 2.4.2 DY vertex Conclusion  45 48 48 50 53 57 57 58 64 68  Orbifolding and the discrete light-cone quantization 3.1 Orbifolding and the BMN limit 3.1.1 String theory on the pp-wave orbifold 3.1.2 .JV 2 gauge theory 3.1.3 Duality 3.2 String field theory on pp-wave orbifold 3.2.1 k 2 Impurity-conserving mass-shift 3.2.2 k = 3 Impurity-conserving mass-shift 3.2.3 4 impurity channel for the k=2 3.3 Giant magnon and the single impurity multiplet 3.3.1 Giant magnon in AdS/CFT 3.3.2 Magnon boundary conditions and the orbifold 3.3.3 Orbifold in gauge theory 3.3.4 Orbifold in plane-wave limit 3.3.5 Near pp-wave limit 3.4 Conclusion .  4  Summary  Bibliography  96  A Fermion representations  104  B Neumann matrices and associated quantities-  107  C More Neumann matrices and relations  109  D Calculational method D.1 Vertices and definitions D .2 Commutation relations D.3 Matrix elements D.4 More matrix elements  111 111 113 113 115  Contents  v  E Energy Shift for DVPPRT Vertex 117 E.1 Derivation of the “Master Formula” 117 E.1.1 The two-string state <ely> and the number <vie>< ely> 117 E.1.2 The two-string state <Q e >< Q 2 ev> 119 2 eiv> and the number <vjQ 2 E.2 4-impurity calculation 125 .  F Hamiltonian and the supercharges of the orbifolded string F. 1 Hamiltonian of the orbifolded string F.1.1 Bosonic interaction Hamiltonian F.1.2 Fermionic interaction Hamiltonian F.2 Supersymmetries of the orbifolded string in near pp-wave limit  .  .  133 133 133 135 136  Acknowledgements  vi  Acknowledgements I would like to acknowledge here immense help of my supervisor, Gordon Semenoff in the research work leading to this thesis. I would like to thank Gordon for ideas, instruction, motivation, patience, funding and good cheer over almost seven years of collaboration. Like wise, I wish to acknowledge huge contributions of our other collaborators, Gianluca Grignani, Shirin Hadizadeh, Martha Orselli and, in particular, Donovan Young. Over the five years of our association Donovan’s output was always inspiring in quality and humbling in its prodi gious quantity. The core of this thesis owes a lot to his superior work [81]. Great thanks also go to the people of UBC physics department, both within the string group and beyond. In particular I would like to thank Don Witt and Kirstin Schleich for being extremely helpful and friendly people as well as great teachers; Mark Van Raamsdonk, Moshe Rosali and Joana Karczmarek for being the backbone of our group; Henry Ling, Brian Shieh, Thomas Kiose, Phil DeBoer and Mark Laidlaw for many useful informal discussions and Janis McKenna for lots of valuable advice and, even more importantly, for her contagious optimism. Not any less, I would like to thank my undergraduate teachers and colleagues at Simon Fraser University’s mathematics and physics departments, in particular Igor Herbut, my undergraduate thesis supervisor, mentor and friend. My thanks go in equal measure to all those important non-physicists in my life who have inspired and supported me, from parents and grandparents who ignited and preserved the spark of my curiosity about the world to the magnificent friends without whom I would have neither the mental acuity nor psychological energy for this work. Special thanks go to the people of St. Catherine’s church of North Vancouver who have quite literally changed my life when they decided to help a family of refugees, with whom they had nothing in common except shared humanity, to immigrate to this wonderful country of Canada. Finally I thank Hilary for the love of twelve years and many years to come.  Foreword  vii  Foreword Since its inception in the 1960ies string theory has had a bewildering array of applications. Original version, intended to explain the “Regge slope “ relationship between spins and masses of strongly interacting bosons, and thus provide theory of the strong nuclear force [1J, fell by the wayside with the advent of quantum chromodynamics (QCD). In the 1980’s a more elaborate string theory was developed, the main feature of which was the fact that the perturbations of the background metric of the theory were quantized objects within the theory itself. Quantization of gravity was an important goal in theoretical physics ever since the Einstein-Hubert action was proposed for general relativity. String theory, in which gravity quanta are natural objects, seemed to offer the best ever hope for the unification of all known physical laws. To this day, the great popularity of string theory and its derivatives stems, to a large extent, from the idea that it is within its framework that such a “grand unified theory of physics” will be found. The “modern” string theory of the 1980’s had its own share of problems. Self-consistency of the theory constrains its background to the seemingly unphysical 10 dimensions. (26 in the case of a purely bosonic theory). Perhaps even more importantly, for a candidate to be the One Theory, string theory appeared to come in at least 5 very distinct flavors. The same underlying principles coupled with some significant differences, such as the number of supersymmetries and their relative chirality, resulted in a number of different physical theories, all of them with quantized gravity. The effort to resolve this “ embarrassment of riches” was at the heart of the second superstring revolution that took place in the mid 1990’s. At this time the mathematical transformations between the different string theories were discovered, indicating that the different theories can be considered as simply perturbative limits of a single underlying theory. These transformations or dualities came to be of extreme importance to the devel opment of string theory. In particular, a relationship was found, so called “strong-weak duality”, that links the strong interaction behavior of one theory to the weak interaction be havior of another. Many physical theories rely on perturbative methods for making sensible calculations, a method which is limited to the weakly coupled sector of the theory in ques tion. Therefore, a tool that could link “inaccessible” strongly coupled regimes to something that can be tackled by a perturbative method in a different theory, was potentially a ground breaking discovery in its own right. While dualities did “unify” string theory into one systemic framework (sometimes referred to as M-theory) it remains plagued by the lack of unique connection to the 4-dimensional world of empirical experience. Specifically, the number of ways in which 6 “extra” dimensions can be compactified, or otherwise dealt with, remains so large as to give an almost infinite number of parameters to any effective 4 dimensional “new physics” coming out of string theory. While this remains an important problem in string theory, the discovery of dualities -  -  viii  Foreword  also opened a different line of research with potentially more immediate benefits. The important question in this line of research is: whether the dualities concept can be used beyond the limits of the string theory and, in particular, if it can be applied to the one physical theory that suffers most from the lack of perturbative method in the relevant regime namely QCD itself. One of the most important recent advancements in string theory came about in late 1990’s when a duality was proposed that links a particular conformal field theory on 4 dimensions to a string theory on a specific curved background [2]. This conjecture, called the “Maldacena Conjecture” by the name of its original author or, more usually, AdS/CFT duality, represents a crucial step in our understanding of, and the ability to perform calculations in, both field theories and string theories, even if it does not yet lead to finding a perturbative dual of QCD itself. While the Maldacena Conjecture has not been explicitly proven yet, it has acquired a considerable body of supporting evidence. Over the past ten years a large amount of work was done testing, expanding and qualifying the conjecture under various regimes. Our own research efforts were a small part of this expansive program. This thesis is a result of over four years of work on the issues concerning the duality between conformal field theory and string theory on a ten dimensional space consisting of 5 x S ). Even more 5 a five dimensional sphere and five dimensional anti-deSitter space (AdS specifically, most of our work was focused on the string side of this duality in the context of 5 x S . 5 a particular particular Penrose limit of the AdS provide the broad context for our work. this will we thesis of In the introduction chapter We will give brief introduction to string theory in the context of Green-Schwarz quantization in the light-cone gauge, relying heavily on [6]. We will also attempt to explain the principle of dualities, in particular, that of AdS/CFT. Finally we will go over the limiting process by which we go from string theory on AdS a very complex theory to the solvable string theory on the plane-wave background. In the second chapter we will discuss the string field theory and the “loop corrections” to the energy of the string states. We will discuss the efforts to obtain the energy shift using those “ioop corrections”, such that it corresponds to the known dual result in the CFT (that dual result being non-planar diagram contributions to scaling dimension of the operators in the conformal field theory). In the second half of this chapter we will present the results of our own work on this problem. Most important of those results is a generic cancelation of the divergencies that were previously present in the theory beyond a certain channel. Other results include the formula for the generic channel contributions to the energy shift (given particular vertex) and the conclusion that all channels appear to contribute in the same degree to the final result. Finally we also present the result of the single channel calculation which up to this point is the one in closest agreement with the desired CFT result. The third chapter will focus on the consequences of the orbifolding of the background of both AdS and CFT theories. We will discuss the supersymmetry breaking that happens as a consequence of the orbifolding and its recovery in the plane-wave limit of the orbifolded theory. Also we will look at the magnon super-multiplet of the conformal field theory, its connection to the orbifolding, and its behavior in and near the pp-wave limit. The scientific work that is at the basis of this thesis was a colaborative one. The work on the string field theory as discussed in Chapter 2. and published in [79] and [80] was -  -  -  Foreword  ix  done jointly with Gordon Semenoff and Donovan Young. We met regularly and shared most aspects of the work with Gordon providing questions and ideas and Donovan and I doing most of the calculations. Our group worked in colaboration with Gianluca Grignani and Marta Orselli who were frequent visitors at UBC but otherwise worked independently on the same set of problems as us. There were frequent discussions between our two teams both in person and in correspondence that were crucial to the eventual results. Results concerning the “master formula” for the DiVecchia vertex were obtained after the joint publications and were largely done by this author alone with motivation and advice offered by Gordon Semenoff. Work published in [85] and refered to in Chapter 3. of this thesis was a colaboration between Gordon and I with Gordon providing context and most of the field-theoretic results and me doing the bulk of the work on the string theory side.  1  Chapter 1. Introduction  Chapter 1 Introduction 1.1 1.1.1  Perturbative string theory Classical bosonic string  Particle path integral The starting point for the discussion of string theory may well be the generalization of the action that describes the motion of a single massive particle in a background gravitational field. Such an action would be given by an invariant length of the world line:  s  =  2 _mfds; ds  =  (1.1)  —g(x)dxdx  where g,(x) is the metric of the background field. If the particle trajectory is parametrized by some coordinate r then we can write the above as: S= _mfdT_gy(x)±±n  (1.2)  with dots signifying the derivative with respect to T. We can write this more generically by using an auxiliary coordinate e(r) S  =  f  (1.3)  — e(r)m ) 2  which reduces to 1.2 if one solves the equation of motion for eQr) and substitutes it back into 1.3. Important to note is that in the original parameterization coordinate r was chosen ar bitrarily and that therefore 1.3 must be symmetric under the re-parametrization of r —* r’. This re-parametrization symmetry enables us to take a convenient choice of e(T) such as eQr) = 1/rn making the solving of the 1.3 a simple matter. We would still have to satisfy the e(’r) equation of motion g,(x)±fz  m 2 + eQr)  =  0; g(x)±±1  =  —1  (1.4)  keeping it as an on-shell condition for the particle. In this basic framework one can then proceed to quantize the particle and construct the path integrals for its propagator, vertexes etc...  2  Chapter 1. Introduction  String action It is possible in principle to generalize this framework to objects of higher dimensionality 1 strings, d = 2 membranes etc...) however, as it turns out, the case than a particle (d advantages over the alternatives, including the symmetry under Weyl multiple of d = 1 has scaling that will prove important in the further discussion. We will therefore focus, for now, on the theory of one dimensional objects strings. As the strings propagate they trace surfaces in space-time. These world-sheets play the same role as the world line does in the particle case. Specifically we can construct the action out of invariant surface area of the world sheet. This action -  SNG  =  _2h1  f  drdudet(g(X)8XP(u, r)8Xv(u, T))  (1.5)  called the Nambu-Goto action is the equivalent of the 1.2. Once again we can introduce the auxiliary coordinate metric of the world sheet: h The resulting action, equivalent of the 1.3 and named after Polyakov, is arguably the foundational equation of the string theory. -  S  = _  f  (1.6)  Where h = det(h) and we introduce the parameter T = (2ira’)’, with the dimension of the inverse square of length, which can be though of as the tension of the string. As in the case of the particle, the parameters a and ‘r used to describe the world sheet are arbitrary. The action will therefore be symmetric under the re-parametrization of those variables. In addition, due to the fact that we are dealing with a 1 + 1 dimensional world sheet, the action will also be symmetric under the Weyl scaling 6h° = Ah where A is an arbitrary infinitesimal function of the coordinates. Again, like in the case of the particle, varying the action with the auxiliary parameter gives us the on-shell condition for the fields. T  =  =  —  0  (1.7)  Where the T is an energy-momentum tensor of the world sheet. It can easily be shown that the condition 1.7 directly relates the expressions 1.5 and 1.6. The three symmetries listed above (two re-parametrization ones and Weyl) give us enough freedom to manipulate the independent components of the world sheet metric and set it for simplicity to a two dimensional Minkowski metric. It is important to note that even with the worldsheet metric gauged away to Minkowski, its equation of motion 1.7 remains a constraint on the theory; so called Virasoro constraint. We can now proceed to write down the equations of motion for the Xs. They will clearly depend on the background geometry gv. For now we can simplify the problem by setting the background to be the flat Minkowski space. With the Minkowski background, and the world sheet metric gauged away, the equations of motions for Xs are very simple: -  -  (3  —  8)X(u, T)  =  0  (1.8)  3  Chapter 1. Introduction  To fully ensure the invariance of the action under the variation of X we also need the boundary conditions. We can choose between X(a, r)  =  cx + ir, T) X ( m  O.X(0,r)  =  8X(7r,r)  (1.9)  and =  (1.10)  0  where in both cases a ranges between 0 and ir. The above two conditions correspond to closed and open strings respectively. Mode expansion and the light cone gauge Focusing on the closed strings we can easily solve 1.8 X(a, T)  X  =  + a’p(r  —  =  (1.11)  X+X  a) +  e_2iT_  n$O  =  +  ‘p(r  +  e_2iT  +  a)  nO  Here we introduced the Fourier components c which can be though of as oscillator coordinates. It is important to note that the reality of X functions implies: =  (iL)t,  =  (1.12)  (ä1)t  and also the fact that x and pP are themselves real. We can get Poisson brackets of the XIL and X from 1.6 [X(a),X’(a’)]p.B.  [)(a),)’(u’)]p.B.  =  [(a),Xv(af)]  =  T’(a  —  0  a’)  (1.13)  which give the Poisson brackets of the oscillators:  [, 1P.B.  [IL =  n1P.B.  =  im6m+nr/’ —n  I IA V1 I m niP.B.  114  Furthermore, Virasoro constraint 1.7 can be written as: =  (  + X’2)  =  0  A very similar procedure can give us the mode expansion of the open string.  (1.15)  4  Chapter 1. Introduction  At this point it is useful to depart slightly from [6] and introduce the light cone gauge framework for most of the subsequent work. What allows us to impose any new gauge on the theory is the fact that setting hW’ = does not fully use up both re-parametrization symmetries and Weyl. Specifically, once we have a Minkowski world sheet metric, we can preserve it while still changing the coordinates though a particular combination of the re-parametrization and Weyl scaling. If the re transforms h’s” as parametrization ° = -  =  (1.16)  —  —  then + combination leaves hIw = work in the coordinates o  =  (1.17)  =  invariant. We can see the power of this symmetry best if we r ± u Then =  0,  =  (1.18)  0  meaning that the all transformations of the form (1.19) keep the metric intact. The main advantage of the light cone gauge is that it makes the implementation of Virasoro constraint trivial and eliminates unphysical degrees of freedom. Its disadvantage is that it makes Lorentz invariance non-obvious. To impose the light cone gauge we begin by singling out two directions and label them as+and X± while keeping remaining the d erably.  —  =  X!_l)  =(X° ±  (1.20)  2 ones intact. We can then use 1.19 to simplify X consid  = —  =  (X 1 ,  —  (1.21)  x)  This relates X directly to the r coordinate of the world sheet by: X(u,r)  =  x  We use this simple form of the X to re-write the Virasoro condition  ( ± X’)  ± X’) 2 =  4a’p  (1.22)  +pT  ± X’) 2  =  0 (1.23)  5  Chapter 1. Introduction  The upshot of this result is that the expression X becomes completely fixed in terms of X making it explicit that the system carries only d 2 physical degrees of freedom. X are fully fixed by the gauge freedom and constraints. Equation 1.23 does not cover all of the Virasoro constraints from 1.15. To do that we use the Fourier expansion of the energy-momentum tensor from equation 1.7. —  r  LI  T —  2  jr  j  I  —  —2imurp ,j £ (hue  —  i  —  ——  V.’ nO  p  Lm  p,r I  2  dg —2im T+  V’ a--+LOm_nan -  =  (1.24)  =  with the requirement T__ 0 0+L use combinations L  = =  2 m  0 translating into Lm = 0 = 0 to obtain: 0—L 0 and L =  =  2 —p  --  (c_a +  Lm  &,)  =  0 for all m. We can then  (1.25)  and (1.26)  =  nO  nO  These two are important results. 1.25 tells us how excitation of the oscillators “creates” their space-time mass. 1.26 is usually referred to as the “level matching condition” and represents the remaining constraint on the kind of states strings can form. Modifications of the level-matching condition will play an important role when we begin considering strings on the orbifolded backgrounds. We are now ready to begin quantizing the light-cone string. The usual procedure is used, of promoting fourier modes into creation and annihilation operators and poison brackets into commutators. However, there are some potential pitfalls. Specifically one can directly test Lorentz invariance of the quantized theory in the light cone gauge. This leads to the addi tional set of requirements on the theory. In particular, the disappearance of the commutator [Ji_, Jj in Lorentz algebra is required if one is to avoid the anomalies. It can be shown that this disappearance can only be affected, within the above framework, if the number of the background dimensions is set to 26. This is an interesting calculation and a harbinger of the important geometric requirements that string theory tends to impose upon itself. I will not pursue this calculation here. The theory we have followed so far is purely bosonic and thus not very interesting for most physical applications. I will therefore follow with introduction of the new world-sheet symmetry relating bosonic degrees of freedom to their fermionic counterparts. The idea of such a “supersymmetry” was one of the most fruitful ones from the early years of “modern” string theory and have lead to a number of applications beyond string theory proper.  Chapter 1. Introduction  1.1.2  6  Supersymmetry  Superparticle Returning to the action of the bosonic particle: S  f  =  (1.27)  )dr 2 (e’± — em  and working in the m —* 0 limit we can extend the action to include more symmetries. Specifically we are interested in a symmetry relating the bosonic coordinates x to a set of a We can introduce N such supersymmetries with N corresponding spinor coordinates 0 Aa where A = 1.. .N. The full description of the supersymmetry uses sets of coordinates 0 A 6 the Grassmann spinors : =  6x  ApIA  öOA  =  A, 6  öe  =  0  (1.28)  where P’ is the representation of the d-dimensional Clifford algebra {F’ ’ 1 , PV}  =  (1.29)  2i’’  A number of generalizations of 1.27 exist that are invariant under 1.28, one of the most straightforward being:  S  =  f  dr 2 €‘(± — irOA)  (1.30)  This action maintains the Poincare invariance along with its super-symmetry. We can write down the equations of motion of the action 1.30  =0, p=0, P•pO=0 2 p  (1.31)  where ,rn  =  —  ApiA  (1.32)  From these it can be easily shown that the matrix P p has only half of its maximum possible rank and that therefore half the components of O decouple from the theory (because 0 only ever appears in the action multiplying P p). This is related to the additional symmetry of 1.30 which gets its name kappa symmetry - from another Grassmann spinor: ic(r) -  gApA  6e  =  4eic  (1.33)  The reader should refer to [6] for proof that 1.33 is in fact a symmetry of 1.30. A peculiarity of the ic-symmetry is that it requires the equationsof motion to close its algebra. Likewise, equations of motion make all the potential conserved charges of ic vanish. Thus the main effect of 1.33 is to keep half the components of O decoupled from the theory.  7  Chapter 1. Introduction  Classical superstring The most obvious generalization of the above principle to the 1.6 action in a flat background is: =  1 2ir  I  •  (1.34)  fl  j  where ri  (1.35)  —  =  While this action does posses N supersymmetries as well as the usual re-parametrization invariance, it does not have ic-symmetry we have noticed in the discussion of superparticle. As a consequence, 0 of this theory would have double the number of degrees of freedom leading to the very complex non-linear equations of motion. It is, however, possible to add a term S 2 to the action such that resulting action S = S +52 has ic-symmetry and thus only half the degrees of freedom of 0 leading to solvable equations of motion. Introducing the S 2 term imposes a number of consistency conditions on the theory. First 2 being of all, it limits the number of supersymmetries to two or less, a construction of S the action: impossible for the cases N> 2. With that condition in mind we can write 2 S  =  f  u{—i 2 d  aax(’F8b8’  —  F8b0 ) 2  (1.36)  +  This term of the action must itself obey the N = 2 supersymmetry. This requirement poses further conditions on the periodicity/handedness of the fermionic fields as well as the dimension of the background space-time. The possibilities left for the dimensionality of spacetime left by the supersymmetry requirement are D = 3,4,6 or 10 with each dimension linked to a particular requirement for spinors being Majorana and/or Weyl. Further conditions imposed by the quantization will limit this to 10 dimensions with Weyl-Majorana fermions. [6] and other standard textbooks give comprehensive proof of the invariance of S = S + 82 under ic-symmetry as well as the detailed discussion of the supersymmetry dependance on the background dimension and properties of fermions. Type I and II theories The restrictions imposed by the supersymmetry: N 2, D = 10, 0 Mayorana-Weyl, still leave a number of distinct choices to be made when constructing the super-symmetric theory. The most obvious choice is between cases N = 1 and N = 2 corresponding to Type I and Type II string theories, respectively. Furthermore, for the case of N = 2, The Weyl condition requires that 0’ and 02 each have definite handedness. That leaves two physically distinct possibilities: either they have the same or opposite handedness. In the case of opposite handedness,the theory will necessarily exclude both open strings = 02 on the string ends) and (because the boundary condition for open strings demands un-oriented closed strings (because the left and right propagating modes will be of opposite handedness). This is the Type hA string theory. -  8  Chapter 1. Introduction  The cases of same handedness can be divided into the cases where left and right moving modes are symmetrized to define a theory of unoriented strings which also ends up allowing the open strings; and not imposing such restrictions thus ending with the theory of oriented closed strings of definite chirality. As it turns out, the first case loses one of the supersym metries reducing to the Type I string theory whereas the later case become Type JIB string theory. There are two further kinds of supersymmetric string theory which combine superstring modes with those of a bosonic string (taking left movers from one and right movers from the other), those Heterotic theories are not discussed here. -  Superstring quantization and the superalgebra The particular supersymmetry formalism discussed in this section is particularly well suited for use with light-cone gauge quantization, with tt-symmetry working along the re-parametrization symmetry. First we define: I±==(F0±F9)  (1.37)  and then use ic-symmetry to enforce the condition: F+OA  =  (1.38)  0  for both. Seeing as exactly half of the eigenvalues of f+ are non-zero, 1.38 amounts to setting exactly half of the 8 components to zero. These are the same degrees of freedom that were afready seen as decoupled from the theory due to ic-symmetry. Counting the fermionic degrees of freedom we start from the generic 32 complex d.o.f. The Weyl-Majorana condition reduces this to 16 real d.o.f and finally 1.38 reduces it further to 8 real d.o.f. for each 8’ We have seen before that the gauge choice together with the Virasoro condition does something very similar for the bosonic degrees of freedom, fixing X± and leaving only 8 fields X as physical degrees of freedom. The only remaining manifest spatial symmetry is therefore the rotational invariance 4 Lie algebra possesses a “triality” sym D SO(8). The Dynkin diagram of the 80(8) metry relating the in-equivalent irreducible representations of the same dimensionality. In particular, there are 3 irreducible representations of S0(8) that are 8-dimensional: the vec tor one 8 and two spinorial ones 8 and 8. The vector representation is manifestly real and the reality of the spinor ones follows from triality. It is then very easy to see the bosonic fields X as the vector representation 8, of the Aà as spinor representations 8 and 8. respectively, or 0 SO (8) and the fermionic ones with the choice of the representation being governed by the chirality. A further effect of 1.38 is to simplify action considerably by imposing OFOa8 for all cases except when between the two Os  =  —.  =  (1.39)  0  This can be seen by inserting the unity 1  =  (FT+FFj/2  9  Chapter 1. Introduction  As a consequence of this, the equations of motion take a very simple form: (8  =  —  (8cr + 8)6  0,  =  0,  (8cr  —  ‘9)9a2  =  (1.40)  0  The bosonic degrees of freedom are solved for and quantized in exactly the same way as in the bosonic theory above resulting in the oscillator modes given by 1.11 and 1.14. Solutions for the fermionic partners for closed strings are given by: gla(jT)  =  a(UT) 2 &  =  (1.41)  where the usual Poisson brackets imply: [i  ij  {3,i3}  = =  ä1 {i#4}  m6m+n6 2 m6m+n5°  J]  möm+nö  {3a  ,ã&}  0 0  (1.42)  after the standard promotion of Poisson brackets into (anti)-commutators and oscillators into the creation/anihilation operators. It is now possible to discuss the actual supersymmetries of the action and their relation ship with other symmetries. In order to preserve the gauge condition 1.38 e and ic symmetries have to be combined in such a way as to ensure that the 60 is always annihilated by the 1’. The resulting transformation has the form:  68a  à 8Xi € 7  =  Xi  =  (1.43)  24€S  are Clebsch-Gordan coefficients for the coupling of the three representations. It can where be seen that the anticommutation of two such transformation gives space-time translation. There is also 8 “trivial” transformations:  osa  ?7a  ox  =  (1.44)  0  The 1.43 and 1.44 are generated by (1.45)  Q 7a8n and Qa  (1.46)  =  respectively. With Qs components of a covariant Majorana-Weyl spinor satisfying the alge bra:  {Q Q&}  ab 6  {Qa,  Qà} =  7aP  {Qã  Qb}  OabH  (1.47)  10  Chapter 1. Introduction  it is possible to combine these generators with the generators of the rotations and boosts to create the full Super-Poincare algebra introducing commutators such as: [Ji_,  a j Q 7  Qa]  (1.48)  Super-aigrebra quantizes trivially with the usual exception of the [J, where J is given by: J  =  Tf du(X  —  XUX +  AFVQA)  J]  commutator  (1.49)  As before, the demand for disappearance of this commutator in the quantum theory will ,select the dimensionality of the space-time background. In the supersymmetric case the required dimension is D = 10, compatible with the supersymmetry requirement in the Majorana-Weyl case. Wider applications of supersymmetry Supersymmetry was originally postulated in the context of string theory as discussed above. However,the idea of a symmetry relating particles of same quantum numbers but different spin-statistics had since found use in regular field theory, and is used as one of the possible modifications of the Standard Model [8] [9]. The main advantage of the super-symmetric theories over the ordinary kind is the fact that the ground state (vacuum) energy in a supersymmetric theory is generically zero. As suming softly broken supersymmetry for our own universe would go long way towards explain ing why is the observed vacuum energy (cosmological constant) is many orders of magnitude smaller than the formal sum of the zero-point energies of all the modes of all the fields of the standard model up to the Planck scale cutoff. Another use of the supersymmetry is to naturally regulate mass of Riggs boson by pro viding cancelations to any mass corrections due to heavy particle loops in the field theory. This enables the Riggs boson to remain at the electro-weak scale in a theory containing particles whose own mass are significantly above that scale without resorting to fine-tuning of the bare Riggs mass. Even more intriguingly, the introduction of supersymmetry corrects the extrapolated coupling constants for electromagnetic, weak and strong nuclear fields, so that they meet exactly at the sufficiently high energy scale a result of significant mathematical elegance. Furthermore, supersymmetry can be used in potential solutions to the problem of dark matter and the problem of matter/anti-matter symmetry in the universe. Lack of supersymmetric pairings within the currently known particles of the Standard Model indicates unequivocally that any potential supersymmetry of the physical universe must be broken. However, considerations of vacuum energy and Riggs mass indicate that this breaking is likely to be soft and that the super-partners of the known particle may well be within the observational scope of the next generation of particle accelerators (including LHC). If that proves correct, within the next decade or so we will see the first experimental confirmation for a piece of “new physics” coming out of string theory. -  Chapter 1. Introduction  1.1.3  11  Closed string spectrum and the background fields  The ground state of the bosonic string is a tachyon due to the presence of the normal ordering constant in the Hamiltonian. Zero mass states are then 8 excited states of the form c4 0 >. In the supersymmetric case normal ordering constants for bosons and fermions cancel out, meaning that the ground state has zero mass. This was considered one of the early successes of superstring theory and the indication that the supersymmetry is somehow “natural” in string context. The grounds states of the superstring must represent the algebra {s, S} = öab. Such representation is given by: ‘a) (1.50) s  (  The representation space is the 8%, + 8 (or alternatively 8%, + 8) 16-dimensional multi plet consisting of 8 fermions and 8 bosons. In the case of closed strings, ground state will be a cross product of two such multiplets, one for right movers and one for left movers. The choice between Type hA and JIB theories (different or same chiralities on left and right movers) is given by choice of different or same spinorial representation. We can thus write the full massless spectrum as either: (8 + 8) x (8 + 8)  =  (1 + 28 + 35, + 8 + 56V)B + (8 + 56 + 8 + 56)F  (1.51)  in the hA case or (8+8) x(8V+8S)=(1+28+35V+1+28+35)B+(8+56C+8+56)F (1.52)  in the JIB case. Where in either case B labels bosonic states and F fermionic ones. The rest of the string spectrum can be built by acting with non-zero mode creation operators on these ground states while obeying level matching conditions such as 1.26. Zero mode operators turn one ground state into another. It turns out that the 35 in the Type II theory corresponds by its quantum number properties to the graviton g. One of the most intriguing early results of string theory was that infinitesimal variations in the background space-time metric were shown to be exactly equivalent to the coupling with this element of the string spectrum, confirming that it is indeed a quantum of gravitational field. All other possible background fields were either shown to lead to anomalies or were themselves found within the spectrum of the string. This incredible self-consistency was part of the reason why string theory appeared, very early on, to hold promise of being an underlying theory of all physics. Supergravity  -  low energy limit of the string theories  It is possible to construct an action whose fields satisfy the same equations of motion as the low energy states of the string theory. It turns out that this effective action was known already as the supersymmetrization of Einstein’s gravity often referred to as supergravity. My discussion of sup ergravity here is based primarily on [5]. The most “natural” supergravity is a 11-dimensional theory. This is a critical dimension because theories of dimensions d> 11 can be shown to contain massless spin > 2 particles -  12  Chapter 1. Introduction  which can not couple consistently in a field theory and are therefore prohibited. There is an immediate relationship between 11 dimensional supergravity and the type II theories in that they share super-algebra. To actually recover low energy Type II theories we can use dimensional reduction of the original 11-d supergravity while keeping only the fields that are independent of the compact directions. This is actually related to the connection between Type II theories (in fact all string theories) and the 11-dimensional M(aster)-Theory. This is a fascinating subject which goes well outside the bounds of this thesis. The bosonic content of the 11-dimensional supergravity is a metric GMN and the 3-form . The action is given by: 3 4 = dA 3 and its field strength F potential AMNP = A = -_  f  x—’ 12 i 1 ci  (R  —  ‘IF  2)  —  f  4 4AF 3AF A  (1.53)  where R is the Ricci scalar derived from C. Here I skip over the details of the action itself including its fermionic component which can be obtained by supersymmetry, and details of the dimensional reduction. A key point of the latter is general metric under the compactification of the 11th dimension. 2 ds  =  GNdxMdxN  GO,,dxLdxu/ + e 2  =  (dx’0 + A(x)dx’) 2  (1.54)  1 gauge field A mapping the original 11-dimensional GMN to the 10-dimensional metric non the along components are three all (if intact remains either 3 and a scalar q5. 3-form A . This spectrum actually corresponds to 2 compact directions) or losses an index becoming B the one we have quoted for the low energy Type II string, with 1 v corresponding to and 28,r, 8 3 respectively. After a fair bit of algebra, including a 1 and A , A 2 v to B v and 56 rescaling of the metric by the factor proportional to the dilaton we end up with the type hA supergravity action: SIIA  SNS  =  —j-  f  SNS + SR +  e (R + 2 d’°x(—G)  SR  =  481La11  2 (IF ——4-- f d’°x(—G)/ 0 4/c’  (1.55)  0 S  —  2  2) HH I 3 I2) 4 + E  F 2 Scs=_—4-fB A 4  (1.56)  ’ can be made explicit in the 5 2 where the overall factor e R with further redefinitions. Similar operations can be performed to produce the spectrum and the action of the type JIB supergravity. ’ 2 e 0 Key point of the above process was to show that the coupling constant of the theory ic is set by the value of the dilaton one of the “particles” of the string spectrum, meaning that it is set naturally by the theory itself in another example of amazing self-referential consistency of string theories. -  13  Chapter 1. Introduction  1.2  T- Duality and Dirichlet branes  Even if the lack of a unique connection with our 4-climensonal space-time prevents string theory from fulfilling its amazing promise, string theory will still contribute greatly to our understanding of physical laws through the concept of dualities. Basic notion of dualities is the idea that a certain regime of one theory corresponds to a different regime of a distinct theory. Most interesting dualities have to do with cases where the perturbative regime of one theory corresponds to the non-perturbative regime of a different theory. The concept is best introduced, however, through the duality between two different space-time geometries as seen by strings. This phenomenon, called T-duality [12] [13] [14], was one of the earliest string dualities discovered and a precursor of the revolution to come.  1.2.1  T-duality  T-duality of the closed strings We have so far considered superstring in the flat 10-dimensional background. Many modi fications of this background are possible, including compactifications of one or more space dimensions. Simplest possible compactification has one spatial dimension forming a circle 9 x S’ Considering only zero-modes of the changing the background geometry from R’° to R string: X(a, T)  =  (XL  +  XR)  + a’(pL  +pR)T  +  under the a —* a + 2ir, X(a, T) transforms as: X(a, T) means that for all the non-compactified directions:  a’(pL  —*  —  (1.57)  PR)a  X(o-, r) + 2lra’(pL  —  PR)  which  (1.58) In the compactified direction, however, X”(a, r) is given by: X’’(a, r)  X’(a + 2K, T) + 2-irRw  (1.59)  where R is the radius of the compactified direction and w is the number of times string winds around the S’ before it closes back on itself. In this direction: Rw (1.60) PLPR a At the same time total momentum along the compactified direction will be quantized in the units of inverse R so PL+PR—  (1.61)  Combined these two equations give us: n R n  Rw a Rw  (1.62)  14  Chapter 1. Introduction  Seeing as n and w are simply integers this suggests the duality between this theory and the with the trivial exchange of n and w; the only change in compactified theory where R’ = this new theory is the replacement PR — —PR or alternatively X(u, t) —÷ XL(U, r) —XR(u, r). With this coordinate change we have just demonstrated the equivalence between two theories with potentially considerably different geometries. This duality holds true for the higher oscilätor modes of the string as well, with —, &,  (1.63)  —  being a general rule. What this means is that Type hA and IIB string theories which we previously thought of as distinct actually represent two different geometries of the same theory [12]. Specifically, compactlfying Type hA string on a circle of radius R and then letting R —* 0, shifts the chirality of right movers and gives the Type JIB string on 10 dimensions. What is more, Type IIA and JIB theories come out as simply the limiting points in the full spectrum of theories governed by the value of R. A similar thing happens between two, hitherto unmentioned, Heterotic string theories. A movement to turn the 5 distinct string theories into one has begun. T-duality of the open strings When we try to apply T-duality to a theory that contains open strings, such as Type I, we run into another interesting result. For the closed strings in the R —* 0 limit, states with n 0 become infinitely massive but the states with n = 0 form the continuum over all values of w because it is very cheap in energy terms to wind around a “small” dimension. Effectively, instead of decoupling from the theory, the compactified dimension re-appears. Open strings, on the other hand, can not wrap around compactified direction. There is therefore no new continuum of states as R goes to 0, and the compactified dimension disappears from the theory. This leads to an apparent contradiction, because all open strings theories necessarily include closed strings and what is more, internal parts of the open strings are indistinguishable from those of the closed string (the only difference being boundary conditions). Apparent paradox disappears when we observe what exactly happens with the open string under the duality transformations 1.63. Originally:  X(u,r)= XL+XR  XL  XL  >  + a’(r + u) +  ane_T  n#O —  1 2  ,fl  XR—-xR+ci—(r—u)+  X(u,r)  1  —in(r--u)  n  R  =  Z  xo + 2a’pr +  ne_Tcos(ng)  n#O  (1.64)  15  Chapter 1. Introduction  T-dual coordinate X(o, r) is given by: (U,T)=L+R=XL-XR (g,  0 r) = x  + 2nu +  ane_ffiTsin(nu)  (1.65)  nO  It can be seen from 1.65 that the boundary conditions on u changed in the compactified direction from Neumann to Dirichiet (cos to sine) and that the actual position of the end 0 points of the string got limited to a single plane in the compactified direction (because x and x 0 + 2irnR are identified). Furthermore, one can carry the same argument to any path connecting two end-points thus showing that all end points of strings end on the same plane. Thus, one dimension is indeed taken out from the degrees of freedom of the end-points of the open string and a new object is introduced to the theory namely the surface on which the open strings may end, Dirichlet-brane or D-brane. D-branes of different dimension can be constructed by compactifying (and T-dualizing) multiple dimensions of the original space. A much more detailed analysis than the one pro vided here can show that T-dualizing Type I string theory on an odd number of dimensions results in a D-brane of appropriate dimension and the Type hA closed string theory away from the brane. Likewise, T-dualizing an even number of dimensions gives us the brane and the Type JIB closed strings away from it. A consequence of this, explained in more detail in [14] is that the states containing the D brane within Type II theories will lose exactly half of their supersymmetries making them 1/2 BPS states [15] [16]. -  1.2.2  Properties of D-branes  D-branes and gauge groups The usual way of introducing gauge symmetry into (open) string theories is through Chan Paton factors [6] [14]. Those are the degrees of freedom associated with the ends of open strings and characterized by vanishing Hamiltonian. Due to the latter they are strictly static terms (string prepared with one set of Chan-Paton states always retains those exact states). Labeling the end states of the strings i and j where i and j run between 1 and N we can write a generic string state as: p;a  p;ij  >=  >  (1.66)  i,j=1  with nxn matrices ) being what is actually referred to as Chan-Paton factors. It can be shown that in the simplest case of oriented strings the non-dynamical nature of Chan-Paton degrees of freedom forces them into trace-like structures (because the two connecting ends of the string must always be in the same Chan-Paton state) creating factors such as: ‘‘ij’jk”’rni =  in each open string amplitude.  ..A) Tr . 2 A 1  (1.67)  16  Chapter 1. Introduction  All such amplitudes are invariant under the U(N) symmetry: —+  (1.68)  UA’U’  that transforms the end points of the string. Thus we introduce an additional gauge sym metry into the theory. More general symmetry groupscan be introduced if one considers the unoriented open strings When we T-dualize a theory we can use Chan-Paton factors to introduce multiple D branes. When compactifying a dimension d we can generically break U(N) —* U(1)N by introducing the matrix = diag{8 , 2, 1 6N}/27rR acting on the Chan-Paton factors. Such a matrix can be thought of as a Wilson line in the compactified direction. Locally can be written down as a gauge:  4  Ad  =  ...,  4  DdA, A 1 —iA  =  diag{eiX  u/271R, ejxd0 /2, 2  eixd91/2}  (1.69)  This would have no further effect on a non-compactified direction but in the compactified direction the string ends pick up phase depending on their Chan-Paton content: diag{e°’,  e2,  ...,  (1.70)  e°”}  Xd + 2irR. under the winding transformation Xd Due to phases 1.70 strings can now have fractional momentum along the compactified direction. This translates into fractional winding number and thus into the ability of open strings to end on different hyper-planes. Specifically, the string in the state ii > will pick and thus its end points will end up being at up the phase _  0 and  R )R=(6— 3 (2irn+6—O )  (1.71)  In other words the arbitrary end point will be given by: jd  =  OR  =  (1.72)  2ircv’A,  Generically then, there will be N hyper-planes at different positions at which open strings can end. Dynamics of D-branes It was noted before that string theory necessarily provides its own background fields. Most importantly, string theory contains gravity. In a theory with gravity it is uniiatural for perfectly rigid objects to exist; therefore we expect to see D-branes fluctuate in shape and position as they interact with other branes and strings. Looking at the simple case of D 2-branes (one compactified direction) and using 1.25 =p M = 2  ([2n+e—o1 2 2iro’  )  (1.73)  massless states will clearly be the ones that are not winding and whose ends are both on a single brane. This makes sense as the string stretched between two branes has a ten sion that contributes to its energy. Furthermore, string excitations along the brane of the  17  Chapter 1. Introduction  form c p, ii > where d is a compactified direction, become, in dual theory, the transverse position of the D brane as we have already seen in the constant gauge case 1.72. More complicated gauge backgrounds will then correspond to the curved surfaces and the quanta of the gauge fields to the fluctuations. Much as the massless closed strings turn out to repre sent the fluctuations of the background geometry we find that certain massless open strings correspond to fluctuations in the shape of the D-branes. The low-energy effective action of the brane fluctuation is the well known Dirac-Born Infeld (DBI) action: S  = —  f  d’edet (Gab + Bab + 2’Fab) +  (1.74)  ...  where -a are the world-volume coordinates of the brane, Gab = G (X)6aXObX” and Bab = BAlw(X)8aXObX’ are the pull-back of space-time fields to the brane and is the dillaton. This action clearly interacts with the massless closed strings as described in 1 55 Considering the U(N) symmetry breaking, as discussed above, the N separated D-branes end up with one massless vector each U(1)”. If m D-branes coincide then 8 = 92 = = and strings are allowed to end on any of the m D-branes while remaining massless. We thus have surviving U(m) gauge group. If all N D-branes are coincident we recover the U(N) gauge symmetry. Stacks of D-branes play important role in the later discussion of AdS/OFT. One of the interesting properties of the stack of D-branes is that the (remaining) supersymmetry of the BPS state ensures that the gravitational attraction between the branes is exactly canceled out by the repulsion due to their form-potential charges. This means that the brane stacks are stable solutions. The DBI action is obtained by promoting the gauge field Aa() to an N x N matrix with components of the matrix corresponding to the end points of the open string and likewise for other fields. The action will be of the form: -  = —  f  ‘eTr [det (Gab + Bab + 2a’Fab) +  . .]  (1.75)  We will soon return to this action.  1.3 1.3.1  AdS/CFT correspondence Origin of the AdS/CFT  Black hole thermodynamics and holography The core idea of the AdS/CFT correspondence goes back to attempts to understand the physical properties of black holes. The original problem was that the presence of black hole may locally break the second law of thermodynamics, if high entropy objects were to be “thrown” into the black hole. The problem was addressed in the mid 1970’s by Bekenstein and Hawking [17], [18] who related the thermodynamic characteristics of a black hole to its surface properties (which are not shielded from the outside by the event horizon and are  18  Chapter 1. Introduction  thus part of the larger system). This concept became known as holography and it turned out to be very common in describing gravitational phenomena. At its core is the idea that complex gravitational phenomena, within a certain region, can be completely understood in terms of the properties of the boundary of that region. Useful as it was in understanding thermodynamics in the context of general relativity, holography was little more than an accounting mechanism until the advent of string theory. This is so because prior to string theory there was never an actual microscopic description of the gravitational system such as a black hole. String theory provided such a description by observing the fluctuations of the branes which were taken to form the bulk of the mass of a black hole. Strominger and Vafa [19] showed that the thermodynamics emerging from the state-counting of those fluctuations corresponds exactly to the holographic thermodynamics of Bekenstein and Hawking. This was the first insight into holography as a duality between two well understood theories, the gravitational one “inside” the black hole and field theory on the surface. t’ Hooft limit and the string gauge duality The Relationship between string and gauge field theories was a foundational issue of string theory. Even after the discovery of QCD the hope remained that a connection will be made between this theory and some stringy equivalent. Very early it was shown by ‘t Hooft [20] that if such a relationship were to exist, free string theory would correspond to an U(N) theory in N —* oo limit with the string coupling constant being given by 1/N. t’ Hooft’s argument is very general, applying to all sorts of gauge theories. I do not replicate it here referring to the [22] for details. We have seen before that string theoretic objects that carry the requisite U(N) symmetry are stacks of N coinciding branes. Therefore the string theory that satisfies the t’Hooft requirement would be one that includes such a stack. Since the stack of branes is by definition a massive object, it is exactly the sort of system that would be engaged in the holographic duality as described above.  Black holes and stacks of branes  1.3.2  As usual, we begin by considering the low-energy (supergravity) limit of the string theory. In this limit the D-branes are described by the DBI action 1.75 with their metric, dilaton and the field form given by [21, 22, 25]: 2 ds  =  (r) [_f(r)dt2 + 12 H’  (dxi)2]  112 [f’(r)dr H 2+r d_] 2 + (r)  (1.76)  and e  =  HY(r)  =  g.,  Ftji...jpr  (1.77)  =  where H(r) =1+  ()7-P,  f(r) =1-  ()7P  (1.78)  Chapter 1. Introduction  19  and p is the dimension of the brane within the 10 dimensional background. . Equations 0 This metric corresponds to an extended black hole with horizon at r = r 1.76 1.78 correspond to the so called p-branes, objects in supergravity theory. To link those exactly to the low energy limit of the D-branes we require the condition of “extremality”; which is to say the equivalence of the mass and the charge Q of the branes, necessary for 9 the BPS preservation of half supersymmetries. As both M and Q are functions of R and r 0 = 0 condition. it can be shown that the extremality is achieved given r It is also fairly obvious from the 1.76 1.78 that the p = 3 has a special status in the theory. It is both, the only dimensionality for which string coupling does not depend on the geometry and the only one in which it does not blow up in the r —* 0 limit. We are therefore interested in the p = 3, and specifically in the near-horizon limit of this theory given by 0 0. The metric of this limit is given by: r — r -  -  2 ds  =  -(_dt2 + dx d 2 2 + dz )+R 2  (1.79)  where z = This is a geometry of the cross product between a 5-dimensional anti-de Sitter space 5 and a 5-sphere of the same radius R. Over all, the geometry can be imagined as the AdS semi infinite funnel which opens into the flat space in the r >> R limit and is non-singular in the r —* 0 limit. To connect this geometric picture back to the properties of the D-branes (which are the source of the geometry), we equate the tension of a given D3-brane (multiplied by N for an N-brane stack) with the total stress energy required to cause the above curving of the space-time. The later property is referred to as the ADM mass and is given by [30, 31]: MADM =  2ir 4 R 3 10 8irG  4 R =  o?’ 8 g 4 32ir  (1.80)  10 and the string where the relationship between the 10 dimensional gravitational constant G coupling comes from the supergravity action 1.55. The tension of the N D-branes is inversely proportional to the string coupling and is given by: =  TND3  N 8ir 3 g /2 a  (1.81)  In all of the above constants such as i’ and 7 are written in terms of their explicit values which can be determined via the amplitudes of the closed string exchange. Equating 1.80 and 1.81 we get the first fundamental relation of the AdS/CFT correspondence, the one connecting the geometry of the background to the string coupling: =  c’V4irgN  (1.82)  We have not yet defined the CFT part of the AdS/CFT correspondence. To do that we start with the low energy DBI action of the brane 1.74. The 3+1 coordinates of the D-brane can now be interpreted as dimensions in 4-dimensional space and the remaining 6 transverse  20  Chapter 1. Introduction  directions can be assigned fields on those dimensions. To do this we use the following kind of embedding of the brane on the target space: X()  =  a  X’(E)  =  (1.83)  keeping in mind that the I field is different from the dilation q5 which, being constant, can be expressed as the string coupling gs = e. a = 0.. .4 corresponds to the worldvolume coordinates of the brane and the I = 5. .9 to the transverse directions. Assuming flat background Ga& = 7 7ab and the low energy limit a’ —* 0 we can re-write the equation 1.74: SD3 =  3 4irg  fd4  ((Fab)2  +  (ôaI)2)  +  (1.84)  ...  with a infinite constant corresponding to the volume of the brane being ignored. This action describes the single brane. To go to a stack of branes we perform the same operation as the one leading to 1.75 promoting the fields to the matrices under the U(N) group:  SND3  4g  fd4Tr((Fab)2 + (Da)2) +  ...  (1.85)  where partial got promoted to a covariant derivative and “...“ includes the interaction terms and fermions. From what we have written we can already see the relationship between the string coupling and the Yang-Mills coupling that represents the second fundamental relationship of the AdS/CFT: 3 4irg  =  g,i  (1.86)  Two fundamental relationships 1.82 and 1.86 together give: 4 R  =  gN 2 2a’  (1.87)  It is very important to note in deriving the geometric description we relied on the limit in which R >> a’ in order to suppress “stringy” corrections to the supergravity model. On the gauge-theory side this limit corresponds to the requirement that gN>> 1. In other words, the low-energy limit of string theory turned out to correspond to the strong coupling regime of the Yang-Mills theory.  1.3.3  Early equivalences  -  Entropy and absorption cross-section  While what we have described above is still not a full fledged correspondence, it is a con struction with testable consequences. One of the early tests was the correspondence between Bekenstein-Hawking entropy in the gravitational picture and the entropy calculated statis tically from the field theory [23]. The other was the absorption cross section for the closed string modes by the system [29]. We discuss both of them briefly.  21  Chapter 1. Introduction  Entropy of the 3-brane stacks Relaxing the extremality limit temporarily we can write the metric of the near horizon r << L region as: 2 ds  [_  (i  2 + df2] + dt  —  2+R dr d 2  (i  (1.88)  —  5 with a limit of a Schwarzschild black hole. We can perform The above is a product of S Eudidean continuation of the metric in order to ‘get its periodicity (and thus temperature = 1/T). For convenience we also change variables: (1.89)  , r=it r=ro(1+R ) p 2 the relevant part of the Euclidean metric is then: 2 ds  2+ dp  (1.90)  2 d 02  which implies, due to the need to avoid singularity at the horizon: (1.91) To figure out the Bekenstein-Hawking entropy we can write down the Area of the horizon (8 dimensional all dimensions except t and r) from the metric: -  Ah  =  2 5 R V 3 (ro/L)  =  (1.92)  V 3 T 8 R 6 K  where V 3 is the spatial volume of the D3-brane. Entropy is then given by [23j: 2 r..r2irrr3  CAdS  1V  2  V31  The entropy on the CFT side is calculated using the usual statistical mechanics techniques 2 scalars and Weyl fermions. Without going into details, which are for massless gas of N presented in [29], it is given by: 02  °CFT  =  iY  V31  The exact scaling in N and T demonstrates that the two pictures in fact represent the same theory. The factor of 3/4 difference is an artifact of the different limits that the two pictures come from. Detailed calculations [24] conclude that the entropy is actually of the form: S  =  3 _N2f(gN)V T 3  where f(g?FMN) seems to vary monotonically between 1 at gN 00.  (1.95) =  0 and 3/4 at g7N  =  22  Chapter 1. Introduction  Absorption cross-sections Another important early test of the duality was the comparison of the two ways for calcu lating the cross-section for the absorption of the closed string modes by our system. On the CFT side (D-brane formalism) these cross section are calculated by the usual field-theoretic method, as in [3], starting from the interaction term of the Born-Infeld action. For dilaton, Ramond-Ramond scalar and graviton this term is given by [21, 29]: S1  =  f  d’x [tr  +  —  ] 3 h3Tcq  (1.96)  where T is the brane stress-energy tensor. From the above we can see that the incoming dilaton would couple to the gyM which is to say it can be converted to a pair of the world volume bosons. Usual methods [29] yield cross-section for the low energy dilaton incoming onto the 3-brane stack: ,c N 3 w 2 32ir  CFT  (1.97)  In the geometric picture the cross-section corresponds to the absorption of the waves incident from the r >> R region by the throat region. r << R. This is calculated by finding the equations of motion of the closed string mode (dilaton for comparison with the previous 5 x S 5 background described by a metric such as case) from the SUGRA action in the AdS 1.88. Those turn out to be simple d’Alembertian equations for the relevant geometry: (X) =  [  (1+  )  +ô+  + r2D ] (X) =0 5  (1.98)  where D5 is the 5-sphere laplacian. Considering only perpendicular modes of the dilaton (in keeping with the D-brane formalism above) this can be reduced to a single-dimensional equation where the single dimension is related to the radial dimension of the geometry: R cosh(2z)] ‘b(z) 2 [0 + 2w  =  0  (1.99)  which is a regular barrier problem in quantum mechanics. This is then solved using the so called matching method whereby the equation is solved in Z —* oo and Z —* —co limits and then matching the overlapping region. The end result is given by [29]: CAds  =  -w3R8  (1.100)  which, taking into account 1.82 ends up exactly identical to the cFT calculated above without even the numeric factor discrepancy as in the entropy calculation. It has been shown [21] that this agreement is not co-incidental, and that for all the absorption cross sections in the large N limit have no perturbative corrections and should thus be identical between strong and weak interaction pictures. The agreement above was replicated for the case of gravitons and a number of other closed-string modes, giving impetus to the idea of the full-blown duality between the Yang-Mills theory and the JIB strings on a curved background.  Chapter 1. Introduction  1.3.4  23  Maldacena Conjecture  We have seen so far that the stack of N D3-branes embedded within the flat JIB string background can be thought of in terms of its geometric interaction with the background or in terms of the field theory given by the D-brane action itself. We have also seen how the perturbative regimes for the two “pictures” end up on the exactly opposite “sides” of the theory: with the supergravity limit of the strings on the AdS (defined by R>> cV) requirement corresponding to the strong coupling limit of the field theory (A = gN>> 1) and vice versa. However, at this level of understanding the two pictures are still just different ways of describing the same phenomenon a stack of D3-branes in a generically flat background. In 1997 Maldacena [2] proposed that it is possible to decouple the flat JIB string back ground from the brane-stack in both pictures while maintaining their correspondence. On the geometry side this decoupling takes place as we take a limit R, cV —* 0 while keeping N fixed. The region inside throat (r < R) is then described by: 8 /cV A = 4irg 4 R 2 -  2 ds  =  (_dt2 +  d) 2 dx) + -(dr2 + r  (1.101)  the cross-section 1.100 goes to zero in this limit making the 1.101 decoupled from the R>> r region which is simple flat space type JIB string theory. On the D-brane dynamics side the same limit likewise decouples the stack of N D3branes from the JIB closed strings that propagate in the bulk. What is left is the N = 4 supersymmetric gauge theory with U(N) gauge group. The Maldacena Conjecture is that 5xS 5 background this theory is exactly dual to the JIB string theory propagating on the AdS 1.101. The key corollary of this conjecture comes from the equation: =A=gMN  (1.102)  which is unaffected by the decoupling limit. It is easy to see that in the case of large A,the ratio of the AdS curvature to the string length becomes large and thus the low energy of the string theory (supergravity) becomes sufficient to describe the dynamics of the system (provided N is taken to be large to avoid the higher genera of the string interactions). The long sought-after method for analytic handling of the strong interaction gauge theory is thus obtained.  1.3.5  Evidence of the AdS/CFT and holography  While there is as of now no actual proof of the full version of the Maldacena conjecture there are multiple reasons it is widely believed to be true. One is the identical symmetry groups of the two theories [22]. The second piece of evidence comes from the equivalence of the two (and higher) point functions between the operators in CFT and the fields on the boundary of the AdS [21, 22]. This later property suggests that the idea of holography is applicable to the AdS/OFT. This is further confirmed by comparing the degrees of freedom of CFT with the surface area of the AdS [21, 28]. -  -  24  Chapter 1. Introduction  Symmetry group of AdS/CFT Conformal field theory is invariant under the usual Poincare group as well as the conformal transformations: Ax  (1.103)  2 + ax 1 + 2a x’’ + a 1 x 2  (1.104)  —*  and  1.103 guarantees that the coupling of the theory does not run and is a free parameter 11 which is necessary for the correspondence to hold. Taking Poincare generators to be M 1 (translations) and the generators of 1.103 and 1.104 as D and K (lorentz rotations) and respectively we can write the whole algebra [22]: —i(7]P jP); [D, P,L] = —iP,.L; [Kr, M] = i(rK iK j; 1  [Mrn,, P]  =  —  —  [M,AV,  irpMvg + permutations; ; 11 [D, K/LI = iK , 1 [Ku, P,} = 2irD 2iM =  —  [D, Mrn,] = 0 (1.105)  This algebra is isomorphic to the S0(d, 2) and can be put into the standard form of S0(d, 2) by defining: uvMiiv; J,Ld=(K,L.—P/L); 4 J 1 =D 1 J J/Ld+l=(K,L+P/L);  (1.106)  Furthermore, the theory has SO(6) symmetry, called R-symmetry, that rotates 6 scalar fields into each other. Bosonic symmetry therefore is S0(2, 4) x 80(6). By their very definition the AdS 5 and 85 have the S0(2, 4) and S0(6) symmetries respectively. To see this consider the embeding functions for AdS and the sphere: (x) R, 2 — 1 x+x =  R (x,) = 2  (1.107)  meaning that the two theories have exactly identical bosonic symmetries. Further inves tigation [22] shows that the supersymmetric extension of the algebra given schematically as: [D,Q] Q; [D,S] S; {Q,Q}P; {S,S}K;  S;[P,S] Q; [K,Q] {Q,S}M+D+R  (1.108)  where Q and S are generators of the supersymmetries, gives the full superalgebra as SU(2, 24) and is likewise identical between the two. Correlation functions and the bulk-boundary correspondence The core test of the AdS/CFT comes from the actual calculations of the corresponding quantities on the two sides. We need, however, to establish what quantities exactly are  25  Chapter 1. Introduction  corresponding to each other. A natural candidate on the CFT side is the operators. As a simple example we can take an operator within the N== 4 super Yang-Mills which changes the value of the coupling constant. This is related by 1.86 to changing the coupling constant of the string theory and thus to the expectation value of the dilaton. The expectation value of the dilaton, for its part, is set by its boundary condition on the boundary of AdS (infinity). In other words adding the operator 0 to the Lagrangian of the CFT will change the boundary condition of the dilaton. Specifically we can write [22]: rh—  , 4 fd x bo(xjQ(x /CFT  \  string VPiX, Zj z=O  =  h  where the left hand side is the generating function of the corelation functions in the field theory with arbitrary cc and the right hand side is the full partition function of the string theory with the boundary condition for defined over all d 1 dimensions of the boundary. Similar equations hold in the general case for fields other then the dilaton as well with the interesting corollary of the scaling dimension Li of the operator 0 being directly related to the mass of the string mode by: —  (d+V1d2+4m2)  (1.110)  because boundary condition in the region close to the boundary for massive fields becomes: b(f  )  _() 6 d  (1.111)  with attendant implications to the dimension of çtc and 0 The correlation functions of the gauge theory can be calculated from 1.109 by differ entiation with respect to o• Each differentiation brings down an insertion 0 and sends the particle into the bulk. The interactions in bulk can then be calculated by the Feyn man diagrams of supergravity (whose external legs correspond to the boundary values o). These could in principle be compared with the field theory except that in most general case the supergravity calculations in the bulk correspond to the strong coupling (and thus the non-perturbative) calculations in the field theory. Nonetheless calculating the correlation functions is a valuable exercise which gives con siderable insight in the relationship between the operators in CFT and strings on AdS. It is done explicitely in [22] for the case of 2 and 3 point functions and the discussion of the 4-point functions was presented. The topic is discussed extensively in literature ([26, 27] and many others). Holography One of the ways to think about the AdS/CFT duality is to go back to the idea of holography which was useful in understanding the entropy of black holes. The basic statement of the “holographic principle” is that for any quantum theory that includes gravity, all physics within a given volume can be described in terms of a different theory on the boundary of that volume, whose degrees of freedom are limited by the area of the boundary and the 3 is naively 9-dimensional but the 5 5 x S Bekenstein bound [22]. The boundary of the AdS small as the 4 remaining dimensions grow compact thus remain and spherical dimensions  26  Chapter 1. Introduction  (as we approach the boundary of the space). This makes any theory on the boundary of AdS x S 5 5 effectively 4-dimensional. The idea behind the holographic treatment of AdS/OFT is that the OFT on 4 dimensions can be considered a theory on the boundary of the AdS that contains all of its physics. The intimate relationship between the operators of the OFT and the boundary conditions of the AdS fields discussed above support this idea but in order to satisfy the holographic principle AdS/OFT also has to meet the degrees of freedom requirement. In the AdS/CFT case this requirement is hard to test directly because any conformal theory has an infinite number of degrees of freedom because it can go down to arbitrarily small scales, and the boundary of the Anti-deSitter space is likewise infinite. It is, however, possible to introduce a cutoff on the number of degrees of freedom of the CFT and see what effect this has on the gravitational theory [21, 28]. 5 is: A convenient way to write the metric of the AdS 2 ds  =  2 R  [4dxidxi —  2 dt  ±  (1.112)  1. As we have discussed above, the correlators of the superwith the boundary set at r )Y(X in 1 Y(X ) ) should be equal to the super Yang-Mills correlators 2 2 gravity fields A(xi)A(x values 2 x X boundary coordinates to their are brought which the bulk the limit in x The way to do this limit is to specify the boundary condition at r = 1 6 and then take ,X 1 2 on the sphere regulated by r < 1 — 6 is 6 —* 0 Geodesic distance between two points X ) and the propagator for the particle of mass m in the bulk will then go 1 0& log(’” —  X 2 , 1 (X )  =  ) 2 x 1 m1o(ix =  xi X 2  (1.113)  This compares with the operator product in the CFT of the form: /  Y(X 2 ) 1 Y(X )  (\ji X 1  1 —  2 X  )  +  ...  (1.114)  j  where is an arbitrary regulator mass scale making sure that the fields Y are dimensionless. Comparing the two equations we see that there is a direct correspondence between the JR regulator 6 in the bulk theory and the UV regulator in the Yang-Mills theory. The large boundary area corresponds to the short distance regulator on the CFT side. The number of degrees of freedom of the super Yang-Mills with the UV cutoff 6 is going tobe A 2 N 2 N (1.115) R6 2 = 6 5 G 8 cg R Which is exactly the Bekenstein bound. Further evidence of AdS/CFT The joint symmetry group and satisfied holographic principle are both strong indicators that the AdS/OFT represents an actual duality. Further evidence comes from certain correlation  Chapter 1. Introduction  27  functions (usually related to anomalies) which do not depend on coupling constant and can thus be calculated on both sides. Further tests rely on equivalence in the spectrum of chiral operators, the moduli space of the theory and a number of qualitative tests such as the existence of confinement for the finite temperature theory. Discussing even a small section of those tests goes well beyond the scope of our research work so far, and of this thesis.  1.3.6  Summary  We end the introductory chapter with a brief introduction to AdS/CFT correspondence one of the most important discoveries in modern string theory. With it, string theory has came full circle from its origin as potential theory of strong interactions. We also learn of the many tests that confirm AdS/OFT without explicitly proving it. In the remainder of this thesis we will focus on particular areas of AdS/OFT and the work, including our own research efforts, on extending the applicability of this amazing correspondence. -  28  Chapter 2. AdS/CFT and the interacting strings  Chapter 2 AdS/CFT and the interacting strings We have seen, in the introduction, how the Maldacena conjecture postulated a strong-weak duality between a string theory on an anti-deSitter space and the conformal super YangMills theory on the 4-dimensional boundary of that space. None of the examples, however, dealt with the full string theory on AdS, but rather with a low-energy limit of it, called supergravity. Supergravity ignores all the oscillator degrees of freedom inherent to string theory and thus all the “stringy” components of it. As long as we are limited to supergravity, therefore, there is little hope of fully understanding the AdS/OFT correspondence by testing and expanding its applicability. The actual string theory beyond supergravity on AdS is not solved. Fortunately, however, it is possible to take a particular Penrose limit [32] of the AdS space to obtain a background on which the non-interacting type JIB string theory can be fully solved [33, 36, 37]. Even more importantly, it has been shown by Berenstein, Maldacena and Nastase [35] that the equivalent limit can be taken on the field theory side and that the spectra of the two limits have the exact correspondence predicted by AdS/CFT. This insight has led to considerable advancement in the understanding of the string/gauge duality and has been extended beyond the BMN limit [40—44] and into the non-perturbative sector [45, 46]. One further extension that would be particularly valuable would be to check the correspondence between the interacting strings and the non-planar corrections in the field theory. This has been a topic of vigorous research [54—62, 65—71, 79, 80] over the past years but the full correspondence is still elusive. Our own work [79, 80] provides some of the hitherto missing pieces. 5 x 85 and its BMN counterpart In this chapter we cover briefly the plane-wave limit of AdS on the field theory side. We then introduce the ideas of string field-theory as a method for handling the string interactions, present the work leading to our research and, in final sections, give a detailed presentation of our own contribution.  2.1 2.1.1  Strings on the plane-wave and the BMN limit 5 x S Penrose limit of the AdS 5  To take a Penrose limit is to consider the trajectory of a particle moving very fast on the space (in this case along one of the 85 geodesics) and focus on the geometry as seen by such a particle. Starting from the AdS 5 x S 5 metric: 2 ds  =  2 p + dp 2 pd + d 2 cos 2 + sin 2 Odf2’] 2 [_dt2 cosh R 2 + sinh 2 0 + dO  (2.1)  29  Chapter 2. AdS/OFT and the interacting strings  5 defined by p = 6 = 0 and parametrized by We focus on the geodesic of S and perform the rescaling: = introduce coordinates: , 2 R  p  9  = ,  = ,  R  —*  .  00  To do that we  (2.2)  obtaining the metric: 2 ds  2  —4dxdx  —  +  2 )(dx) + dyT 2 + dr il 2  (2.3)  Which can be written in the form of the pp-wave metric 2 ds  =  2 ,22dx+  —4dxdx 1234 F+  =  5678 F+  =  +d 2 const x  (2.4)  Where t is a mass term usually set to 1 but which can be re-introduced by scaling: x —> UX+. x/ii and x+ This background has a well-defined Green-Schwarz superstring action in the light-cone gauge and solved string equations of motion. [36, 37]. As in our flat space discussion, we quantize the strings on this background in a light-cone gauge ending with 8 bosonic degrees of freedom corresponding to the 8 transverse directions. Coupling to the RR background gives 8 fermionic degrees of freedom. Out of the standard 32 supersymmetries, half end up being linear (again just as in the flat space case) and commuting with the Hamiltonian. This ensures the same mass between the bosons and the fermions. The Hamiltonian itself is of the form: =  —+  =  1 H  =  NTh2  +  2 (‘p)  (2.5)  where N is the usual number operator: (2.6)  (aa +  with o and /3 being the bosonic and fermionic oscillation modes. In this notation, the left moving modes are labeled with positive n and the right moving modes with negative n.  2.1.2  Strings on the pp-wave limit  Here we summarize the results concerning free string theory on the pp-wave background from the [36] and [37]. For detailed derivation of these results we refer the reader to those papers. The Green-Schwarz Lagrangian for the superstring in the background 2.4 is obtained using the supercoset method in [36]. In the standard lightcone gauge (2.7)  F9’=0  x=pr,  it is given by: LB  (Ox’O_x’  —  (2.8)  Chapter 2. AdS/CFT and the interacting strings  =  i(e’a+O’ +  —a_o & 2  —  30  (2.9)  , 2bIO”—FW ) 2  where the x and 0 bosonic and fermionic fields respectively. The former are given in vector notation and the later in the suppressed spinor notation. The same spinor notation is used for the ‘y and which are are 16 x 16 Dirac matrices. 2.8 and 2.9 give rise to the equations of motion: ‘  1 X 2 88_x’ + [t —  =  =  302  0  (2.10)  0, + i110’  =  0.  (2.11)  which, for the closed string boundary conditions, are solved by:  x’(u, r)  =  coswx +  ‘  ±((u, r)a’ + (a, r)a’)  sin irp + i  (2.12)  n0  0’(u,-r)  r0 cos u 1  + sin  tr  Hog +  cn((o,r)0  + ipso(u,r)H0)  (2.13)  ipso(u,r)H0,)  (2.14)  exp(—i(wnT + nu))  (2.15)  nQ  (u,r) 2 0  =  cosuTOg  UT 1 Sfl  —  fi&j +  —  n# 0  where the basis functions (u, r)  =  o2 (u, T)  exp(—i(wn’r  are —  (u, r)  nu)),  =  and  w=n2+[I22,  ,  c,=  12  n=+1,±2  (2.16)  Before quantizing it makes sense to do the following change of basis: a’  =  =(p + i/Lx),  =  =(p’o  a=  a_=  —  itx),  (2.17)  n=1,2,...  (2.18)  pfT)bJ  (2.19)  for the bosons, and: 0 0  =  1 c  =  1  [(1 + pH)b + e()(1  c[(1 +pfT)b_  —  —  e(c)(1  —  for the fermions. Furthermore, the choice leading to the zero vacuum energy breaks the apparent symmetry of the background from SO(8) to 80(4) x S0(4) (which is in fact the real  31  Chapter 2. AdS/CFT and the interacting strings  symmetry of the background when the Ramond-Ramond field is taken into consideration). It will be easier to change the spinor-basis once more to account for this and thus give ; whereby 2 1 x (SU(2) x SU(2)) the fermionic oscillators the indices under (SU(2) x SU(2)) of representations 1/2) 12 and b b 12 would transform in the (1/2, 0, 1/2, 0) and (0, 1/2, 0, , respectively. 2 1 x (SU(2) x SU(2)) (SU(2) x SU(2)) One final change of variables is needed to reconcile the notation between [36, 37] and our work in [79, 80]. This final notation will be used in the remainder of this thesis.  = 3c1a2  i3’ + i\/b1a2  = _/31a2  +  —  ia 2 ã 3  /312,  3ia2  (2.20)  +  for n> 0, and =ai 0 a —  —  0  forn=0. The usual Poison brackets then yield, after promoting a and ,8s into operators: [a,at]  =  öömn,  3 {  2t} 2 3 3 ,  666mn(2.22)  6aömn,  The details of the spinor notation are given in Appendix A. The Hamiltonian 2.5 can be written in this notation as: 2 H  (c’a +  cia 2 n 3 12t/  +  ) 2 12t/3nic  (2.23)  =  as usual lead to the level-matching condition which can  The equations of motion for x be written as: n (a’ta  2.1.3  2 +12ta +1a2tna ) 2 a 1  I)  0.  (2.24)  The Supersymmetry group of the pp-wave  and J’’ where Isometries of the pp-wave background are generated by H, ]J+, J+I, 5, 6, 7, 8. The latter two are angular momentum generators of the i, j = 1, 2, 3, 4, i’j’ transverse SO(4) x SO(4) symmetry. There are 32 conserved supercharges Q+, Q+ and Q, Q. These generators are divided into two groups, kinematical generators: F’,  p+  j+I  jii  ji’i’  Q+ Q+  which act locally and are thus not corrected when the string interactions are introduced, and the dynamical generators: H,Q,Q  32  Chapter 2. AdS/CFT and the interacting strings  which will get corrections from interactions. The parts of the super-algebra that differ from those of the flat space are then given by: [P’,  [H, F’] = , 1 J 2  {Q, }  =  Q—]  =  f’Q, 1  [H,  Q+]  =  [LHQ,  2H + i u7jHJ + 1  (2 25)  It will prove convenient to define a linear combination of the free super-charges such as to separate Hamiltonian in the anti-commutator of the supercharges. In the notation of [79] this combination is given by: (2.26) where = The dynamical constraints in this notation will then be equal to: , Qi3i, 2 2} { Qcià  {  3i Q 2 3 1  = {a12  (ii)  , 2 Qaiã  12} =  }  212 e 2 a j3 H +  22  (itii)  J’’  aii  (2.27)  The free dynamical supercharges are given by: =  -  -  +  +  e(a))  [o1a2  12 k  —  +  ie(ak) Vwk  12 kk  —  k#O  —e(a) =  (v’ck  +a  —  ‘(1 + e(a)) +  +  + —  ie(ak)  —  12 k )]  aO12]  ie(ck)  —  k#O  +e(a)  2.1.4  (Vwk  +2 kf3ia 3 ckiI  —  ie(ck)  —  )] 2 /3k/3ia  (2.28)  BMN limit  5xS The above background is a continuous limit of the AdS 5 background. It is therefore to be expected that the analogous limiting procedure exists on the CFT side of the correspondence. To identify it, we focus on the energy and angular momentum along the relevant geodesic in 1 and the angular momentum global AdS coordinates. The energy there is given by E = i8 1 We have seen in the previous section that the angular momentum in AdS by J = iD, corresponds to charge under R-symmetry (SO(6) symmetry identified with the symmetry of the 85) which we can label J. Energy, as we have also seen is equal to the conformal dimension of the operator in the Yang-Mills theory. We can then write [35]: = —+ =  iO÷ =iO+ =i(D +D)  =  L— J  (2.29)  33  Chapter 2. AdS/OFT and the interacting strings  C)  +  — —  P-  5-  —  —  1  •  — —  1 LC)  —  —  —  ‘  —  —  Focusing on the fast moving particle in the geodesic around and taking the R —* 00 limit /iSi and sending N —b 00. The operators corresponding 2 is equivalent to setting: J R to string spectrum will be ones with fixed L — J. We can then combine 2.29, 2.30 with 1.82 and 1.86 to get: ‘  (2.31) [35] then goes on to define the spectrum of operators with finite Zi J. The single state operator corresponding to the vacuum p = 0 is given by Li the only single-trace operator satisfying this is: —  Ovac =  r  J/TZ]  =  J, and  (2.32)  °,P+)i.c.  where Z = q + ic? . with plane 5 — 6 being one rotated by J. 6 We start building the spectrum of the field theory by considering the super-multiplet of the °vac We act on the °vac with the generators of the symmetries. For example acting with the generators of SO(6) that are outside of SO(2) defined by J we get: +h/ T[ZZ 2 Nj/  1  =  Zj l/ + 2 NJ/ T  (2.33)  where r = 1...4 are dimensions that do not rotate under J. Going through the Poincare symmetries and supersymmetries we find that all the other states of the super-multiplet can likewise be represented by the insertion of a new field or a derivative into °vac It turns J eigenvalue [35]. Modes out that these inserted fields separate according to their Li J = 1 and they are: contributing to the original supermultiplet are then ones with Li DZ = 9Z + [At, Z], with i = 1..4 and eight fermionic operators x!• We can act 1 with these generators multiple times every time turning one of the Zs into a Li — J mode. On the gravity side, this is equivalent to multiplying a vacuum state with the zero-modes of the string oscillators c and b thus once again affirming the correspondence between supergravity and CFT. One of the principal contributions of [35] was to extend this correspondence to the higher “stringy” modes of the oscillators by associating the mode number n in c to a position dependent phase in such a way that, for example, c corresponds to: —  —  J  1  f—f  V7NJ/2+1/2  Tr[Z14ZJ_h]e#al  (2 34)  This, combined with the cyclicity of the trace assures that the level-matching conditions 1.26 are automatically satisfied for example the one “impurity” states automatically disappear. More impurities can be added with the following identification: -  atz_+DjZ  fori==1,... ,4  34  Chapter 2. AdS/OFT and the interacting strings  forj=5,.•. ,8  4 ai—  (2.35) This covers the entire spectrum of the free theory on the string side.  2.1.5  Interactions in the BMN limit  Expanding the square root from 2.31 we can write: (t—J)=w=l+  n 2 N 3 2irg +.l+j+. 2 J  (2.36)  . 2 4 is the t’Hooft coupling and goes to infinity at the same rate as J R where the ) BMN Such operators. the between for the interactions This suggests a new coupling )‘ = a coupling would be a perturbative parameter in the expansion of the free string energy. Field theoretic methods can be used to reproduce the entire square root from 2.31 [35, 47— 51]. To give idea of those methods we glance briefly at the leading order calculation from [35]: Consider the two impurity state / operator: ‘-  a  8t  IO  Tr[3Z14ZJ_1]e’  P+)i.c.  =  78  (2.37)  ’ 2 N/  and the quartic scalar interaction in the DBI action: gYMTr([Z, ][Z,  ])  (2.38)  (x), 078(0)) the above interaction connects the 78 If we look at the two point function (0 Q78 (0) in which impurities are moved by one spot causing the 78 with that in 0 term in (x) change in conformal dimension: (x), 078(0)) 78 (0  (2.39)  (2)J++fl2’+  Details of ,\’ + 2 Which, for N = 2 gives the appropriate leading order: L — J = 2+ n this calculation are given in Appendix A. of [35] and are too involved to be reproduced here. So far we have discussed one possible finite tunable coupling: ....  1  =  gMN  2 J  ‘  N, J  ,  ‘  oo  (2.40)  and this is in fact the only one that is relevant in the free string theory, being connected only to the string tension. 2.5 is the equation of free string theory so the fact that it can be replicated with a perturbative expansion in )‘ is not a surprise. Dimensional analysis, however, suggests that there is another possible coupling in the field theory that is also finite and tunable: 3 (ita’p) 4irg 2  =  ,  N, J  —*  oo  (2.41)  35  Chapter 2. AdS/CFT and the interacting strings  This one depends on the string coupling as well and therefore on the string side corresponds to the full-blown interacting string theory. The A’ coupling depends on the gy and N only through the t’Hooft coupling gN and therefore the interactions that involve only that coupling correspond to the planar limit or large N t’Hooft limit of the Yang-Mills theory. The theory involving interacting strings then corresponds to the non-planar diagrams of CFT. As mentioned in the preamble to this chapter, the planar limit has been thoroughly tested against the free string theory and their correspondence affirmed [35, 38, 39]. The non-planar limit has been studied extensively on the field theory side and a fairly reliable 2 has been calculated for a number of examples [47—53]. double expansion series in A’ and g The one that we will focus on in the remainder of this chapter describes a particular two impurity state of the string, and is given by: -  -  —  J  =  ’ n A 2+ 2  —  ‘n4At2  +  ‘n6A3  +  (  + 3222) (A’  —  ‘A!2n2)  +... (2.42)  Confirming this series on the string side would be a major triumph for AdS/CFT correspon dence but to this day despite the considerable effort [54—62, 65—71, 79, 80] this has not been fully accomplished. The remainder of this chapter will focus on understanding this problem and on our own contribution towards its solution. In order to even attempt the g 2 expansion on the string side, one needs a working theory of string interactions. For this we turn to string field theory.  2.2  String field theory on the pp-wave background  String theory as presented in chapter 1. and in the preceding sections is a first-quantized theory. Objects in the Lagrangian are not strings themselves but rather modes of oscillation within a string. This means that it is relatively difficult to construct robust theory of interacting strings from the materials we have provided so far. This is not to say that string interactions are not at all considered in the standard string theories. The idea of vertex operators uses the conformal symmetry of the strings to reduce the interaction to a point operator on the world-sheet of the interacting string [4—7]. This is a fruitful technique but ultimately a limited one and, as it happens, one that is hard to use in a light-cone gauge quantization. This makes it useless in our quest to find the interacting string correspondence to the non-planar aspects of the CFT. The most successful alternative rests in fully second-quantizing the string by introduc ing the multi-string Hilbert space with operators that act to create/anihilate entire strings (contrary to the usual c and a which only affect excitations on a given string). Such a model was developed first in the 1980’s by Green, Schwarz and Brink [10, 11] for the type JIB superstrings in a flat background. It was relatively dormant for two decades but got revived due to its relevance to AdS/CFT correspondence. In 2002 it was generalized by Spradlin and Volovich [54, 55, 65, 69] to a pp-wave background. In this context the energy shifts due to string interactions are equivalent to the anomalous dimension of their corresponding operators. Calculating these energy shifts exactly for the case of particular  Chapter 2. AdS/CFT and the interacting strings  36  2-impurity states was the subject of a number of papers, most notably by Pankiewitz [57, 60, 68, 70]. Our own work, which will be focus of the later sections of this chapter, continues the research from the papers listed above. Most of what will be presented in this section comes directly from one of the above references.  2.2.1  Bosonic particle in the plane wave background  Following [54, 55] we start by presenting the particle field theory on the pp-wave background and then generalize it to strings. This is a logical method because, contrary to the flat background case, there are no qualitative differences between the two cases (even a particle on pp-wave lives in a harmonic oscillator potential). Writing again the pp-wave metric 2.4: 2 ds  =  —2dxdx  —  (2.43)  . 2 (dxj + d f 2 1L  we can write down the action of the free ‘massless’ field: S  f  = —  DI8  =  f  dxdxdx t9+O —  f  (2.44)  dxH,  with the free Hamiltonian H  =  f  2+2 [(a) . (a_) x it ]  dxdx  To quantize this we find the canonical conjugate of the We can then write: [1(x, x), (y, y)]  =  iö(x  —  ,  which turns out to be ‘I’  yjö(x  —  y).  (2.45) =  (2.46)  and in the Fourier basis: (x, x)  =  f  (2.47)  dp_dp  we get: [(pp)(qq)]  + qjS(p+ q).  =  (2.48)  where (2.49)  0.  =  The latest inequality will not hold in the actual bosonic case, because it stems from the supersymmetries, but seeing as we are actually interested in generalizing to a supersymmetric case we can assume it. Since 1 is real (as a classical scalar field), the corresponding operator is Hermitian, which means that =  (2.50)  —p).  The Hamiltonian can now be written as 2 H  =  f  dp_dp  t(2  ) 2 + (x) =  f  dpdp  (2.51)  37  Chapter 2. AdS/CFT and the interacting strings  where h is the single-particle Hamiltonian h  l(2  w  , +w ) x 2  (2.52)  =  The single-particle Hamiltonian may be diagonalized in the standard way: a  =  (p  —  iwx),  p  + at),  =  x  —  ar),  (2.53)  so that h  =  e(pj(ata  +  ),  (2.54)  It is important to note that h is a Hamiltonian of the one particle Hilbert space and that the generators a and a create/anihilate excitations of a single particle. The states of a single particle can then be expressed as: N;pj  ;p), (a) O 1  N  =  (2.55)  0,1  In addition, there exists a multi-particle Hilbert space with operators A(p+)t/A(pj that create/anihilate particles in the state N; p). These operators satisfy (A(pj)t = AN(—pj and (2.56) [AM(pj, AN(q)] e(p)SMNö(p + qj. Once we move to the string theoretic equivalent, the Hubert space of the single particle (whose Hamiltonian is h) will be the world-sheet Hilbert space and the multi-particle one will become the “space-time” Hilbert space. In the standard field-theoretic fashion we can write the expansion for F: =  (2.57)  N;pjAN(pj.  noting how it is an operator in “space-time” Hilbert space and the state in the “worldsheet” space. In this basis we can write the space-time Hamiltonian as: 2 H  fdP+ENAN(_P+)AN(P+)  EN  =  (N+  ).  (2.58)  An important generalization of 2.51 exists that relates symmetry generators on the “world-sheet” space with their equivalents on the “space-time”: 2 C =  f  dpdp p+tg.  (2.59)  To the above we can add interaction by introducing the cubic part of the Hamiltonian: 3 H  =  sf dXdX  17,  where V is some cubic function of 4 and its derivatives.  (2.60)  Chapter 2. AdS/OFT and the interacting strings  38  In terms of modes this can be written as: 3 H  = f  (2.61)  dpdpdpo(pt +p +p) N, F, Q=O  here cNpQ(p, p, pt) encode matrix elements of the interaction written in the basis of the harmonic oscillator wave-functions. From here on we will use the convention that p be the momentum whose sign is opposite of the remaining two. Thus index 3 will always be labeling the initial state of a splitting transition 3 —* I + 2 or the final state of a joining transition 1 + 2 —* 3. 3 with a vertex In the Hilbert space of 3-particle states it is possible to identify the H state V: V)  . Q;-p 3 j 2 cNpQ(p,p,pN;p)IP;p j  =  (2.62)  N,P,Q=O  The above construction fully describes the cubic interactions for this bosonic particle theory.  Bosonic string field theory  2.2.2  The analogy from the previous section is very direct. Instead of the field I (x) representing the particle we now have a functional [x(a)] of the string embedding x(u). Consequently the integrals over dx are replaced by functional integrals Dx(u). Delta functions are replaced by delta functions over all the Fourier modes of x(a) which are written as delta functionals The action is then given by: dxdx_Dx(u) 6+6_  S  —  fdx+ H,  (2.63)  f  where H  =  3+ 2+H H  The formula (2.59) is replaced by 02 =  f  dpDp(u) p g. t  (2.64)  And the worldsheet Hamiltonian is then: h  2IpI e(pj 1  [4p2  +  ((Ox)2 +2x2)]  =  (2.65)  where =  /2  +  (/+)2  (2.66)  which is very much in keeping with the bosonic part of the Hamiltonian on the pp-wave as discussed in the previous sections with a and a having usual stringy meanings. As before we have to impose the level-matching condition: nN  =  0.  (2.67)  Chapter 2. AdS/CFT and the interacting strings  39  States of the second quantized theory will now be labeled I’) where the component N of the vector N gives the occupation number of oscillator n. The second quantized Hilbert space 7-t is then built from a vacuum 0)), which is acted on by the operators A (p), which for p <0 create a string in the state N; p). The free Hamiltonian is then: (2.68)  =fdP+E4(PiA(Pi, 2 H 0  To introduce the interactions we need a vertex state equivalent of 2.62 in this formalism. The main constraint on the interactions is the principle of continuity and conservation of momentum that imply that the vertex must satisfy: -  (pi(a)  +p2(u)  +p (a)) V) 3  =  (x(a) +  X2(a)  —  (a))IV) 3 x  =  0.  (2.69)  To construct such a vertex it is easiest to work in the basis of Fourier modes of one of the strings. We arbitrarily choose the string 3 and then construct the matrices Xr)mn which express the Fourier basis of the string r in the basis of the string 3. These matrices are obtained by simple Fourier transforms, —  mn\  flsm7rm!3 1 1( 2 2mn\ x ’ 2 n—m,i3  ) _pm+n+ism(m! i 3 1 ‘ n—m/3  —  where 13a pt/ P3 is the ratio of the width of string a to the width of string 3. The by definition. We can now write the following equations that must hold for each m: XpnrIV) r=1  =  0,  e(p)Xxn(r)V)  ri=—oo  =  0.  2 70  3 x  1  (2.71)  r=1 n=—oo  Those are most easily solved if we assume an ansatz solution: V)  =  f(p,p,p)exp  [  r,s=1  m,n——oo  (r)Na)] O(1))IO(2))O(3)),  (2.72)  and expand the x’s and p’s appearing in (2.71) into creation and annihilation operators. Solving the resulting matrix equations we get:  j(rs)  srso —  2 V’Wm(r) Wa(s)  (x(?’)TF_lX(8)  )mn.  (2.73)  where 3 (Fa)mn =  r=1  00  p=—oo  Wp(r)XXJ.  (2.74)  as a unique solution modulo undetermined function f(p, p, pr). Exact calculation of the Neumann matrices Nab is a non-trivial exercise that is done in detail in [62]. We will return to it briefly in the next section.  Chapter 2. AdS/CFT and the interacting strings  2.2.3  40  Fermionic contribution to the vertex  Generalizing the procedure listed above to the superstring involves two major considerations. We address them in this section and the following one. First of all, the continuity/conservation requirements given by (u)) V) 3 (a) +p2(u) +p 1 (p  =  (Xi(U)  (u))V) =0. 3 +x2(o) —x  (2.75)  need to be supplemented by the equivalent requirements for the fermionic coordinates 0 and their canonical conjugates A  (u)) 3 (Ai(a) +A (u) +A 2  IV)  =  (Oi(u)  +02(J)  —6s(u))IV) =0.  (2.76)  We do so by introducing an additional factor to the vertex, to be annihilated by the fermionic operators. Thus: V) = EaEbIO> where Ea is given by the exp  EaIO>  [  (2.77)  0), r,s=1 m,n=—oo  as defined above and E is constructed to satisfy the 2.76. As before we can work from an ansatz: EbIO)  =  :  exp  -  ] 2 mr)Qmn13n(s)ài  0),  (2.78)  r,s=1 m,n=O  Taking into account the transformations 2.19 we can write: =  —-—v  [örsö  (r)  (  —  +  ((r)Tf_lY(s)) 2  ‘n(r)  ]  V.  (2.79)  T y(r)j7(’i  (2.80)  where Y  Ymn  =  =  r)  =  \/i(1)3/2  (l—pH) Pb  and p is defined in 2.16. A fair bit of information is skipped in this brief overview. Most specifically the method of Gaussian integrals which originally motivates the form of ansatz that we take. Also, methods for solving the matrix equations and certain subtleties involving the zero modes of the matrices Nmn and . mTh For all these details we refer the reader to the [54, 55, 69] from Q which the results presented above are taken.  2.2.4  Interaction super-algebra and the pre-factors  When we discussed the supersymmetry group of the pp-wave we made a distinction between the generators that act on a point on the string world-sheet (local or kinematical operators)  Chapter 2. AdS/CFT and the interacting strings  41  and are thus incapable of joining or separating strings, and those which act on the whole string and thus can include string interactions. The latter, called dynamical generators , are the Hamiltonian H and the half of the supercharges, labeled Q and Q (not to be confused with Neumann matrices Q). All the dynamical generators will be of the form: (2.81)  • i +tiG 2 G=G + 3 + 4 G  where k is proportional to the string coupling constant. All the G 3 terms will have to contain the V) as derived above, because the Qs have the same relevant symmetries as the Hamiltonian. However, the Qs and H also have to satisfy their own relationship within super-algebra, namely: ,Qii Qai } {2  =  (2.82)  H 2 ç , 1 —2e  Note how we can ignore the potential contribution of the J3 generators to this anticommutator because they can be factored away for a given linear combination of Qs but also will give no contribution at all to the non-zero orders in ic due to being kinematical. In fact, the equation 2.82 is the only one in the entire super-algebra that will have higher ic . 2 corrections. We will be interested in it at orders tc and ic . In the state-language 3 H At order ic the 2.82 can be written schematically as {Q2, Q} defined above it can be written as:  +  =  , 2faiiE ) 3 H 22  (2.83)  +  =  a , 22 € 1 2€ ) 3 H  (2.84)  2 ) 3 Q 2 Q(r)aià  =  0  (2.85)  where Q(r) 12 and Q(r) 12 are the quadratic, free string supercharges Q2 as defined in 2.28. Simply using IV) for the Hamiltonian and generators Q will obviously not suffice to satisfy the super-algebra. We do, however, have a free function of the momentum floating in front of IV) in each of the above cases (see 2.72). We can utilize that to introduce the pre-factors to V) that would differ depending on the generator:  )V), IH I 3 =h  Q)=qV),  Q)—qV)  (2.86)  Actually solving for the pre-factors is complicated and the result turns out to be non unique. The first and most widely used result is due to Pankiewitz and that is the one will we be using in the following sections. We state result here and refer the reader to [68] for the derivation:  42  Chapter 2. AdS/OFT and the interacting strings  ) 3 1H  , 3 2 f(ia g  a 1 3 a  —  —  k, 1 (K  Z) — K,1Ká2a2S* (y)Sià K1a1Kã2a2S,a ( —2  =  (ta 3 f 2 g ,  1  1 a  —)—  —  Z)] (Y)xi ( 2  1 (2.87) (Y)t/3272 72 + is$,  =  (JLa 3 f 2 g ,  1 a  —)-  1  Iv),  (Z)K272)  V),  -  V(7’  (Z)K’7’  (Y)t 2  Z(Y)k272) v). 27 + is$172 (Z)t$ .  Where a number of additional notational definitions are in order, starting with Further  ’ 7 K  l 7 Ktui  , I<’’ 7272 K’o-’  7272 K  ,  =  ,  i  7272 k’a’  7272 K  . 3 2 1 a  (2.88)  where the u-matrices are given in the appendices and are used to convert between vector and spinor notation. Furthermore, (y4 + z ) + TY4z4] 4 ii( 2 i[y 1 + Z) — Z2ii (1 + +  =  —  =  —  o’’ [i.  —  i [ 3 1 i 2  (y4  (1  +  21 ‘[y2Zj  ij  ) + Y4Z4] 4 +z  i 2 — z) — Z  (i  __Y4)] + 12  —  ] 2 [Yz  ‘‘  where i 2 y  ii (y ) Z 2  ,  ,  Uai$,Y  y2k(iz2i)k  (2.89)  and = —  ylW2 ycl2 $i  ‘  Y/32 31 Y, 4 y where  —  2 = y  a,7l”  271 i 3 /  — vala2  yf 2 = c12/32  =  (2.90)  $2  2 y 2 Y 132a2 C1  1 c12$2  2 —Y  =  272  (2.91) 272 Y /32  — —  4 Y $ 2 a  (2.92)  Chapter 2. AdS/CFT and the interacting strings  ‘,‘2  —  ,r2a131  ‘cj2  —  =  43  ‘jj222  2I32’  —  The spinorial quantities s and t are defined as t(Y)=_e+iY2_Y4.  (2.94)  Analogous definitions can be given for Z. Finally: K’  K’  , Kfl(S) ) 3  =  ) 3 Kfl(S)’( s=1 nEZ  s1 nEZ  2 Y’  (2.95)  (2.96)  Z1ã2 =  =  s=1 nEZ  s=1 nEZ  where: n,  ,  =  +a3sln(nlrr)V /r(1_r)A—A  K  (2.97)  (2.98)  ‘r  sin(Inirr)  1 Gq-,  G  /wq  A  =  e(q) 4/wq +  /3rI3,  (2.100)  A;  =  e(n)/wn +  . 3 /L  (2.101)  —  —  299  and =  =  —  , 3 /3riua  —  It is also important to get the exact form of the Neumann matrices 2.73 and 2.79. This was done in [621. The key element of that calculation is inverting the infinite matrix a as defined in 2.74. This is done by finding a differential equation for the function of this inverse with respect to the mass parameter i and then solving with the initial conditions of i = 0 corresponding to the already known fiat space solution. This solution requires a particular inverse integral transform but can and has been done successfully to all orders in p. For our purposes, however, we are interested primarily in the match of the string theory (2.40) what to the gauge theory expansion in A’ that we listed earlier. Seeing as A’ the happy By matrices. Neumann of the limit we are really interested in is the large coincidence in this limit matrices take a rather simple form. We present these forms here and the full ones in the Appendices. For the details of the derivation we refer the reader to the [62]. In the large u limit:  Chapter 2. AdS/CFT and the interacting strings  sin(n7rr),/ (AA + AA)  =  2ir./LZ3 (q 3r  —  riq  2iriZ  Q  2.2.5  =  Qr  Qrq  (q  —  /797 (AAE +AEA) 47/3 (/3swq + !3rWp)  i3rn)  isin(nIirr) (wq + /3rn)  —  where  —  =  ‘  rfl) 3 i  rs qp  —  —  i  (/3q  r! 3 P) 47r/ (!3swq + I rwp) 3 —  44  (2102)  (2 103)  1  —  The contact interaction term  Equations 2.77 to 2.102 give a description of a three-string vertex in the large u limit (and can be complemented with the equations in the Appendix for the full u dependency). This . In order to calculate the one-loop energy shift we use a world-sheet 2 vertex has weight of g diagram containing two such vertexes. One string propagates, splits at the first vertex, two string propagate from there and rejoin at the second vertex. This means that the lowest order in g 2 such a shift would exhibit would be g, in correspondence with the 2.42. From 2.81 we can see that the quartic Hamiltonian (as well as quartic super-symmetry 4 generators) also carries the factor of g. Therefore, it is to be expected that the single H 3 term. Such a term will contribute to the same extent to the energy shift as the two H term is called contact interaction term or contact term and physically represents the limit in which the propagators of the two strings vanish and the two vertices are brought in contact with each other. The contact term is the Achilles’ heel of the calculation we are attempting to perform. There is to this point no generic quartic vertex state equivalent to V) and the conservation laws such as we used in constructing V) do not seem sufficient to construct one. What understanding we do have of the contact term comes from the super-algebra. At the second order in g 2 the {Q, Q} = H anti-commutator can be written as: , 2 {Q2czjà  ,+ 2 i 4 Q }  , 2 {Q4oic  131 + } 2 Q  {Q3ciã,  12 } 3 Q  4 H 22 2fiiEa  =  (2.104)  ,} 12 meaning that we can calculate at least one part of the contact term specifically {Q3 312 Q using the available components. The existence and contribution of the Q4 components remains one of the most important unsolved problems in the string field theory. The current state of the art has it that Q is not necessary to close the algebra and can thus be set to zero without causing inconsistencies [70] but that there is no physical reason why it would have to be set to zero. In our work we follow the convention and set Q to zero for the lack of another option, resulting in the contact term vertex given by: -  -  4 H ‘These expressions are also valid for q, p  ‘.QfJ  — —  ,j3r —iT —la  F’r”np  n,p=O.  =  (2.105)  = Q12Qaia . 2  0, except in the case of NM  =  —N Iq,p0, and in the case of  Chapter 2. AdS/CFT and the interacting strings  45  Luckily, the divergence cancelations that are one of the main results we are presenting in this thesis come fully from this part of the contact term, making our work relevant despite the possible incompleteness of the equation 2J05. Nonetheless, the lack of understanding of the Q term is quite possibly the greatest stumbling block in establishing the correlation between the non-planar CFT and the interacting string. We will return to it briefly in the conclusion to this chapter.  2.2.6  Impurity conserving channel  We now have the entire machinery needed to perform the energy-shift calculations. We will conclude this section by presenting the single-impurity calculation both as an example of applying the above formalism and also because it was the state of the art calculation before our work which will be presented in the next section. This calculation and a number of other important insights presented here are due to Pankiewicz and collaborators and can be found in [57, 60, 68, 70]. The first question to be addressed is the string form of the two-impurity state whose energy shift is being calculated. The result 2.42 is valid for a generic two Is-impurity operator independent of the space-time index of the impurities. This is a consequence of the conformal nature of the CFT. This means we have a choice between the scalar two-oscilator states on the string side. The natural choice and one that minimizes the calculation is the state: löiiatkatk) (2.106) [9, i])(i) = t (atat2 + Ia) —  Its advantage is that it is a unique representation of the state under SO(4) x SO(4) and the one that can be expressed solely by the bosonic oscillators. It is fairly easy to see that while it is possible to write the fermionic representation of the state  I[’ 1])  =  afkafk)  (2.107)  in the form:  [“‘1)  (2.108)  =  Fermi statistics actually prohibit such a dual representation for [9, i])(ui). Using the later, therefore, prevents the mixing of states as we turn on the interactions, making for the much easier calculation of the energy shift. The actual calculation of the shift starts from the standard perturbation theory: =  ([9, l]1H 3  —  Ht’  l]) + ([9, 1]1H 1[9, i])( 4  (2.109)  where the E° is the free energy of the string, P is the projection operator on the space of the two-string states and Ht is a free Hamiltonian acting on the internal strings. At this stage, it was customary to restrict the definition of P to the two string states which between them carry only oscillators. This was justified by the argument from gauge theory in which the number of impurities is in fact conserved. The principal reason it was actually  46  Chapter 2. AdS/CFT and the interacting strings  done though, was that the “impurity non-conserving” channels appeared to result in the non perturbative terms and unphysical divergences. In the next section we will be addressing the non-impurity conserving channels and dealing with most of the problems associated with them. For now we use the impurity conserving channel and write the equation: 9 (ii)([  (l+ö) 6E,cL ) 2  1fiQ  9, H [ 3  3 1]1H  F 1  Q3[9  ])(ii) 1  (2.110)  with:  ‘B =  L 1 K=  2r(1-r)  (  tKafai)  + 1 ‘F E 2 , 1 E  r(1-r)  (  a2)(2  fL  (aia  I2)@2I  1I  (2.111) tKtE12  I) Ia2)(a2  a2a  P  +a  ) fE1E2  a2)(a21  1E2  (au  1 and 2 are a generalization of spinor indices to include both dotted and un where E dotted ones. This distribution of the oscillators is a product of the level-matching condition, something that will prove important when we begin looking at the orbifolded backgrounds where the discrete quantization changes the level-matching. Furthermore we note the normalization of the string vacuua: (alKa2a2)Iai)  =  r(1  —  r),  =  (2.112)  1.  /a 2 —a . 3 and 1 r 3 —au/a where, once again, r In general terms there is also a double fermion term in ‘B but it is irrelevant for traceless state such as [9, i])(ui) due to Fermi statistics. We then need the matrix elements between the internal string states and IHr)/Q >. The 3 calculation of those is relatively straightforward and the results are given by [70]: —  9 (ii)([  (ii)([9  (a a (ajj a 1 2  113)  =  —2r(1  ) 3 ifi (a2 (auIaa H  =  -2r(1  1]  —  -  r)  (1(3)  r)(  I  + -  )  ijk1  (2.113) 3131ijk1  and 9 (ii)([  1 2 ill (a  (,3)12  (au a  12 ) 3 Q  =  ) l’0 K_( ) ) +3 3 —2i C Go( ) (K( 2 9 (ii)([  1 (au 2 ‘]I (a  (_)12  a  ) 312 IQ  iik1(1)ôi  2 114  =  N’_) uik1(l) K_( ) ’+3 —2i C (u) (K( ) 11 3 G  2  47  Chapter 2. AdS/OFT and the interacting strings  jk — löi3ökl} 1 //r(1 and O + 6 intermediate steps of this calculation are presented in the Appendices. The energy denominator is given by:  where  ijkl  .1{öiköil  —  —  2 (w  —  r).  Some of the  (2.115)  r’w)  We have now all the elements from the equation 2.110. All that is left is the conceptually simple, although technically challenging task of performing a sum over the mode number p. This is accomplished through the technique of contour integrals whereby the the sum is calculated by the following substitution: f(p)  =  (2.116)  -dzf(z) cot(z).  a z, u 1 —i r We then rotate and scale the integration variable through the substitution z —* 3 If the large . 2 limit one in the set to which can into coth(iricriz) be cotangent the p to turn 3p summand f(z) has no poles on the real axis, the procedure simply replaces p by p’ = rci and integrates  =  (2.117)  f:dP’f(po)  yielding the large behaviour. If there are poles on the real axis, one must evaluate their residue using the integrand in (2.116) and then integrate along any cut which f(z) may possess along the imaginary axis. Specifically, the sums in our calculation will fall into two general forms: 1 F  P(p)  2 F  =  2+ Q(p)p  2.118  —  2’ (r ) 3  1 will therefore have both poles and a cut and where P(p) and Q(p) are polynomials in p. F 2 just the poles. generically then, result will be: F 1 F  =  —  ) 1 Res (cot(P  ,pi  {pQ(p)=  PiJ  +  f  [Q(ixrz)]*  dz  1  2 F  =  o})  —ir Res (cot(irPi)  P(ixrz) + c.c. —1 IQ(zxrz)I 3,Pj  e {p  Q(p)  o})  (2.119)  (2.120)  Obtaining the result is then just a matter of careful computation. The original result from [70] turns out to have a calculation error that was corrected in our paper [80]. The terms neglected by this approximation are of order 2  ). exp(—Ic I 3  Chapter 2. AdS/OFT and the interacting strings  48  The correct result is given by:  Ti  =  65 (± + 2KJ A’ F(± 312 A’ + + 16 \ir [\\24 64ir J n 2 2 --  4-2 2 —  \48  +  64  89 (J + 5 A’ 2 A’ 32 2 2irJ J n 2 128’ir 45\, / ‘59 339 \, +Q(A 2 / 7 )A ) 4 +n 3 )A ( 4 + 256ir 2 16Or n 2 512ir —  --  (2.121)  There are notable similarities between the result 2.121 and 2.42 but also some rather dramatic discrepancies. Most importantly, the half power terms in A’ appear to be native to string theory results and yet absent in principle from their gauge theory counterparts. Also, despite notable similarity, the actual numeric terms differed even in the full powers of A’. Our own work, presented in the following sections was motivated in large part by the desire to reconcile these discrepancies.  2.3  Higher impurity channels  Two major assumptions were made on the string side to get the result 2.121. One was that the Q4 supercharge can be ignored for which there is necessary, but by no means sufficient, justification. The second was that the only channel contributing to the energy shift will be one where the intermediate states have the same number of oscillators as the end states so called “impurity conserving” channel. We could not do much about the first assumption but were interested in testing the second. One of the primary motivations for excluding the higher impurity channels was that the contribution of the four impurity channel appeared to diverge if the large limit was taken before summing over the mode numbers. On the other hand, if the sum is taken before the appeared in the limit then the divergence is regularized but the non-perturbative term energy shift causing the result to be fundamentally different to the leading order from the gauge-theory result. In our paper [79] we analyze the relationship between the potential divergencies and the 3 non-perturbative terms and then show that divergent terms actually cancel between the H contributions. We generalize this to all and the contact term, taking with them the impurity channels. This section draws heavily on the work of the author and his collaborators in [79] and a substantial part of it is taken directly from that paper. -  2.3.1  Trace state  The simplest example of the behavior we are interested in can be observed in the careful calculation of the two-impurity channel contribution to the mass shift of the normalized bosonic trace state [1, 1])  =  crjc)  (2.122)  49  Chapter 2. AdS/OFT and the interacting strings  In [66], this calculation was performed by taking the large i limit first, then summing over the mode numbers. That procedure found a finite result. If is kept finite, however, there are logarithmically divergent summations which must be dealt with before the large limit is taken. We calculate the following matrix element for the state [1, 1]) (cccL  I(aiI 2 (  ) 3 H  T 2 -g  [8  (1)  (3)  +16  =  (3)  (1 r)  -NN KL +16 —NN’’  kl  (_ +  ]JKL  (2.123)  where the index i = 1,. , 4 is summed over. Note that K, L KL is given by only for k = I = 1,. , 4. The matrix 11 .  .  (1)  .  =  1,  .  .  .  ,  11 is non-zero 8, while ö  .  KL 11  =  diag(1, 1,1,1,—i, —1,—i, —1)  3 contribution to the mass shift it is only the very last term in (2.123) When calculating the H which is divergent. Singling-out its contribution, one finds (using the two impurity channel as defined in 2.110 and 2.111): r  6E=fdr  r(1_r)) 2 (g  2  00  2r-r)  L  Wp  1  -  1 flK 1 33 N N (2.124)  A quick inspection of the forms of the Neumann matrices (see Appendix B.) reveals that the numerator in (2.124) goes like a constant for large p, and thus the sum as a whole goes like l/jpt for jpj >> l,La I. This is a logarithmically diverging sum. In [661 the strict 3 2 behavior large limit was taken for the energy denominator, leading to a convergent 1/p instead. Here we will stick with the finite i expressions and show that the divergence is removed by the contact term. Note that a double fermionic impurity intermediate state also 3 piece, however it does not display any divergent behavior. Further, contributes to the P1 the ccii)c4Ic ) intermediate state is unimportant to us as it does not contain the sum over 2 mode numbers. The contribution from the contact term stems from the following matrix element,  (92w  =  )  N+ 3 (G K  ) 2 i 3 (3I$n (2IK1I2IQ KNP)  (k)diâ2  + 4G? KN(a6 (2.125)  along with a similar element with ). 312 Here K = 1,. , 8 while the E and 3 indices are Q either dotted or undotted as required by the particular SO(4) representation indicated by .  .  K.  The last term in (2.125) gives rise to a log-divergent sum. For large positive p, (K)2 goes as a constant, and so the sum is controlled by (G?)2 which goes as l/p, and hence  Chapter 2. AdS/OFT and the interacting strings  50  diverges logarithmically. For p negative, the sum converges. Thus, the divergent contribution ) is found to be: 2 to 6E(  öEv  =  8fdr  r) 2 (g  (2.126)  )  The leading factor of 8 comes from the sum over K. Note that two factors of 2 from the delta function (in Pauli indices) and the (squared) Pauli matrix trace cancel the two factors / 31 1 Q 2 Again and 3 of 1/8 coming from the two terms of the contact term, the intermediate state crIai)/3!a > is unimportant to the convergence and is ignored here. 2 In taking the large p limits of the summands in (2.124) and (2.126), one finds, 1  6EV  f  dr  g—r)  +fdr21  r ra i 3  (2.127)  2 (p33) (N)  p  (2.128)  3 contribution the divergence is found for both positive and negative Noting that in the H 4 contribution the divergence occurs only for positive p, and hence a relative p, while in the H 3 term, one sees that the logarithmically divergent sums cancel factor of 2 is induced in the H 3 and contact terms, leaving a convergent sum. identically between H 3 and contact terms to that employed This cancellation fixes the relative weight of H in [70]. It differs by a factor of 1/2 from the weight originally given in [65], where it was argued to be a reflection symmetry factor. This last statement is important because the purported factor of 1/2 provided for the leading and sub-leading order agreement between the two-impurity [9, i])(ui) energy shift and the gauge-theory. Despite this inviting coincidence, the above argument, together with the reasoning in [70] seems to exclude the possibility of such a factor.  2.3.2  Four impurity channel  I[  i])(ui) string state due to intermediate states We now consider the mass shift of the which contain four impurities. In the explicit expression for the matrix element quoted below, we see that the parameter [LQ3 occurs only in combinations involving w and there is a duality between the large p 3 limits. Therefore, since a logarithmic divergence in the sums indicates and the large /tc that the summands have as many (inverse) powers of the summation variables as there are 3 dependence for this contribution summation variables, this translates into a vanishing ,uc / \/). It is thus seen that /) behavior is simply the result of log 2 leaving 6E to divergences, which should, if the pp-wave light-cone string field theory is to make any sense, cancel out entirely.  51  Chapter 2. AdS/OFT and the interacting strings  3 contribution to the mass shift. We consider the following interme We begin with the H diate state, ‘B =  f  2 a 1 Ia2)K2I (a  aai) 4!r(1-r)  (2.129)  P1P2P3P4  where the sum over the mode numbers is restricted by the level matching condition p =0 —3(1 r). r, 2 3 —cr and a 1 ), 3 Although there are many possible contractions of this state with the oscillators in 1H we will only be concerned with those which lead to log divergent sums. These are the ones where the c in the prefactor of 113) contracts with one of the oscillators in 1 B• We find 2 to be , 3 this contribution to SE —  \2 r 3 —a r(1—r) ) I 1g2 4 Jo 4!r(1—r) \. LiJnrEi_iwpi p 2 1 4  £div_ Ij  UL’JH  (2  —  dr  1P2 1P2)  /  —..  ‘  ‘  {8. 12  (31)2  x  +  where Pi = —(P2 + P3+ P4). The factors of 6 and 12 are combinatoric and count the number of ways equivalent contractions can be made. The factor of 8 comes from a sum over the spacetime indices of ‘B and only affects squared terms. It is easy to see that in the above, the sum over P2 is log divergent. In fact, it is the very same form as appears in (2.124). The matrices with one leg on the external string have a common form: NT  e(n)  /  (7)  iWp  sin nrr  (3) Wn +/ r 3  (2.131) P  2ir /w3) L$  —  ((r) -  -i  sin(nr) (r)  /3r fl  + (2.132)  P  —  i3r Ti  The sums over mode numbers involved with these matrices will be dominated by the poles u. We are therefore back exactly to the situation rTi and will be of the order zero in 4 3 p = / described in (2.124) as far as the power-counting of u is concerned. The remaining summand, Ip’ and the integral is taken over p’ as per the instructions for the 3 P2 is replaced by z = a contour integral technique given above. u dependance then disappears from the square of and the energy denominator cancels out the measure of the integration the matrix )/ 2 oE( giving: 6Ev constant, and therefore 1 There are also contributions from intermediate states which contain two bosonic and two fermionic impurities, however these produce convergent sums and Q)’) contributions to 2 /. 6E he normalization 1 + 3 T  of the external state has been suppressed here.  Chapter 2. AdS/OFT and the interacting strings  52  We now show that the contact term contribution stemming from the following interme diate state, ‘F  =  3!r(1- r)  0  P  Ia’) I2)(2I (  P2aP3P4  (2.133)  P1P2P3P4  3 contribution, leaving an QQ’) contribution cancels the divergent piece coming from the H to 6E( )/t. In the above a is an SO(8) index and thus represents both dotted and undotted 2 indices in the language of [70]. The log divergent piece comes from contractions where the c in the prefactor of Q) is joined with one of the bosonic oscillators in ‘F One finds, (92)2  6EV 3!r(1— r)  (313)2  K_ 1 (2G ) 2  (2.134)  +34 N_npaNnp 3 Nrip N } _np  In the above one sees the very same pattern as was seen in section 2.3.1. The sum over P2 is divergent on the positive side, and cancels the divergence in (2.130). The remaining (convergent) expression gives an Q1)’) contribution to öE( )/jt. Again, there is a non2 divergent contribution from the intermediate state with three fermionic and one bosonic impurity which is not considered here. The cancellation exposed here is also found for the following remaining pairs of interme diate states,  113! r(1- r)  ‘B  0  ‘F  =  11 2!r(1- r)  p ‘B  L  N  l2)(2I  (aI  P1P2P3 N  1a2)(a21  (& aP3 aP2P1 (2.135)  P1P2P3  0 and,  2. (2!)2r(1  =  ‘F  =  af ) 1 a,) a  0  where  a K atL atM  2!r(1- r)  -  r)  pi  lai)  P2  1a2)(a21  P2  1 (  P1P2  aPi  Pi  Iai) aP2  P2  1a2)(a21  P2  (2.136)  P1P2  and so we find that the entire contribution to óE( )/t from the four impurity channel is 2 convergent / leads as )/. It is not hard to generalize the above argument to 1 B’ contain ing an arbitrary number of bosonic impurities and no fermionic impurities. The divergent expressions cancel against contact interactions with iF’S containing one fermionic and the same number (less-one) of bosonic oscillators as ‘B Adding fermionic impurities is far less trivial because the full forms [68] of I1I3) and IQ3), given in Appendix B. must be used for the calculation . In the next section, however, a more elegant argument is presented which 4 claims the absence of log divergences for arbitrary impurity intermediate states. We remind the reader that the I [9, 4 impurity channel.  1]) () state receives no contributions to its energy shift from the zero  Chapter 2. AdS/OFT and the interacting strings  2.3.3  53  Generalizing to arbitrary impurities  3 portion of the It is possible to formally manipulate the contact term in such a way that the H energy shift is cancelled entirely, leaving a convergent expression, which does not contain any )/t. The manipulation proceeds through supersymmetry algebra. 2 contributions to SE( we have At order 92  { {  Q 312 , 2 Q2ia } Q 312 , 2 Q2àia }  + 2131132 ,Q 2 a 1 {Q3a } + 12 {Q3a } 212 ,Q  = =  2 H 2 ç 11 —2 3  (2.137)  analogously to order g one has + { 2+ ,Q 2 Q3w 1 , 3 {} Q ,iji 2 Q3criâ i 3 }  ij ,Q 2 {Q2ai j 4 }  +  =  2 H 2 ç 1 —2€ , 4  2 } 1 , 4 ,Q 12 {Q2  + 212 ,Q 2 {Q4j }  =  a n 122 6 2 — r 1 H4.  (2138)  3 and H 4 the first of the equations in both (2.137) and (2.138) should be multiplied To get H by e’ e22 and the second by ‘‘ f22• On the left hand sides of the equations the epsilons just raise indices, on the right hand sides they give -4. We thus have: Q1f32} 23 {Q  ,Q1/32} 3 , 2 {Q  , 3 =+4H  3 —+4H  (2.139)  and  4 H  =  1 Q 3  2 3 Q1/  +  ‘Q  13 Q 2 3 1/  +  + I  .  f31/32  1 2 ’4 / 2 ’ 8 m 3 f  2 140  L  2 can be rewritten as a sum of a term 4 to SE Using these formula, the contribution of H 3 contribution plus other pieces which all contain Q2 acting on one of which cancels the H the external states. Taking the expectation value of part of (2.140), and introducing P as a representation of unity, we have  K  1I 3 Q/ 2  +  3 Q  Q12) 2  pEO  =  ) 2 H2QØ  lQ  +  1  /Q  \  3thi3  2 —_H 1 , 2 0 H E 2 “  (2.141)  2 142  —  It could be that the energy denominator which we have introduced here will have a zero. In that case, the projector P is a reminder to define the singularity using a principle value . Equation (2.141) can be written as 5 prescription There is one additional subtlety, the intermediate states must each obey the level-matching condition. 5 This condition can be enforced by inserting a projection operator. For example, for two-string intermediate  54  Chapter 2. AdS/OFT and the interacting strings  [H Q12]) , 2  H 0 E 2  -H 0 E 2  --  , 2 [H  Q1/32])  (2.145)  Up to order g 2 the following equation holds  [  Q/312l ,H 2 3  j  ] 3 [Q/31i2H  —  (2.146)  so that (2.145) becomes  , 3 [H  P  Q12])  +(Q —  --  Q1132]). ,[H 3  (2.147)  H 2 _ 0 3ifl2E  2 one has Since Q2 commutes with H Q/312  +  =  2  +  0 E 2  —  —  \\ P / + 3 H —H 0 E 2  )  3Q2 H H 2  P  Q1/32______ 23  0 E  +  —  -  3 H H 2  >  3Q2) 2EO_H H 2 (2.148)  KH3EPHH3) -  3 contribution to the energy shift. The final expression for and the last term cancels the H the energy shift is  =  + + + +  (Q/31132 E 3 Q 21312 0  —  —  KQ KQ  2 H  / +K:Q2/ +  —  1132  0 E  —  2 H  ) 3 H  H 3Q2) _ 0 2E H KQ3131/32  +  (Q.Q1132)  1+  (Q.Q1132)  13112\  413122  P  1/ 3 Q/ 2  H 3Q2) 2 _ 0 2E H  —  -  P  (2.149)  states, we can combine such a projector with the energy denominator as P —H 0 E 2  =  I  dreE  Jo  r  2 dO 1 d8 /7 2hT r27ni_ J_ r  T+j9212) e_ ) e_1’T+i911°) 2  (2.143)  where N(T)  =  n (aT)ta’  + bb))  (2.144)  with r = 1, 2 are the level number operators for the two intermediate strings. The net effect of the operators in the above equation is to make the replacement (at, bt) —* (e nT+ino(r)a$[)t, e_T+9r)b) for  all creation operators which lie to the right of the projector. Then, after the matrix element is computed, eE0T and integrate over r and r Any potential divergences come from the region near 8  we multiply it by  N  T  =0.  55  Chapter 2. AdS/CFT and the interacting strings  It is amusing to note that the vanishing energy correction for a supersymmetric external state is manifest in (2.149), since if Q annihilates the external state, all of the terms are identically zero. Q is unknown and it is consistent with the closure of the super-algebra to set it to zero. iJ)(ui) external state, we can check that what is left is manifestly convergent Using the for the four impurity channel, and then show that the addition of impurities will not disturb this, leaving O(A’) contributions at every order in impurities. We have two sorts of terms in (2.149), which we can represent schematically as follows  I[  (((‘lQ3)) 1  =  I  E  )* 3 (IIlH (W111Q ) 3  ((Il(JlHa))*  —  2  1 E  = —  I  1 E  external state, 1W) = Q211), and II) is a level-matched, twowhere t) is the [9, u behaviour of string intermediate state. In order to evaluate the convergence and large 1 these terms, we can be entirely schematic. We take (see (A.22) for the expression of Q2 in the BMN basis) i])(ui)  1W)  /Za3  ) 3 I) c4 aLIa  / Ia3)  (2.151)  while for the purpose of evaluating convergence we can take i  constant  qp  1 p+q  (2.152)  1 in (2.150), we have two where we take all integers to be positive. Let us begin with 6E choices for four impurity intermediate states A a a I’-c I P1’’P2 73 j4  I’)  L  (2.153)  a1/32/313/34Ia1)la2).  We can proceed with the first one, which will give  1 6E P1P2P3P4  2rw  a 3 (a —  (a 1ap 2 a_n(a 1 1  P2  3a a 4  IQ3) (2.154)  i=1  x  ((a3Ifl a_  1(ai la 2 (a 1  p2  4 3a a  113))  3 and where x = —a pj = 0. There are two general ways in which we can contract the 3(r)s. They can connect to factors of ) and Q3), or they 3 Gm/3 in the prefactors of H can pair-up to bring down a factor of Q, from the exponential. As far as convergence and (s) large x power-counting is concerned however, Gm G is equivalent to _•_rs and so we will simply use the former. When contracting 3(3) ‘s there is a fundamental difference between and Q, as far as large x behaviour is concerned, because of the pole in the latter. Zm  (r)  .  .  Qmp’  56  Chapter 2. AdS/OFT and the interacting strings  is essentially equivalent to N, and therefore the two can be interchanged in In fact this analysis. Because K_ goes as a constant for large p, the worst convergence will always be realized ) and 1Q3). 3 by contracting the intermediate bosonic impurities with the prefactors of H These contractions will yield 17(1) j31 ) I31 1 r( I Vflp 3 V_.jplk_p 4  4 =  17(1)  j7(1) 31  ikp4 3 I_p  1 IV_p  (r31 “flP2  J  —  G’ 3 G P2  r13 “P2fl .  )  w.  2rw—.  P1P2P3P4  Taking  X  ), and using (2.152) we see that 3 —(pi + P2 + p 1 P1P2PS  1fr2  P 2 (P1+P2+P3)  11  (2.156)  where all p are considered absolute valued, or equivalently the sum considered over positive integers. This is manifestly convergent. Continuing on to evaluate the leading x dependence, for the top choice in (2.155) we have poles for all three summation variables, while in the 1/\/ and the energy denominator is linear in large x limit the K’s go as constants, G 1/x. For the bottom choice in (2.155), Pi and p 3 have poles, while x, thus giving 6E 1 the sum over P2 must be executed using (2.117). The scaling turns out identical however. Thus 6E //L is convergent and Q(A’). One can repeat this argumentation for the second 1 intermediate state in (2.153) and find the same behaviour. Also the entire exercise may be repeated for 6E 2 in (2.150) using the following intermediate states ‘  rsJ  I’I  cP1 aP2 I3P3 /3P4  1,  a 2,  (2 157  and one discovers the same behaviour. The essential point is that we will always have at least 5 (inverse) powers of the summation variables, while the number of summation variables is 3. Alternate positionings of the oscillators in the intermediate states such as 1) €41 €42 I €i) 2) only improves the convergence, since level matching removes one more summation variable in these cases. We can now consider adding additional pairs of fermionic and bosonic impurities to the or two factors of GWG (or intermediate state I’). This will add two factors of Either way the number of powers of summation variables equivalently two factors of increases in concert with the number of summation variables, preserving the convergence. Similarly the leading behaviour in A’ is unaffected. So it would seem that there are Q(A’) )/jt at every order in impurities, however any non-perturbative 2 contributions to 6E( behaviour is absent.  Chapter 2. AdS/CFT and the interacting strings  2.3.4  57  Conclusion  The principal result we have presented in this section is the cancelation of the logarithmic divergences that appear in the sum over the excitation numbers for the intermediate states with multiple oscillators. This cancelation is not unexpected as it is reminiscent of similar behavior in string field theory on Minkowski space. We have also shown that the non-perturbative term \/X that appeared in the calculations involving these “impurity non-conserving” channels are an artifact of those divergences and thus disappear in a careful calculation leaving the O’) as the leading order in the expan sion. These statements were generalized to an arbitrary number of impurities estabilishing generically both the absence of Q(\/) and the presence of O’) in the calculation at an arbitrary number of impurities. The upshot of the later result is that, barring some further insights, it would be very difficult to perform the full calculation on the string theory side that would match the CFT result given in 2.42. Furthermore, our results seem to imply that the intermediate states of arbitrary energy would all contribute to the same order to the energy shift. This is a position that our physical intuition argues heavily against but that, so far at least, seems to be born out by the mathematics. There are several possible ways in which this problem could be solved. The most obvious is the potential contribution from the hitherto ignored Q4 factor. Unfortunately we know very little about this factor and so it remains the black box of this theory. The second potential solution is the existence of hidden cancelations that would suppress low order terms in A’ from the high impurity channels. We were interested enough in the possibility of such hidden cancelations that we actually performed the full 4-impurity calculation under two different formalisms (both will be presented in later sections and Appendices) as well as the rudimentary 6-impurity calculation. In neither of those cases are there any cancelation of the 0(A’) terms. The final consideration of potential significance is the fact that the string vertex presented above is not a unique solution satisfying the required symmetries and conservations. Two other distinct vertices can be constructed which fuffill the same requirements. In the remainder of this chapter we will discuss the string field theory using these alternate vertices.  2.4  Alternate vertices  The vertex given by 2.87 (and referred to as SVPS vertex after Spradlin, Volovich, Pankiewicz and Stefanski) is not the only solution that satisfies the super-algebra and conservation laws. The alternative vertex becomes apparent if one gives up on smooth fiat space limit as —÷ 0 and instead focuses on the simplest possible pre-factors which still satisfy the super-algrebra. This vertex was first proposed by Di Vecchia, Petersen, Peterini, Russo and Tanzini [67] and is referred to as the DVPPRT vertex. It was then shown by Dobashi and Yoneya that the weighted averages of DVPPRT and SVPS vertices also satisfy the necessary conditions. They, in particular, argue that the equally weighted average of the two gives a vertex which is the most physically relevant of the three. This vertex is referred to as DY vertex. The simple form of the DVPPRT vertex enabled us to develop a generic calculational  Chapter 2. AdS/CFT and the interacting strings  58  method for higher impurity channels which we used to test for “miraculous cancelations” of low order terms at higher impurities. This, previously unpublished work, will be presented in this section. The full 4-impurity calculation and cursory 6-impurity calculation have not yielded any indication of cancelation of low order terms or “dampening” of higher impurity contributions. We have also confirmed the divergence cancelation studied in the previous section for the DVPPRT and DY vertices and calculated the 2-impurity DY energy shift which, for the reasons which may or may not have a physical basis, is in so far the closest result to the CFT one given by 2.42. These results were published in our paper [80] and will be repeated in this section as well. -  2.4.1  -  DVPPRT vertex  The cubic term in the supercharge and Hamiltionian in the DiVecchia proposal are  =QIv> IQ > 3 Where  1  -  v >= e(tt  =HIv> IH > 3  1 > tt)i&  ®2  > Ø3>  (2.158)  where 3 0 and the Neumann matrices Q a + 1 a + 2 Q are diagonal in the SO(4)xSO(4) indices, which we have suppressed in the above formula. These cubic terms are designed to satisfy the sup ersymmetry algebra  {  H 22 E, 11 2f  ,Q, 2 {Qria }  ,  =  EáiiEa 2 H 22  (2.159)  for the full supercharge and Hamiltonian operators which are defined as the series of string coupling constant, g , 2  g gQ ... 4 + Q=Q + 3 Q 2 +  ,  g gH ... 4 + H=H + 3 H 2 +  (2.160)  It should also satisfy [Q, H] = 0. We will consider the corrections to the energy of a state of two bosonic oscillators,  Ie  >  (2.161)  >  Its energy at zeroth order in 92 is 1e >= 2 H  EoIe>  ,  n+ 0 = 2c(3) = 2 E  2&  (2.162)  We will compute the leading correction to this energy coming from string loops. First order perturbation theory vanishes. At second order, the contribution is  > <H e E = —g 3 2 6  2 H  0 E  3 > +g <eIH <eIH e> 4  (2.163)  Chapter 2. AdS/CFT and the interacting strings  59  4 is an operator which can contain either two-string states or four-string Here, we note that H states. In (2.163) above, we are using only its two-string part, where one of the strings is incoming, the other outgoing. (2.163) can be written as ö2 E  1 2 Eo g ( 2 e 2 <vie >< ejv> + [< viQ + <vie  > + <eIQ v Q2 2  >  >. Q2  vQe  <  <ely> +  ><  Qev  >])  (2.164)  The intermediate states in the above formula are not automatically level-matched and must be projected onto level-matched states. We will accomplish this by inserting a projec tion operator onto the level-matched states. The operator is 2  P  =  fl f  d9 -—  (2.165)  ) exp 5 (i9 N  where the level number operators for each string are N(s)  Ti  =  (ota + (s)f(s))  and we have suppressed the SO(4)xSO(4) indices. One might worry that the supersymmetry algebra is only valid once the level matching condition is applied, and we have used it to derive the equation above. It is easy to see that, if level matching is applied earlier, at the derivation of the equation for second order 2 and supercharge Q2 commute with the perturbation theory, since both the Hamiltonian H constraint, and the external state e> is automatically level-matched, all steps remain valid and in the end we obtain (2.164) with the operator P inserted in intermediate states. The calculational method of Appendix E can then be used to obtain the full formula for the second order energy shift:  —  —  ia3I/I J  • (öitSik  2ir  1 + UQUQ) det ( 8 (1 UNUN) 8 det —  1 [Nu  NUNU  -  1 [EONU  N]  -  uNu  1 +QQUQU +oj [Nu 1 -  NUNU  N]  1 [EONU -  1 +QUQU +óiköul [Nu 1 -  NUNU  N]  1 + QUQUQN  NUNU  + QUQU  1 [EONU -  1 N + QU  NUNU  -  UNUNUN]Th  N + QQU 1 QN -  + QUQU  + UQU  UNUN  N + QU 1  UN] -n -n  + QUQU  60  Chapter 2. AdS/CFT and the interacting strings  1 +QUQU kóu + j 6  1 [Nu -  1 [EONU  N]  NUNU  -  1 +QQUQU Where N and by:  Q  + QUQUl NUNU  -  UNUNUN]  N + QU 1  + QUQUNl  -  + Q’UQU  UNUNUN]166)  are usual Neumann matrices as defined in the appendix B and U are given  U U  r, .s r or s = 3  =  s 8 ööpqeW  =  0  =  (2.167) (2.168)  1, 2  2.166 is a “Master Formula” that holds exactly for the DVPPRT vertex. It is from the specific ways of solving this equation that we obtain particular impurity channels and the u limit. The existence of this formula and its potential counterparts limits such as the large 4 potentially be very useful in investigating the behavior of the could vertices for the other channels with the arbitrary number of impurities. 2.166 can also be used to calculate the higher impurity channels more easily then the usual methods like the ones in [80]. We give the 2-impurity calculation here and the 4-impurity one in the Appendix. The two impurity truncation To find the 2-impurity truncation, we keep all terms that are of the second order in the matrix U. Then, integration over the angles just enforces the level-matching condition, which we find more convenient to write explicitly in this case. A useful identity is =  3 2w,  —  2n  ,  3)  & 3 u 1 —2  =  In the 2-impurity approximation, we get öE,.  —  —  4  2  co  sin (nirr)  ç- s—’ L••d L..d  l°3I/ r=1 p=—oo 2n  4 (p13rfl)  —  (3)2  W  2n 4 (p+i3rri)  (r)2  LiJp  ((n) ‘  j-  R fl  P  )  (3)  2 169  —  (p+/3rn) 2 (p/3rn)  Here, both of the impurities must be on one of the two internal strings and the summation over r = 1, 2 counts the cases where both impurities are on one string or the other string. The sum over p is the sum over the opposite world-sheet momenta of the pair of internal strings. Since the integration measure is symmetric under the replacement r —* 1 r which interchanges the two internal strings, this sum can be replaced by a factor of 2. Also, note that the first two terms (which arise because of the averaging over n and —n in the master formula) cancel when we change the sign of the summand, p. —  61  Chapter 2. AdS/CFT and the interacting strings  To perform the sum over p we use the contour integral formula d f(p)  cotirzf(z)  =  p—--OO  where the contour C is the sum of infinitesimal circles surrounding the integers on the real axis, z with counter-clockwise orientation.  f  SEr  2  IL  k3J/-  =  L  dz sin (Inirr) 4 w) 2iK4w) 2  /  (r) +  (3)\4 /3rW  COt lrZ(  )  —  3)2(  P’3 + 3)2  (2.170)  The contour can then be deformed to encircle (with clockwise orientation) the singularities of f(z) and goes to zero sufficiently rapidly at z —* oo that there is no pole there. In our case, f (z) has both pole and cut singularities. The cut singularities are square-root cuts which occur in the linear and cubic terms when the quartic in the numerator is expanded. The poles are double poles. Taking into account the reversal of the orientation of the contour, we get the following contribution from the poles. SEr IL  4 =  3 [w  —  +  nIr) sin ( 4 32  —  r”2  (z  ‘  + /5’r)  2  cot rz  +  ...  (2.171)  fl Z1 r 3  This is not an approximate expression. The three dots denote contributions from contour limit of the cut integrals is integrals around the cuts. It can be shown that the large smaller than the order of interest, 1/i2 and thus for our considerations, it can be neglected. (Another way to see this is to take the large t limit, keeping the contour fixed. ) The dominant terms in the large t limit are the terms produced by the derivative acting on the cot irz and the 1/(z + ,6rn) . The result is 2 4  öEr IL  —  InIr) sin ( 4  4 (W+rW)  3  Ic3i  wO’) ) 3 2  (z  + /3rn) 2  1 2 irz sin fl Z1 r 3  cotirz (z  +...  + /3rn) 3  (2.172)  Z— /3 rfl  Taking the large  limit, we get  r [L  •2 =  7rlo3IIL 2  2 2  (InIKr) +  - 2 2— IILrfl 3 lrIa 3  .3  sin  (nirr) cos(nirr) + ...  (2.173)  In the first term on the right-hand-side, a factor of 7t as well as a minus sign was produced by 2 which cancels two powers taking the derivative. The derivative produces an inverse of sin produces a minus sign denominator of the the derivative term, second of the sine. In the  62  Chapter 2. AdS/CFT and the interacting strings  . The final 3 and a factor of 2. In taking all of the limits, we have assumed that n << !3ita expression should be integrated over r from 0 to 1. The result is:  1 , 2 SE =  r(1—r) SEr 2  dr g2 1  8  —  —  2 K  32[2  11 (1 + 64ir n 2 4  2174  This, however, ignores the channel whereby one impurity is on each string. In that case though, the level-matching condition limits the sum to the zero modes. Therefore the missing part of energy shift will be given by: , 2 SE  =  [2NN] [—2(  =  5cI2z2  —  ] a)N N 0  =  (2.175)  ] [N N [NNp,] 0  factors canceling each other with p and s here being different strings and with as they do above. Taking the expression for the zero mode Neuman matrix we can write the above in the large t limit as: , 2 SE  —— g 22 1’ d r  r(1  — r)  2 (nir(1 2 (nirr)sin 2 sin —8n r(1 r) n 4 K 22 3 Ia  2  — r))  —  —  2 —8n  —  2 & 3 I 2 K 2 I  (  6  64n2n2  2 176  Therefore, the full value of two impurity contribution is: 2 SE  —  1 , 2 SE  —  +  , 2 SE  — —  8  (1  2 K 2 3 2  4  +  n 2 64  2 177  Which is in keeping with the full 2-impurity result we used in [80]: SEDvIT_._ — 4ir 2  64qr2n2)  (  1287r2n2)  2 +n +  5  —(-—+ 24 +  +  512K2fl2)  16  —n 2  2K)  ir2  (32  +  5/2  (2.178)  2+ ’ 7 A’ n + +4 (12 256K)  Four impurity truncation The result derived from the “master formula” confirms the absence of divergences and agrees with the usual method of calculation. However, this result is still different from the CFT one given in 2.42. We were therefore interested in performing the 4-impurity calculation with hope of either observing some cancelations or perhaps getting closer to the 2.42 result. The details of the 4-impurity calculation (which can also be used as a template for all the higher impurity calculations are given in the appendix E.2. The end result for the 4-impurity truncation is: 64 \ 1 4 SE /1 (2.179) + =  3 2 K  64K2n2)  63  Chapter 2. AdS/CFT and the interacting strings  4 2 .6E 6E —+——= j  2  15  J 22 J 2  \24  57 1+• j n 2 64ir  (2.180)  The conclusion of this calculation is that the 4 impurity channel contributes to the same orders in t as the 2 impurity one, and that it does not lead to any noticeable convergence to the 2.42 result. The skeletal 6-impurity calculation which we do not give in detail in this thesis appears to lead to the same conclusions. Divergence cancelation term. The reason for We have seen explicitly that the DVPPRT vertex does not carry 3 and the contact term. this is once again cancelation of the divergent terms between the H This cancelation can be seen exactly in the derivations of 2.166 but it is useful to derive it explicitly in the language of [79]. We did that in [80]: In the language of [79], DVPPRT vertex is given by the following expressions [67],  IH)  =  ) 2 IQfl,  =  92  =  g2f(a3,  92  [K2  , 3 f([ta  +  2  —  2 4ya12y, —  K 12 (z  f(,  —  ] 12 4Za1a2Z  IV>,  K) v>, 172 iY (2.181)  V). -iZ K 2  3 divergence does not stem from the two-bosonic-impurity Unlike the SVPS case, the H 3 2 +K 2 for K K in the H intermediate state. This can be traced to the substitution of K prefactor. There is, however, another divergence that was not present in the SVPS case. It is due to the contribution coming from matrix elements with two fermionic impurities in the intermediate state. In particular, the relevant matrix elements are given by I  i  4g2r  (1  i  I  /  riD  (12t  3 n—n \2 \Q1 /p(1) /-—p(1)/3j/32 113 \(Y (i)\ / (3) —  r)  (\- +  (2.182)  Nö62 (‘i  and similarly for the intermediate state with dotted indices. The divergent contribution to the energy shift coming from these matrix elements is found by taking the large p limits of the summands. One finds: 6E  r’I  —  f  dr  2 gr(l—r) (p33)  (2.183)  The contribution from the contact term stems from the following matrix element, -1  (1 (92  j  2I(a1’’  ) 12 E2Iq  =  Chapter 2. AdS/OFT and the interacting strings  2 (G? KK + G? K3)N) (uk)62 + 8 G’?  64  (2.184)  The divergent contribution to the energy shift is found to be, +fdr  5E  1 2 g  (.185)  (NL) 0 p>  Noting that in the H? contribution the divergence is found for both positive and negative p, 3 contribution the divergence occurs only for negative p, and hence a relative while in the Hj factor of 2 is induced in the H? term, one sees that the logarithmically divergent sums cancel identically between the H? and contact terms, leaving a convergent sum. This result can be generalized to arbitrary impurity channels, as was done for the SVPS case in section 2.3.3  2.4.2  DY vertex  Seeing as both the SVPS and DVPPRT vertex satisfy the super-algebra it is to be expected that every linear combination of them will do likewise. We can thus create any number of vertices by taking weighted averages of the two. One of those, namely the one in which weights of the two vertices are equal is of special importance. This was noticed by Dobashi and Yoneya in [71] and we refer the reader to that paper for the exact details of the reasoning. In summary, they claimed that the cubic Hamiltonian ought to only count excitations of the one SO(4) of the SO(4) x SO(4) background, specifically the one that inherits from . The reasons for this are subtle and have to do with isolating the 5 the geometry of the S contributions of various fields to the three point functions on the CFT side of the duality. 3 part of the vertex The end result was that in order to maintain the AdS/OFT duality the H 2 symmetry and ignore the contributions from one of the SO(4)s. must explicitly break the Z -even prefactor of DVPPRT and the 2 This is accomplished by taking an average of the Z the second SO(4) zero modes cancel-out. The DY -odd prefactor of SVPS. In this way 2 Z vertex is then given by:  H)  )+ T (H’  =  HVPS))  (2.186) —  “3  -  I r)DVPPRT\/1  /_‘3  ,SVPS “3  Divergence Cancelation The argument from section 2.4.1 can be extended to the DY vertex. More generally it can be extended to any linear combination of the SVPS and DVPPRT vertices. We give this generalization in [80]: An arbitrary combination of the SVPS and DVPPRT vertices: He” QN  = =  QH QSVPS  +H’ QDVPPRT +3  (2.187) (2.188)  65  Chapter 2. AdS/CFT and the interacting strings  similarly yields a finite energy shift. We calculate the mass shift of the trace state. The 3 divergence (2.124) 3 term is simply a 2 times the SVPS H divergence stemming from the H /32 times the DVPPRT divergence. The reason is simple the SVPS divergence stems plus from an entirely bosonic intermediate state, while (2.183) results from an entirely fermionic one. This precludes any divergences arising from cross-terms. We note that the SVPS divergence is exactly equal to (2.183), therefore we have -  —(ag +  32)  f  dr  2 r(1 g  r)  2 (j33)  (2.189)  The pieces of the SVPS Q relevant to a two-impurity channel calculation are exactly QVPPRT with K .— K, therefore, from (2.126) 1  /r (1  (a3aa  (92  (a2  N) + +3 K$ )  2G? ([a  (aj Ia’  E E QDVPPR ) T  =  4? (KN + K3)N3)] (uk)62  + 4 (4?K + aK)Nn(uK))  (2.190)  The last term in (2.190) gives rise to a log-divergent sum, the large-p behaviour of which is +(a2  +  /32)11 dr  g(i-r)  ()2 .  (2.191)  0  Thus, by the usual arguments , the energy shift is finite for arbitrary a and 3. The DY vertex uses a = 4? = 1/2. Again, as for the DVPPRT vertex, the generalization of these arguments to the impurity non-conserving channels is a straightforward application of the treatment given in section 2.3.3. 2-impurity channel DY energy shift In [80] we reported the results of the calculation of the 2 impurity channel contribution to the energy shift in the case of DY vertex. As it happens it is the closest result thus obtained to the one on the CFT side. While it is possible that this indicates that something is fundamentally “correct” about the DY vertex it bears remembering that the correspondence is still very much inexact and, perhaps more importantly, we have every reason to believe that the higher impurity channels contribute in equal order to the energy shift making every correspondence from a single channel potentially just a numerical coincidence. On the topic of numerical coincidences, we also find that the unjustified factor of two 3 and the contact term contributions still improves the result in the DY change between H case making it even more closely aligned to 2.42 The calculations undertaken in [80] are practically identical to those in [70], using the DVPPRT and DY vertices in place of the SVPS vertices used there. The external state for which we are calculating the energy shift is still  66  Chapter 2. AdS/CFT and the interacting strings  ij)(ui)  [9,  (atioti +  =  löotkctk) 3).  —  For this particular state, individual 113 and contact terms are not divergent in the two impurity approximation. It should be further noted that for this state, and for the impurity conserving channel, we shall find that use of the DY vertex, rather than the SVPS vertex, is equivalent to making the replacements of the quantities (K, K) as K —* (K + K)/2 and (K + K)/2 in the SVPS vertex. This is the simplest way of reproducing our results. K The separate 113 and contact term contributions to the energy shift for each of the three vertices are given below. We find that the DY energy shift agrees with gauge theory only at the leading order, while also enjoying the vanishing of the 3/2 and 5/2 powers of A’. The 2 term is of the correct form, but suffers from an overall factor of 4/3. The SVPS and order-A’ DVPPRT results do not agree with gauge theory at the leading order. By multiplying the contact terms by two (an unjustified operation), one can recover the correct gauge theory result up to A’ 2 order with the SVPS (including vanishing of its ,\13/2 term) and DY vertices. Further, this operation does not spoil the vanishing 3/2 and 5/2 powers of A’ for the DY result. —  3 terms H oEPS  15  =  ,\I  +3  (1  +  1)  [ +  A2 27  A’ 312 +  ,2 —  512 A’  (_- + 2  (2.192)  :2::fl4  (  SEVPPRT  =  32ir 2 —  +  EDY 3 H  —— +  \3  —  +  A’ +3  4ir 2 n 2  2+ ’ 3 A’ + (2irJ ‘sir  2  ---  — j 8ir n 2  \  21 32ir n j 2  5n2(2+3A/5/25fl4(1 \4 4irj \ir  (1  j 2 5ir  \167r  (2.193)  ] O(A’ ) 4  g2 2  rQ  / 1  L 12  4  2 A’  (  +  32ir2n2) 10247r2n2)  A’  3+n A’ 4  \  / 1  ( ‘ k96  2 —  (2:6  256ir2n2)  72 + O(A14)] (2.194) + 6407r2) A’  Contact terms öESVPS 4 H  — — +  g  F (1 + A’ [ 87r2n2)  327r 2 2 (11 \41r  +  3 (1  +  1  N A’ 32  -  8  19  (1 —  —  52 + A’ --(\ 8irj  2 —  —  105  j n 2 8ir  3 A’  — (--- + \32ir  N A2  16ir2n2) J 2 20K  67  Chapter 2. AdS/CFT and the interacting strings  (2.195)  ] OA’ ) 4  +  DVPPRT E 2  DY ‘  4 -‘--‘H  [n2  42  ( (  +  ZSVPS  —  (2.196)  U -‘-‘H 4  \  35  2562n2)  A  n /1 4 5 2 ,  +  29 8K2n2)  3 A, (2.197)  712 + O(A’4)] A’  +  Energy shifts The results for the complete energy shifts are as follows,  SVPS 6 E  ni  g  =  +  65 642n2)  89 2(1+) “  -  4 n  +  (  +  339  5i222)  /1  DVPPRT 6 E —  \‘  42  +  —  —  +  2 A’  (  A’ +  +  1 “5 1282A2 + n J 2  +  / 9 105 \ 4 A’ + n \—+ 3 51222 ( 4 n )  _[(+ 142 4 79 n  1  512 A’  45 N 59 712 + O(A’4)] (2.198) A’ 3+n A’ 4 + (i6o2 256)  +  4.24  +  2 / 1 n  —  5 n 2 64ir  35  SEDY  A’ +  2 —  NI  327r2n2)  (32  +  2ir j 21  1  512 A’  165 303 + 2567r 160ir  4 4n , 2 N n 2 --A +  \  “  2 ’ 3 A’  )  /  )  712 + O(A’.i99) A’  255 \ A + i67rn) (2.200)  Recall that the leading 3/4 is irrelevant and can be scaled away by fixing the overall f factor which multiplies the vertices (and which has not been written in the above formulae, where it would appear in each as an overall factor of If! ). We see that the gauge theory result 2.42 2 2 term being is matched only by the DY result, and only at leading order in A’, with the A’ miraculous the absence also We see of 4/3. an with factor overall of the correct form but 312 and A’ 512 terms which are clearly generic in the string field theory. The result of the A’ (2.200) represents the best matching of this quantity to gauge theory so far, and thus is an indication that the DY vertex is an improvement over its predecessors. Mysteriously, if the contact terms are scaled by a factor of 2, the agreement with gauge theory is enhanced for both the SVPS and DY results,  Chapter 2. AdS/CFT and the interacting strings  2  SESVPS 4 2H  42  —  4  +  — EDY 6 4 2H +  /  —  Q .-, g2 2  /1  (  iJ  \.12  (  +  ‘  +  +  2 ‘‘__i5/2  2 16ir (2.201)  —  fl )I2 \ 2  )L’  2  )  ‘  4  /7  “  1  288  7682n2)  (4)  however, the DY result is still superior in that the  2.5  + +  2561r2n2)  \/ I I 32irn}  )  2  32ir2ri2)  .J’J  (_1 + 2 \107r 327rJ  2 \ ‘2  \/ I (;‘  68  ),/5/2  (2.202)  power is absent.  Conclusion  The string field theory program is very ambitious. In the best case scenario it could provide one of the most important remaining confirmations of the AdS/CFT duality - one that has to do with the interacting strings and non-planar limit of the CFT. In attempting to pursue this program, however, we have encountered a number of important obstacles. Perhaps most importantly we still do not have any reason for setting Q4 to zero and therefore may well be missing a fundamental factor in our calculations. There is currently no obvious way to address this issue. Secondly, in the aftermath of our own work we are facing an unpalatable result which would have intermediate states of arbitrary energy contribute equally to the energy shift. Our proof of divergence cancelations removed the a-priori reasoning used in excluding them and the explicit calculations of the 4 and 6 impurity contributions for the case of DVPPRT vertex confirmed their existence at all relevant orders in )‘. This is a difficult problem but one that can possibly be tackled. Constructing a DY version of “master formula” 2.166 that enables us to work formally in all impurities, and working to develop a way of manipulating the inverse matrix products such as the ones that appear in that formula, is a possible line of approach. This author is quite interested in continuing that line of research. The fact that, despite these serious problems, forms of the stringy result still closely resemble 2.42 is both tantalizing and fascinating.  Chapter 3. Orbifolding and the discrete light-cone quantization  69  Chapter 3 Orbifolding and the discrete light-cone quantization There are a number of interesting modifications to the pp-wave/BMN limit. Most broadly, it is possible to investigate what happens as one goes to higher orders in the original coupling constant ). = g thus going beyond the strict pp-wave limit. This was done by Callan and collaborators and published in [44] as well as by a number of other authors [40—44]. Equally interesting is a change of geometry of the background space in the string theory whereby some of the dimensions get orbifolded by the periodic identification. Early work in this direction is due to Takayanagi and Terashima [82] and Mukhi, Rangamani and Verlinde [83] and was followed by extensive efforts including [73] and our work in [85]. The important point about orbifolding is that it, if it is done in particular directions, leads to discrete light-cone gauge quantization of the string and to changes to the level-matching conditions. One of the consequences of this is that the string field theory calculations of the sort discussed in the Chapter 2. become somewhat easier. Starting from the observations of [83] where the gauge theory analogous to the orbifolded strings is described, De Risi, Grignani, Orselli and Semenoff calculate the generalizations of the double expansion energy shift formula 2.42 to the DLCQ case for various values of the total string momentum. As part of our string field theory research we performed string theoretic analogues of that calculation. The results, published in [81] suffer from most of the problems already discussed with regards to the string field theory calculations, but the methods employed may well prove valuable in later attempts to resolve those issues. We will discuss those methods and results in the following sections. Another interesting result is that the DLCQ inducing orbifolding is an inevitable result of imposing the “magnon boundary condition” on the string, which is necessary for finding the string theoretic equivalents of a “single magnon” multiplet in CFT theory. This connection was the main result of [85]. In the last section of this chapter we will discuss the relationship between the magnon multiplet and the single impurity string multiplet in and near the pp-wave limit.  70  Chapter 3. Orbifolding and the discrete light-cone quantization  3.1 3.1.1  Orbifolding and the BMN limit String theory on the pp-wave orbifold  . We begin from the metric 5 5xS We can take the plane-wave limit of the orbifold of the AdS given by [83]: 2 ds  =  2 R  —  2 p dt cosh 2pd + 2 + dp 2 + sinh  a (d7 2 22 22 da adO + cos +sin 7dX +cos  +sin2d2)]  ,  (3.1)  Here, the first line is the AdS metric in global coordinates and the second is the 55 metric embedded in the 6-dimensional space. If we want the metric to be orbifolded we treat it as 5 x Ss/ZM where ZM is contained in the same 6-dimensional space as 55. Somewhat AdS different form of the metric was used here from that in 2.1 in order to emphasise the period 5 and S icities of the AdS . The coordinate frame given here is also one in which orbifolding 5 is most natural. It is a simple exercise to perform the coordinate transformations relating 2.1 to 3.1 We identify the scalar fields acted on by the orbifold group with the coordinates of this space. At this point, all the directions of the embedding space are equivalent so our labeling 5 x S 5 at is arbitrary. However, we will soon be focusing on a plane-wave limit of the AdS which point one direction will be singled out as a light-cone direction. It will then become important whether this direction was chosen for orbifolding or not. Throughout, we will point to the consequences of those two possible choices. Picking two’ arbitrarily labeled angles we can then write the periodicity condition for the orbifolded 55: -*ç—. (3.2) x—*x+, We now take a pp-wave limit. As stated above, we can choose whether the light-cone direction will have orbifolded periodicity (light-cone in the direction of x or q) or not (light-cone in the direction of 8). The main consequence of this choice will be whether or not the x coordinate ends up being quantized and therefore whether the end theory is quantized with DLCQ or with the regular light cone gauge quantization. The choice, however, does not affect the number of supersymmetries in the near pp-wave limit as we shall show later. As we are interested mostly in the DLCQ case, we proceed with taking limit with the light cone in the direction of x It will be relatively easy to follow what happens in the alternative case. Still following [83] we introduce new coordinates: r=pR, w=aR, y= R. 7  (3.3)  1t turns out not to be possible to orbifold just one direction and end up with a supersymmetric theory, 1 the gauge theory discussion will provide some indication as to why this is the case  Chapter 3. Orbifolding and the discrete light-cone quantization  71  however, seeing as we will be interested in near-pp wave limit and want to avoid periodicity in x+ we choose slightly different light-cone coordinates then usual: —  x  t,  (t  =  —  x)  (3.4)  Making the substitutions the metric (3.1) becomes 2 ds  =  2 R  [  2 sin In the limit R 2 ds  =  —*  —4dxdx  -  2 cosh  2+ (dxj  2 + cos d6 2  (  —  00  —  d2 +  2 + sinh  2 + cos  (dx+  —  +  2 dx_) + sin  d2)]  (35)  the metric reduces to d0 + dy w db (3.6) y 2 + dr 2+r d + dw 2 2+2 2+2 2+w (r 2+y ) dx 2  which is the standard pp-wave metric. To see that, we make substitions: cose,  =  8 y cos q , x 5  1 through x 4 from r and And similar for the x usual: 2 ds  =  6  =  —4dxdx  —  =  wshi8  (3.7)  =  y sin q  (3.8)  angles. The metric can then be written as  2 (x)  8  2+ dx  , 2 dx  (3.9)  It is important to note here, that we are interested in the near-pp wave limit and should therefore be keeping the next order in as well. However, a detailed analysis shows that for the purposes of analyzing the effects of orbifold on the spectrum of one impurity states and supersymmetries higher orders in the metric and the subsequent corrections to the action and Hamiltonian do not contribute and that the only relevant contribution to the order will come from orbifold identifications. To establish this we have re-done all the near-pp wave limit calculations, originally done in [44], but with the orbifolding condition imposed. Looking at the periodicity equations coming from the orbifolding 3.2 and the definitions of the pp-wave coordinates, we can see first of all that the light cone gauge coordinate x acquires periodicity condition: 2 irR (3.10) x —*x+-j5 and S , is given by: 5 , the radius of AdS 2 R 2 R  =  /2NM 8 4q-g  (3.11)  where MN is the total number of units of 5-form flux through the 5-sphere with M being the number of copies of fundamental domain that are identified by the orbifold group and  Chapter 3. Orbifolding and the discrete light-cone quantization  72  N the number of flux units per fundamental domain. The rules of the Penrose limit taken, 3 small demand that R 2 be made large by scaling both N and M to infinity while keeping g can defined: quantity finite be and finite so a is kept fixed but finite. Also, the ratio  R  (3.12)  =  It can be seen therefore, that even in the full pp-wave limit the null direction becomes periodic, resulting in the light-cone momentum 2p being quantized in the units of This is exactly the discrete light-cone gauge quantization (DLCQ) which will lead to the introduction of wrapped states and a change of the level-matching condition in comparison to the standard pp-wave string theory and ultimately to the introduction of the central extension into the superalgebra. All this has been discussed, elsewhere in our work, as well as in [73]. Discrete light-cone quantization of the string on the pp-wave background is a slight generalization of [36]. One component of the light-cone momentum is quantized as .  2p=—  (3.13)  k=1,2,3,...  ,  The other component is the light-cone-gauge Hamiltonian, ,/4n2(R_)2 =  (8  Nn 3 47rg Ji + 2  +  (3.14)  t are the annihilation and creation operators for the discrete bosonic 6 , where c, a and and fermionic transverse oscillations of the string, respectively. They obey the (anti-) com mutation relation ij faa iti 6 F 3 15 nini —  —  —  ,  V’ni’t’niJ  —  In the last line of 3.14 we have written the compactification radius in terms of string back ground parameters. There are also wrapped states. If the total number of times that the closed string wraps the compact null direction is m, the level-matching condition is (8  n  km  +  ,  (3.16)  =  The states of the string are characterized by their discrete light-cone momentum k and their wrapping number m. The lowest energy state in a given sector is the string sigma model vacuum, 1k, m) which obeys  Ik,m)=O=/3Ik,m)  ,  Vn,i,a  Chapter 3. Orbifolding and the discrete light-cone quantization  73  Other string states are built from the vacuum by acting with transverse oscillators,  fl  fi  a;,lt 13  k,m)  (3.17)  n+n’=km.  (3.18)  ji  j’=i  The level matching condition reads  j=1  j’=i  There is, however, one more simultaneous periodicity condition for the bosonic variables. One way to express it simply is to do the Namely, one coming from the periodicity of following change of variables: .  , 4 2 z’=x’+ix x ix — 2 z + 3  (3.19)  7 + ix x 8  (3.20)  1 y  =  , y 6 5 + ix x 2  =  which explicitly shows the breaking of the S0(8) symmetry into S0(4) x 80(4) with the 5 and (y)s - the first 50(4) coordinates (z) being descendent from the coordinates of the AdS . The orbifolding periodicity condition is then simply: 5 second 50(4) - coming from the S 2iiw  lI2  —*  , 2 e7vTy  112 —*  e  —2iriw M  (3.21)  Y2  If we chose the light-cone to be in non-orbifolded direction, we do not get the DLCQ quantization but we obtain the same periodicity condition on Yi as we do on Y2 It should be noted that contrary to the light-cone direction periodicity, these periodicity and therefore are not present in the full pp-wave limit. However, conditions are of the order it is the contribution from them that will cause energy splitting and the change in the number of supersymmetries that occurs in the near pp-wave limit. Fermionic fields also acquire periodicity from the orbifolding. The superstring on the pp-wave is described by the 8 worldsheet scalars and 8 worldsheet fermions which are free 1 as a representation of and massive. Bosonic fields are described by the functions y and z will likewise be a representation of those, with 1 x 80(4)2. Fermionic fields S0(4) under S0(4) and 80(4)2 respectively. transformining indices spinorial and second the first The dot on the spinorial index represents the splitting of the S0(4) into SU(2) x SU(2) as per the Appendix A. The orbifold transformation will always affect exactly one half of the fermionic fields. This can be seen from the index analysis of bosons in the spinorial representation (once again, using the conventions of the Appendix A) or more generally, by analysis of the breaking of the 80(4)2 under the orbifold group. Half of the fermionic fields affected will split into quarters, each consisting of a pair of 2ir, 2ir fields and those pairs will acquire J’ —* e M i,L’ and L’ —+ e M b , periodicities respectively. Which fermionic fields end up acquiring the periodicity ends up depending on the choice of the orbifolding direction. .  Chapter 3. Orbifolding and the discrete light-cone quantization  74  We will argue that in the DLCQ case two pairs of fermionic fields will have different chiralities and will therefore be from the different SU(2)s: 27riw  , 2 eMTh1aii  a112  122,  /122  _7yj M  (3.22)  —+ ‘/112  (3.23)  I’a122 —+  2 i 1  e  To see that this is necessarily the case, consider the following argument: The effect 2 will depend on the directions on the original SO(6) being of orbifolding on the SO(4) orbifolded. There is exactly three distinct choices there. Either the orbifolding directions will be two transverse directions (one option) or they will be a light-cone direction and one of the transverse directions (two options). Assuming that one fermionic pair affected by orbifolding belongs to an arbitrary SU(2) the other pair will then either belong to the same SU(2) (one option because two pairs would constitute entire SU(2)) or to the other SU(2) (two options). This “singlet” and “doublet” are theories with physically distinct symmetries and must match each other between the two cases. The conclusion is that the periodicity conditions on the fermions differ, depending on whether or not we orbifold along the light-cone direction, even though no fermionic field “disappears” out of the action in the fashion of x+ and x. This argument will become even more clear when we write down explicit orbifold trans formations in the gauge theory case. The periodicity of the DLCQ fermions is the one given above in 3.22 and 3.23 whereas the other case of orbifolding in two transverse directions yields: 2,rjw  2 j 1 ?/  , 12 eT  112  “  /ai12,  l122  —p  e  M  2 L’aj2  2 ‘hIlIt2  (3.24) (3.25)  The results 3.22, 3.23, 3.24 and 3.25 can also be obtained in rigorous but not particularly illuminating fashion by index analysis starting from the bosonic results. We will return to the Hamiltonian and the super-charges of this theory later in the context of the near-pp-wave limit.  3.1.2  .N  =  2 gauge theory  Most of the treatment in this subsection follows closely that in [73]. The dual gauge theory is constructed by replacing the N’ coincident branes of the original U(N’) gauge theory with N coincident branes located at the C /ZM orbifold point. In this 3 new theory we can therefore talk about M copies of N branes. This breaks the gauge group in the following way: U(NM)  —  U(N)’) x U(N) ) x 2  .  .  .  U(N)(M).  (3.26)  Chapter 3. Orbifolding and the discrete light-cone quantization  75  U(N)(M+l) is identified with U(N)( ). The orbifold group will be the cyclic group ZM whose 1  generator -y acts on the six scalar fields of N -v  iq 4 + 3 2 qS (q’+i I  \ i 6 + 5 ç ’ I  =  •\f’f’/)  4 theory as  =  2 ( W ‘+i I  /‘  ‘\  w  4 + 3 1 _  \ iq 6 + 5 q ’ I  W  =  e  M  (3.27) If we want to express the orbifolding in terms of the R-symmetry we can assemble the scalar fields into the anti-symmetric bi-spinor 0 =  coab  p2c°3  —ço  0  —ç02  S03  3  2 0 S  (3.28)  cp 0  0 —  which transforms as b’  FT aTT  Pab  —ia “ 1  -‘&  Pa’b’  under the SU(4) R-symmetry and which satisfies the self-dual constraint ab  This constraint ensures that Weyl spinors transform as  Zab  abcc,,  abcd  =  çojçoj is invariant. In the same basis, the four  =  —a  Ua Xca’  Xoa  (3.30)  ,  X,  —a’  ta  Xa Uai  (3.31)  The orbifold transformation is implemented with  Uorb =  100 0 010 0 0 0 w 0 0 0 0 1 w  (3.32)  The U(l) symmetry corresponding to the conserved charge J is implemented by  Uj =  0 0 0 e° 2 e 0 12 0 0 0 0 e 0 12 0 0 e 9 12 0 0 0  (3.33)  The four Weyl supercharges transform in the conjugate representation. They are schemati cally like Qa ab (3.34) Q = If Xa, Qa transform like a 4, a, Q transform like 4. The orbifold projection is the constraint UorbQQ  ,  QUrb=Q  0. In this way the N = 4 supersymmetry will be which will set Q, Q r- 0 and Q, Q reduced to N = 2 supersymmetry with eight supercharges Qi, Q2, Q’, Q . 2 Importantly, however, in the double scaling limit BMN limit M —* oc and so w —* 1 and the N = 4 supersymmetry reappears.  Chapter 3. Orbifolding and the discrete light-cone quantization  3.1.3  76  Duality  Following the treatment of BMN we identify the energy in the string theory with the con of the operators in the gauge theory. formal dimension Likewise, we would like to identify the J angular momentum with the charge under under SO(6) R-symmetry. With the R-symmetry broken, however, where there was one angular momentum J, now there are two angular momenta [83]:  J=—-(—a),  (3.35)  J’=—(a+8)  of which J’ generates U(1) group which is in the remaining SU(2) of the R-symmetry. There is also U(1) generated by J whose eigenvalues are integer multiples of M. Light-cone momenta are then expressed as p =i(a+a)=z.—MJ—J’ 2 L + MJ + J’ (0 + 2p=’i = 2 R 2 R .  (3.36)  —  It is useful to label the scalar fields of the gauge theory as: 1 A  1 ), B 2 +i  =  ), 4 +i  Where the fields are N x N matrix blocks and the index I gauge group. , B 1 , and 4? fields are [83]: 1 The charges of the A L  A 1 1 B I  1 1 1  MJ 1/2 —1/2 0  ). 6 +i  =  =  (3.37)  1... M refers to the corresponding  J’ 1/2 1/2 0  With a similar table for the super-partners being given in full in [83]. , 2 In analogy with the BMN case, we would like to take z and MJ + J’ to infinity as R while keeping their difference finite. (MJ + J’) = 0), with The desired operators are long chains of Aj’s (for which L ,B 1 1 as the fundamental impurities which have Li = 1 while having insertions of , , B MJ + J’ = 0. The other 80(4) impurities are constructed via insertions of derivatives of the . In order that the operator be gauge invariant, the product must be over all M copies of 1 A S0(N). The vacuum is then given by: —  Ik= 1,m=0)  ...AM) 2 (AlA  where for the string vacuum we must have m direction). For general k, the operator is  =  (3.38)  0 (a string must exist in order to wrap a  Chapter 3. Orbifolding and the discrete light-cone quantization  k, m  =  . 2 A 1 ((A  0)  .  .  AM)k)  77  (3.39)  . Apq 1 2 are traced, over. Adding impurities we see a novel so that k copies of the string A feature as compared to the standard BMN picture. Consider the addition of a single impurity to the operator (3.38) .  .  kM  (at + ia) k, m)  ) e M 2  -*  . 1 Tr (A  .  .  11 _ A . 1  .  .  1 AM (A  .  .  .  AM)k_1)  (3.40)  We have superposed over positions at which the impurity could be inserted. The momentum in the insertion n coincides with the world-sheet momentum of the oscillator state. The level matching condition comes from realizing that the actual periodicity of the operator is I —* I + M, rather than I —* I + kM, which the plane waves anticipate. This requires that n = km, where m is an integer; this is the level matching condition. The integer m is identified with the wrapping number of the world-sheet on the compact coordinate. In [73], the DLCQ analogue of (2.42) was computed for one and two-impurity operators 2 may be expressed in terms of N, built upon k = 1,2, and 3 vacuua. The couplings A’ and g M, and k  —  2R_  —  2 R  2 ‘+2 g2=gYM(ap  where  t  =  gy M 2 k N  V  1 2 (a’p+)  N  —  gN M 2 k  (3.41)  has been scaled out of the metric. The results of [73] may be summarized as follows.  should 1. The single oscillator state is no longer a protected operator. Its dimension get radiative corrections beyond the tree level in Yang-Mills theory, even for planar diagrams. In fact, it must get such corrections if it is to match the string spectrum,  =  V/i +  2 gNn  (3.42)  However, the op for planar diagrams. It produces this spectrum to one order in erator is quasi-protected in that, in the double scaling limit, all non-planar corrections to (3.42) vanish. Yang-Mills computation predicts that the spectrum of this state in string theory does not receive string loop corrections. 2. String states with one unit of light cone momentum and any number of oscillators are free states in that they do not get string loop corrections. 3. For string states with two units of light cone momentum and two impurities there are two possibilities: states for which both the world-sheet momenta are integer multiples  Chapter 3. Orbifolding and the discrete light-cone quantization  78  of k = 2, namely are even, have a free spectrum; states for which both the world-sheet momenta are odd get only the one string loop correction given by:  J’= (24(n+n)A’+...) —N J 2 2  n odd 2 , 1 n 2 even n,n  (343)  which truncates at O(g). The states with even-odd world-sheet momenta are excluded by level matching. 4. String states with three units of light cone momentum and two impurities for which both the world-sheet momenta are integer multiples of k = 3, have a free spectrum. k = 3 states for which both the world-sheet momenta are not integer multiples of 3 get computable corrections to all orders:  —J’= (2+(n+n’÷...) —N J 2 +  gA’ 2 16ir  1+  \\ 2 \ 1 /lrfl \ 2 flrfl / lrfl / / (cos(—)sinI-———)—cos(———Jsin(——-— \3 \3/ 3 )\ 2 ir(flifl  6  .  .  (3.44)  +...  ,n 1 2 where all non-planar corrections (not just the leading term shown) vanish for n multiples of three.  3.2  String field theory on pp-wave orbifold  The construction of the string field theory in the DLCQ case is parallel to the construction in the regular light-cone gauge, with two obvious but important distinctions. Firstly, mo mentum p+ is quantized and secondly the new level-matching conditions take effect that, among other things, allow single impurity states to exist. Q uantization of the momentum imposes several constraints on the energy shift that correspond to the results of the previous section. 1. Every Neumann matrix with a “leg” on an external string carries the term proportional to sin(1) where k is the momentum of the external string. This means that the states with mode numbers which are a integer multiples of the external k will always have zero non-planar corrections. This corresponds to result 1. from the previous section and also explains riders on the results 3. and 4.. 2. k 1 string can not split because, by conservation of momentum, there is no lower momentum number for it to split into. It will therefore be described solely by the non-interacting string / plane level diagrams regardless of the number of impurities. This corresponds to result 2. from the previous section.  Chapter 3. Orbifolding and the discrete light-cone quantization  79  To calculate the actual energy-shift we apply exactly the same methods as in the regular light-cone gauge with the exception of the integral over p being replaced by a sum. The results quoted here were calculated by our group after our work on [79] and [80] and were first published in [81].  3.2.1  k  2 Impurity-conserving mass-shift  =  1 and n 2 satisfying The mode numbers of the external [9, 1]) state have distinct, odd values n 1+ n  2 fl  =  (3.45)  2m  where m is the external wrapping number. For the impurity-conserving channel, we may either place the two intermediate-state impurities on the same string (string #1), or one on each string. In the former case string #2 is in its vacuum state and necessarily has wrapping , where 1 1 +q number m 2 =m 2 = 0. The level-matching condition for the excited string gives q 1 = m. In q are the internal mode numbers; conservation of wrapping number then gives m 2 m. Thus the two choices 1 +m 2 = m 2 while m , and q 1 the latter case we have q 1 = m both leading to are indistinguishable, state impurities intermediate for the distribution of 2 where the same condition which is introduced into the amplitudes via the factor öqi,mq )/2 E Z. We begin with the SVPS result for the 113 term 2 1+n m = (n  3EPS =  r(1  r)  ri r2  ql q2  [()  (2)2  +L N 2 1  1L 2 ‘Vfl 1q 2N q 1 q +L 2 +L 1L 1N  1 Wn  +  2 Wn  36 a q1,m—q2 1 + (n 2 1 i3 Wq /3;:; 1 Wq  —  .‘  (3.46)  ) 2 n  —  where now instead of an integration over a continuous r E [0, 1], r is fixed at 1/2. The result is  2 (oES)k  =  2 16ir  (—-  [8  —  +  (n + n) +  512 A’ (3.47)  ((n + n) +  52 +... A’  The contact term contribution is as follows  6EPS  2 (Gqj)  -  rj r2  1 +(n  €—*  Nn Nmiq ] q 2 (Gqi) 1 )+K 2 (n 22  q1,m-q2  q q  ) 2 n (3.48)  Chapter 3. Orbifolding and the discrete light-cone quantization  80  giving  (6E)k  [  2 16ir  =  +  1 (n  -  / 1  2 +fl  +  2  ) 2 +2(nj+n  4  2)j  (1  1 /23  1 /13 -  i  2  2 (n-bn (3.49)  Combining the results we find 2 g 2 )k_ 55 (6E  =  162  [A’ +  (fh +fl2  +  2  /1 2)J  +)(1+1))A12 i+n2+ ( 2 -  —  2 (n+n  +  1 —  4  ((n +n)+  )AI5/2 +n)+f ) 2 n (n 1  +  +...]. (3.50)  This result does display a leading agreement with the gauge theory result (3.43). However, it also suffers maximally from half-integer powers of A’. We will see that the DY vertex will do better, in analogy with the standard case. First, we present the results for the DVPPRT 3 term is vertex. The expression for the H  —  ETPPRT —  i2 2 r(1—r)64c$  )  rl r2 qi q2  Z3n1  ±  fl2 qi  2  2 (j3 r2 fl2 q2)  Z  Z3n1 ni  3r2 j3r2 fl2 q2 ni q2 qi  3  x  +  +  2 Wn  —  r fli qi  ±  r flj q2  Z’’  fl2 qi  ,m_q 1 öq  qi ! r 3  1 Wq 2  j3 r2  j3 rl  fl2 q2  Z3r2 ii q2  1 + (n  fl2 qi  2 N r 3 fli q2  1 ni qi  j  ) 2 n (3.51)  with result  2 (EIPPRT)k =  2 g [_2A’ +8 162  —  (  +  ((n + n) +  2 2 + (n + n)A’ A’  512 ) A’ 12  —  1  ((n + n)  —  2 ’ 5 ) A’ 12  +...]  (3.52)  Orbifolding and the discrete light-cone quantization  Chapter 3.  81  while the contact term gives  SEPPRT  —  —  —  1  2 a’ 2 3r2 21 [(K ) 1 22 1 (Gqj) 3r2 (Gq I3r2 ) +K fl22j 6 qi ,m—q n2q2) nlq2 33 16cr rlr2qlq2  1 +(n  —*  ) 2 n  (3.53) with result 2 g 167r2[  (6E’T)k=2  —  1 In  + 2  2  312 2+2)) A’  +4  1 1 n (+))A’2 2 + 1 _2(fl ) —  (n+n 1 /23  +(+  (fl+fl+fl2fl1)  (3.54) Combining these results we obtain the mass-shift for the DVPPRT vertex  2 (6EDVPT)k_  =  2 16ir  L  A  —  11  2 +n 2  + (  + ((n+n+2(n1+n2)  +(  +n  1 /29 2 2)+ it —(n’+n 8  ——  2  2ir / j 1  2 ’ 3 A’  1 /53 i) n n 2  +n)  —  —  ))A1512 2 1 n  +  .  (3.55)  which fails to agree with the gauge theory result even at the leading order, as the sign is incorrect. Finally, we compute the extra cross-terms required to assemble the DY result. 3 cross-term is given by The H 6EV  =  (2 H?vlTIe)(eIHrPs) ZE 2 2 g a’ r(1 — r) 32a ri r2 qI q2  =  [  r 3 Z3n1 nqi njqi L I  ’ Z ’ 3  n2ql  2 (3r2 n2q2)  +  Z3n1  njqj  3 3r2 3r2 + L n2q2 niqi nlq2  r2 3 L  Z3n1 n2ql  q1,m—q2 6 _ 3 —cr  Wni  +  2 Wn  1 Wq  —  j3r2 j3ri  niq2  n2q2  T2 3 L  nIq2  + (ni  n2ql  j3r2 nlq2  “  3ri  1  fl21j  ) 2 n  2 Wq  (3.56)  82  Chapter 3. Orbifolding and the discrete light-cone quantization  resulting in  2 (oEV)k  162  =  [8A’  —  16  (  A’  +  —  4(n + n)A’ 2+  (n + n) +  +(n+n)A!2+...].  (3.57)  The contact cross-term is given by  SEV  =  l(QDvPPRTe>(eIQsvPs)  [KniK_ni  —  ri r2  2 (Gqi)  =  (3T2)2  ] 2 K_ (Gq 1 K 2N ) 1 +2  + (ni  “  qi q2 (3.58)  with result  2 (oEV)k =  2 16Tr  [_2A’ +8  (  +  2 + (n+n)A’  _(  + (n+n+nln ))A15/2+...]. 2  fl+fl)+fl2fl1)  (3.59) Assembling the final result 6EDY  2 (6EDY)k =  1622  [A!  —  =  EDVPPRT 6 (  n + n, 2  —  )/4, we find 55 1 + 6E 5 + 5E  1 +n (n ) 2  (  +  2+ / 5 A  .  .]  (3.60)  This result matches the leading order gauge theory result (3.43) if we re-scale the unde The result is superior to termined function f (appearing in front of the vertices) by the SVPS result (3.50) as it does not contain the 3/2’s power of A’. It would be interest 2 term also agrees with gauge theory, however the gauge theory ing to know whether the A’ computation of this term has yet to be done.  3.2.2  k  =  3 Impurity-conserving mass-shift  For the k = 3 string, the splitting and level-matching are more involved. There are two 1 = 1. We can then distribute the two distinct cases, the first is when string #1 has k  ) 2 n  Chapter 3. Orbifolding and the discrete light-cone quantization  83  intermediate state impurities both on string #1, both on string #2, or one impurity per string (of which there are two equivalent configurations). The next case is when the assignments of 2 = 1). 1 = 2 (and so string #2 has k light-cone momenta are reversed, so that string #1 has k  1 = 1 case again, leading to a factor of two. The level-matching is This just counts the k therefore achieved via the insertion of the following operator  r  =  2 2 (SrI16r21öqim_q +  ,  +  l, m_ql)) 2 26n11ör22Sq (  (3.61)  where the intermediate-state impurities have mode-number/string label configurations (qi, ri) 1 and n 2 )/3 E Z is the external state winding number while n 2 1+n ), and m = (ri 2 and (q ,r 2 are integers and not multiples of three. The expressions given for the k = 2 case in the previous subsection are equally valid here, however with the replacement of the k = 2 delta function with (3.61). The results are difficult to obtain for high order in A’, and so we present leading order results only. Since the calculations are straightforward, we will be brief and simply state the results  (SEs\s)  EDVPPRT) 6 (  (6EDY)  ‘  =  k=3  +  2 16ir  2 g  =  k3  =  2 16ir  gA’  —  k—3  22 167r  9 [cos  3 [cos  —  (9-’) sin (9’) ir(ni  2  [cos  +  —  lr(ni  2  —1 +  1  (9’) sin (9’)  (9’) sin (91)  —  ir(ni  2  —  cos (9’) sin ) 2 n  —  —  (9’)]  cos (9’) sin ) 2 n  (9’)]  cos (9’) sin (s)] ) 2 ri  +...  +...  ••  (3.62)  (3.63)  (3.64)  Comparing with the gauge theory result (3.44), we see that although the dependence on the external mode numbers is of the correct form, the coefficient of the second term is not matched by any of the vertices. Further, the first term of the DVPPRT does not match on account of the sign.  3.2.3  4 impurity channel for the k=2  The relative simplicity of the DLCQ calculations also makes it possible for the “standard” method calculations to be carried to the four impurity channel. The principal simplifications have to do with the fact that the k = 2 string splits necessarily into two k = 1 strings. Unfortunately the calculation still leads to an integral: 1 5E  —  —  gA’ l6ir  [°° j_,  1 dq  [°° j,  3 dq  (At + A) A [AAt c..’ (1 w 1 q 4 c 3 —  —  —  AA] ) 4 w  (3.65)  —  where (3.66)  84  Chapter 3. Orbifolding and the discrete light-cone quantization  which is not analytically solvable. Our collaborator D. Young solved this integral numerically and the results are presented in [81]. His results confirm the higher-impurity behavior that we have observed in the masterformula 2.166 calculations for the DVPPRT vertex and generalize them to SVPS and DY indices. Specifically, there is still a contribution to all orders in A, no miraculous cancelations are taking place and the added results are still functionally same but numerically different from the gauge theory ones.  3.3  Giant magnon and the single impurity multiplet  It is possible to draw a parallel between the string theory on the orbifold and the so called “giant magnon” states in the CFT. This was noticed in [86] and [87] and in particular in [84]. A strong version of the statement was made in our paper [85]. In this section we follow [84] to provide the context and background and then state results of our work as published in [85].  3.3.1  Giant magnon in AdS/CFT  The term “magnon” comes from solid state physics and refers to the collective excitation of the electron’s spin structure in a crystal lattice. In the CFT context it refers to an impurity in the chain of Z fields in the large J limit. The name is chosen because the problem of diagonalizing the planar Hamiltionian can be reduced to a type of spin-chain. The large J limit that we are interested in here is defined by: J p  —  =  =tfixed A=g N 2  oo, fixed,  E—J=fixed  (3.67) (3.68)  This differs from the BMN/plane wave limit in two ways. First, here we are keeping A fixed, while in BMN it was taken to infinity. Secondly, here we are keeping p fixed, while in BMN n = pJ was kept fixed. The reason this limit is of interest in the first place is that it decouples the quantum effects which are governed by the A from the finite effects governed by J. This distinction persists even after we eventually take the large A limit. Despite stated differences with BMN limit we proceed in a similar fashion as there by studying the states with finite E J. The state with E J = 0 corresponds to a long chain (or string) of Zs, namely to the operator Tr [Z]. We can also consider a finite number of other fields W that propagate along this chain of Zs. In other words we consider operators of the form (3.69) ZZZWZZZ...) 07, —  —  .  .  where the “magnon” field W is inserted at position 1 along the chain. Using supersymmetry, Beisert has shown [90] that these excitations have a dispersion relation of the form: E—J=  V+sin2  (3.70)  Chapter 3. Orbifolding and the discrete light-cone quantization  85  Note that the periodicity in p comes from the discreteness of the spin chain. The large ‘tHooft coupling limit of this result is  E—J=z sin  (3.71)  3.71 is a strong coupling result and we therefore expect it to have a dual in the perturbative regime of the string theory. Hofman and Maldacena in [84] determine the conditions needed 5 x S . 5 to recover the 3.71 result in the string theory using the usual strings in AdS (which is to say a They show that the strings corresponding to the “giant” magnon magnon on a long chain of Zs) will be a closed string with an open boundary condition, where the azimuth angle spanned by the two ends of the string corresponds to Pmag with the side result of introducing the central charges to the super-algebra that matched ones noticed by Beisert [90]. [86] argued that the open boundary condition led to a modification of the level-matching condition and gauge parameter dependence of the spectrum was a result. In [87] it was suggested that the single magnon is well-defined as the twisted state of a closed string on an orbifold where the orbifold group acts in such a way that it identifies the ends of the string, resulting in a legitimate state of closed string theory. In [85] we took this reasoning a bit further. —  3.3.2  Magnon boundary conditions and the orbifold  The main observation of [85] was that if we consider the single magnon state in the Type JIB string theory, with the boundary condition that the string is open in the direction of magnon motion, we are inevitably led to an orbifold. To get the gist of our argument, consider the following (drastically oversimplified) ex ample of the closed bosonic string on fiat Minkowski spacetime where we legislate that one of the string coordinates is not periodic, but obeys the “magnon” boundary condition X’(T,cr 2ir) = X’(r,O) +Pmag and all other variables, including 8X’(r,o) are periodic. Then, a solution of the worldsheet equation of motion (ô 8) X’ = 0 with the appropri T + Pmag + oscillators. One of the Virasoro 1 ate boundary condition is [4] X’ = x’ + c’p 0 L 0 0 which takes the form constraints is the level matching condition L —  —  N_N+p1=0 2n-  (3.72)  c.cr and N = where N = n.n• Since the spectra of the operators N and N are integers, there is no solution of the level-matching condition unless P Pmag = 2ir• integer, 1 i.e. the momentum p’ is quantized in units of integer.2ir/pmag. This is identical to (and 1 is compactified with radius indistinguishable from) the situation where the dimension X R = itr and where we consider a wrapped string with fixed momentum which is then We see that the magnon boundary condition leads us to string quantized in units of theory on a simple orbifold, a periodic identification of the direction in which the magnon boundary condition was taken. We shall observe a similar fact for the more complicated case 5 x S background. of a single magnon on the AdS .  Chapter 3. Orbifolding and the discrete light-cone quantization  86  The reverse of the above argument is a simple exercise in T-duality, applied to the string sigma model in a direction where the space is not identified. The T-dual of a string with momentum p is one which is no longer closed, but has the magnon boundary condition. The original giant magnon of [84] is a solition solution of a bosonic Type IIB sigma model 5 x S. We can write the Lagrangian of this theory: on AdS  ( {_  aZ.ãaZ (1Z O ) aToaT+ 2 ) a 2 2  2  a}  +  +  (3.73)  supplemented by Virasoro constraints. The eight fields and transform as 4-vectors under . We will impose the magnon boundary condition on the angle 4 SU(2) SO(4) x SO(4) coordinate (3.74) x(T, a = 0) + Pmag x(T, a = 27r) -  If  xQr,  a)  =  periodic,  (r, a) + pmaga/27r with  £[T,  =  i  £[T,  - (()  (  +  (375)  The effect of the magnon boundary condition is to add terms to the action. These, as well as similar terms which appear in the Virasoro constraints, will break some of the (super 2x 2 x SU(2) )symmetries of the background. The last term in 3.75 has the symmetries SU(2) The bosonic part of the level-matching condition 2 are translations ofT and 2 where the R R is .  =  o  Pmag  da {HTT’ + HZ’ + ll’ + llyY’} +  (3.76)  where H 8.1/OXI are the canonical momenta conjugate to coordinates X and the charge J is the generator of translations of , x x + const.  —  J =  f  2ir  da11  =  Y K  f  /  2ir  da  2 y2\ 4 2  )  x  (3.77)  2 Furthermore, being generators Since x x + 2K, the eigenvalues of J must be integers. of the fields, and the fields involved being cr-argument worldsheet the of of translations 3 Since periodic in a, the first four terms in (3.76) must be integers plus a possible constant. 2 W hen fermions are included, they could be half-integers. ,(o-). Consider eigenstates Ic > and 4 i Consider the operator which has the property [, ço(o•)) 3 la >= ea’) <a’lç(O)lcr >, the matrix 1 a’ > where Ia >= ala>. If < a’I(o)la >=< a’le(O)e element obeys <a’i(o)la >=< a’(o + 2ir)Icr> only when a a’ = integers. The eigenvalues are equal to integers plus a constant which is common to all eigenvalues. If, there is a reflection symmetry o —f 2ir u under which —> —, the constant must be either an integer or half-integer. —  —  Chapter 3. Orbifolding and the discrete light-cone quantization  87  the theory has a symmetry under a —* 2K — a, the constant must be either zero or onehalf. Thus, the spectrum of the first terms in (.76) is either integers or integers+. To eliminate the second possibility, we shall see that, in the plane wave limit, we can solve for the spectrum explicitly and there we find that it is integers. Then, since the spectrum should not change discontinuously as the plane wave limit is taken, we conclude that it should always be integers. Since J comes in units of integers, and the first four terms in (3.76) are integers, (3.76) Then, J is quantized in units of is a rational number, will only have a solution if M. This is identical to what should occur for a rn-times wrapped string on a ZM orbifold of 5 x S AdS 5 where the orbifold group ZM makes the identification x —p x + 2ir. To get the superstring, we must include the ferrnions. For this, we must decide what their boundary conditions will be. It is clear that, at large J, we will obtain the correct magnon supermultiplet if we add them in such a way that, in the modification of the Virasoro constraint (3.76), J also contains the appropriate fermionic contribution J —* J = f(ll5’ + ,Th/”). This gives the magnon boundary condition for the fermions 1 H,  9  =  where  diag  (.  2K)  — , ,  =  —)  eu13ma/,(r,  a  =  O)e_2PmaJ  =  em8(r,  a  0)  (3.78)  and the orbifold identification is  (x)-  (3.79)  All of the fermions have a twist in their boundary condition. With this identification, all supercharges transform non-trivially under the orbifold group and all of the supersymmetries will be broken (in fact, the supercharges are set to zero by the obtifold projection). This twist in the fermion boundary condition and concomitant breaking of supersymmetry is well known from orbifold constructions in string theory [88] and was outlined in detail in a context similar to ours in [89]. Some supersymmetry can be saved if we impose a slightly more elaborate identification: ) +iY , (,Yi 2  ,eiPma) +iY ) (X+pmag,e_iPm(Yl 2  (3.80)  = diag (0, 0, 1, —1). This contains the previous identification of the angle x where, now as well as a simultaneous rotation of the transverse Y-coordinates. Half of the fermions are un-twisted and this identification preserves half of the supersymmetries. The giant magnon can still be considered a wrapped state of this orbifold where the identified Y-coordinates are not excited. This double orbifolding geometry turns out to be exactly the one whose plane-wave limit leads to the DLCQ quantization of the string, as described early in this chapter.  3.3.3  Orbifold in gauge theory  The gauge theory dual is likewise one described above. It is obtained by beginning with the parent theory, N = 4 super Yang-Mills with gauge group SU(MN) and coupling constant gy. Then, we consider a simultaneous R-symmetry transformation by a gen erator of the ZM orbifold group and a gauge transform by a constant SU(MN) matrix  Chapter 3. Orbifolding and the discrete light-cone quantization  88  wM_l) where is the M-th root of unity. Each diagonal element of diag(1, w, the MN x MN-matrix 7 is multiplied by the N x N unit matrix. The projection throws away all fields which are not invariant under the simultaneous transformation. This reduces a typical field which was an MN x MN matrix in the parent theory to M N x N blocks embedded in that matrix in the orbifold theory. For example, consider a field Z of the parent theory which is charged under the orbifold group and transforms as Z —* wZ. The orbifold projection reduces it to a matrix which obeys (3.81) Z7=c-yZ ...,  =  By similar reasoning, a field I which was neutral in the parent theory commutes with 7 once the orbifold projection is imposed, (3.82) Given any single-trace operator of the parent Al = 4 theory, for example, a single magnon mZJ with 7 state such as TrZ, there are a family of M states of the orbifold theory rfr M — 1. The operator must be neutral under the orbifold group transformation m = 0, 1, M1 into the trace and use the in the parent theory. To see this: we could insert 1 = 7 commutators such as 3.81 and 3.82 and cyclicity of the trace to show that the trace of any operator which is not a singlet under the orbifold group must vanish. In our example, if mZJ 7 is neutral, this requires quantization of J in units of M, J = kM, in the state rfi. This gauge dual of the quantization of the momentum J in units of M.integers, rather than integers after the orbifold projection, is imposed in the sigma model, discussed above after 3.77. In addition, the single-trace operator of the parent theory descends to a family of M operators which are distinguished by an additional quantum number, m. It is easy to mZJ changes the operator by 7 was inserted into rp see that moving the position where an overall factor of wm. This implies that this trace is already an eigenstate of magnon momentum, Pmag = 2ir. The integer m is the gauge theory dual of the wrapping number of the string state on the orbifold cycle. There is a theorem to the effect that, in the planar limit of the orbifold gauge theory, un twisted operators (with m = 0 in the above examples) have the same correlation functions with each other as those in the planar parent Al = 4 gauge theory with the only difference being a re-scaling of the coupling constant by the order of the orbifold group [91]. For this reason, in the planar limit, the gauge theory resulting from either of the orbifold projections (3.79) or (3.80) is a conformal field theory. In the non-supersymmetric case (3.79) non-planar corrections would give a beta-function, whereas in the Al = 2 supersymmetric case (3.80) the beta function would vanish in the full theory. On the orbifold, the spectrum of states in the Al = 4 magnon super-multiplet are expected to be split according to the residual symmetries. In the two cases we considered, the first 4 x R 2 bosonic symmetry. We would expect (3.79) has no supersymmetry but has SU(2) that the fermionic states gain different energies than the bosonic states and that the SU(2) multiplets within the bosonic states also split. In the other case (3.80), there remains Al = 2 2 x R . The 2 supersymmetry and the spectrum should represent the super-algebra SU(21) ...,  —  Chapter 3. Orbifolding and the discrete light-cone quantization  N  =  89  4 magnon supermultiplet becomes mDZZkM_l 7 rfr mZkM 7 rfr mXZkM 7 rpf  ,  mkM+l 7 rp.  mZkM 7 Tr  m2ZkM 7 rpj  mXZkMl 7 rpf  m1jkM_l 7 rfi. < 4 m 7 rfi. Z kM+l  (3.83) (3.84) (3.85)  Here m gives the number of units of magnon momentum Pmag = m and k is the number of units of space-time momentum J = kM. There are two limits where the operators in — J = 1: One is when we turn off the set (3.83)-(3.85) are degenerate and have energies the ‘tHooft coupling A = g,-MN —* 0 so that the operators have their classical conformal dimension. The other is when magnon momentum vanishes, m = 0. In the latter, the “untwisted operator” with m = 0 is known to have identical correlation functions with the operators in the parent N = 4 theory and therefore have an exact conformal dimension = J + 1. The spectrum away from these limits will depend on both A and m.  Orbifold in plane-wave limit  3.3.4  We re-define the string coordinates as: T that  —  j  =  -  (  =  X+,  x  =  —  X+. This has been chosen so  In addition we re-scale the transverse coordinates —  9  —f  =  The appropriate plane-wave limit [35] then takes A —f cc simultaneously finite. From (3.76) we see that the limit with L — cc and J —* cc with L — J and that This implies is finite. taken so that PmagJ should be —*  (3.86)  Pmag  The magnon boundary condition (3.74) implies X(u  =  7r)  =  X(u  0) +PmagV 5  (3.87)  The scaling (3.86) then gives a finite radius for X—. should be quantized in integral units. = We have already argued that J = In fact, in the magnon sector, we have argued that the level-matching condition (3.76) has a solution only when Pmag = 27r where m and M are integers and J is quantized in units of M, J kM with k an integer. To get the correct scaling of Pmag we must therefore take is held finite. the plane wave limit by taking M to be large so that What is effectively the same limit was discussed in [83] where it was shown to result in a plane-wave background with a periodically identified null direction, X r X + 2irR (To be consistent with (3.87)), the integer m which appears in Pmag is where Th = interpreted is a wrapping number.) The resulting discrete light-cone quantization of the string on the plane wave background is a simple generalization of Metsaev’s original solution [36]. Here, we are interested in a wrapped sector where X(u = 27r) = X(cr = 0)+2irRm. In [83] the spectrum of the JIB string theory in this plane wave limit was matched with the appropriate generalization of the BMN limit of the N = 2 Yang-Mills theory which is .  90  Chapter 3. Orbifolding and the discrete light-cone quantization  obtained from .N = 4 by the orbifold projection corresponding to (3.80). It was also used to study non-planar corrections [123] and finite-size corrections at weak coupling [117]. Periodicity of X Together with the limit, we take the light-cone gauge, X = p+ = k/R. We obtain the sigma model as a free massive worldsheet field theory quantizes _{ãaO+8a+(p+)2(Y2+Z2)}  (a +  ---  (3.88)  + 2ipbfl)  with U = diag(1, 1, 1, 1, —1, —1, —1, —1). In this limit, the magnon parameter Pmag does not appear in the Lagrangian or the mass-shell condition which determines the light-cone Hamiltonian: p  2 + (p+)  =  (a1a1ta  +  Lac1c2tf t-’naa  &‘fl  I  22 2à2tana  p-”  38  &‘flL2I1  Its only vestige is in the level-matching condition. km  fl  +  (a1á1ta  (3.90)  +  +  aa22t  where k are the number of units of J = kM and m is the wrapping number. The bosonic oscillators have the non-vanishing brackets and fermionic /3i/3iti C j  Lmcxicxi,  , 22 [amc  /32/32t]  —  —  ,ç  ,çai,cai  3 1 VmflU  j  V  lPmcrlcx2,  /1 3i/3fi J /32/31t} 13  =  ,  mi, 3 {/ à 2 cr  —  —  ,c  1 UmnJ/3  /32 = ómn5’  (3.92)  . We confirm in (3.90), which is the plane wave 4 and bi-spinors of SO(4) x SO(4) SU(2) J =integer. Here, solution of the level matching constraint unless limit of (3.76), there is we can think of the null identification as the vestige of the orbifold identification. The level-matching condition (3.76) allows 1-oscillator states and the magnon supermul tiplet is the sixteen states cmLIII I  >  a 2 akma  I  >  kmo1ã2 3 /  I  >  Ikmo21  I  >  (3.93)  These states are degenerate with spectrum given by =  (km)2 +  (+)2  =  i + (Rj m 2  =  1+  m2  (3.94)  The degeneracy of the states in (3.93) can be attributed to an enhancement of the supersymmetry which is well known to occur in the Penrose limit. One would expect, and we shall lndices are raised and lowered with 4  E°  and  respectively, always operating from the left.  91  Chapter 3. Orbifolding and the discrete light-cone quantization  confirm, that the supersymmetry is broken when corrections to the Penrose limit are taken into account. Before that, we recall that in [90], [98] Beisert argued magnon states form a 3 ) 1 sixteen dimensional short multiplet of an extended super-algebra SU(212) x SU(22) x (R where the spectrum (3.70) is the shortening condition. The superalgebra SU(22) has gen and the algebra of SU(2) x SU(2), supercharges Q erators R. and 1 and  3711  jci  = i  i j71  —  Ji’2] =  f (-)acvi’ $131  1  —  ó/ 1 3 ci  -‘  -a (132 1. I1 aj’ 13if -  —  c12/32  —  2 J  132  : —  2’  13zf  52 —  5/3i 5(Y2(  5(l27j31 ai 132  ri J’ c 3  V  —  -  Jà2  ai  2 12 3 I  /32  a1131  3 represents any generator with the appropriate index, AC, P and C are central charges. In J p and our application, C = —  —  Là2  {a’c + {at2à2a  313 t172  +  /fl  }  ‘1/32  {t7ii .  }  —  {  i/wfl /w  + ie(n)  —  —  if 12  / 3 2 7 p+/3l 2  12 p’’ +p  /3172/37172  +  —  {—e(n)/wn + p+  =  +  —  + j/w  + e(n) —  —  }  12!3,r2  }  2 3 p+ana/3f71/  a722 72 + p+nai  }  ie(n)w + p’’ 3+ 1  c 172  —  —  }  p+/9172at  (3.95)  2+n .J(p+) We have used Metsaev’s [36] conventions for 2 and e(n) = where the supercharges (those called Q and Qj. Computing their algebra, we find that the plane wave background supercharges indeed satisfy Beisert’s extended superalgebra with the central extensions set to the plane-wave limits of those found by Beisert [90] .  p  =  .VPmag —  1)  AC  =  ./XPmag  V’  (eiPma —  i)  (3.96)  The existence of the central extension follows directly from the fact that the unextended algebra closes up to the level matching condition and the level-matching condition (3.76) kM Pmag contains the term with km 5 sigma model was 3 x S A derivation of Beisert’s superalgebra in the context of the AdS first given in [118] and developed in [119]. They worked with the un-orbifolded theory by “relaxing” the level-matching condition. Then, there is a central charge in the superalge bra which depends on the level miss-match. The idea is that, once the resulting algebraic structure is used to study magnon and multi-magnon states, the level-matching condition  Chapter 3. Orbifolding and the discrete light-cone quantization  92  should be re-imposed so as to get a physical state of the string theory. They work in the “magnon limit”, where J —* cc, but magnon momentum is not necessarily small (in our case it relaxes the plane-wave limit by taking M not necessarily large). They obtain the full central extension, rather than the form linearized in Pm that we have found in (3.96). In r = (1 — a)T + ax, x = x — T their work, they use a generalized light-cone gauge a = 2ir) — x_(T, ci = 0) = Pws x_(T, identification, with a a parameter. They also use the with Pws an eigenvalue of the level operator and z = -r trivially periodic in a. For the variables in (3.73), this amounts to using the boundary condition x(r, a = 2ir) x(-r a = 0) = —(1 — a)pws and T(T, ci = 27r) — T(r, 0) = apws which is different from the one which T(r, ci = 0) and 0 (they primarily use a = ) where T(T, a = 2ir) we use when a the effect of where infinite T J x(r, a = 2ir) x(r, = 0) = Pmag. This makes no difference at a is diluted by scaling. However, it matters at finite size. In fact, the same gauge fixing was used in Ref. [86] and the a-dependence of the one-magnon spectrum found there (away from the infinite J limit) can be attributed to this a-dependence of boundary conditions, rather than the gauge variance which is claimed there. —  -  —  3.3.5  Near pp-wave limit  To see how the spectrum will be split in the near plane-wave limit, we must include cor rections to the Lagrangian and the Virasoro constraints that are of order . A systematic scheme for including these corrections in the usual Pm = 0 sector are outlined in the series of papers [120]-[44] and nicely summarized in [122]. There they find that the corrections terms to the Hamiltonian add normal ordered terms which are quartic in oscillators. They also adjust the gauge by adjusting the worldsheet metric in such a way that the level-matching condition remains unmodified. We have shown, and will present elsewhere, that the modifi cation of at procedure in the magnon sector are minimal. The corrections to the free field theory light-cone Hamiltonian are of two types, quartic normal ordered pieces from nearplane-wave limit corrections to the sigma model identical in form to those found in [120], [122] and terms such as the last one in 3.75 which arise from the orbifolding. To leading order in perturbation theory, the normal ordered quartic interaction Hamilto nian cannot shift the spectrum of 1-oscillator states. Furthermore, none of the extra terms displayed in Eq. (3.75) contribute in the leading order in 1/”. However, recall that, to preserve some supersymmetry, the orbifold identification (3.80) that we have been discussing also acts on the transverse direction and this action must also be taken into account. This generates simple correction terms in the Hamiltonian to order . The derivation of the interaction Hamiltonian is given in the appendix F. The relevant part of the interaction Hamiltonian is =  .Pmagl  1111 + jp fda (  +  (3.97)  With this orbifold identification exactly half of the supersymmetries are preserved in the near plane-wave limit. Specifically, out of the 16 supersymmetries Q, S’ only Sr’/Q and S / Q survive. This leads to a splitting of the energies of the single impurity states. The original multiplet had 16 states (8 bosons 22 0 > and 8 fermions 0 >, c /3a1I2l0 >, /3l0 >). In the near plane-wave, it breaks up into 4 super-multiplets of -  -  Chapter 3. Orbifolding and the discrete light-cone quantization  93  the residual superalgebra: one with 9 elements (5 bosons and 4 fermions) and two with 3 elements (2 fermions and a boson in each) and one boson singlet. The following table illustrates the breaking of the original super-multiplet: Q/S1  ’/Q 2 Sj at •  2i2j  10>  —  2112  I  1/$2 2 Q  /3212210> S2/Q  —  I aI0>  21j  at  —  lj2i  lO>  —  I3  1112  0>  2122  I  22221  I  I l0>  at  —  11j  0>  I  13112210>  —  I  I at  0>  0>  1122  (3.98) 0>  Q /S —  aI0>  I0> 2 a  I0> 1112 /3  —  S /Q  Here, columns and rows with dashes represent the surviving supersymmetry transfor mations: S 1 / Q and S / Q. Columns and rows without dashes represent the broken 1 and S / Q. sup ersymmetries: S / Q2 The energy degeneracy of the original multiplet is likewise broken by the interaction Hamiltonian in the near plane-wave limit. One of the triplets gets positive energy shift, its energy becoming:  2 M  2\/  V The other triplet gets an equal but negative energy shift: 1  !i+m2  V  2 M  2 M  2V’X  11+.\!2 2 M  V  The s-inglet and a 9-multiplet are annihilated by the interaction Hamiltonian and thus retain the energy of the original multiplet:  I  m  + 1 v 3.4  Conclusion  We have made an number of observations about the giant magnon solution of string theory. We noted that the previously noted resemblance of the magnon to a wrapped string on a 5 x S 5 seems to be the only solution of the Virasoro constraints in the ZM orbifold of AdS string sigma-model. We argued that this point of view is consistent with AdS/CFT duality as single magnons are physical states of the orbifold projections of .A/ = 4 supersymmetric Yang-Mills theory. We also argued that this point of view is consistent with the planewave limit, where the sigma model is solvable. In that limit, the orbifold identification appears as a periodic identification of the null coordinate and the magnon is a wrapped  Chapter 3. Orbifolding and the discrete light-cone quantization  94  string. There, we can see explicitly how the wrapping modifies the supersymmetry algebra 2 supersymmetry of the orbifold and is consistent with the magnon spectrum. The N is enhanced to N = 4 supersymmetry in the plane-wave limit, so that the full sixteen dimensional magnon supermultiplet appears there. We end with a question. We have shown that the supersymmetry is broken again by near plane-wave limit corrections to the sigma model by showing that the energies of the magnon multiplet are split. However, there is another limit, the “magnon limit” which is similar to the plane wave in that A and J are taken to infinity but it differs in that Pmag remains of order one, rather than scaling to zero. It would be interesting to understand whether the supersymmetry is also enhanced in this limit so that the orbifold quantization of the infinite volume limit has more supersymmetry than the orbifold itself.  Chapter 4. Summary  95  Chapter 4 Summary This thesis is a result of five years of research in the area of string-gauge duality with the particular emphasis on the string theory in the plane-wave background. Two main lines of inquiry we pursued were the question of string interactions and the consequences of placing the theory on an orbifold. The following is the list of the most important original results we achieved which are presented in detail in the earlier chapters. In the context of string interactions 1. We investigated the impurity non-conserving channel of the interacting string, in the context of string field theory, and found that the divergences, which were previously thought to disqualify this channel from having a meaningful physical contribution to the energy shift, actually cancel out at all orders. This introduces an important new consideration for the theory of interacting strings. 2. This cancelation also settled the issue of symmetry factors, and relative weight between the interaction term and the contact term in the energy shift calculation. 3. We calculated the energy shifts of a particular string state of interest, under the differ ent vertex regimes, up to a variable number of impurities. In doing so we have achieved hitherto closest agreement ever between the interacting string calculation and its gauge theory dual. We were also interested in the string theory on the orbifolded plane-wave. In this context: 1. 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Orselli and G. W. Semenoff, JHEP 0411, 053 (2004) [arXiv:hep-th/04093 15].  104  Appendix A. Fermion representations  Appendix A Fermion representations In greater part of this thesis we work in the representations of SO(4) x 80(4)2 labeled 2 spinor indices. Fermionic creation operators are 1 x (SU(2) x SU(2)) by (SU(2) x SU(2)) and ! then given by 12 which transform in the (1/2, 0, 1/2, 0) and (0, 1/2, 0, 1/2) , respectively; ak,ak being two2 1 x (SU(2) x SU(2)) representations of (SU(2) x SU(2)) component Weyl indices of SO(4)k. The 80(8) vector index I splits into two S0(4) x S0(4) vector indices (i, i’) so that we use vector index i = 1,. , 4 and bi-spinor indices a, à = 1, 2 for the first 80(4) ) for the second S0(4). Vectors are constructed in terms of bi-spinor indices 2 ,a 2 and (i’, a as (an)aii = 22 = u (an)a a/V’ and transform as (1/2, 1/2,0,0) and a 2 (0, 0, 1/2, 1/2), respectively. Here the u-matrices consist of the usual Pauli-matrices together with the 2d unit matrix .  .  (A.1)  =  {i]t = —u. These properties are also and satisfy the reality properties [u] = u, 131 ( = —/3; the 3’ 2 and )t satisfied by the fermionic oscillators, so that 2) same relations are obeyed for the dotted-index fermions. Spinor indices are raised and lowered with the two-dimensional Levi-Civita symbols, ,0 iN for example = _i 0)’ ( ) =  e 13 A’=A  (A.2)  Aa=AEa  and e  =  (A.3)  e u.  The u-matrices satisfy the relations + uo4  =  25’ö,  Some other properties satisfied by these matrices are  +  =  2S6.  (A.4)  105  Appendix A. Fermion representations  =  UU  uu uo  (A.5)  —  =  +  =  +  [i  (u  uo  3 (ae à3  {i  g  j]a  (A.6)  =  j]a  i  (A.7)  =  (A.8)  =  s+ 7 ( 4  =  (A.9)  o€-y),  (A.10) 2uu  k(i  j)k  . /3 U U 1 s + 13 6  =  (iJ)t f3 k/3ii 0 7  =  —  (A.11)  —  =  (A.12)  iki 5  jki 3  —  (AJ3)  c3  In this basis the gamma matrices have the following representation /  (  =  o  0  0  l2/32  =  0  .  2 cr  5a2\  cii3i /32  Q•Z(’/3’5/32  0  a2  “  i 3 /  1U)  (A.14)  )  51g2  0  =  1  0  (_oun!a2/32 ‘  (_-5/3’uj’  (  ‘  7aa  0  —  I )  0  /32  —  aa 7  (  5/32\  2/32  )  .  (A.15)  and the projector reads  ab  2 (i 3 i 5 )th 234  0  (12  0  (o’1u2cr3a4)62  )  —  /5/315/32 a  5(115(12  0  —  \  0  I  ,  (A.16)  —  so that (1 ± ll)/2 projects onto (1/2, 0, 1/2, 0) and (0, 1/2, 0, 1/2), respectively. is a (0, 1/2, 1/2, 0) representation. The supercharge Q cxi is a (1/2, 0, 0, 1/2) and /32 In this notation it is convenient to define the linear combinations of the free supercharges , /Q (All) + iQ e(c)(Q±)t. On the space of physical states they satisfy the  Q  = e’ , and 4 where dynamical constraints  =  Q/3 12 , 2 a 1 {Qa }  { and similarly for  , 2 Qcxiã  12 Qa  } 2 Q/3l/  and  -  =  =  (gui) =  3 2 / 1 Q .  2/32  cu  i3i  3 12 H / —2e 2 ç 1 +  (‘)  J’’  (A.18)  (12/32  The free supercharge with raised indices is understood as  106  Appendix A. Fermion representations  t, (Q ) 1 a e(a) 2  QW2  Qà1c2  (A.19)  e(c) ) 12 a 2 (Q  and this gives f12f  ‘2  2 à 1 “2c  —  A  U  —  r1a2  — —  for these operators in the single string Hubert space ‘H . For states in the three-string Hubert 1 space ?-i , i.e. jQ), the e(c) is already encoded into the construction so that it should be 3 dropped in the adjoint =  ) 2 (Qai  ) Q 2 Q2aj 2 and similarly QI supercharge Q2a 12  . ,a 2 a 3 (Q )  =  ) 3 +4H  (A.21)  In the BMN basis, the full expression for the quadratic  k  2 a 1 Q2c =  ) 2 IQ3cz,a  (c4  2 + i e(a)  ak  k#O  :  i /32 + I k 3  (t  e(c) c  e(a) ak  2  +  /32  + e(a) co  1L  /32)  (A.22)  O/3i a  /L  /32)  where k is defined in (C.6). Among states that are created by two oscillators, the state with quantum numbers (1, 1, 0, 0) and (0, 0, 1, 1) which are created by two bosons have no analogues amongst the two oscillator states containing either one or two fermions. Thus, they are not mixed with other members of the supermultiplet. These states in the main text are denoted [9, iJ)(ui) })(i’i’) in SO(8) notation. and [1, 9  ‘Note that c4’  =  and similarly for the other SO(4) since [u’]t  =  Appendix B. Neumann matrices and associated quantities  107  Appendix B Neumann matrices and associated quantities .  .  In this section we present the explicit expressions for the quantities appearing in the pref ) and 1Q3) (2.87). Following the notation of [70], the 3 actors and exponential part of H written as Neumann matrices can be  {  St  mn  ljst 2 ImIIn  ( \  ‘Na t ImIO  ,m,n  + Um(s)Un(t)) m0  0 (B.1)  with’ N3t  mn =  —(1  —  [CUG2NS]  4K)’ 0 N  =  —2w(1  [CU)1C2Nt]  m  asn(t) + atwm(s)  t  —  e {1, 2}  (B.2) (B.3) (B.4)  s,tE{1,2}  (B.5)  e {1,2} while m,n rs ‘%  mn  =  e(m)Io,  m  0 (B.6)  0  — rs  Qoo {e(m)Q”fl’, where [68]  “  C) ‘  mn  mO =  =  e(ar) [U2Ch/2Nr3C_h/2Uh/2l (S)j  1 _a ( 3  e(a) —  mn  m,n>0  ‘  [(U(s)C(S)G)1/2JVS]m  m>0  (B.7)  /_ 3r —  -Q  /:-E, —  3 a  Q=0,  r,s= {1,2}  c( 5 ) 1 (—1)( ‘To have a manifest symmetry in 1 —* 2 we additionally redefined the oscillators as ) oscillators. the fermionic for analogously Z, s 1,2,3 and for n e en(s)  Appendix B. Neumann matrices and associated quantities  and we note that  Q  Cn  =  108  0, while  =  ) 8 C(  fl,  un(s)  =  = n2  1  71 U ) 5 ,(  U,)  —  (B.8)  , 2 ) 3 + (a 1 = —(W( ) 8  (B.9)  ua 1 ) +8  and [62] 1 —  3 N  —  sin(nirr) irr(1 r) —  —  4irr(1  1 (w( + —2fLa ) ) 3 3 W(  —  2irr(1  O(e3)  —  =  (B.11) Ua 1 ) 3  2  r)  —  (B.12)  ) 8 ic3  For the bosonic constituents of the prefactor 3  3  K’  (B.1O)  —  1  N(/3 a ) N =3 3 up to exponential corrections one has  1  4/IKK  , Kfl(S)c ) 8  K’  =  (B.13)  ) 3 Kfl(S)’( s=1 nEZ  s=1 nEZ  where ) 8 Ko(  =  (1— 4cK)V2_(1  ) 3 Ko(  =  (B.14)  0  —  and (1  Kn() =  —  8 c 1 v’  /N (i 11 ) 3 w( + /ta 3 ( 2 4K)” )  —  (B.15)  ) U( ) 8  For the fermionic constituents of the prefactor one has 3  3  yc1a2  tala2  S 3 G1nIs/  ZalQ2  n(s) 3 ZZG InI(s)/  t1’2  =  (B.16)  s=1 nEZ  s=1 nEZ  where ) 8 Go(  (1  —  ) 3 Go(  /f— i3, K)” . 1 2  =  (B.17)  0  and  e(a) =  /j (1  —  2 4K)’I  where in the above expressions we have used /3  (B.18)  ) + as)wn(s)N 8 (w( 11 r and  To compare with the definition used in [62] note that ‘here 2  /32  =  1  —  r (with  3 —at/a  there (_1)8(18ns 1 2  and  Appendix C. More Neumann matrices and relations  109  Appendix C More Neumann matrices and relations The above matrices can also be written in the following compact form:  j3r  —  nq  sin(n7rr)/ (AA + AA) 27r/7(q—/3n)  isin(Inirr)(wq +/3rwn)  jrs  qp  Q  =  Q  —  QT  swq+/ 4ir/JJ(/ r 3 wp)  rs =  qp  2ir/(q—8rn) where  —  r1 3 P) 4ir./(/3swq+/3rwp) —  (c.1)  (C.2)  We also find A+  sin(nr)/T(1T)_A—_ 3 Kn+  Jk  (C.3)  V ira’  Kq  ‘r(1-r)AE-A V 71°’/3r 2/  4 3 a  =  I  1  (C.4)  sin(Inirr) =  e(n)(A  —  (C.5) (C.6)  A)  where, =  /wq  —  3 !3rY  AE  =  e(q)/wq + / riLa3 3  (C.7)  =  e(n)-../w + /  (C.8)  We will also find use for L ” 3 nq = KnK_q + K_nKq The following relations may also be proven —  K8)  KT)  Lrq  2 a r (1 + K K(r) —q —  (s)\  (r)  —  (C.9)  KnKq + K_nK_q.  r) (Wq  a’  K3)KT)+KK=2a3T(1T)  +  rs  qp  (C.10)  iqV  (C.1fl  (r)  (__w3))  Appendix C. More Neumann matrices and relations  I  ç(r)  V  =  (3)  /r(1—r)  =  ‘“q  /  a(s)  (r) rs  rs  P  KT)  (C.12)  1 —K 3  (C.13)  —  L’  G(3)  1  /3a’ r (1 r)  110  a’  *K(T)G(8)  3 a  q  /  a q 3r nq( 1 N 7 nq+ )  (C.14)  (C.15) 3  /  ç(r)  q  3r  3rç(3)  NThq=(l)  nq  2  2  (‘v;) (3)2  —  ‘%\  +  qpj  (37.\2  \  nq)  =  (c;) G?)  q  ‘  (C.16) (C.17)  = —  (G;? G)  (C.18)  111  Appendix D. Calculational method  Appendix D Calculational method While most of the work presented in this thesis was done in collaboration, particular credit for systematizing the calculational methods presented in this appendix goes to D. Young who was one of our co-authors in [79] and [80]. This appendix itself is taken largely intact from [81].  Vertices and definitions  D.1  IQ)  ) and 3 Beginning with the known construction of H  in (2.87):  3  V)  =  lEa)  (D.1)  )6(ar) r=1  ) are exponentials of bosonic and fermionic oscillators respectively 3 where Ea) and E, /  lEa)  =  3  00  exp  a rn(s)  \  st atK) rnn n(t)  (D.2)  123 a)  r,s=1 m,n=—oo  and /3  lEa)  =  exp  00  (  (1a2tt() rn(r)  rS  —  rn(r)  n(s)r1à2)  \r,s=1 rn,n=—oo  where a) 123  ) 3 H  =  —  =  I  123 a)  Qi  Ka1Kdl2a2S  a’ 8a  (Z) 12 (Y)s  ala2 \  17f(/UY 2 g , 3  —  al) 3 a  —  —  —  (Ki?  Ka1a1Ka2a2s*  —  (Z)] 2 (Y)si  ala2 \  iv>,  l(Y)’7’ 17 (Z)t/3 12 ___(s  (D.4) (Y)t/3272 (Z)K22)iV), 12 +is/3  ) 2 /3 31 IQ  =  (D.3)  ) 0 0;a 2 ). We then have 3 lO;ai) 0 0;a  , 3 2 f(ic g  =  rnn  f(a 2 g , 3  1 a  k(*(y)t*(z)171 —) 3 a  (Y)K22)IV). 272 +is/3172 (Z)t/3  112  Appendix D. Calculational method  where 3  3  K’  K(s)c,  =  K’  (D.5)  Kfl(S)a’(S)  s=1 nEZ  s=1 nEZ  3  3  Za  t12  3 3 i 8 G1flI  =  t1c2  n(s) 3 InI(s)!  =  ,  (D.6)  s=1 nEZ  s=1 nEZ  and ‘‘7272  7272 K  (D.7)  where the a-matrices are defined in appendix A. We also have =  (Y +  o’ [1 +  ) 4 z  + Y4Z4]  21 i _[Y2ii(l+±z4)_z2ii(l+±Y4)]+[Y2zj  s’’ (i —  i  —  ——  12  4+ (Y  [i1i —  ) + -J_Y4z4] 4 z 144  Z)  (1  —  y4)]  —  [y2Z2]i’Y  +  Here we defined Zi 2 y  1 3 iiy2a1I  2 Z  ,  Z2a11  o  y2k(iZ2i)k  )ii Z 2 (y  &/3i  (D.8)  and analogously for the primed indices. We have also introduced the following quantities quadratic and cubic in Y and symmetric in spinor indices 2 y  = —  yala2 ya2 /3  2 y  ‘  =  C2/32  —  2 y  =  /322  yc1a2 yai  (D.9)  /32  ya  (D.1O)  ai  and quartic in Y and antisymmetric in spinor indices 4 y  = —  2 y  171  27 y ’  = 2/32  272 g 2 y  =  y, 62132  (D.n)  where —  y2al/31 2 y aii3i  —  —  y22/32 2 y a2/32  (D.12)  The spinorial quantities s and t are defined as s(Y)Y+Y  ,  t(Y)e+iY2_Y4.  (D.13)  Analogous definitions can be given for Z. The normalization of the dynamical generators is not fixed by the superalgebra at order 0(92) and can be an arbitrary (dimensionless) function f(u , ) of the light-cone momenta and t due to the fact that P is a central 3 element of the algebra.  113  Appendix D. Calculational method  Commutation relations  D.2  Rules for (anti)commutation of /3 annihilation operator with Y and Z elements in the pre factor: 3 /t’1’2 =G ml (r)P (r)  r=1  2I {/371 72 rn(s), Y’’ {/3y  72  rn(s),  Yai  G ImI (s)  —  —  y cr2) 2 1 7 6 Y 2 + /3  (D.18)  (671  {/37  =  m (s) 1 G  (6722y12  aiY (f i 7  2 m(s), 7 i 7 {/3 } 4  (D.16) (D.17)  s) G ( m 1  Gimi(s)  (D.15)  72  iYci 72 i 7 € 72 + ) aiYi  =  {/37172m(s),2} =  y  2 72 Girni (s)ai 7lEC  (2}  {/37172m(s),  72m(s), } 132 c  (D.14)  n  22 71 Y 7 E 1  2  (D.19)  ) 2 Yyi 72 +Yai  (D.20)  72 4Glml(s)1’  =  And exactly the same for Z and the dotted indices  Matrix elements  D.3  Some useful matrix elements are (where =  ) (c I 3  (31  (Ku”7’  (D.21)  (31V)  Where s and any other internal string index is restricted to run over 1, 2 only. We also have,  (31 a$ KkKIIV)  (31  (3)j fli  2  =  Kk k iv) 1  (KA  (3)  =  (K 2 K_ik o  +  3s j j 1  A (p33  o  1 +K  ct(s)ióik  fl2P  1 Kk +K + K,,  (s)i’\  ik 1 + KkK R6 1 + K Kk S  P (s)j  2P  +  np  ) (31V>  (D.22)  il öjk  +  n’p  2 Kk +K  3s 3r  p  (s)i p  k f(s)i j 6  1 j5 fliP  P  (r)j\ aq  ))(3IV)  (D.23)  fl4  Appendix D. Calculational method  (31  71 (Y) 1 t  (S  IV) =  —  Jfl  {  {  72 (Y) 1 s  {  (3I a  Q2  +  s3  7 f  ) 2 g 1 Y  (s)  PflJ  (22  P12  2 + Y 7  272 EU  sf(s)  —  Pfl s3)  flP  PU1U2  (6272  —  ITI  3  +  1U2  }  (31V)  —  ) + 2 Yl  72 ( 1  +  272 T1U2  3  ‘72)  }  (31V)  +G  U272  (D.26) (c Y 2  +  71 Y4) 1 e  71 + + iY,  (D.25)  (3s  IV)  —  flP  + iG  I’8 flUU2  (6th71  PU1Q•2  jU2  (as =  ()  (D.24)  + iG  72 (Y) V) (31 1a2 t  Qs3)  (l  ala2 =  72 ) 1 Y + Yg Y  —  (n2  j/1  1 a  th2  (i  +  }  (3W)  (D.27)  ) n ) 2  i ni  We now construct the matrix elements we need in later calculations, for instance,  (3l( to( iQ 2 2 e1thEa2I ) ) 3 .  {  KL,  +  1 +G  €,  c4  +  s)i  ) 312 lQ  —  =  QT  p  j  tlUl(Y)  z  1 +G 2 (Y) 1 s  QTh  4a  (s)i nip  {cz:qI3t(r)  712 + 1 ( + iG 22 Z  where  92 =  2 à 1 q y  22  { —  ) 2 2+z (z i2I32  I —  f32U2  2 ) 1 Z  }]  aii 6 a2/32  2 z  â 1 y  iZ3 2U2  (31V)  +  +  3r  f(r) q71a2  n2q  2 z) z  } (3)  a Z4) 1 e  —  (i  1 n  1 G 2  1322 E  ) 2 n (D.28)  712  115  Appendix D. Calculational method  (3(3)ia(3)i  ’ g a 2  ) 3 H  ifl2  (r),  öij+3s3r nq  p  3  K’  —  3 K + (K(3) —fl2 flu  ik 6 5 fi  —  a’  —  2 (Z) 1 2s (Y) 1 s  K2P2  3 K 3 +K —flu fl2  j 6 3 jk  KP2P2 3s  (ii5iPi  fl2P  —  (iiii  2 K 1 2 K  +  P  3s 722P  iIiP1  K’  —  i  fl2P  P  c4  u1 5  2 (Y) 1 s  KP2  2 Kk +K  ) flip  af(s)iöjk  ö1) flip  v’l  p  2 (Y)s 1 s (Z) 112  (s)i)  K2P2  i? I  +  2 K 2 3 K fl2 2 K  ’ 1 Sk’l’j v’’  —7— a  i(s)  + K$)  +  P  {Kk’i?l’  —  1 Kk +K —  1  vkt  x ({Kkkl  Z)] s ( 22 (Y) 1 s  a’p  (31V)  (D.29)  More matrix elements  D.4 Consider  (3’ a ’ 3 fl  fl2  K’r7iIV)  =  j7i’Y13s  1 (K  at’8  fl2P  P  2 + KL  j7i7i3s flip  +k”’  a’’ p  (D.30)  flu fl2  and therefore i 2 g fl2  I’31/2)  =  4c$V  2  Np a ’ x {s i(Y)[Ki 7 17 (Z)t 12  ±K’ 13172 (Y)t 2Y2 (Z)272 + is  flifl2  p  c5  /33 flifl2  135 7 3 uj ±K nip  1 r 7 +  +  niq  q  q  fl2P  q  fl2P  (D.31)  a81)]  as)i)  }  We will need the following expressions:  fl/31a2  = —  fl/312’  — —  +  G{oi  [Y2iiZ i2 3 /  —  (  3 &2  /3ir —  ±y4 /3ic2 36  (i + + ‘z’ a2) /  ) 1y4)]  —  1 y2k(ii)k z 1’y1 3 /  a2f  (D.32)  116  Appendix D. Calculational method  — [n/3 ’ 2 à 7  =  =  +  [  2  (Z3/ icz 3 3  G 3 I’  2i’i’3 1 y /312 3  ‘‘  —  P272  ly43 /31a2j 36  —  z+ 1 6 2  672zP2) a2  —  (i  Y4)] f3  272  (D.33)  We’ll also need the following Q2  a2  3 a  k  l at( ) 3  ) -*n ) (k-l, n 1 2  +  ucot 711 (c  k3)  i  3)  (D.34) or = Qi/2  l a )kc 3 a( ) ) 3  ‘ =  1 l n  (k  L3 +  c  (‘“fli°  ,,/]j  fl2’ 71171  ) Ti ) 2  Ica)  (D.35)  allowing us to calculate, 2 a’ g  ) 3 (,\1H  3r ‘ 3s niqj Th2P q(r)f2  (nh’ —  +  ].9  K’’  KP2P2  at(8{  fl2P  —  P  G +3 In’ I fl2 (K(3)  S2  (Y)  L- S]  K2P2  1P1 KI2P2  {  ’ (w ) 1  Kp  —  YI 1  8  P1P2  P1P2”  ] o,yi  /  jia.  —  —  1  ia  —  v  1P1  —  2P2 3 1  [  —  —-  3r ‘ niq) q(r)42  (Z)  ,,\71  }  (w  3 G IniI  Z  q(r)2  2P2 6  f32p2 6  2p2 3 1  z  2 z’)] z +1  12 z)] z’ + z  }  }  3r fhq)  (o’  2P2 3 1 —  3 \  P1  2 Z  12 P2  —  f322  z’  +  4)]  +  2 4’)] z  3r  i (_ t 8 q(r)/  P2  —  “si’ z 2  PI 3 k  €  [  2P2 3 1  ) 1 v + (wf  (_ti  3 G mu  (Z) 2 ” sP1P2(Y ){si 2 K  P1  3  +  ri (_, t 3  (Y){s*(Z)  —  +  (Z)  2 (Y) 1 s  —  T7(3)  P1P2 8  — !-  s(Y)  2 K o’i) K.7 6 + K  — a’’ K K2P2 s 3 712  — -1 P1 P1  [Kk  )2  K’’ 3 +G Ill  +  at(8)1{ P  +  niq)  i 3 P  2P2  2 2  ) 2 (k—*l,ni<--*n (D.36)  }  Appendix E. Energy Shift for DVPPRT Vertex  117  Appendix E Energy Shift for DVPPRT Vertex This appendix contains the hitherto unpublished work by the author. It is therefore some what more detailed then the earlier appendices.  Derivation of the “Master Formula”  E.1 E.1.1  The two-string state  <ely>  and the number <vie ><  ely>  The simplest quantity that we shall need to compute is <ely>  oi 3r + = <3I_n i >= Fjc [ -nfl  (r)it(s)itfs3l < qfl] q  Iv >  (E.1)  In the last expression, p and q are summed over all integers and r and s are summed from 1 to 2. This is a state in the two-string sector of the Hubert space. The state < ev > has Neumann matrices with creation operators for the two strings. Now, from Eqn. (E.1) <ely>  =  ca?iv> <& l 3  Ijçr33  +  L  ]f3r  c(r)ita(s)ifJrs3]  qn  q  v> <o I 3  we have  <vie > P <  ely  >=< vi&3>  .p  [ir33 öki  [33  —fin ó  +  3r —fip  r)ifa(s)jts3l qfl] q v > 3 < a  +  =  T N? +J —nn —np <vi&3 >  (P6ui(1kta(s)tt  r ]TW 3 f  < vö > ai  —np  <vie>P<elv>= +/nfl  —n  I  r)i(s)i Sktp)  <vNqia3><a3ivMp>  <  v>N+ <ö I 3  NS NS 8)jPaktcrtt < °3i > 3 qn tn  2 dO dO  f5iii(qOs+PSi) —flp’  jçr3r  +  (s)kf(r)if  JT 90 < +I3r —np —n e  > + <  f 133  2  U  (r)i(s)j  >  (r)i(s)jO)kt(t  (E.2)  k1) 5  >  ]cr:+  jr 3 jrs  qn  flj  (E.3)  118  Appendix E. Energy Shift for DVPPRT Vertex  Now, we have the formulae for contractions: <  s)i(r)j  <  >= 5(1 (e NeOr +  (s)if((r)Jt  eiP°sJcT,1 e1Ot1  (.iSi + N  >=  Nj? eZl2Ot2 t2riqSr + 12q  eul1Ot1N2eul2&t2N  (E.4)  (E.5)  +  If we define (T3T  pq  ” 8 u pq  =  SsrSpqeePos  =  0  (E.6) (E.7)  r, s = 1, 2  r or s = 3  The first two lines of the equation above combined with the <aa> and tions of the third line give the result: ij k 5 l  {[N + NUNUN + NUNUNUNUN  crci  >  contrac  det ( 8 1 + UQUQ) = (1 UNUN) 8 det  2 i—nnj  ••  <  —  2  =  1  Suio {  [1  —  UNUN]  _}  det ( 8 1 + UQUQ) (1 UNUN) 8 det —  The other two contractions of the last term produce 1 [N  UNUNUN]  —  det ( 8 1 + UQUQ) (1 8 det  1 UN] 1-UNUN  [  —  and ik j 5 l  1 [N -  N 1  UNUNUN]  [  1  -  det ( 8 1 + UQUQ) (1 UNUN) 8 det  UNUNUN]  -  Combined, these give <vie > P < ev >= 1 d det (1 + UQUQ) 8 (1 UNUN) 8 2det =  J_  33  33  k1 5 (öi  1 [N  —  1 [N  U1NUN] -nfl  —  —  UNUN] 33  33  +obo  1 [N -  IN 1 L  UNUNUN  -  1 UUN UN  UNUNUN]  —nfl  33  33  1 ö [N 5 +  J  [N nn[_  (E.8)  1— UNUNUN] )  119  Appendix E. Energy Shift for DVPPRT Vertex  This can be decomposed into a contribution to the singlet <vie> P < eiv >1(1,1)  T  =  1 + UQUQU) det ( d8a 8 (1 UNUN) 8 det  33  (  1 [N  -  -  [N 1  UNUN]  JN  —  1 [N —  UNUN] -nfl  IN 1—UNUN UN] ] L  UN]  -  IN  1 JNUNUN  inn  “  33  r 33  1 + [N  +  L  (E.9)  1—U NUNUJ_n,—nJ  the anti-symmetric state <vie > P <eiv  >1(6,1)  =  1 1 + UQUQ) (1 det ( 8 UN] 1 N 1— UNUN (1 UNUN) 2 8 2det  33  33  1 [N  L  —  —  ,33 _  [1  -  UNUNUN]Th  1 UN] [N L 1— UNUN  JNUNUNI  33  )  (E.10)  and the symmetric traceless state <vie > P <ely >1(91)  = ,  1 FN UNI 1— UNUN  (1 + UQUQ) (1 8 da det (1 UNUN) 2 8 2det  =  33  33  1 [N  L  —  —  33  +  [1  -  UNUNUNI  IN ,  [  1  —  UNUNUN]_nTh  1 UN] UNUN J  33  )  (E.11)  The two-string state <Q eIv> and the number 2 eIv> 2 e>< Q 2 <vIQ  E.1.2  Now, we must consider the contribution of the states of the form  <Q2eiv  >=<  n  Iv>  —n 00  k(3)  <  —fl  (  (3)  3/iit a 1 2 â 1 kf  k=-oo  = =  11 —  67llJi * ( ) 3  —  j(3)  <a3i_n  8(r) ai/1’ q712  iv> +(n 12  3) Qr ( 3  <aaiv>  +(n  “  )i  —n,i —n, i  >  ,‘  j) j)  (E.12)  120  Appendix E. Energy Shift for DVPPRT Vertex  where this expression is summed over p, q, r, s, j3. The minus sign comes from transposition of Q. Then:  QeIv >= (s) 5 1< vI > {ii(3r)(r)i 3 P  <vQe =  112a2  P  <  ‘  k3I P  >  (3f) (tjkf [u1*N_npap  ,()t  (3)  n  9 2 d 1 dO  —np  eo2o N r 3 nb  a)’/3 12 3 I  1)]  <&aIv >}  >< 3VMP>  (f)kf()f  <  ()f q31a2  s)t (r)lt/ ( 3  r)i,3(8)  >  nq  ijn  —nq  n  3s3  >  uq  12  8) (r)j, ( +uu1*]3nI3fei < 3 P P  3s  q/31c2  (r)kt (r)j(8) P ’ 1 qthc2  <  —fl  +u1u1*1/3? i  <  —n  flp  1  -*  i)].  ‘  f  3 ]r  —up  n, k  —  —n,i  <VNQ3  3 a 1  + (n  nq  ! + (n  _€1a1Ea22_.  • [J-i/ 1Ulth*N3r 3  (3s)  qi3ia  clf()f  q/31c2  3s  >  —nq  qthc2  —fl  3 c?’  1  (E13)  •..)  (E. 14)  q—ni  Fermionic contractions are: <  I  q/3i/32 > = 3 21 <  ai / a2/32 i6 3  a(r)f  (s)t  pa1a2q12  (ewes —  >  1fl122  %pq  1 e1 t 9  0 e’  —  Qpiie  tl t 2  1112  il29t t2riqer 6 2 12q  q 2 e2Qj 112 1Qj  +  +  (E.15)  Using those and the properties 2ö5  +  (E.16)  we get: >< <vjQ e 3 • (öJlöik  IV > Q e 3  1 [Nu -  4  I 3 Icr  —  1 [QQu  + QUQU  Q] 33  33  1 [Qu 1 [Qu  33  lö + j 6  1 + UQUQ) det ( 8 (1 UNUN) 8 clet 33  1 N] NUNU  1 N] 1 +o [NU 1- NUNU 1 +öulöik [Nu N] 1- NUNU  f  dO 2ir  r 1 Ni 1 JQQU [Nu 1-NUNU ]L +  + QUQU  QQ]  1 Q] + QUQU 33  ‘uQuQ]_,)  (E.17)  Remaining terms are: <ve> Q P 2  <  eIv 2 Q  >  P <ely> 2 + <vIQ e> Q 2  (E.18)  121  Appendix E. Energy Shift for DVPPRT Vertex  First of those is:  P 2 Q  <vje>  1  11çr33 L —nfl  > 3 <vI&  Iv >= Q e 2  <  +  qnj  q  00  m(t)  mai m/31a2 (t8tt  m=-oo (3)  •P  ()kt)f  + (n  ‘  3 t) 1 1 t  +  mi  —n, k  ‘  m/31a2)  (E.19)  v> 1)] 3 < (  12  Disregarding for now the trace component, we are left with: <vje>  P 2 Q  Hm  <  1  v >= 2 ejQ  1 dO dO 2  Vk3IIsI  —n’,fl —np JSqn JS’  l/ *f-3r i 3 Oi  <  j3) —)(3)  1/3i * j r 3  —np qn n r) 3 qn —np fs3f(  (p3)  js3j(3f)  T 3 _j2m(t)0a1N  (3)  <  VNQIa3 ><  moj  m1  3jVMp>  (r)kt()f  (T)t (t)i 5  m/31a2  r)i(s)j (t) 1f  <  —  r 3 _m(t)U1N 1 a  f (r)i(s)j  >  q/31c2  ()kf(.)f > mf31a2 q/31c2  (t)  (f)1t()t r)ia,s)i(t)fIB(t)t ma m/31c12 q/31a2  <  (r)i(s)ja(t)/1tB(t)  00 < _e  mc  ()1f()f  m/3lcr2P  q/31a2  >  >]  Taking traceless Wick Contractions: <vie> [_c2m(t)0i1*1d1J np  P 2 Q  <  1  Iv > Q e 2  js3jçr(3f) qn  ()r  (< r)ia(f)kf ><  i(9+9)  + < +<  p  Qm(t)Ua.’  (3)  N_np  qn  0 _e’  (< + <  ?flclj  +  <  )j(f)kf 3 (  ><  (t)f  ()t  m/31a2  (t)f  >  q/31c2  $)f  m/31a2  >  q/31c12  ()t  (t) m/31cl2  (s)jU)1t  (s)j(t)/it  md  > J  q/31a2  (t  m/3lc2  ><  ><  ><  s)j(t)fi >< (t)Imdl  ><  >< r)ia(t)flf mdl  )j(ijkf 8 (  ><  mcli  j <  ><  >< a3iVMP >  (t) ()t > >< >< a(8)ja(t)1t mcli i2 q/3ici  (r)j()1t  p  <V NQIa3  mai  ><  mal  r)i(f)kt  +i < 3r  2ir 27r  \/Ia3iiosI  >  q/31a2  ><  (t)t (f > m/31c12 q/31C2  ><  (t) ()t m131d12 q/3la2  ><  (t) 0 mJ3lo2  (t q131c12  >  >1 JJ  122  Appendix E. Energy Shift for DVPPRT Vertex  Which is equal to: 4  <vie> Q P < eiQ2v 2 33 [öilöik  1 ([Nu -  Vi3ilsi  >  I”’ d8a 2  1 + UQUQ) det ( 8 (1 UNUN) 8 det —  33  1 1 1 UQQ FNUNU 1-NUNUTh+UQUQ  N’UNUN]  33  33  1 -i [Nu -  NUNUN]  [1  -  UNUN1 + UQU Q  UQl  inn  33  1 + [Nu  -  FNUNU  N1UNUN]  -i [NU 1 -  -133  NUNUN]  1— NUNU1 +  -  33  33  jki1 5  1 ([Nu -  NUNU  FNUNU  N]  L  n-n  1 1 UQl NUNU 1+ UQUQ j  1  33  33  1 -i [Nu -  1 NUNUN]nn [N  -  33  1 + [Nu  -  1 FNUNU  N’UNUN]  —n—n  1 Qc UNUN1 + UQUQU i -n-nj  1 [N fin  UQUQUQII  -nfl  1 N] z[Nu 1- NUNU  -  1 UQl UNUN1 + UQUQ i-nfl  1 UQl NUNU1 + UQUQ 33  33  1 [N -  -.  UNUN 1+ UQUQ  33  \1  I  UQl i n_nj]  (E.20) Similarly: <viQ2e> 1  > 3 <via  L  Q2P  <  ev >=  P  ‘  00  •  m(t)  t (ta t 1 ‘ (t) 8 a(t)hult mfll2) m/31cz2 mai  m=-oo [33 .  /cl 6  +  > <a v 3  Appendix E. Energy Shift for DVPPRT Vertex  123  For traceless part this is:  e> 2 <vQ j/3i*j-3r  [c  m(t)  s3  qn  —np  al  Q2P <  ev  r d8 1 dO 2  1  16 cr2a2 ã 1 a 6  >  2r 2ir  HosI 3 /Ia  c r-•fJ 3 N( ) )N n q,n  <  r)i)kt  ><  ><  ><  ><  ><  ><  i< fr)i(f)kt  + < + < ?I9 3 +m(t)O1N i  np’q—n  —n5  7,n  ><  mai  (j <  ><  +i <  ><  + <  ><  + <  <VNQI3  ><  ()1f 1 (t).à (f)kf 1 (t)/  (t)f 0  3() qP1c2  m131Q2  (t) 5  (s)  mf31c2  q/31a2  ><  (t) 8 ,  q1a2  m/31c2  $s)  (t)f 8  q/31a2  3(3)  ><  ><  mcfl  (t)t 8  q131a2’ m/31cx2  3()  (t)  q/31a2  /3()  m13Jc2  (t)  qI31c2  m/31a2  <ely  >=  4  1 dOa 2  det ( 8 1 + UQUQ) (1 UNUN) 8 det  I J_ 8 a 3  —  33 [óiióik  (  1 [Nu —  NUNU  N]  ,33 1 UNI 1— UNUN inn  + UQUQ  ,  FQU QUl+UQUNlJNU N UN1  +[NulUNUNl  ]  i  33  1 -i [Nu  NUNUN]  -  1 + [Nu jk i 6 l  -  + UQUQ1  nn  FQUQU  NUNUN]  33  L  -  NUNU  UNUNUN]  N]  1 UN] 1— UNUN ]  + UQUQ  n-n  -133  1 + [Nu  -  1 Fu  UNUNI  L 33 1N] 1 _j[NU in—n  1 + [NU  -  NUNUN] nn  i —n—n  1 1 + QUQU QNiUUNUNl] —n—n/  33  (_ [Nu 1  —  + QuQul  -  33 -nn  33 1 UNI UNUN J  33 1 UN] [Q1+JQUQ1_UNUN in-n  FQUQU 1  1 QUQUN 1 +  -  UNUNUNI  I  i n_n) i (E.22)  Taking into account that the expressions in square brackets which are odd in  Q  are anti-  >  >  >)  m/31c2  Or: e> Q 2 <vQ P 2  >  q/31a2’ m/312  ><  mai  (t)f 3  a(s)  ,(s)  ><  5  >< a3IVMp>•  > > >  >.21) ,‘j  124  Appendix E. Energy Shift for DVPPRT Vertex  symetric in change of indices and that ones that are even in the two cross terms to get: <Q v>= >Q <ev>+<ve>Q e <vQ P P 2 e  Q  are symmetric we can add  (1 + UQUQ) 8 dOa det det8(1_UNUN)  8  /lc3IkS  —  33  33  [Silöik  1 ([Nu -  1 FQUQU  NUNUj  L  + UQU1  -  UNUNUN  33  1 + [Nu  -  1 FQUQU  NUNUN]  L  + UQUl  -  UNUNUN] -fl-fl)  -i33  1 jk i 5  1 ([Nu -  NUNUNI  L_  1 FQUQU L  + QUQU1  -  33  J  NUN  1 UN  J  33  33  1 + [Nu -  1 FQUQU  NUNUN  j fin  33  L  + QUQUNl  -  UNUNUN])] (E.23)  We can then combine all the above to get the full formula for traceless part of energy shift.  /J  4 3 a (Si1k  f  1 + UQUQ) det ( 8 (1 UNUN) 8 det  dOa 2ir  —  33  1 [Nu  1 [EONU  NUNUN]  -  -  QQU NUNUN + 1 + UQU 33  1 +QQUQU  + QUQU  1 N -  UNUN  UN]  33  +öulöik [Nu 1  -  1 [EONU  NUNUN]  -  QU NUNUN + 1 + QUQU 33  1 +QUQU  + QUQUNl  -  U’NUNUN]  33  +öikSut [Nu 1 -  N’uNuN]  1 [EONU -  -nfl  QQU NUNUN + 1 + QUQU 33  1 +QQUQU  + QUQU  1 N -  UNUN  UN]  33  jk 6 + 1  1 [NU -  NUNU  N]  1 [EONU -  NUNU  N + QU 1  + QUQU 33  1 +QUQU  + QUQUNl  -  UNUNUN]  )  (E.24)  125  Appendix E. Energy Shift for DVPPRT Vertex  E.2  4-impurity calculation  At the 4-impurity level we have to take into account the determinants. Two posibilities for the 4-impurity contributions are one where two factors of U come from determinant expansion and two from the rest of the formula and the one where all four come from nondeterminant part of the formula. To second order in U the determinant factor can be written as: det ( 8 1 + UQUQ) =1+ 8[Tr(UQUQ) + Tr(UNUN)] + (1 UNUN) 8 det  ...  (E.25)  -  will be of the form:  so 4the order in U contribution of  [d& Lj 2ir 3 fr. /1, 33 [NUN + QUQ] + ...) 8[Tr(UQUQ) + Tr(UNUN)] (öiköil[NUN] [NUNUNUN])[EoNUN + QUQ1] + ...) 1 +(6ö +[NUN][EoNUNUNUN + 2QUQUQUQ1 + QUQU2NUN] + ...) 4  .  (E.26)  where once again, integration over 0 is substituted for by direct level-matching condition that all indices on Us in in any given product add up to 0. To simplify the above lets make the following substitutions: —  sin(nIirr)(w+/3rw,)  1  2is/  r 3 P/  1  —  rfl 3 P/  T 3 A  (E27)  flJJ  1  =  (E.28)  Wsn1 2 47/Wsn1W2(,3r  +  riWt’ 3 / ) 2  then T_ 3 N  ‘E2  —i4  =  2 NW  = /2  (E.30)  3 + /Lari ,/Wt + IIar (w 2 + e(st)J Iiri 2 !3rit)Bsrthl  =  —  —  )BW [Lr 2  (E.31) (E.32)  In cases where both s and t are much smaller then u this can be written more compactly as: —  —sin(Inhirr)/  1  (E 33)  7t  1  (E.34)  (/3ri/3r 8ir(iu) ) 2 7j3r_  r 3 I A _e\n) =  —iA,  (E.36)  126  Appendix E. Eneggy Shift for DVPPRT Vertex  Trr1r2 .LVSt  C)  — —  a a Lst I.La3IJri,Jr . 2  12r1r2  flT1T2 ‘st  •f,q  —  —  4KUO V’! ri /r2 3  a  8 I-’r2  —  —  —  #pr1f2  We can then generalize the cancelation from 2-impurity calculation by noting: UN + QUQ1] [E N 0 UN + QUQ] [E N 0  =  UN + QQUQ] [E N 0  =  0 (—E  UN + QQUQc2] [E N 0  =  0 (—E  0 (E  =  0 (E  —  —  f1J[AUA]  [AUA]  —  —  2n[AUA]  (E.39)  —2n[AUA]  (E.40)  =  =  =  —2(w  =  —2(w  —  —  a)[AUA]  (E.41)  ,ua)[AUA]  (E.42)  In all cases (to all impurities), the above are multiplied by a term proportional to [AUAImm so the sum can be written as: -n [AUA] [AU.. .A]) +  2n( [AUA] [AU.. .Aj  -  --  3 [AU. .A]+ [AUA] [AU. .A]) ([AUA]  (E.43) It is obvious that the parts represented by 3 dots can be moved from one square bracket to the other so the first two terms cancel out and the second two can be writen compactly: (E.44) rather then NUN + QUQ2] will be of order 0 Therefore all the terms proportional to [E of order uc In 2-impurity case this brought the leading term from being of order 1 to being In 4-impurity, as well as all higher impurity cases, this makes all terms of order \ = NUN + QUQQ] term subleading. 0 containing [E To see this and to identify the leading order behavior we need to look into the ua dependance of the individual terms. We also have to take into account the indices that are To see that ua terms for any being summed over that could in principle blow up to rival 1 at 4-impurities, all indices are controled and kept finite by the poles we can look at a generic term:  .  [AUA]I [AUBUBUAjmi A;1 p  q  + q + s + t =0=  t  S  1 =  p  q  t  1  1  1  tr lqr 4 f 1 prjflpr m 2 m 3r2Br2ra .A3rlArl A 3 np  pn’ mq  q(—p—q—t)  3 tm’ (—p—q—t)t A”  Br3T4  E 45  Clearly the poles will controll the sum and the term will be of the form: [AUA], [AUBUBUA]mi  =  A;’ A;A B pqt) Bq_t)tA  Where p,q and t are constants set by the values of the poles.  (E.46)  127  Appendix E. Energy Shift for DVPPRT Vertex  With all the sums thus controlled only factors of A in the large i limit will come from . Expanding w in large ,u limit we have: 2 factors of c and w = n2 + (au) 1  (E.47)  We can then count the highest possible order (baring any cancelations) in A coming from any term in the expansion so as to identify those terms which may contribute at order A. 2 which are its multiples. B’ is clearly of order 1 and so are therefore N and 2 has the and so is Q is of the order 2 (as long as the sums are controled) but the N 0 is of order ci and 2 Furthermore, E 2 term which makes it overall of order (/w + c) NUN + QUQc2] will be of the order 0 Lastly, as we have seen above [E is of order Therefore: The highest order possible contribution from the various terms will be .  UN + QQUQQ] +...) NUN][E 0 [ 1 (S6 —-8[Tr(UQUQ)+Tr(UNUN)] N  O()3) (E.48)  ‘(ikSi1{NUNUNUN1)33[E NUN + QUQ] + <  O()  (E.49)  O()  (E.50)  ...)  O()  (E.51)  ...)  O()  (E.52)  +  ...)  ±[NUN][2QUQUc1NUN] + ±[NUN133[EONUNUNUN1 +  ±[NUN][QUQUQUQJ  ...)  Note how here fact that the internal N matrices are order in oqi higher then internal Q ma trices actually plays a pivotal role making it impossible for the two-impurity style cancelation to happen. Therefore, to find the order A contribution at the four impurity level we just have to calculate:  I  uNuNUN]+ ([NuN][E N 0 NUNUNUN] + 0 +[NUN] [E NUNUNUN]+ 0 +[NUN][E +[NUNj[EoNUNUNUN])  (E.53)  128  Appendix E. Energy Shift for DVPPRT Vertex  This can be written as: 4 SE  /2V/q’ +  4(2w)  ç—  /LL 3 a  L.d  sjn ) 2 [Lar ( n1rr)(w4 i’ +2 W 4 /3r ( n) + /3rin)(W 3 + /3r n) 3 + fr3(’3 + 4 2 ’W ww 6 256i w 4 ) (/3r3 2 2 (I3rW’ + /3rj w 3 + 13r2) 1 1 1 1 1 + nt—1 1 q—I n r nt+/ 1 q+/ n fl’ 3 4 P+/3r fl’ 3 4 PI3r 1 11 1 1 1 1 1 1 p+np—nq+flt—fl  /tri \/-‘t  In the case when all the summands are bound by poles and/or level matching we can take a simple large t limit: 1  SE 4 1L  r4 3 V’1 1 4in7rr —s’in 2 • .  —  r 6 / 3 2(cr 7 2 .L)  pqt /9r2  1 1 1 1 •_1 +_1 t/3r n’q/3r 4 p+/r n 1 n l 3 lt+/ flq+/ 4 p/ f 1 f r 3 1 1 1 1 1 1 1 1 p+flp—flq+flt—fl p+flp—flq—flt+fl  (E.55)  With level matching condition to be imposed by hand. When all the impurities are on the same internal string then (E.56) = / r3 3 r4 = r 8 , = and the level matching condition is (E.57) Furthermore, at the poles, = (±rn)2  +  Wp =  2 = rw (ra)  =  Wt  (E.58)  Using contour integral method we can then write it in large ii limit as as: (nr) 4 sin  4 (r) SE  2 (nr) [a 2 [cot(p)]prcot (—nr) + ã [cot(p)]p_nrcot  2 ()2 +  rncot2(_nh)c0t(nh)  + -_cot2(n1rr)cot(_n7rr)](E.59)  or, even more simply: (r) 41 5E  1 -2  2 3 /L1t 3 Q  I_I  Taking the large 1 , 4 SE  — —  i  —cos 2 (nirr)ir  —  1 1 —sin(nirr)cos (nirr) Tn J  (E.60)  limit and integrating over r we get:  g2 2  1  dr  r(1  —  2  r) 5E ,i(r) 4  —  —  1 2&3I22  (—1  +  3 n 2 64  ( E 61 .  129  Appendix E. Energy Shift for DVPPRT Vertex  We also have to consider the cases where the impurities split between two strings. There are three such combination one being: r1 = 3 /  =  r2 3 1  r; I ra 3  T4 3 1  =  =  (E.62)  r’  where r+r’=l  (E.63)  s=—q;t=—p  (E.64)  In that case, level matching gives us:  In the same way then: rw; Wt  =  =  =  (E.65)  r’w  Using this and taking large u limit we get: 4 (nrrr) r’ sin 2(&) r 6 it 1  r) 5E ( 2 , 4  (p  —  —  (p  —  1 (p + r’n)(q 2 r’n)  1 (q rn) 3 (p + r’n) 1 (p r’n)(q + rn) 2 (p + r’n)  1 (q + rn) 3 r’n) —  rn)  —  —  —  (E.66)  —  —  Integrating over p and q and being carefull of the signs of r we get: rI r  f £1’ u.c rj 42  1 i  —  -  ii-) 2 t 3 (a  —2cos 2 (nirr) +  sn(n-irr)cos(nirr)  2r’nir  (E.67)  with the single-pole contributions canceling out the contributions from Finaly integrating over r: 2 , 4 SE I’  2 f’ —g 21 —  7’  r(1  —  r)  r) 5E ( 2 , 4  2  —  (2  —1  —  2221+ [L \12 it  k3  27 128ir 22 n  (.68)  remaining two combinations are: /ri  =  /r4,/3r2  =  r3  (E.69)  /3r4  (E.70)  and =13r3;/3r2  First of those is identical to the one evaluated above and the second gives: 43 SE /1  1  21  1 ii2 ) 3 2(c 6  .  4,  -  1  1  pq  21  1  1 1 1 1 1 1 1 2 + p+r’np—r’n ( q+rn ) + p+r’np—r’n ( q—rn )i j  (E.71)  130  Appendix E. Energy Shift for DVPPRT Vertex  1 _sin(n7rr)cos(nlrr) (E.72) sin(nirr)cos(n7rr)] + 2rnir 2r’n7r 7 8 •\ 1 ,s(r) 4 r(1—r) 6E 43 =g2f 1 E (E.73) dr 23I22 642n2) 2 As in the two impurity case we still have to account for 0 modes, specifically for the case where one 0 mode exists on one string and 3 remaining ones are left on the other. Consider first case where: (E.74) T !3T4 = 1 , I3T = 13r2 = /3T3 = r 1 p) 3 ( 2  (r) 3 , 4 6E  [  —  s=—q—t ;  (E.75)  p=O  It should be noted that for some of the poles =  (±2rn)2 + (r[Ic) 2  (E.76)  rw,  However, as long as the both t and q are limited by the poles =  will still hold in the large  t  rw  =  (E.77)  r/ta  limit so we can continue to use it. We can then write: 1 ir 2 u) 34 2(ä 6  (r) , 4 6E  r(1  —  2 r)n  1 1 1 1 1 1 1 [1 —0 + + + q—rnt+rn] — q+rnt +rn q—rnt—rn q+rnt—rn  (E.78)  Second 0-mode case is one with: I3ri  =  !3r4  p  =  =  —q  —  r  t ;  13r2 s  =  1  =  —  (E.79)  r  (E.80)  0  In this case we have:  [  r  1 j)ir 3 2(ä 6  (r) 5 , 4 öE  1  —  r  sin  (Inrr)  1 1 1 1 21 21 q+t+rn q+rnt+rnq+t—rn q—rnt—rn 1 1 1 1 + + q+t+rnq+t—rnq+rnt—rn 11 1 1 1 +q+t+Tnq+t_ rnq— rnt+rnj  (E.81)  We can simplify:  [  1  2  1  r  1 ir 2 2(&) 6  (r) 5 , 4 6E 1  1 1  —  2  r  sin ((n(irr) 1  1  • ( q+t+rn q+rnt+rn +( q+t—rn q—rnt—rn +  1 1 q+t+rnq+t—rn  (  1 1 1 1 + q—rnt+rn q+rnt—rn  )1]  (E.82)  131  Appendix E. Energy Shift for DVPPRT Vertex  then using partial fractions: 1  r  2(t)26  1 —r  r) 6E ( 5 , 4  sin Gnhirr)  ‘\ 1 1 1 çq+rn — q+t+rn) q+t+rnt+rn 1 1 1 1/1 +-( t q—rn q+t—rn ) q+t—rnt—rn 1 1 1 1 1 (q+rnt_rn+q_rnt+rn)j +t+rn)  Fl/i  L  —  1. 17 2rn q+t—rn  +—(  r  r) E ( 5 , 4 11/ 1 q+t+rn  •[ +  1—r  ) 26 t 3 2(  IL  (E.83)  (Inrr)  sin  1 1 1 1 ‘\ )2i q+rn) t+rn — q+t+rn t+rn  — rn 1 (_ — rn) t—  q+ti_rn)2t_lrn  1 1 1 )(i 1 1 1 + q+rnt—rn q+t—rn — q+t+rn q_rnt+rn)]  84)  It can already be seen that with some shifting of variables like q’ q + t and such large parts of the above dissapear. Only thing that is left is: (nIr) 4 r__sin (r) 45 öE V’ L.a’ ) qt 1 r u 3 2(& i 2 ,  •  —  The pole at t  =  _lr 1_)2_i [(q q — rn t — rn] T + rn t + rn +(_1_)2_1 0 cancels out and we are left with 1__sin(nr)cos(nr) —l (r) 5 , 4 6E n1 r 3 u)ir 31 (&  (E.85)  —  (E.86)  öE 5 , 4 I-I  =  g  /  1  dr  Jo  r(l  —  r)  2  /  1  r) 6E ( 5 , 4 =  L 1&31 K I 2  8  642n2)  (E.87)  Finaly there is two cases where q Or t is equal to zero and is on the separate string. There: 2(&)26  IL 2  [p—rn q+rn(i—r)n +rnp —rn q +rn (1 —r)n  +  +(  JnJr) e(q)sin ( 4  1 1 )2 1 p+rn q—rn(1—r)n  p +rnp —rn q —rn (1 —r)n]  (E.88)  132  Appendix E. Energy Shift for DVPPRT Vertex  r) tSE ( 6 , 4  1  1  ) (1 r 5 3 2(ã i 2  L  —  p  1  1 1 1 1 1 )2 )2 + + + p+rnp—rn p+rnp—rnj p—rn p+rn 1  1  r) SE ( 6 , 4  r (1 5 i 2 ) 3 2(& 1  L  (nhirr)cos(Inirr) 3 sin  r)n  )2  p—rn  p+rn  (1  —  )2  +  (nIirr)cos(Inirr) 3 sin  r)n  p  1  1 1 1 1 + p+rnp—rn p+rnp—rn]  1  1  1 (&3)23  1  +  —  r)n  6 , 4 SE  (E.89)  2  (E.90)  2 (InIr)cos2(InIr)] jsin(jnr)cos(InIr) + —sin nirr  (E.91)  L  =g2 2  dr  r(1  —  r)  r) 5E ( 6 , 4  (E.92)  2  We can then put it all together: 4 SE  E 5E ,l+ 5 3 + 6 2 + 5 4 6 , 2 SE  4 SE  2  /5  -2I22  (\  / 1  1  =  & 3 l j 2 +  57  \  641r2n2)  64 \ + 642n2) 93) (E.94)  Appendix F. Hamiltonian and the supercharges of the orbifolded string  133  Appendix F Hamiltonian and the supercharges of the orbifolded string F.1  Hamiltonian of the orbifolded string  The action of the pp-wave string is derived using the formalism of Cartan forms [? ]. This formalism uses vielbeins and spin connections associated with the metric 3.9. As the actual form of the metric remains unchanged by the orbifolding, the entire mechanism discussed in [? ] applies for as long as no boundary conditions on the scalar and fermionic fields are used. As we are interested in the near pp-wave limit, we are in fact following formalism of are concerned, however, there the [? 1. As far as the effects of orbifolding to the order is no difference between the results derived from [? I and [? I. In both cases we derive the same Hamiltonian density: H =  f  d  [(xi)2  (pi) + + 2  (xu’)2]  +  +  + 2ifl]  (F.1)  Where the prime indicates u derivative and the addition over all possible fermionic indices is assumed. Furthermore, the bosonic and fermionic fields are governed by the usual equations of motion: (F.2) 0 —ii”+x (P.3) i(+’çb’)+fl=0 However, the periodicity conditions of non orbifolded theory which demand that the xs and bs are periodic over the a coordinate are in our case replaced by the equations 3.21, 3.22 and 3.23. (Or 3.21, 3.24 and 3.25 if we are dealing with a case of two transverse directions being orbifolded). Before we proceed with solving the equations and integrating the Hamiltonian it is useful to note that for any x(a, ‘r) and (a, r) that solve equations F.2 and F.3 with standard periodicity conditions we can construct the (a, T) = e’x(o, T) and ‘zb(o, r) = e(o, r) where the phase governs the periodicity of the boundary conditions. This lets us use the known solutions of the F.2 and F.3 and simply making sure to insert phase where appropriate. the eAI or  F.1.1  Bosonic interaction Hamiltonian  It is obvious that the only part of the Hamiltonian that will be affected by the orbifolding will be a derivatives of the functions that have acquired the phase from the new periodicity condition. We can then write the interaction part of the bosonic Hamiltonian: =  fda {0( 7 )8( 8 +i  —  ) 8 i  —  77 8(x )O(x 8 +ix  —  ] ix ) 8  (P.4)  Appendix F. Hamiltonian and the supercharges of the orbifolded string  134  which, following the equation 3.21 can be re-written as:  =  f  dcr  [8[e(x7  + ix8)]8 [eW(x  f  ] ix ) 8  —  7+7 8(x )8,(x 8 ix  —  ix8)]  (F.5)  is equal to:  Which to the first order in =  —  7 2 dcr  + ix)(x’  [(x7  or: =  f  —  (x” +  ) 8t ix  da [x7x81  —  ix8)]  (F.7)  x7’x8]  —  (F.6)  We can now use well known solutions of the F.2: x(u, T)  =  x(r)  *(ae_T  —  teiwnT)  (P.8)  Substituting solutions P.8 into the F.7 we get:  =  f  Z-  —  (F.9)  nxsme_im 7 m)x  n,m  =  2iriw  i(n  da  4iriw  [(7atneT  —  (F.1O)  aae_iL1T)  , (a a +7  —  at8cr7)]  (F.11)  After quantization, a are promoted to creation operators and the a into annihilation oper ators The part carrying the T phase then mixes the states of various number of impurities, second part however acts on the single impurity multiplet (as well as any other multiplet of n impurities) and breaks the energy degeneracy of that multiplet. For the multiplet  aIO >;i we diagonalize  =  (F.12)  1..8  with the new set of eigenstates: aO >;i  =  O ic4 8 + 7 1..6,(a )  ic4 IO> 8 — 7 >,(a )  The first 6 clearly remain degenerate with energy, E splitting of order — —  Wfl  2irwn  Mw’  —  —  =  i  w  2irwn Mw  (F.13)  whereas latter two get energy F 14  In the case of orbifolding along the two transverse directions, an additional term would be of the same form but with the oscillators in directions 5 and 6 replacing added to the the 7 and 8 ones respectively. In this case, each new energy level would get two states rather then one.  135  Appendix F. Hamiltonian and the supercharges of the orbifolded string  F.1.2  Fermionic interaction Hamiltonian  A very similar argument holds for fermions. Fermionic part of the interaction Hamiltonian can be written as: t 1 [a112a +t12a t +122Og12 +tD (F.15 int 11  —  —  fd —  ‘a 12  fdu’  [cr112  ‘  _a122  t +tah1 , 2  _ta122t c122j  2 1 à  112  (F.16)  Solutions of the F.3 are: 00 =  (F.17)  12(r)e2fl fl= —00  (T) = //n1/ 2 3  1 A 13 1 2e_twflT + 2/ V’n/n  Bn12e1T)  00 (g, /jf1I 2 3  T) = fl=—00  =  A  =  1 (HBn12e_?2T 2/  (i/n— \/w+nH), B  Substituting back into the F.15 1 —71W F(AThA_ + int 11 4M  =  —  HAn  2eiWnT) 3 f  (/w+n+/w—nH)  2 BnB_n)(/3hl2/3_naii  —  ,t +(AA_ + BB_)(/3hl2 —n ‘flc112  +2(AnB_n H1 —  flall2  71W =  —  f 5 atc122  2 1 —nà  ‘fl  flC112  Qcz122  2 1 l-’—Th’1  fl  —  l /f 1  —  l / f 3  (F.18)  )e 1  t122  BA_)(3h’2!3  [2(ih12a  4M  /fla122)j  e_ IT j 2 3_na )  (F.19)  n  t 213 tãi 13 fl  —  —2iwT  aj22  —  T 2 e  —  8n(  u12/3n12  )] tc122i 3 .  1 —na122  As with the bosonic case, there are two -r dependent parts that mix states of various number of impurities. For the energy splitting in the single impurity multiplet Hamiltonian is already diagonalized and we can see that 4 states break degeneracy into a pair of two-fold degenerate states: 5 E  6 E  —  =  2,rwn  IVIC*)n  7 E  =  8 E  =  wj,-, +  Mw  (F.20’)  8 to states 7 and E 6 correspond to states th122 0> and /3t2122 10> and E E and E Where 5 t 3 , 2112IO> ,t3t111210> and In the case of two transverse directions the fermionic result is very similar except that the index structure on the fermionic oscillators matches that of the boundary conditions that is to say all indices are dotted. -  Appendix F. Hamiltonian and the supercharges of the orbifolded string  F.2  136  Supersymmetries of the orbifolded string in near pp-wave limit  We can now examine the supersymmetries of the pp-wave and see which are broken by the orbifolding. Supercharges can be written as: 00  (s) 2 à 1 Q2a  ) (it 8 1  ak(s)  kai  12 k/3  +  (s)t c’ ie(& ) 5  lcczi  k=-oo  =  (s)/32[  ko2  (  00  (s)  ç k(s)  (s)13j °ka  (s)  ai/3 2 k 3 I  (s) k/31cr2 3 /  —  +  (5)132 (s)f ‘\ e(aS)ak 2 2 13 1 k 3 I )  (F.21)  (s)j3i (s)f ie(ã5)ckci k/31a2 13  k—oo —  (s)f2  —e(c) aka 2  t/3(S) ia(8)/ 2 3  c1i32  (s)f ‘\ ki13 ) 2  (F.22)  where we define =  —  —  e(k)/w +  ,  e(x)  x/Ix  =  (F.23)  Obviously, in the strict pp-wave limit H 1 disappears and thus all supersymmetries are we have to consider which of the above commute with preserved. To the first order in Ignoring the terms that carry the r phase we can write: 2iriw M  fl  w  Using the (s)  I [an  —  ’ 4 a n n13  1  i(s)  —  fRt’1l2  it-’n  (s) an c 2  “  —  2 Pnc,,1  3 ta122 13  n  ncsl22)]  (F.24)  1 —  a2a2  fl  It is relatively easy to show that ] 2 Qaij  ] 2 Qai  =  0, 0,  [H, 1 Qa 2 , Qaii 1 {H l 2  =  0 0  And that therefore exactly half the supersymmetries survive.  (P.25)  

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