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Applications of path integral localization to gauge and string theories Gordon, James B. 2018

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Applications of Path Integral Localization to Gauge andString TheoriesbyJames B. GordonB. Sc. (Hons), University of Cape Town, 2008M. Sc., Uppsala University, 2011A THESIS SUBMITTED IN PARTIAL FULFILLMENTOF THE REQUIREMENTS FOR THE DEGREE OFDoctor of PhilosophyinTHE FACULTY OF GRADUATE AND POSTDOCTORALSTUDIES(Physics)The University of British Columbia(Vancouver)March 2018© James B. Gordon, 2018AbstractIn the first part of this thesis we exploit supersymmetric localization to study as-pects of supersymmetric gauge theories relevant to holography.In chapter 2 we study the 12 -BPS circular Wilson loop in the totally antisym-metric representation of the gauge group inN = 4 supersymmetric Yang-Mills. Wecompute the first 1/N correction at leading order in ’t Hooft coupling by means ofthe matrix model loop equations for comparison with the 1-loop effective actionof the holographically dual D5-brane. Our result suggests the need to account forgravitational backreaction on the string theory side.In chapter 3 we solve the planar N = 2∗ super-Yang-Mills theory at large ’tHooft coupling again using localization on S4. The solution permits detailed in-vestigation of the resonance phenomena responsible for quantum phase transitionsin infinite volume, and leads to quantitative predictions for the semiclassical stringdual of the N = 2∗ theory.The second part of the thesis deals with the Schwinger effect in scalar quan-tum electrodynamics and in bosonic string theory. Chapter 4 presents a detailedstudy of the semiclassical expansion of the world line path integral for a chargedrelativistic particle in a constant external electric field. It is demonstrated that theSchwinger formula for charged particle pair production is reproduced exactly bythe semiclassical expansion around classical instanton solutions when the leadingorder of fluctuations is taken into account. By a localization argument we provethat all corrections to this leading approximation vanish and that the WKB approx-imation to the world line path integral is exact.Finally, in chapter 5 we analyse the problem of charged string pair creation ina constant external electric field. We find the instantons in the worldsheet sigmaiimodel which are responsible for the tunneling events, and evaluate the sigma modelpartition function in the multi-instanton sector in the WKB approximation. We fur-ther identify a fermionic symmetry associated with collective coordinates, whichwe use to localize the worldsheet functional integral onto its WKB limit, provingthat our result is exact.iiiLay SummaryThe physics of fundamental particles is described by quantum field theory (QFT).QFT calculations relevant to experiment generally become challenging wheneverparticle interactions are strong. We apply supersymmetric localization, a power-ful technique for strong-coupling calculations, to several problems in QFT andholography.The latter is a conjectured mathematical equivalence of certain QFT’s to higher-dimensional string theories. If true, it constitutes another invaluable tool for strong-coupling; but testing its validity and scope requires an independent source of strong-coupling results, which localization provides.We first study two unrealistic yet mathematically important QFT’s. We uselocalization to derive predictions for their holographically dual string theories,allowing for precision testing of holography in interesting parameter regimes.We then investigate the Schwinger effect – spontaneous particle production inbackground electric fields – in QFT and string theory. We compute the particleproduction rate and using localization prove that our results are exact.ivPrefaceChapter 1 is the sole work of the author, as is chapter 2. A version of the latterappears on the physics ArXiv as 1708.05778 and was recently accepted for publi-cation in Journal of High Energy Physics (JHEP).A version of chapter 3 was published as “N = 2∗ super-Yang-Mills theory atstrong coupling”, JHEP 1411, 057 (2014). It was a collaboration between theauthor, Konstantin Zarembo and Xinyi Chen-Lin.Chapter 4 is a version of J.Math.Phys. 56 (2015) 022111, a collaboration be-tween the author and Gordon W. Semenoff.A version of chapter 5 appears on the ArXiv as 1710.0331, and was recentlysubmitted to JHEP for publication. It was a collaboration between the author andGordon W. Semenoff.vTable of ContentsAbstract . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . iiLay Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ivPreface . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . vTable of Contents . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . viList of Figures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ixGlossary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xAcknowledgments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xiDedication . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xii1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.1 Introductory discussion and motivation . . . . . . . . . . . . . . . . 11.2 Supersymmetric gauge theory . . . . . . . . . . . . . . . . . . . . . . 41.3 Wilson loops . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81.4 Gauge/gravity duality . . . . . . . . . . . . . . . . . . . . . . . . . . 91.5 Supersymmetric localization . . . . . . . . . . . . . . . . . . . . . . . 121.6 The Schwinger effect . . . . . . . . . . . . . . . . . . . . . . . . . . . 151.7 Outline of thesis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17vi2 Antisymmetric Wilson Loop inN = 4 Super Yang-Mills . . . . . . . . 192.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 192.2 Antisymmetric circular Wilson loop . . . . . . . . . . . . . . . . . . 222.2.1 Planar approximation . . . . . . . . . . . . . . . . . . . . . . 242.3 1/N expansion from the loop equations . . . . . . . . . . . . . . . . 272.3.1 Solution of the loop equations . . . . . . . . . . . . . . . . . 292.3.2 Free energy I: F as generator of ⟨( trM2)l⟩ . . . . . . . . . . 312.3.3 Free energy II: ρn from Wn(p) . . . . . . . . . . . . . . . . . 332.3.4 Modification for SU(N) . . . . . . . . . . . . . . . . . . . . 362.4 Numerical checks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 372.5 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 393 N = 2∗ Super Yang-Mills at Strong Coupling . . . . . . . . . . . . . . . 433.1 Critical behavior of N = 2∗ SYM and predictions for holography . . 433.2 Planar limit of the matrix model . . . . . . . . . . . . . . . . . . . . . 463.3 Solution at strong coupling . . . . . . . . . . . . . . . . . . . . . . . 483.3.1 Wiener-Hopf problem . . . . . . . . . . . . . . . . . . . . . . 483.3.2 Exact solution . . . . . . . . . . . . . . . . . . . . . . . . . . 503.3.3 General structure of solution . . . . . . . . . . . . . . . . . . 553.4 Decompactification limit . . . . . . . . . . . . . . . . . . . . . . . . . 573.4.1 Small ξ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 583.4.2 Oscillatory behavior . . . . . . . . . . . . . . . . . . . . . . . 603.5 Strong-coupling expansion . . . . . . . . . . . . . . . . . . . . . . . 663.6 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 694 The Schwinger effect from worldline instantons . . . . . . . . . . . . . 714.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 714.1.1 Worldline path integrals . . . . . . . . . . . . . . . . . . . . . 764.2 Effective action from semiclassical worldline path integral . . . . . 814.3 No more corrections . . . . . . . . . . . . . . . . . . . . . . . . . . . 864.3.1 Another proof . . . . . . . . . . . . . . . . . . . . . . . . . . 904.4 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 94vii5 String pair production in a background field from worldsheet in-stantons . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 965.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 965.2 Semiclassical evaluation of the cylinder amplitude . . . . . . . . . . 1015.2.1 Worldsheet instantons . . . . . . . . . . . . . . . . . . . . . . 1025.2.2 Fluctuations about the instanton . . . . . . . . . . . . . . . . 1055.2.3 Determinants à la Gelfand-Yaglom from contour integration 1125.2.4 Result . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1205.3 Exactness of semiclassical approximation: Proof by localization . . 1225.4 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1276 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 130Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 132A Exact expression for g(ξ) . . . . . . . . . . . . . . . . . . . . . . . . . . . 143B Anomalous contribution to B . . . . . . . . . . . . . . . . . . . . . . . . . 145C Large M limit of scaling function . . . . . . . . . . . . . . . . . . . . . . 146D Infinite products and special functions . . . . . . . . . . . . . . . . . . . 150D.1 Infinite products and zeta-function regularization . . . . . . . . . . . 150D.2 The Dedekind eta and Jacobi theta functions . . . . . . . . . . . . . 152E Proof without scaling: order-by-order cancellations . . . . . . . . . . . 154F A toy model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 156G Gaussian integral computation . . . . . . . . . . . . . . . . . . . . . . . 161H Fluctuation prefactor from explicit mode expansion . . . . . . . . . . . 167viiiList of FiguresFigure 2.1 Strong coupling expansion of (logW)planar versus (logW)exact 26Figure 2.2 1/N correction to the eigenvalue density . . . . . . . . . . . . . 36Figure 2.3 Numerical versus analytic results for F1, the non-planar freeenergy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38Figure 2.4 Linear dependence of F1(k) on λ . . . . . . . . . . . . . . . . . 40Figure 3.1 Phase diagram of N = 2∗ SYM . . . . . . . . . . . . . . . . . . . 45Figure 3.2 Singularities of the inverse kernel, Sˆ−1(ω) . . . . . . . . . . . . 52Figure 3.3 Scaling function g(ξ) for mass M = 0.5. . . . . . . . . . . . . . 55Figure 3.4 g(ξ) evaluated numerically for M = 10 . . . . . . . . . . . . . . 58Figure 3.5 Scaling function g(ξ) and endpoint of eigenvalue density, show-ing resonances . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62Figure 3.6 Cusp-like structure of g(ξ) in the regime ξ ∼O(M) for largeM, compared with numerics. . . . . . . . . . . . . . . . . . . . . 63Figure 3.7 Numerical endpoint behavior of the eigenvalue density, for M =10 and M = 100, compared to leading order semi-circle . . . . . 65Figure 3.8 Fitting the curve µ(M) to the numerical data . . . . . . . . . . . 67Figure 3.9 Density close to the endpoint, showing matching condition(M = 0.5) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68Figure 4.1 Feynman diagram for the vacuum energy of a scalar field . . . . 76Figure 5.1 The worldsheet instanton . . . . . . . . . . . . . . . . . . . . . . 103ixGlossaryCFT conformal field theoryQED Quantum ElectrodynamicsQCD Quantum ChromodynamicsQFT Quantum Field TheorySYM Supersymmetric Yang-MillsADS Anti-de SitterVEV vacuum expectation valuexAcknowledgmentsI would like to thank my research supervisor Gordon Semenoff, whose wealth ofexpertise and enthusiasm for the subject were always a source of inspiration. I verymuch appreciated our discussions and our work together.Thanks also go to Kostya Zarembo for his expert and generous supervisionduring my time at Nordita, and to Xinyi Chen-Lin, my collaborator and officematethere. It was an intellectually enriching and thoroughly enjoyable period. I ac-knowledge the EU GATIS network which provided the funding for this fellowship.To my wonderful parents, Anne and Roy, I am profoundly grateful for yourlove, encouragement and support over many years. In short, thank you for makingthis possible.My dearest wife Katja, your love and companionship is an immense source ofstrength. Thank you for your support and not least forbearance over the last fewyears.And finally to our darling baby girl Sofia, whose arrival in the midst of this PhDmarked a definitive phase transition in our lives: you remind us of truths beyondfundamental physics!xiTo my beloved girls, Katja and Sofia.xiiChapter 1Introduction1.1 Introductory discussion and motivationIt is a truth universally acknowledged, that a high energy theorist in possessionof intellect and ambition, must be in want of a quantum theory of gravity.1 Lesswidely appreciated, however, is the intimate relation of this project to that of reveal-ing the intricate structure of quantum gauge theory, and the formidable difficultiesrelated thereto.Our understanding of physics at the most fundamental level is formulated inthe language of gauge theory, as encapsulated in the Standard Model of particlephysics. The seemingly modest goal of elucidating the behavior and spectra ofordinary nucleonic matter depends on a quantitative understanding of the strongly-coupled dynamics of non-abelian gauge theory, and especially of confinement.2To this day, an analytic derivation of confinement in Quantum Chromodynamics(QCD) has not been found.However, even the more esoteric motivations alluded to above now animatethe study of strongly-coupled gauge theory. It is understood, thanks to the pro-found discovery of gauge-gravity duality, that gauge theory is an integral part of1Jane Austen[1], paraphrased.2 Incidentally, let us remind the reader that the popular claim that Peter Higgs’ (or God’s, if youprefer) eponymous particle provides an explanation for the mass of all matter is misleading; a paltry1% of nucleonic mass is directly attributable to the Higgs. More important by far is the phenomenonof chiral symmetry breaking, and the ensuing complicated internal dynamics of the nucleon.1the quest for a unified theory. This connection was first intimated by ’t Hooft, andlater made precise by Maldacena [2], who outlined a precise duality, or equiva-lence, between superstring theory – the leading candidate for a unified theory –and certain supersymmetric gauge theories. Maldacena’s discovery is known asthe Anti-de Sitter (ADS)/conformal field theory (CFT) correspondence, and is aconcrete realization (in terms of conformal field theory) of the more general con-cept of gauge/gravity duality. We shall give a more technical description later insection 1.4.A key feature of the correspondence is that it is a strong-weak duality. In otherwords, difficult (strong-coupling) questions in one theory are mapped to easier(weak-coupling) ones in the other theory. In principle, this makes it a powerfultool in the study of both gauge and string theory. On the other hand, the exactequivalence of the dual theories remains a conjecture, albeit a well-substantiatedone at this point, and the strong-weak nature of the duality makes it very difficult totest quantitatively. On the one hand, it is not yet known how to formulate string the-ory non-perturbatively3; on the other hand, strongly coupled gauge theory, while inprinciple well-defined, is not amenable to the standard Feynman diagram approachto Quantum Field Theory (QFT). Any technique that enables one to perform ex-plicit gauge theory calculations at strong coupling is consequently of great value, assuch results can then be compared against manageable string theory calculations,serving as a precision test of the correspondence.A perennial challenge in the study of gauge theories, and indeed theoreticalphysics as a whole, is that of moving beyond the perturbative paradigm. Pertur-bation theory is a systematic calculational scheme for obtaining approximations,in terms of some small parameter(s), to quantities that are not amenable to precisecomputation. While it is frequently possible to write down a precise mathemati-cal formulation of a physical theory or phenomenon, say for example the defininglagrangian of QCD, it is rare that one is then able solve the theory, in the sense ofcalculating, exactly, all interesting physical quantities flowing from it. In QFT, theonly exceptions to this are trivial and uninteresting, and certainly do not describethe real world. Fortunately, a limited number of strong-coupling/non-perturbative3 In fact the conventional philosophy is to take ADS/CFT as a definition of non-perturbative stringtheory2techniques do exist. This thesis deals with some of these methods and their appli-cations to certain gauge- and string-theoretic problems in high energy physics.Dualities, as described above, are one way of learning non-perturbative infor-mation about a theory. But as mentioned, the cases when duality might be useful inthis way are precisely the cases that are difficult to test. One way to make progressis to study simpler constructions – theories which capture important features of thecorrect “physical” theory, yet do away with some of the confounding complexity.An example is supersymmetric theories. While supersymmetry may certainly be ofrelevance to particle phenomenology, its utility in the study of gauge theory doesnot depend on its being realized in the real world. Supersymmetric gauge theoriesare in many ways much simpler than their non-supersymmetric counterparts, forexample thanks to non-renormalization theorems that mitigate the effects of quan-tum fluctuations on certain observables. Yet in general they are still far from trivial,presenting a rich array of phenomena. Furthermore, it is precisely these types oftheories that feature in known examples of gauge/gravity duality.Supersymmetric localization is a technique that allows one in some instances tocompute exactly the partition function, as well as certain special (supersymmetry-preserving) observables of a theory, for all values of the coupling. This remark-able state of affairs transpires when the theory under consideration has sufficientsupersymmetry, and was first demonstrated by Pestun for the case of four dimen-sional N = 2 supersymmetric gauge theory in [3]. In this thesis we exploit local-ization to obtain non-perturbative results for two important supersymmetric theo-ries with known gravity duals, namely N = 4 and N = 2∗ Supersymmetric Yang-Mills (SYM).Besides the problem of strong coupling, there exist phenomena that are in-trinsically inaccessible to perturbation theory even at weak coupling (the term“non-perturbative” is sometimes understood in this narrower context). Considerthe functionF(λ) ∼ e−a/λ . (1.1)All of its derivatives evaluate to zero at λ = 0, so its perturbative (i.e. Taylor) expan-sion in λ is identically zero to all orders. Consequently, any physical phenomenacharacterized by this functional dependence will be entirely invisible in pertur-3bation theory. A canonical example is that of quantum tunneling, which finds afascinating manifestation in the form of the yet-undetected Schwinger effect. Thelatter refers to an electric field-induced vacuum instability of Quantum Electrody-namics (QED) – and other gauge theories – which leads to spontaneous productionof charged particles out of the vacuum. In the latter part of the thesis we studythis effect in the context of both scalar QED and bosonic string theory. Surpris-ingly, we will discover that a form of supersymmetric localisation has a role toplay in our calculations even here, despite both of these theories being explicitlynon-supersymmetric.1.2 Supersymmetric gauge theorySupersymmetry has enjoyed much acclaim, and in latter days notoriety, for itspromise of resolving, or at least ameliorating deep conceptual and technical issuesin particle physics. Notable among these are the hierarchy problem, gauge cou-pling unification and dark matter. The notion that nature might be fundamentallysupersymmetric is a tantalizing proposal, not least for its elegance, yet experimen-tal evidence remains conspicuously absent.But the relevance of supersymmetry extends beyond the soteriology of particlephenomenology and its practitioners; it is of crucial importance for purely theoret-ical reasons, both as a source of insights into the structure of gauge theories andas a central concept in string theory and quantum gravity. Supersymmetric gaugetheories such as those studied in this thesis serve as important toy models of realworld physics, permitting many detailed analytic calculations that could not oth-erwise be accomplished by non-supersymmetric methods. Furthermore, all knownquantitative examples of gauge-gravity duality involve supersymmetry. Indeed itis a basic feature of any realistic string theory4; so in that sense, to the extent thatone might hold string theory to be correct as a fundamental description of nature,supersymmetry would also have to be true.In essence it is a symmetry relating fermions and bosons. To be more precise,supersymmetry is a spacetime symmetry that extends the Poincaré invariance en-joyed by relativistic QFT by the addition of spinor “supercharges” to the Poincaré4That is, any string theory whose spectrum contains bosons and fermions but no tachyon.4algebra. These are denotedQaα and Q¯α˙a. The index a runs from 1 toN , the numberof independent supersymmetries, while α, α˙ = 1,2 are Weyl spinor indices corre-sponding to the left and right SU(2) factors of the Lorentz group, respectively.These charges commute with all internal symmetries, and satisfy the followinganti-commutation relations{Qaα ,Q¯α˙b} = 2σ µαα˙Pµδ ab (1.2){Qaα ,Qbα˙} = εαα˙Zab (1.3)The central charge Z commutes with all other generators, and can only be non-zero in the case of extended supersymmetry, i.e. N > 1. There is an additionalglobal “R-symmetry” that comes along for the ride, namely an automorphism ofthe above algebra that rotates the supercharges. (This is just a U(1) phase rotationwhen N = 1, but becomes non-Abelian for N ≥ 2).The presence of spinor supercharges has important implications for the allowedfield content of the theory. The details can be worked out by studying representa-tions of the full super-Poincaré algebra separately for massless and massive par-ticles. The usual protagonists appear, namely scalar, fermionic as well as spin-1gauge fields.But as a consequence of the commutation relations, these fields nec-essarily occur in multiplets of the given supersymmetry algebra. It would be su-perfluous to present here the rather extensive, known taxonomy of supersymmetrymultiplets and theories in various dimensions, but let us give an overview of thetheories that will be investigated in this thesis. Chapters 2 and 3 deal with super-symmetric Yang-Mills gauge theories, namely theN = 4 andN = 2∗ theories, withgauge group SU(N), respectively.N = 4 SYM is a very special theory in many regards. Not only is it the gaugetheory with maximal supersymmetry in four dimensions (if one excludes gravity),it is also conformal, even at the quantum level. This implies, inter alia, that the the-ory lacks any inherent mass scale, and its beta-function is zero to all orders. Theconformal and supersymmetries are subsumed in the larger N = 4 superconformalgroup PSU(2,2∣4) which again is unbroken even at the quantum level. As a conse-quence of the large degree of symmetry, the theory is integrable, permitting manydetailed calculations and making it an ideal toy model. But an equally important5fact is its conjectured equivalence to type IIB string theory on AdS5 ×S5, a topicwe will take up in subsection 1.4.An elegant and concise construction of this theory proceeds by dimensionalreduction of the ten-dimensional N = 1 SYM theory5. For the most part we willfollow the conventions of [3]. The N = 1 “gauge multiplet” consists of the gaugefield Aµ and Majorana-Weyl fermion Ψ in the adjoint representation of the gaugegroup SU(N); the relevant action is thus6S = 12g2Y M∫ d10x Tr(12FmnFmn−ΨΓmDmΨ) (1.4)Here Fmn ≡∂mAn−∂nAm−i[Am,An] is the gauge field strength; Γm are ten-dimensionalgamma matrices; and Dm is the gauge-covariant derivative. This action is invariantunder the following supersymmetry transformationsδεAm = εΓmΨ (1.5a)δεΨ = 12FmnΓmnε, (1.5b)where ε is a constant spinor. We now “dimensionally reduce” along six of theten dimensions, say m = 0,5,6,7,8,9, i.e. we restrict to configurations that aretrivial along these directions. This breaks the ten-dimensional Lorentz symmetrySpin(9,1) down to the Lorentz group acting on the remaining coordinates of space-time (which is now Euclidean), times an R-symmetry inherited from the reduceddirectionsSpin(9,1)→ Spin(4)×Spin(5,1). (1.6)The resulting field content, from the four-dimensional point of view, consists of• the gauge bosons Aµ (µ = 1, . . . ,4),5 One can similarly obtain different supersymmetric theories of dimensionality d < 10 by thisapproach.6 We suppress the color indices (i j), per convention. Recall that the gauge field, for example,being in the adjoint representation, is given explicitly by (FMN)i j ≡ ∑a = T ai jFaMN , where T a aregenerators of the gauge group. The trace in (1.4) is taken over these indices.6• six real scalars ΦI (corresponding the reduced components of the 10d gaugefield)• four chiral fermions, Ψ = (ψL,χR,ψR,χL). The ψ’s and χ’s differ from eachother by their R-symmetry transformations, the details of which we omit forbrevity.These fields are all massless and transform in the adjoint of the gauge group. Theresulting flat-space action isSN=4 = 12g2Y M ∫ d10x Tr(12FµνFµν +DµΦIDµΦI+12[ΦI,ΦJ][ΦI,ΦJ]−ΨΓI[ΦI,Ψ]−ΨΓµDµΨ) (1.7)For the purposes of localization it will be necessary to compactify the theory. Inorder to define the theory on the four-sphere, S4, one is forced to include a couplingto the scalar curvatureR, such that conformal invariance is preserved. The relevantterm is δS = R6 ΦIΦI . We will find that it is precisely this term that survives on thelocalization locus, leading to a Gaussian potential in the resultant matrix model.The N = 2∗ theory is the unique massive deformation of N = 4 SYM that pre-serves half of the rigid supersymmetry. One restricts to anN = 2 subalgebra of theN = 4 algebra by a suitable restriction on ε , the spinor-valued transformation pa-rameter in (1.5). From theN = 2 perspective one then has two different multiplets,namely the N = 2 vector- and hyper-multiplets. The former consists of the gaugefields Aµ , their scalar superpartners Φ and Φ′, and two Majorana fermions. Thehypermultiplet contains four real scalars Φ5,...,8 (which it is convenient to combineinto two complex scalars Z1,2) plus two Majorana fermions. All the fields are inthe adjoint of the gauge group which we take to be SU(N). The Lagrangian of theN = 2∗ theory is then obtained from that of N = 4 SYM by giving common massM to the hypermultiplet fields and adding certain Yukawa couplings necessary forsupersymmetry; see [3] for more details.7Supersymmetric VacuaAs is typical of such gauge theories, N = 4 and N = 2∗ SYM possess a multiplicityof supersymmetric vacua. These are parametrized by vacuum expectation values(VEVS) of the scalar fields7. The quartic scalar potential V ∼ [ΦI,ΦJ]2, which canbe traced to the commutator in the covariant derivative of the N = 1 action beforedimensional reduction, has flat directions which give rise to a continuous modulispace of vacua. On the so-called Coulomb branch, a linear combination of thescalar fields from the vector multiplet acquires a VEV and the gauge symmetrySU(N) is broken to a product of U(1) factors. Diagonalizing this matrix-valuedscalar VEV by a gauge transformation,⟨Φ⟩ = diag(a1, . . . ,aN), (1.8)we obtain the Coulomb moduli ai. This Higgsing generates masses mi j = ∣ai−a j∣for all of the massless adjoint field components, while in N = 2∗ SYM the hyper-multiplet masses are shifted to mhi j = ∣(ai−a j)±M∣. As we will discover in chapter3, intriguing resonance phenomena occur whenever these mhi j approach zero, whichtends to happen in the strong coupling regime.1.3 Wilson loopsIn the study of non-abelian gauge theories, the Wilson loop operator plays a fun-damental role. It is defined as the following trace of a path-ordered exponentialW(C) ≡ TrRP exp{i∮CAµdxµ} , (1.9)where Aµ is the gauge connection and C is any closed curve. In addition to servingas an order parameter for the confinement-deconfinement phase transition, it pro-vides a naturally gauge-invariant formulation of the theory which, while inherentlynon-local, is quite natural from the point of view of the correspondence to stringtheory. It can be understood as the phase acquired by a probe particle as it tracesout some closed path C. As well as this contour, the Wilson loop is labeled by a7Lorentz invariance of the vacuum state precludes the possibility of higher spin fields obtainingVEVS.8representation R of the gauge group, describing the charge of the probe particle.InN = 4 andN = 2∗ SYM, the natural (supersymmetric and UV finite) Wilson loopobservable is the Maldacena-Wilson loop, which includes a scalar field coupling:W(C) ≡ 1NTrR(P exp{∮Cdτ(iAµ x˙µ + ∣x˙∣nIΦI)}) . (1.10)For the special case where C is a circle and nI a constant unit vector, this preserveshalf of the supersymmetries, permitting its localization after compactification onS4. In chapter 2 we study the 1/N expansion of the circular Wilson loop of N = 4SYM in the totally antisymmetric representation. In chapter 3 we will evaluate(1.10) in N = 2∗ SYM for the fundamental representation, at strong coupling.1.4 Gauge/gravity dualityThe intuition behind gauge-gravity duality emerged as early as the 1970s and waslargely due to Gerardus ’t Hooft. In the so-called ’t Hooft limit,N →∞, gY M → 0 with λ ≡ g2Y MN held fixed, (1.11)of a Yang-Mills gauge theory, the large-N expansion can be organized into a string-like topological expansion. Successive classes of Feynman diagrams generate tri-angulations of two-dimensional surfaces – i.e. strings – of increasing genus, withthe dominant contribution at large N coming from the “planar” diagrams. Suchconsiderations led to a rather general expectation of gauge-string equivalence inYang-Mills gauge theories at large N. Of course string theory is inherently a grav-itational theory, containing as it does a massless spin-two graviton in its spectrum;hence the nomenclature.This idea was given a precise formulation by Juan Maldacena in the landmarkpaper [2]. He proposed an exact, quantitative duality between type IIB string theoryon the curved AdS5×S5 spacetime andN = 4 SYM on four-dimensional Minkowskispace. We will briefly review the now well-known argument leading to this con-jectured duality.Consider a stack of N coincident D3-branes in type IIB superstring theory in9ten dimensions8. The effective expansion parameter for string perturbation theoryaround flat space is gsN, where gs is the string coupling constant, and N comes froma Chan-Paton factor for each worldsheet boundary. Stringy perturbation theory isvalid as long as this quantity is small, gsN ≪ 1. It is well-known that the low-energyeffective theory describing the stack of D3-branes in perturbative string theory isprecisely N = 4 SYM, where one identifies the Yang-Mills and string couplings asg2Y M = 4pigs. (1.12)In addition there is a decoupled sector of closed string modes propagating freelyaway from the branes.Now we note that classical supergravity provides a complementary perspec-tive on the same system. One can identify certain (explicitly known) solitonic“black brane” solutions in supergravity with the D-branes, and this description isvalid when the combination gsN is large. The branes behave as a heavy classicalsource, generating an extended black-hole type geometry. The characteristic cur-vature scale L of the solution is expressed in units of the string length ls = √α ′byL4α ′2 = 4pigsN (1.13)The regime of validity is precisely the opposite of the perturbative string theorylimit. If we again take a low-energy limit, the massless closed string modes oncemore decouple. However, due to red-shifting by the warped geometry, massivestring states survive the low-energy limit in the near-horizon region. Furthermore,the geometry of the near-horizon region turns out to be AdS5×S5.In summary, we have outlined two complementary descriptions, albeit in dif-ferent parameter regimes, of the same physical system. They have in common a de-coupled supergravity sector in flat ten-dimensional spacetime (the massless closedstring sector). Factoring out this common piece, it is therefore natural to conjecturethat the remaining theories are in fact equivalent, for all values of the parameters.This will be true if the operation of varying the coupling λ = gsN commutes with8 These are dynamical, non-perturbative objects on which open strings can end. In general aDp-brane has p+1 worldvolume directions: p spatial and one temporal.10that of taking the low-energy limit. The conclusion is summarized asN = 4 SU(N) SYM == Type IIB String Theory on AdS5×S5where the parameters of the two theories are identified according to (1.12) and(1.13).Numerous other dualities can be derived in this manner, by studying differentbrane setups. The equivalence extends in principle to all correlation functions inthe theories. Indeed, we can formulate it as an equivalence between generatingfunctions, schematically⟨e∑i ∫ φiOi⟩CFT== ∫Φi(z,x⃗)∼φi(x)DΦe−Sstring[Φ]. (1.14)Here O are local operators in the CFT, Φ denotes supergravity fields in ADS, andon the right hand side we specify boundary conditions on these fields at the bound-ary of ADS in terms of the CFT sources. It is frequently useful to visualize thefield theory as living on the boundary of ADS. Since this thesis does not presentany calculations on the gravity side of the correspondence, we shall not present adetailed discussion of the dictionary. More details can be found, for example , in[4].The observable that will be of particular interest to us in this context is the su-persymmetric Wilson loop, described in subsection 1.3. In [5] it was argued thatthe fundamental representation Wilson loop defined in (1.10) is dual to a funda-mental open string in ADS. The string worldsheet ends along a contour on the ADSboundary, described by the Wilson loop contour C. In the ’t Hooft limit, wherethe string worldsheet is semiclassical, the problem of evaluating the Wilson loopexpectation value is then equivalent to that of determining the area of a minimalsurface in ADS with specified boundary curve.The situation is rather interesting in the case of Wilson loops in higher repre-sentations of the gauge group. The general prescription is detailed in [6, 7]. Forthe totally symmetric and antisymmetric representations of rank k (correspondingto a Young tableau of k boxes with a single column or row), the dual object is aD3 or D5-brane carrying k units of worldvolume electromagnetic flux. In N = 411SYM, several of these Wilson loops have been successfully matched with theirholographic duals at leading order [8, 9, 10, 11, 12, 13]. However, the importantproblem of matching at the one-loop level remains unresolved in all cases; thiswill be the subject of chapter 2, where we study the antisymmetric Wilson loop ofN = 4 SYM.An important generalization of the duality described above is to non-conformaltheories. Any dual geometry must be a consistent string background, which makesit challenging in general to write down such a correspondence. A natural startingpoint is to consider a continuous deformation of the conformal setup described sofar. Suppose we deformN = 4 SYM, the boundary theory, with a relevant perturba-tion. On the gravity side this amounts to imposing certain boundary conditions onthe bulk fields, which are then evolved into the bulk subject to their equations ofmotion. In fact we have already discussed such a theory, namely the non-conformalN = 2∗ SYM, the only relevant perturbation toN = 4 SYM preservingN = 2 super-symmetry. The holographic dual of this theory on flat space has previously beenconstructed in [14] and is known as the Pilch-Warner geometry.1.5 Supersymmetric localizationSupersymmetric localization is a method to compute path integrals in interactingfield theories directly, without making any approximations [15]. It can be used totest gauge/string duality in a very precise way, while also giving us insight intopossible dynamical effects in strongly coupled gauge theories. The essential ideais that certain path integrals are given exactly by their semiclassical approximation.In the finite-dimensional context this was originally formulated in the Duistermaat-Heckman and Atiyah-Bott-Berline-Vergne theorems.Suppose we have a collection of fields, collectively denoted φ , governed byan action S[φ]. The expectation value of an observable O in the correspondingquantum theory is defined by the following path integral⟨O⟩ ≡ ∫ DφOe−S[φ]. (1.15)The partition function is simply Z ≡ ⟨1⟩. Suppose furthermore that the theory pos-sesses a non-anomalous fermionic symmetry δ , which squares to a bosonic sym-12metry of the action, denoted L. Now let V be any L-invariant, Grassmann-oddfunction. If the observable O is itself invariant with respect to δ , then we maydeform the path integral by a δ -exact term in the following way⟨O⟩(t) ≡ ∫ DφOe−S[φ]−tδV [φ], (1.16)and note that the result is independent of t! That is,ddt⟨O⟩(t) = −∫ DφO(δV)e−S[φ]−tδV [φ] (1.17)= −∫ Dφδ (O ⋅V ⋅e−S[φ]−tδV [φ]) (1.18)= 0. (1.19)We have assumed that there are no surface terms, which is usually the case, but ingeneral need not hold. Thus we can compute ⟨O⟩ by evaluating (1.16) at any valueof t. In the limit t→∞, the path integral localizes onto the critical points of δV : theonly contributions that survive come from this locus plus the one-loop (quadratic)fluctuations around it. Higher-order fluctuations are suppressed by inverse powersof t.In order for this argument to work, the bosonic part of δV should be positivesemi-definite. Also, the resulting functional determinant is formally infinite andrequires regularization. Typically one compactifies the theory on a sphere, therebyobtaining a discrete spectrum; supersymmetry then leads to large cancellations ofputative divergences.An obvious class of candidates for the above procedure is supersymmetricgauge theories, where the role of δ is played by a combination of the supersym-metry and the BRST symmetry associated with gauge fixing. It was successfullyapplied toN = 2 supersymmetric theories compactified on the four-sphere S4 in theseminal work [3]. (It turns out that in four dimensions at least N = 2 supersymme-try is needed). Since then an impressive array of similar results have been obtainedfor different theories, dimensionalities, and even background geometries; see [16]for an extensive review.There are numerous technicalities that we necessarily skim over here, not leastthe actual evaluation of the 1-loop determinants using index theorems or spherical13harmonics, or such as the need to introduce auxiliary fields to ensure off-shellsupersymmetry. But let us summarize the results of such a calculation for thecases that will interest us. Our work exploits localization on S4, in which case thefield-theory path integral reduces to a finite-dimensional matrix model [3]. Witha canonical choice9 for the localizing function V , the path integral localizes toprecisely the scalar field zero modes that parametrize the Coulomb branch modulispace. The result is a finite dimensional integral over the Lie algebra g of the gaugegroup ⟨O⟩ = ∫g[da] Z1−loop(a) ∣Zinst∣ e−S[a]O(a) (1.20)For gauge group SU(N) (or U(N)) this is a Hermitian matrix model. In this waywe make contact with the venerable field of random matrix theory. ∣Zinst∣ is the in-stanton partition function, generated by singular gauge field configurations; it willnot be relevant to us in this work and we henceforth set it to unity10. The fluctu-ation determinant Z1−loop is equal to 1 in N = 4 SYM but becomes a complicatedfunction for N = 2∗.An interesting regime that can then be explored in detail is the planar, large-Nlimit. The strong-coupling behavior of a planar theory is generally believed to havea simple (weakly-coupled) string description. The problems we study in chapters2 and 3 involve an expansion around this limit, in which the integral (1.20) isin general highly non-trivial, and sophisticated techniques are required to extractresults that can be compared to holography.An important characteristic of the theory at large N is the master field, charac-terized by the eigenvalue density:ρ(x) = ⟨ 1NN∑i=1δ (x−ai)⟩ . (1.21)Localization allows one to compute some special correlation functions, for ex-9Schematically of the formV = ∑λ∈{fermions}∫ Tr(δλ)λ10 ForN = 4 SYM there are no instanton corrections when one computes expectation values. In theN = 2∗ theory, instanton corrections can be consistently neglected at large N as they are exponentiallysuppressed in N.14ample the VEV of the Wilson loop for the big circle of S4. The localizable loopoperator couples to the scalar Φ of the vector multiplet, in addition to the usualpath-ordered vector coupling; see equation (1.10). If C is the equatorial circle ofS4, the Wilson loop can be computed at large N by just substituting the constantclassical value (1.8) forΦ and averaging over the eigenvalues with the weight givenby the partition function of the matrix model:W(C) = ⟨ 1N∑ie2piai⟩ = ∫ µ−µ dxρ(x)e2pix. (1.22)1.6 The Schwinger effectThe spontaneous production of charged particle-antiparticle pairs out of the vac-uum in the presence of an external electric field is known as the Schwinger effect.First predicted by Euler and Heisenberg [17] in 1936, it was derived in the frame-work of QED by Julian Schwinger in [18]. The QED vacuum behaves as a non-linear optical medium – a modern reconception of the old-fashioned “luminiferousæther” idea – exhibiting both dispersive and absorptive properties with respect tothe propagation of light. In particular, the vacuum becomes unstable in the pres-ence of a background field due to the possibility of particles tunneling out of theDirac sea. Intuitively, virtual electron-positron pairs are accelerated by the elec-tric field, becoming “real” or “on-shell” when the energy imparted reaches theircombined rest mass 2mec2.This is an important non-perturbative phenomenon, and one that is of relevanceto other field theories such as QCD, as well as string theory. Furthermore, it is veryclose to being probed experimentally. The field strength required to produce adetectable effect is extreme, of the order of 1018 V/m; but given recent advances inthe area of focused, high- intensity laser beams, it is hoped that direct experimentalverification of this phenomenon is not far off [19].On the theoretical side, there are various challenges. Precise quantitative predic-tions are essential if one is to test the theory against experiment. Given the labo-ratory difficulties, considerable effort has been expended in understanding how toenhance the effect through non-trivial background field configurations involving15both space- and time-dependence.Mathematically, the vacuum persistence amplitude (or equivalently the pairproduction rate) can be obtained from the imaginary part of the effective action Γ,since ∣⟨0∣0⟩∣2 = ei(Γ−Γ∗) ≈ 1−2ImΓ. (1.23)A useful tool in the study of effective actions is the worldline path integral. It is afirst-quantized formulation of field theory, originally written down by Feynman forscalar QED in an appendix to [20]. Essentially it is a quantum mechanical path inte-gral over particle trajectories (worldlines) parametrized by proper-time, with addi-tional integration over one or multiple modular parameters (see chapter 4 for moredetails). For many field theory problems this formulation is traditionally spurnedin favor of the second quantized approach, yet it has found numerous useful ap-plications, including but not limited to the computation of quantum anomalies andeffective actions.The worldline path integral is a particle theory analog of the Polyakov pathintegral of string theory11. We will therefore be able to take a common approachin our analysis of string and particle pair production in chapters 4 and 5. Theadvantage of the worldline method in computing the Schwinger production rateis that it allows a semiclassical treatment which in principle is straightforwardlygeneralizable to inhomogeneous and/or dynamical electromagnetic fields, as wellas to higher loops.We will evaluate the path integral using an approach due to Affleck, Alvarezand Manton [22], namely a semiclassical approximation wherein both the targetspace coordinates and the modular parameter are treated as dynamical variables.One then expands around certain Euclidean instanton solutions on the worldline.We will find that the localization argument discussed in subsection 1.5, albeit insomewhat simpler form12, can be brought to bear in this analysis. In the constant-field setup that we study, the quadratic approximation, with a suitable treatment ofzero modes, turns out to give the exact result for the production rate. Remarkably,11 It is interesting to note that the infinite string tension limit of string theory reduces to QFT.This connection was investigated notably by Bern and Kosower, resulting in a set of rules for thecomputation of certain scattering amplitudes in non-Abelian gauge theory; see for example [21].12In particular, no localizing deformation “δV ” is needed16this magic also transpires for the more complicated case of the charged bosonicstring (chapter 5). This ultimately will be explained in terms of an accidentalfermionic symmetry unrelated to spacetime supersymmetry.1.7 Outline of thesisIn the first part of this thesis (chapters 2 and 3) we study N = 4 and N = 2∗ SYMwith gauge group SU(N), two supersymmetric gauge theories that are of impor-tance to holography. Supersymmetric localization reduces the partition functions(as well as expectation values of certain observables) of these theories to matrixmodels (and certain matrix model expectation values). We study these matrix mod-els and their Wilson loop expectation values at large N.In chapter 2 we focus onN = 4 SYM. Here we study the 12 -BPS circular Wilsonloop in the totally antisymmetric representation of the gauge group. An outstandingproblem in the literature is to match the 1-loop fluctuations of the holographicallydual D-brane to the corresponding field theory calculation. The latter amountsto computing the 1/N correction to the Wilson loop expectation value at strongcoupling. We will accomplish this by developing a systematic 1/N expansion basedon the matrix model loop equations.In chapter 3 we investigate the vacuum structure of the planar N = 2∗ SYMtheory at strong coupling. Again, supersymmetric localization on S4 reduces thepartition function (and circular Wilson loop) to a matrix model, whose strong-coupling expansion we study at large N. Our goal is to understand the nature of theresonance phenomena responsible for quantum phase transitions - and in particularhow a signature of such behavior might emerge holographically. We also derive aquantitative prediction for the semiclassical fundamental string dual to the circularWilson loop.In the remainder of the thesis we study the Schwinger effect in scalar QEDand in bosonic string theory. In chapter 4 we study the semiclassical expansionof the worldline path integral for a charged relativistic particle in the presence of aconstant electric field. After computing the pair production rate in the semiclassicalapproximation we identify a fermionic symmetry which allows us to construct alocalization argument proving that corrections to this approximation must cancel17and that the result is therefore exact.In chapter 5 we generalize the preceding analysis to the charged bosonic stringin a constant external electric field. We find the instanton solutions in the world-sheet sigma model which mediate vacuum pair production, and evaluate the par-tition function in the multi-instanton sector in the semiclassical approximation,yielding the vacuum decay rate. Once again, we identify a fermionic symme-try (distinct from the BRST symmetry) of the gauge-fixed path integral. We arethereby able to localize the path integral and prove that our semiclassical result isexact.18Chapter 2Antisymmetric Wilson Loop inN = 4 Super Yang-Mills2.1 IntroductionSince its inception, the AdS/CFT correspondence has held out the promise ofa fully non-perturbative definition of quantum string theory in non-trivial back-grounds. Testing this strongest form of the conjecture is, however, very hard.Progress can be made in this direction by considering controlled deviations fromthe large-N, large-λ limit. In an exciting development, the techniques of super-symmetric localization and integrability have in recent years generated a profusionof exact gauge theoretic results, enabling such quantitative testing and ushering inan era of precision holography.Supersymmetric localization reduces the partition function of N = 4 super-Yang-Mills to a gaussian Hermitian matrix model [3]. Furthermore, a certainsupersymmetry-preserving sub-sector of the theory is completely captured, for ar-bitrary N and λ , by matrix model expectation values of appropriate insertions (seeeqs. (2.3,2.4) in the next section). For the fundamental Wilson loop this was antici-pated in the prescient work [8] (also [9]) where an infinite class of planar diagramswas explicitly re-summed, generating what was correctly conjectured to be the ex-act planar result. Localization formulæ for more general correlators and Wilsonloops have since followed.19To date the correspondence has withstood over two decades of sustained scrutiny.It is therefore noteworthy when tension, let alone disagreement, is found betweenputatively dual quantities. We can expect such cases to reveal important subtletiesor misunderstandings of the dictionary, or indeed to elucidate the limits of its ap-plicability.In this chapter we study an as-yet unresolved mismatch in the most scruti-nized example of AdS/CFT, namely the duality between N = 4 supersymmetricYang-Mills theory with gauge group SU(N), and type IIB superstring theory onAdS5 ×S5. The discrepancy occurs in the 1-loop correction to the 12 -BPS circularWilson loop in the rank-k totally-antisymmetric representation of the gauge group,with k ∼O(N). We compute this quantity on the gauge theory side by solving theloop equations for the corresponding matrix model obtained from localization1. Infact, the antisymmetric Wilson loop was evaluated exactly using orthogonal poly-nomials in [25]; however, it is not clear how to extract from their result the 1/Nexpansion, which is needed for comparison with holography. Their formula willhowever be useful for numerical verification of our result.Apart from its intrinsic interest, an understanding of the structure of higher-rank Wilson loops may also yield insight into the analogous, longstanding 1-loopmatching problem for the fundamental Wilson loop [26, 27, 28, 29, 30]. See [31,32, 33] for recent progress on this problem.According to the AdS/CFT dictionary, the antisymmetric Wilson loop is dualto a probe D5-brane with k units of electric flux on its AdS2 × S4 worldvolume[6, 7], which “pinches off” along the circular contour described by the Wilson loopat the boundary of AdS. At leading order in large N and λ the on-shell action ofthe D-brane was successfully matched with the gauge theory [11, 12].The D-brane tension is of order N, and its 1-loop effective action captures non-planar contributions. The spectrum of fluctuations and 1-loop effective action were1 A similar approach to deriving Wilson loops in higher representations from the loop equationshas been advocated for example in [23] where general k-loop (fundamental) correlators, from whichgeneral representations can be constructed, were computed building on the work of [24]. However,their results are not directly applicable here as we will be interested in the limit k→∞, with k/Nfixed.20derived in [34, 35] and [36] respectively, with the result2Γ1 = 16 lnsinθk, (2.1)where θk is defined by (θk − sinθk cosθk) = pik/N. A first step towards reproduc-ing this from the gauge theory side was taken in [37]. They obtained the samefunctional dependence on k, but a different overall constant:Γ˜1 = 12 lnsinθk (2.2)The mismatch is not surprising, as the computation neglected the backreaction ofthe Wilson loop insertion on the equilibrium eigenvalue distribution of the matrixmodel. Here we will systematically take this into account. However, our result,which withstands convincing numerical testing, still does not match with (2.1);even the power of λ is different. As we mention in the conclusions, this contribu-tion to the free energy likely corresponds to the gravitational backreaction of theprobe D-brane, i.e. our result is of very different origin to (2.1). It would still be in-teresting to try to match (2.1) with a gravity calculation by a careful determinationof strong-coupling corrections on both sides.The layout of the chapter is as follows. In section 2.2 we summarize the local-ization result [3] for the Wilson loop, and set up the problem. In section 2.3 wederive a sequence of loop equations for the gaussian matrix model perturbed bythe Wilson loop insertion, and solve them for the resolvent up to the second sub-leading order. We then derive from this the free energy, by two different means. Wealso calculate the correction to this result due to considering gauge group SU(N)instead of U(N). Section 2.4 presents some numerical checks of our answer, bycomparing to the exact result of [25]. Finally we end with some conclusions andopen questions in section 2.5.2The answer of 112 lnsinθk quoted in [36] was updated by the authors in the subsequent publica-tion [37] to incorporate a missing normalization factor.212.2 Antisymmetric circular Wilson loopLocalization ofN = 4 SYM reduces the full partition function to that of a HermitianGaussian matrix model [3]ZGauss = ∫ [dM]e− 2Nλ trM2 , (2.3)while the expectation value of the circular Wilson loop is mapped to an expectationvalue (denoted ⟨⟩0) in this matrix model:⟨WR(Circle)⟩ = 1dim[R] ⟨trR eM⟩0 (2.4)The representation R of the gauge group is completely arbitrary at this stage. Wewill be interested inR =Ak, the totally anti-symmetric representation of rank k, inthe large-N, large-λ regime withf ≡ kN∼O(1) (2.5)held fixed. The generating function for the character of this representation isFA(t) = det(t +eM) = N∑k=0tN−k(Nk)WA (2.6)so that we can write the Wilson loop expectation value as [12]⟨WA⟩ = d−1A ∮Ddt2pii⟨FA(t)⟩0tN−k+1 , (2.7)where D encircles the origin and dA = (Nk) is the dimension of the representation.The following change of variables, which maps the complex t-plane to the cylinder,will prove convenient:t = ez. (2.8)22It will also be useful to view the expectation value of FA as defining a family ofperturbed partition functions parametrized by z,Z(z) ≡ ∫ [dM]exp{−2Nλ trM2+ tr log(1+eM−z)} . (2.9)and to define a corresponding “free energy”F(z) ≡ − 1Nlog[Z(z)Z−1Gauss] . (2.10)Note the unconventional factor of Z−1Gauss here. In this manner we obtain the follow-ing exact expression for the Wilson loop:⟨WA⟩ = d−1A ∮ dz2pii exp{N( f z−F(z))} (2.11)The free energy of the purely gaussian matrix model isO(N2) and has a genusexpansion in powers of 1/N2, ie.− logZGauss = ∑n=0,2,4,...N2−nFGauss,n ; FGauss,n ∼O(1) (2.12)Since Z(z) differs from Z0 by a perturbation to the action of O(N), its logarithmgoes in powers of 1/N, with leading term identical to that of logZGauss. Conse-quently, F(z) defined in (2.10) is O(1), and we writeF(z) =F0(z)+ 1NF1(z)+ 1N2F2(z)+ . . . ; Fi(z) ∼O(N0) (2.13)As the exponent in (2.11) isO(N) we can evaluate the z-integral in the saddle-pointapproximation, which yields⟨WA⟩ = d−1A2pii eN( f z∗−F(z∗)) ⋅ i[ 2piN ∣F ′′(z∗)∣]12 (1+O( 1N)) (2.14)where z∗ solves the saddle-point equationF ′0(z∗) = f . To the order in N given here,there is no backreaction on the saddle due toF1(z). The i prefactor is from analyticcontinuation of the “wrong-sign” quadratic form. Finally, plugging in (2.13), we23havelog⟨WA⟩=N[ f z∗−F0(z∗)]+[−F1(z∗)− 12 logF ′′0 (z∗)−log(dA√2piN)]+O(1/N)(2.15)The leading order result was already obtained in [11, 12] and agrees perfectly withthe D5-brane on-shell action [11]. The logF ′′0 term was obtained in [37]. Interest-ingly, the latter turns out to have the same functional dependence on k as the 1-loopeffective action of the D-brane, as computed in [36], but with a different numericalcoefficient. One might anticipate that F1(z∗), at leading order in 1/λ , should givea similar contribution, so as to correct the numerical mismatch. In fact this turnsout not to be the case: the log term is subleading in λ .We review the planar solution in the next subsection. What then remains is tocompute the non-planar free energy F1(z) of the matrix model (2.9). We do thisin section 2.3 by calculating the resolvent W(p) ≡ ⟨ tr 1p−M ⟩ order-by-order in Nusing the loop equation method. With the resolvent in hand, the free energy can bedetermined in either of two ways.1. λ -integral: W(p) is the generating function of monomial expectation values,⟨M j⟩ = N∮ d p p jW(p). But ⟨M2⟩ is also just the derivative of logZ withrespect to λ−1. Therefore F is obtained from the resolvent by an integralover p and λ .2. Eigenvalue density: At large N the eigenvalues condense into a continuousdistribution ρ(x). The O(1) perturbation to ρ due to the Wilson loop inser-tion is encoded in the discontinuity of W1(p) across its single cut. Fluctua-tions around the large-N saddle-point of the ∫ [dM] integral are not neededas they cancel against the same contribution coming from ZGauss.Naturally, we find exact agreement between these methods.2.2.1 Planar approximationWritten in terms of eigenvalues the matrix integral (2.9) isZ(z) = ∫ (∏idmi)e−N2S[mi;z] (2.16)24where S = S0+ 1N S1 andS0 = 2λN∑i m2i − 2N2N∑j=1j−1∑i=1 log ∣mi−m j∣ (2.17a)S1 = − 1N∑i log(1+emi−z) (2.17b)The double sum in S0 is the usual Vandermonde determinant. Recall that at large N,the eigenvalues mi condense into a continuum distribution described by a spectraldensity ρ ,ρ(x) ≡ 1NN∑iδ(x−mi), (2.18)with support (−√λ ,√λ) ⊂R. This is normalized to unity, and has an expansionρ(x) = ∞∑n=0N−nρn(x); ∫ ba ρn(x)dx = δn0 (2.19)where ρ0(x) is just the Wigner semicircle distributionρ0(x) = 2piλ√λ −x2, −√λ < x <√λ , (2.20)since at N =∞ the Wilson loop insertion does not backreact on the eigenvalues.Thus the expectation value of the generating function reduces to its average withrespect to ρ0(x), hence⟨WA⟩ ≃ √λdA ∮ dz˜2pii exp{N ( f√λ z˜+∫ 1−1 dxρ0(x) log(1+e√λ(x−z˜)))} . (2.21)To facilitate the strong coupling expansion we have re-scaled z according to z ≡√λ z˜. From now on we will drop the tilde. This integral was evaluated in [12],to leading order in large-λ , using a saddle-point approximation. There is a singlesaddle-point z∗ ∈ (−1,1) on the real axis, determined by the equationarccos(z∗)− z∗√1− z2∗ = pi f (2.22a)250 50 100 150 2005010015020025080 90 100 110 120255260265270Figure 2.1: Strong coupling expansion of (logW)planar versus (logW)exact ,for N = 200, λ = 35. The right hand plot is a close-up of the middleregion. The solid blue is the exact result logW evaluated numerically(see section 2.4). The orange, green, red and purple lines show, in order,the successive approximations to (logW)planar given by (2.81). Theyclearly converge to a fixed residual with respect to the exact result, andthis residual should be well approximated by the second square bracketin (2.15). We confirm this in section 2.4. In this and subsequent plotswe omit the constant prefactor dA.or in an angular parametrization defined by z∗ = cosθk:(θk − sinθk cosθk) = pi f (2.22b)The Wilson loop (2.21) then evaluates to(logWA)planar = 2N3pi√λ sin3θk (2.23)This coincides with the on-shell action of the dual D5-brane [11]. Subsequentterms in the strong coupling expansion are obtained by expanding the logarithm in(2.21), which is like the anti-derivative of the Fermi-Dirac distribution, in inversepowers of λ [38] - see section 2.4. In figure 2.1 we compare the strong-couplingexpansion of the planar approximation obtained in this way with the exact numer-ical result.262.3 1/N expansion from the loop equationsWe now set up a systematic expansion around the N =∞, 1-cut solution of the ma-trix model, using the well-known loop equation approach [24]. The matrix model(2.9) is rather exotic - it involves a non-polynomial O(1/N) perturbation to theGaussian potential. Consequently the usual genus expansion familiar from thestudy of polynomial-potential matrix models (see eg. [24]) becomes an expansionin 1/N.We begin with a few definitions. Our main object of study will be the resolvent,defined byW(p) ≡ 1N⟪ tr 1p−M⟫ (2.24a)= ∞∑n=01NnWn(p). (2.24b)The double angle-brackets mean the expectation value is with respect to Z(z), ie.such expectation values are always functions of z. More generally, the “s-loopcorrelator” is defined asW(p1, . . . , ps) =Ns−2⟪ tr 1p1−M . . . tr 1ps−M⟫connected (2.25)The so-called loop equation for the resolvent follows from invariance of thepartition function under the infinitesimal change of variablesM→M+ε 1p−M . (2.26)The Jacobian for this transformation is ( tr 1p−M)2. By the following simple manip-ulations1N2⟪( tr 1p−M)2⟫ = ( 1N ⟪ tr 1p−M⟫)2+ 1N2 ⟪ tr 1p−M tr 1p−M⟫conn.= W(p)2+ 1N2W(p, p) , (2.27)27and1N⟪ tr (G′(M)p−M )⟫ = ∫Σdmρ(m)∮C dω2pii 1ω −m G′(ω)p−ω = ∮C dω2pii W(ω)G′(ω)p−ω ,(2.28)where the positively-oriented contour C encloses the singularities of W but ex-cludes the point p (and possible singularities of “G”), we obtain the followingequation for W(p):∮Cdω2piiV ′(ω)p−ω W(ω)= (W(p))2+ 1N ∮C dω2pii W(ω)p−ω φz(ω)+ 1N2W(p, p). (2.29)Here φz(ω) is defined byφz(ω) ≡ ∂∂ω log(1+eω−z) = 11+ez−ω . (2.30)Our problem is specialized to a Gaussian potential, V(x) = 2λ x2. This is almostidentical to the well-known loop equation for the Hermitian matrix model withpolynomial potential V(ω), except for the 1/N term on the right-hand side[24].Plugging in the expansion (2.24b) we find a series of equations which can be solvediteratively in n. The n = 0 equation is unaffected by the Wilson loop insertion:∮Cdω2piiV ′(ω)p−ω W0(ω) = (W0(p))2 . (2.31)For a Gaussian potential the solution is well-known (see e.g. [39]):W0(p) = 2λ (p−√p2−λ) (2.32)This has a single branch cut along −√λ < p <√λ . The higher order equations are{Kˆ−2W0(p)}Wn(p) =∮ dω2pii φz(ω)p−ω Wn−1(ω)+⎧⎪⎪⎪⎨⎪⎪⎪⎩0, n = 1∑n−1n′=1Wn′(p)Wn−n′(p)+Wn−2(p, p), n ≥ 2 (2.33)28where we have introduced a linear operator Kˆ, defined as in [24] byKˆ f (p) ≡ ∮Cdω2piiV ′(ω)p−ω f (ω). (2.34)The contour C is defined as before. Note that the RHS always involves correlatorswith smaller n than the LHS. Thus one can in principle solve iteratively to obtainany Wn(p) 3.In the rest of this section we shall solve (2.33) up to n = Solution of the loop equationsThe n = 1 equation is{Kˆ−2W0(p)}W1(p) = ∮Cdω2piiφz(ω)p−ω W0(ω) (2.37)From now on we specialize to V(ω) = 2λω2. Note that all the p-dependence on theRHS is in 1/(p−ω). By deforming the contour C to infinity (assuming f (p) hasno singularities outside of C) we find{Kˆ−2W0} f (p) =M(p)√p2−λ f (p)+∮∞ dz2pii V ′(z)z− p f (z) (2.38)where ∮∞ means we pick up the residue at infinity, and M(p) is given byM(p) ≡ ∮∞ dz2pii V ′(z)(z− p)√p2−λ = 4λ (2.39)3In [24], the general iterative solution beyond leading order relies on the fact that, unlike our eq.(2.33), the RHS there is always a rational function of p (proof by induction), so by a partial fractiondecomposition can be written as a sum of powers of (p− x)−1 and (p− y)−1, where x, y are theendpoints of the cut. The solution Wn is thus easily expressed in terms of a set of basis functionsχ(n)(p), Ψ(n)(p), determined explicitly there, with the property that{Kˆ−2W0(p)}χ(n)(p) = (p−x)−n (2.35){Kˆ−2W0(p)}Ψ(n)(p) = (p−y)−n (2.36)In general the operator {Kˆ−2W0(p)} can also have zero modes; in such cases, assuming a singlecut, this freedom is constrained by the large-p asymptotics of W(p).29Therefore {Kˆ−2W0(p)}⎛⎝ 1(p−ω)√p2−λ ⎞⎠ =M(p)p−ω . (2.40)(The integrand in the last term of (2.38) goes like z−2 for large z). Thus (2.37) issolved by W1(p) = λ/4√p2−λ ∮C dω2pii φz(ω)W0(ω)(p−ω) (2.41)Shrinking the contour to lie along the real axis, and rescaling ω so the cut extendsfrom −1 to +1, we haveW1(p) = −12√p2−λ ∫ 1−1 dω√ω2−λ(p−ω) 11+ez−√λω (2.42)We now proceed to the n = 2 equation:{Kˆ−2W0(p)}W2(p) = ∮ dω2pii φt(ω)p−ω W1(ω)´udcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymod¸udcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymod¶a+(W1(p))2´udcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymod¸udcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymod¶b+W0(p, p)´udcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymod¸udcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymod¶c. (2.43)By linearity we can write the solution asW2(p) =W2,a(p)+W2,b(p)+W2,c(p) (2.44)where W2,i solves (2.43) with only term i on the RHS. In fact we will only need tosolve for W2,a. We do not need W2,c since the corresponding contribution to thefree energy F2,c is precisely the genus 1 (n = 2) component of the Gaussian freeenergy FGauss,2, which cancels in (2.10). Nor is W2,b relevant, because the asymp-toticsW1(p→∞)∼ p−2 imply that it does not contribute to ∂λF ∝∮∞ d p2pii p2W(p)4.Here the p-dependence of the RHS is the same as for the n = 1 equation and the4 It is however easy to show using (2.38) that W2,b is justW2,b(p) = λ/4√p2−λ (W1(p))2 (2.45)30solution is therefore analogous:W2,a(p) = λ/4√p2−λ ∮C dω2pii φ(ω)W1(ω)p−ω (2.46)= (λ/4)2√p2−λ ∮C′ du2pii φ(u)p−u 1√u2−λ ∮C dv2pii φ(v)u−vW0(v) (2.47)(2.48)The contour C′ encloses C but not p. Only the singular (square root) part ofW0(p)contributes. Again shrinking C, C′ around the cut, we can writeW2,a(p) = −λ8pi2√p2−λ ∫√λ−√λ du φ(u)(p−u)√λ −u2 ⨏√λ−√λ dv φ(v)√λ −v2u−v (2.49)where ⨏ denotes the Cauchy principal value.2.3.2 Free energy I: F as generator of ⟨( trM2)l⟩The resolvent (2.24a) is the generator of expectation values of monomialsW(p) = 1N∞∑k=0⟪ trMk⟫pk+1 . (2.50)This allows us to obtain the free energy as an integral over λ and p, sinceλ 22N2∂λ logZ(z) = 1N ⟪ trM2⟫ (2.51)= ∮∞ d p2pii p2W(p) (2.52)The potential V(M) = 2λ M2 acts as a source for insertions of trM2. Thus we haveF(z) = −2N∫ dλλ 2 ∮∞ d p2pii p2W(p)+C0(z)−FGauss (2.53)31with some integration constant C0(z). Due to the subtraction of FGauss, the leadingorder is determined by W1. The p-integral is easily doneResp=∞ p2(p−ω)√p2−λ = −ω, (2.54)as is the λ integral, and we findF0(z) = −2Npi ∫ +1−1 dω√1−ω2 log(1+e√λ ω−z)+C0(z) (2.55)in agreement with (2.21). Proceeding in the same way with W2,a, we getF1(z) = −14pi2 ∫ dλ ∫ 1−1du⨏ 1−1dv u√1−u2√1−v2u−v 11+ez−√λu 11+ez−√λv (2.56)Recall that z here is to be substituted with z∗ =√λ cosθk. At strong coupling theFermi-Dirac function can be approximated by a step function - this is the first termin the “low-temperature” expansion. Thus∂λF1(z) ≃ −14pi2 ∫ 1ζ du u√1−u2 ⨏ 1ζ dv√1−v2u−v (2.57)where ζ ≡ z∗/√λ . We find for the v-integral, taking care with the principal value,that1√1−u2 ⨏ 1ζ dv√1−v2u−v = uarccos(ζ)+√1−ζ 2√1−u2 − log 1−uζ +√1−u2√1−ζ 2∣u−ζ ∣ .(2.58)For the u integral we get∫ 1ζduau2+bu√1−u2 = 12 arccos2(ζ)+(1−ζ 2)+ 12ζ√1−ζ 2 arccos(ζ) (2.59)and∫ 1ζduu log1−uζ +√(1−ζ 2)(1−u2)u−ζ = 12 [1−ζ 2+ζ√1−ζ 2 arccos(ζ)](2.60)32resulting in∂λF1(z) = −18pi2 [arccos2(z∗/√λ)+(1− z2∗/λ)] . (2.61)Finally this can be integrated with the help of Mathematica to give∫ λdλ [∂λF1(z)]= −18pi2 [λ −2z√λ − z2 arccos( z√λ )+λ arccos2( z√λ )]+C1(z).(2.62)In terms of the√λ -scaled parameter we then haveF1(z) = − λ8pi2 [1−2z√1− z2 arccos(z)+arccos2(z)]+C1(√λ z) (2.63a)What about the integration constant C1(z)? The leading λ -dependence obtainedhere suggests it should be of the form C1(z) = az2. Then the requirement thatF1 = 0 at z = 1 (corresponding to k = θ = 0) fixes a to be 1/8pi2:C1(z) = z28pi2 (2.63b)In terms of θk the result isF1(θk) = − λ8pi2 [sin2θk −θk sin2θk +θ 2k ] (2.63c)In the next section we will show that with this choice of C(z), F1 agrees with thedirect evaluation of the matrix integral using the eigenvalue density.2.3.3 Free energy II: ρn from Wn(p)An alternative route to the free energy is via the eigenvalue density. The first1/N correction, F1, will require knowledge of the backreacted eigenvalue density,which is encoded in the resolvent5.5Fluctuations around the large-N eigenvalue saddlepoint, which would contribute a factor of12 logdet[∂ 2S/∂mi∂m j] to the free energy, do not contribute at this order as they cancel againstthe equivalent contribution to FGauss.33The action (2.17) in terms of ρ isS0 = 2λ ∫ dxρ(x)x2−2∬ dxdyρ(x)ρ(y) log ∣x−y∣ (2.64)S1 = −∫ dxρ(x) log(1+ex−z) (2.65)Expanding around ρ0 givesS[ρ0+ρ1/N +O(1/N2)] = S0[ρ0]+ 1N ⎧⎪⎪⎨⎪⎪⎩∫ ρ1 δS0δρ ∣ρ0 +S1[ρ0]⎫⎪⎪⎬⎪⎪⎭+ 1N2⎧⎪⎪⎨⎪⎪⎩12∬ ρ1 δ2S0δρδρρ1+∫ ρ1 δS1δρ ∣ρ0 +∫ ρ2 δS0δρ ∣ρ0⎫⎪⎪⎬⎪⎪⎭+O( 1N3 ) (2.66)The terms involving first derivatives of S0 are identically zero, by the equationof motion. We can also eliminate the awkward double integral by means of theO(1/N) equation of motion:0 = ddx⎡⎢⎢⎢⎢⎣∫ δ2S0δρ(x)δρ(y)ρ1(y)dy+ δS1δρ ∣ρ0⎤⎥⎥⎥⎥⎦ (2.67)Thus integrating the quantity in square brackets against ρ1(x) (a trick used in [40])gives∬ ρ1 δ 2S0δρδρ ρ1 = −∫ ρ1 δS1δρ ∣ρ0 (2.68)Therefore the action finally reduces toN2 S[ρ] =N2S0[ρ0]−N∫ dxρ0(x) log(1+ex−z)− 12 ∫ dxρ1(x) log(1+ex−z)(2.69)The first term is canceled by FGauss, the second is the planar result in (2.21), andthe third is the one we are after. After scaling of x, z by√λ as before, we haveF1(z) = −√λ2 ∫ 1−1 dxρ1(√λx) log(1+e√λ(x−z)) (2.70a)≃ −λ2 ∫ 1z dxρ1(√λx)(x− z), (λ →∞) (2.70b)34We now determine ρ1(√λx) from W1(p). Recall that the continuum form of(2.24a), namely W(p) = ∫ ρ(m)p−m dm, implies that the eigenvalue density is given asthe discontinuity across the cut:ρ(x) = 12pii(W(x− iε)−W(x+ iε)) (2.71)Using the relation1x± iε =P (1x)∓ ipiδ(x), (2.72)where P denotes the Cauchy principal value, and recalling (2.42), which we repeathere, W1(p) = −12√p2−λ ∫ 1−1 dω√ω2−λ(p−ω) 11+ez−√λω ,we findρ1(x) = 12pi2√λ −x2 ⨏√λ−√λ dω φ(ω)√λ −ω2(x−ω) (2.73)As usual we re-scale ω and z by√λ . Then at strong coupling we can approximateφz(√λω) ≃ θ(ω − z), i.e.ρ1(√λx) ≈ 12pi2√1−x2 ⨏ 1z dω√1−ω2(x−ω) (2.74)Using (2.58) we thus findρ1(√λx) = 12pi2 {xarccos(z)+√1− z2√1−x2 − log 1−xz+√1−x2√1− z2∣x− z∣ } (2.75)This function is plotted in fig. 2.2. Note the logarithmic singularity that arises atinfinite coupling, located on the cut at x= z. (At finite λ , (2.73) is a smooth functionof x). ρ1(x) is correctly normalized to zero: ∫ 1−1 dxρ1(√λx) = 0.The free energy (at strong coupling) now follows from (2.70a) and (2.75). Theresult isF1(z) = −λ8pi2 (1− z2−2z√1− z2 arccos(z)+arccos(z)2) , (2.76a)35-1.0 -0.5 0.5 1.0-0.2- 2.2: ρ1(√λx), the 1/N correction to the density, at large λ and withz = 0.2. The analytic expression is given by (2.75).or in terms of θk: F1(θ) = −λ8pi2 (sin2θ −θ sin2θ +θ 2) , (2.76b)in precise agreement with (2.63).2.3.4 Modification for SU(N)AdS/CFT is generally held to describeN = 4 SYM with gauge group SU(N). Thisis motivated by considering the Kaluza-Klein spectrum of IIB supergravity. On theCFT side, for a U(N) gauge theory, the U(1) and SU(N) components decouple,up to global identifications. On the AdS side, dimensional reduction of SUGRAon the internal S5 does indeed give rise to a free U(1) multiplet, but this comprisespure gauge modes which can be set to zero in the bulk (see e.g. [4, 41]).Since we are studying 1/N effects here, the difference between the U(N) andSU(N) theories is potentially important; so far we have only considered the former.For SU(N) the integral (2.3) is over traceless Hermitian matrices. It is not hard tointegrate out the trace degree of freedom explicitly6, as described for the funda-6 Alternatively we can keep the integral over all of u(N) and impose the tracelessness constraintwith a Lagrange multiplier Λ. This adds N2SΛ = Λ∑Ni=1 mi to the action, resulting in a perturbationδρΛ(√λx)= Λ2pi x√1−x2 to the density. The tracelessness condition ∫ 1−1dxx [ρ1(x)+δρΛ(x)]= 0 then36mental case in [9]. Write M = M′+mI, where M′ is traceless. The measure is just[dM] = [dM′]dm. From the definition of WA in terms of the generating function(2.6) we have ⟨WA⟩U(N) = ⟨ekm∂N−kt˜ det(t˜ +eM′)⟩0 ∣t˜=0 (2.77)where we made the replacement t˜ = te−m. Integrating out m we obtain the followingexact relation between the Wilson loops of the two theories:⟨WA⟩U(N) = ek2λ/8N2 ⟨WA⟩SU(N) (2.78)In terms of the free energies we have FSU(N)(z) = FU(N)(z)+ 18λ f 2. Our finalresult for F1(θk) is remarkably simple:FSU(N)1 = − λ8pi2 sin4θ . (2.79)2.4 Numerical checksThe antisymmetric Wilson loop was evaluated in [25] using orthogonal polyno-mials, yielding the following exact result for the generating function (2.6) of theWilson loopFA(t) = det[t +Aeλ/8N] , Ai j ≡ Li− jj−1(−λ/4N), (2.80)which we evaluate numerically for large values of N and λ . In order to comparethis with our F1 we must subtract off the planar result at strong coupling. Asdetailed in [38], the latter is obtained as an expansion in large-λ using the “low-temperature” expansion of the Fermi-Dirac function which appears in the planarfixes the multiplier to piΛ = z√1− z2−arccos(z). Using the saddlepoint equation (2.22) for z (or θk),we then find precisely the result (2.78) above.37100 200 300 400-14-12-10-8-6-4-2100 200 300 400-4-3-2-1Figure 2.3: Numerical versus analytic results for F1 as a function of k (withN = 400, λ = 100). This was defined via equations (2.10,2.13,2.22). Thesolid blue line is our analytic result for F1, and its numerical approxi-mation Φ(k) (defined in (2.83)) is given by the orange dashed line. Theplot on the left is for gauge group U(N) (eq. (2.76)) while that on theright is for SU(N) (eq. (2.79)). (As in figure 2.1, we have replaceddA = (Nk)→ 1 here).saddle-point equation. We find(logWA)planar = 4piNλ [λ 32 sin3θk6pi2 +√λ sinθk12 − 1√λ pi21440 (19+5cos2θk)sin3θk− 1λ32pi4725760(6788cos2θk +35cos4θk +8985)sin7θk+⋯] , (2.81)where we have corrected a numerical error in the last two terms of eq. 2.10 of[38]7.Finally, to compare precisely with the numerics, we need all other contributionsup to O(N0). This includes the prefactor (Nk)−1 √λ2pi from (2.11), as well as the 1-loop contribution from the z-integral:¿ÁÁÀ 2piNF ′′0 (z∗) =¿ÁÁÀ pi2N√λ sinθk. (2.82)With these factors included, we find good numerical agreement with the exact re-7I thank Kazumi Okuyama for correspondence on this point.38sult (2.80). In figure 2.3 we plot F1(θk) versus Φ(k), defined asΦ(k) ≡− log(W exactA )+(logWA)planar,3+ 12 log√λ4Nd2A +⎧⎪⎪⎪⎨⎪⎪⎪⎩0 , U(N)λ8 ( kN )2 , SU(N) (2.83)where (logWA)planar,3 contains the first three terms of the planar strong couplingexpansion (2.81). The parameter values used are N = 400, λ = 100. We then ex-pect the next correction to be O(10−1), since for the “higher-genus” and strong-coupling corrections we have respectively (√λN )2 logWA ≈0.5 and (√λN ) 1λ logWA ≈0.2. This is indeed borne out by the numerics: from the plot we see that the residualis approximately ∣Fnumeric1 −Fanalytic1 ∣ ≈ 0.2. If instead we take N = 700, λ = 30, sothat 1λ ≫ √λN , we get∣Fnumeric1 −Fanalytic1 ∣ ≈ (√λN ) 1λ logWA ≈ 0.25,whereas (√λN )2 logWA ≈ 0.05.Finally, as a check of the λ dependence, in figure 2.4 we plotFnumeric1 (k) versusλ , for fixed N and several different values of k. This plot clearly illustrates thelinear behavior at large λ .2.5 ConclusionsWe computed the first 1/N correction to the 12 -BPS circular antisymmetric Wilsonloop of rank k, with k of order O(N), in N = 4 SYM, at leading order in ’t Hooftcoupling λ . The result is given in equations (2.76) and (2.79), for gauge groupU(N) and SU(N) respectively.The holographic dual of this object is a known probe D5-brane configurationwith k units of electric flux on its worldvolume. Interestingly, the results obtainedhere and the D-brane 1-loop effective action computed in [36] do not match. Therethey found F1 ∼O(N0λ 0). In contrast, our calculation yielded F1 ∼O(λ), imply-ing an expansion in√λ/N. The obvious explanation for this discrepancy is thegravitational backreaction of the brane, which so far has not been accounted for.3920 40 60 80 100 1202468λ∣F1∣Figure 2.4: Linear dependence of F1(k) on λ : Each line corresponds to aparticular value of k in the range 100 < k < 300, with N = 400. Thedashed rays are included simply as visual aids. For large enough λ wesee precisely the linear behavior obtained in (2.76).Integrating out the bulk action in the Gaussian approximation would indeed givea result O(λ), although the problem is not so simple as one needs to account forthe infinite tower of Kaluza-Klein modes and their couplings to the brane8. It alsoremains desirable to resolve the numerical mismatch at O(N0,λ 0), by studying1/λ corrections.It is worth mentioning that several important properties of the heavy probesstudied here follow directly from the Wilson loop expectation value, including theso-called Bremsstrahlung function9 [45]BAk(λ ,N) = 12pi2λ∂λ ⟨WAk⟩ , (2.84)and the additional entanglement entropy ∆S , relative to the vacuum, of a spherical8I thank Kostya Zarembo for comments on this point.9see also [42, 43, 44] for a similar formula in the context of ABJM theory40region threaded by the probe10 [47, 48],∆S = (1− 43λ∂λ) log⟨WAk⟩ . (2.85)Also intriguing is the relation of the antisymmetric Wilson loop to a supersym-metric Kondo model [35].The plethora of gauge theory localization results in the literature opens the doorto a number of natural extensions of the present work. Firstly, there exist exactresults for various gauge theories with generally richer structure than the highlysymmetric N = 4 SYM. Expectation values of higher rank SUSY Wilson loopshave been studied in the planar limit in N = 2∗ SYM [49], N = 2 SQCD [50] andalso ABJM theory [51]11. On the gravity side some of the corresponding probeshave been studied in eg. [52, 36, 34, 53]. The problem of 1-loop matching remainsopen in all these cases.Continuing in this vein, we could also consider more general correlators, againbeyond the planar limit. The resolvent (2.24a), which we have obtained from theloop equation, encodes expectation values of monomials in the presence of theWilson loop, ⟨ trM j⟩WA . For the Hermitian matrix model these have no directphysical interpretation. However, the analogous quantities in the normal matrixmodel describe correlators of the Wilson loop with chiral primary operators inN = 4 SYM [54, 55], and it would be interesting to extend our analysis to this case.On the gravity side, the corresponding “backreaction” calculation may prove moretractable than that of the Wilson loop expectation value itself.This story generalizes still further to a larger subsector of Wilson loops andchiral primary operators inN = 4 SYM. For example, one can consider the generi-10 Incidentally, it should be possible to calculate ∆S holographically using the approach of [46],whose authors studied the additional holographic entanglement entropy due to the presence of probebranes. The leading order effect arises from the backreaction of the probes on the geometry, and theconcomitant distortion of the Ryu-Takayanagi minimal surface. This was shown to be captured bya compact “double-integral” formula, where the integrations are taken over the brane worldvolumeand unperturbed minimal surface respectively, obviating the need for a full, backreacted solution. Asargued in some detail in [46], complications due to fields other than the metric being sourced by thebrane may be avoided, thanks to the particular worldvolume gauge field configuration relevant to thisproblem.11In the latter work, a partial 1/N contribution, analogous to the logarithmic term in (2.15), wasalso calculated.41cally 18 -BPS configurations of multiple loops and chiral primaries supported on anS2 submanifold of R4. It is believed that correlators of such observables reduce tobosonic 2d Yang-Mills theory [56, 57, 58, 59, 60], which in turn can be mapped tocertain multi-matrix models. (This is still at the level of conjecture, as the 1-loopfluctuations around the localization locus have not been explicitly evaluated. See[61, 62, 63] however for several non-trivial checks of the conjecture). Aspects ofthe matrix model machinery we have employed can be generalized to the study ofmulti-matrix models.42Chapter 3N = 2∗ Super Yang-Mills atStrong CouplingSupersymmetric localization, as a method, has obvious limitations as it requiresa sufficient amount of supersymmetry. Gauge/string duality is believed to have amuch broader scope, but it too is formulated precisely only in a limited numberof cases. The model studied in this chapter, N = 2∗ SYM theory, is special in thisrespect. Its partition function on S4 and some select observables are calculable bylocalization [3], and at the same time it has a well-defined holographic dual – thetype IIB string theory on the Pilch-Warner background [14].3.1 Critical behavior of N = 2∗ SYM and predictions forholographyAs described in section 1.5, the localized partition function of N = 2∗ SYM on S4is a matrix model which at large-N can be studied by standard methods [64] ofrandom matrix theory [65, 66, 67, 68, 69]. A number of observables computedwith the help of localization have been successfully compared to string-theory pre-dictions at strong coupling. These include Wilson loops for asymptotically largecontours [66], and the free energy on S4 [70].Away from the strict strong-coupling limit, localization leads to somewhat un-expected results. It turns out that the planar N = 2∗ SYM has a very complicated43phase structure, undergoing an infinite number of quantum phase transitions as the’t Hooft coupling changes from zero to infinity [67, 68]. While these are a commonphenomenon in large-N theories [71, 72], the behavior in N = 2∗ SYM is uniqueand differs in many respects from phase transitions in ordinary matrix models [68].Similar phase transitions have also been found in QCD-like vector models [68, 73]as well as in three [73, 74, 75] and five [76] dimensional theories.The existence of an infinite number of phase transitions raises the question ofhow the non-trivial phase structure of N = 2∗ SYM is reflected in its holographicdual. Ideally, one would like to tune the ’t Hooft coupling to its critical value, butgoing to finite coupling is notoriously difficult in holography, as it requires quan-tizing string theory on a complicated supergravity background. Here we propose adifferent route. The transition points accumulate at infinity, and rather than varyingthe ’t Hooft coupling we can instead approach the accumulation point by varyingthe compactification radius while keeping the coupling strictly infinite (fig. 3.1).The string dual then always remains in the classical supergravity regime. More-over the supergravity dual for the theory on S4 is explicitly known [70], so thisrange of parameters is potentially accessible to standard holographic calculations.Our goal is to study the exact master field of the N = 2∗ theory on a four-sphere of radius R at large ’t Hooft coupling λ ≡ g2YMN ≫ 1. Up to some point, wewill keep the full dependence on the dimensionless parameter MR, and will thenseparately study the decompactification limit R→∞, where most of the interestingphenomena occur. The resulting theory can be viewed asN = 2∗ SYM in flat spacein a particular vacuum state selected by compactification. The same vacuum issingled out by ADS/CFT duality for the Pilch-Warner background (see [69] for amore detailed discussion of the vacuum selection in this context).Localization reduces the path integral of N = 2∗ SYM on S4 to an (N − 1)-dimensional eigenvalue integral [3]:Z = ∫ dN−1a∏i< j(ai−a j)2H2(ai−a j)H(ai−a j −M)H(ai−a j +M) e− 8pi2Nλ ∑i a2i ∣Zinst∣2 , (3.1)whereH(x) ≡ ∞∏n=1(1+ x2n2)n e− x2n . (3.2)44Figure 3.1: The phase diagram of the planarN = 2∗ theory [68]. If the strong cou-pling limit is approached at strictly infinite radius (on R4), as shown in blackhorizontal arrows, the theory undergoes an infinite number of phase transi-tions. In this paper we approach the same corner of the phase diagram alonga different direction, by varying the compactification radius while keeping thecoupling strictly infinite, as shown in green vertical arrows. Although the re-sults may depend on the direction along which the critical point is approached,we do find the structures that cause phase transitions in the strong-couplingfinite-volume solution.At large N we can disregard the instanton contribution, Zinst = 1, as it is exponen-tially suppressed.The leading-order strong-coupling solution of the matrix model that describesN = 2∗ SYM on S4 was obtained in [66], and is essentially equivalent to the so-lution of the Gaussian matrix model. This result is way too simple to capture thecritical behavior observed at infinite radius. Our aim here is two-fold. First, wewould like to develop a systematic strong-coupling expansion beyond the leadingorder. Second, we want to study the approach to the critical point in the decompact-ification limit. The strong-coupling corrections are equivalent to quantum correc-tions on the string side, therefore, they are potentially calculable by semiclassicalstring quantization in the dual supergravity background. As we shall see, irregularstructures responsible for phase transitions in the decompactification limit are al-ready present in the first-order approximation, which opens an avenue to study the45critical behavior of N = 2∗ SYM within semiclassical string theory.3.2 Planar limit of the matrix modelIn the limit N →∞, the eigenvalue integral (3.1) is dominated by a single saddle-point. Introducing the continuum eigenvalue density (2.18) as in chapter 2, we canwrite the saddlepoint equation as a singular integral equation⨏ µ−µ dyρ(y)S(x−y) = 8pi2λ x, (3.3)where the kernel is given byS(x) = 1x−K(x)+ 12K(x+M)+ 12K(x−M). (3.4)The function K(x) is the logarithmic derivative of H(x):K(x) = −H′(x)H(x) = 2x ∞∑n=1(1n − nn2+x2 ) = x(ψ (1+ ix)+ψ (1− ix)−2ψ(1)) . (3.5)ψ(s) here is the dilogarithm function (the logarithmic derivative of the Gammafunction). In writing the above, we have chosen units such that the radius of thesphere is one. In this manner the decompactification limit R→∞, which we shallstudy later, is traded for the infinite mass limit M→∞.Let us briefly review the infinite coupling solution of (3.3), which was derivedin [66]. The right hand side of (3.3) constitutes a restoring force on the eigenvalues,which otherwise tend to repel one another. At large λ , clearly this force is verysmall and the width µ of the distribution grows, leading to µ ≫M and µ ≫ 1. Thetypical distance between x and y is of order µ , thus we can approximate the lastthree terms by a second derivative:12K(x+M)+ 12K(x−M)−K(x) ≈ 12M2K′′(x) ≈ M2x(3.6)and the function K(x) by it’s asymptotic large-x behaviorK(x) ≃ x lnx2 (x→∞) . (3.7)46Consequently the saddle-point equation reduces to that of the Gaussian matrixmodel ⨏ µ−µ dyρ(y) 1+M2x−y = 8pi2λ x, (3.8)with the well-known Wigner’s semicircle solution:ρ∞(x) = 2piµ2 √µ2−x2, (3.9)whereµ = √λ (1+M2)2pi. (3.10)Taking the decompactification limit M →∞ leads to the Wigner distributionof width√λM/2pi , in precise agreement with the D-brane probe analysis of thePilch-Warner solution [77].Using this solution in (1.22), we would obtain for the Wilson loopW(C) ≃ √2/pi(1+M2)−3/4λ− 34 e√λ(1+M2). (wrong)While the leading exponential scaling here is correct, and indeed matches preciselywith the area law for the holographic Wilson loop in the Pilch-Warner geometry[66], the prefactor is in fact incorrect. This is due to the fact that the Wilson loop,being an exponential function in the matrix model, is dominated by the contributionof eigenvalues very close to µ (i.e. µ −x ∼O(1) ). In this region the density is verysmall, and corrections are O(1) rather than O(1/√λ).An obvious approach to calculating the next order in the 1/√λ expansion ofρ(x) would be to continue the expansion (3.6) of the kernel to higher order in 1/x.However, one obtains precisely the same equation again, with only the boundarybehavior of ρ(s) changing. Note that the general solution to this equation isρ(x) = 8piλ (1+M2) √µ2−x2+ βµ√µ2−x2 . (3.11)The second term, which we normalized to obtain a 1/√λ correction to the Wignerdistribution, satisfies the homogeneous form of the integral equation (3.8). Thecoefficient β is thus not fixed by the equations; nor is it fixed by the normalization47condition of the density, which is supposed to determine the endpoint position µ .This freedom to choose any coefficient for the second term may look worri-some. Moreover, the density vanishes at the endpoints for any finite λ , while thesolution above blows up at the edges of the interval. A resolution of these apparentcontradictions lies in the fact that the second term in (3.11) cannot be treated as asmall correction near the endpoints. The two terms in (3.11) become comparableat µ −x ∼ 1, where the naive strong-coupling expansion breaks down. We shall seethat the most interesting phenomena, responsible for the phase transitions in infi-nite volume, happen precisely in this regime. We will eventually fix the constantβ and then µ by matching the exact solution in the near-endpoint region to theasymptotic solution (3.11) in the bulk of the eigenvalue distribution.3.3 Solution at strong coupling3.3.1 Wiener-Hopf problemAs we concluded above, the strong-coupling expansion breaks down near the end-points of the eigenvalue distribution. The integral equation, therefore, has to beanalyzed separately in this regime. We recall that the support of the eigenvaluedensity becomes very large, µ ≫ 1, at strong coupling. We are interested in thebehavior at x close to µ , with µ − x ∼ 1. To gain some intuition we can start withthe leading-order solution (3.9). Introducing the variableξ = µ −x, (3.12)we find at ξ ∼ 1:ρ∞(x) ≃ 2 32piµ32√ξ . (3.13)We thus expect that the exact density near the upper endpoint has the formρ(x) = 2 32piµ32f (ξ), (3.14)where f (ξ) is a scaling function that does not depend on λ .48Since ρ∞ is a good approximation in the bulk of the eigenvalue distribution,(3.14) should approach (3.13) away from the endpoint. This fixes the boundaryconditions on f (ξ) at large ξ :f (ξ) ≃√ξ (ξ →∞) . (3.15)At ξ ∼ 1, f (ξ) deviates from √ξ by an O(1) amount.An integral equation for f (ξ) can be obtained by the following trick: the exactsaddle-point equation (3.3) can be re-written as⨏ µ−µ dy [ρ(y)S(x−y)−ρ∞(y) 1+M2x−y ] = 0, (3.16)where we have made use of (3.8). This step entails no approximations. We can nownotice that for x close to µ the weight in the y integral is peaked near the endpoint.We can thus introduce the scaling variable (3.12), replace ρ(y) and ρ∞(y) by theirscaling forms (3.14), (3.13), and extend the limit of integration over the scalingvariable to infinity:⨏ ∞0dη [ f (η)S(η −ξ)− (1+M2)√ηη −ξ ] = 0. (3.17)The integral over η converges at the upper limit due to the asymptotic form ofK(x) given in (3.7), (3.6) and the boundary condition (3.15).The integral equation (3.17) is of the Wiener-Hopf type. It can be broughtto the standard Wiener-Hopf form by introducing the regularized scaling functionwith a better behavior at infinity:g(ξ) = f (ξ)−√ξ . (3.18)This function satisfies the equation⨏ ∞0dη g(η)S(η −ξ) = F(ξ), (3.19)49whereF(ξ) = ⨏ ∞0dη√η [1+M2η −ξ −S(η −ξ)] . (3.20)Both integrals here converge – in the first case because g(η) ∼ 1/√η at large η ,and in the second case because the kernel behaves as 1/(η −ξ)3 at large η , as aconsequence of (3.6).3.3.2 Exact solutionIn order to solve the integral equation (3.19), let us first assume that the functiong(ξ) is defined on the whole real axis, but is equal to zero for ξ < 0. Then, theintegral on the left-hand side of (3.19) takes the convolution form:S∗g(ξ) = F(ξ)+θ(−ξ)X(ξ), (3.21)where X(ξ) is an arbitrary function whose appearance reflects the fact that theoriginal equation only holds for ξ > 0.After the equation is written in the convolution form, it can be brought to analgebraic form by Fourier transform:g(ξ) = ∫ +∞−∞ dω2pi e−iωξ gˆ(ω). (3.22)After the Fourier transform we get:Sˆ(ω)gˆ(ω) = Fˆ(ω)+X−(ω), (3.23)where X−(ω) is a function that has no singularities in the lower half-plane of com-plex ω , because its Fourier image vanishes on the positive real semi-axis. Thesubindex + will be used similarly, but to indicate analyticity in the upper half-plane. In fact, gˆ(ω) = gˆ+(ω). The explicit expressions for the remaining terms50are:Sˆ(ω) = ipi signω ⎛⎝1+ sin2 Mω2sinh2 ω2 ⎞⎠ (3.24)Fˆ(ω) = ipi signω ⎛⎝M2− sin2 Mω2sinh2 ω2 ⎞⎠ i32√pi2ω√ω + iε , (3.25)which can be obtained from the definitions (3.4), (3.5) and (3.20), as well as thefollowing representation of K′′(x):K′′(x) = 4∫ ∞0dωω2 sin2xωsinh2ω. (3.26)Here, the sign function should be understood in terms of analytic regularization:signω = limε→0√ω + iε√ω − iε , (3.27)where the square root√ω ± iε has a branch cut extending from ∓iε up to infinityalong the negative/positive imaginary semi-axis.The solution of the Wiener-Hopf problem (3.23) is based on the factorizationformulaSˆ(ω) = 1G+(ω)G−(ω) . (3.28)In our case, the functions G± have the form:G±(ω) = (M2+1ipi )12 (ω ± iε)∓ 12 e∓ iφω2pi Γ(M∓i2pi ω)Γ(−M±i2pi ω)Γ2 (∓ iω2pi ) , (3.29)whereφ = 2M arctanM− ln(M2+1) . (3.30)These formulae follow from factorization of trigonometric and hyperbolic func-tions in terms of the gamma function:Γ(x)Γ(1−x) = pisinpix(3.31)51Figure 3.2: The singularities of the inverse kernel, Sˆ−1(ω), are two cuts along posi-tive and negative imaginary semi-axes, and simple poles at ω =ωn and ω = ω¯n.The functions G− and G+ inherit singularities in respectively the upper andlower half-planes.The phase term e∓ iφω2pi is introduced to cancel the bad asymptotics of the combi-nation of gamma functions at large imaginary ω , which can be inferred from theStirling formula. Without these factors the functions G±(ω) would exponentiallygrow in their respective domains of analyticity.The inverse kernel of the integral equation Sˆ−1(ω) has a cut along the imagi-nary axis, due to the sign function, which we break in two parts by regularization(3.27). It also has simple poles at ω =ωn and ω = ω¯n, n ≠ 0, whereωn = 2pi (Mn+ i∣n∣)M2+1 . (3.32)The factorization (3.28) assigns the cut in the upper half-plane and the poles atω = ωn to G−, while G+ has a cut in the lower half-plane and poles at ω = ω¯n(fig. 3.2). The poles of G± lie on the straight lines that make an angle α = arccotMwith the real axis. After the Fourier transform back to ξ space, the poles will createresonances. For generic M the resonances are damped because Imωn ∼ Reωn, butwhen M becomes large, the poles pinch the real axis and cause oscillations in the ξ52space, with periods that are integer multiples of M. This behavior is a manifestationof the nearly massless hypermultiplets that we discussed in section 1.5. We shallstudy the large-M behavior of the solution in much more detail in the next section.Returning to the Wiener-Hopf equation, we can use factorization of the kernelto rewrite (3.23) asgˆ+(ω)G+(ω) =G−(ω)Fˆ(ω)+G−(ω)X−(ω). (3.33)Defining the projection on the positive/negative-frequency part of a function viacontour integration:F±(ω) = ±∫ +∞−∞ dω ′2pii F(ω ′)ω ′−ω ∓ iε , (3.34)we can project out the − term in the last equation, and thus find the solution to theWiener-Hopf problem:gˆ+(ω) =G+(ω)(G−Fˆ)+ (ω). (3.35)It is important here that G+(ω)−1 is also an analytic function in the upper half-plane of complex ω , and thus the left hand side of (3.33) is a + function.In order to compute the positive projection of (G−Fˆ)(ω), let us discuss theanalytic structure of this function first. Since the branch cut of G− cancels in theproduct G−Fˆ , the latter is a meromorphic function with simple poles at ω = ωn,n = ±1,±2, . . . and double poles at ω = −2piim, m = 1,2, . . .. All the double poleslie in the lower half-plane, and the integral in (3.34) can be done by closing thecontour of integration in the upper half-plane and picking up the poles at ω andωn:gˆ(ω) = Fˆ(ω)Sˆ(ω) −G+(ω)∑n≠0 Fˆ(ωn)ω −ωn resz=ωn G−(z). (3.36)The first term is the naive Fourier transform that would solve the integral equationon the whole real axis. Since we need a solution identically equal to zero for ξ < 0,its Fourier transform must be analytic in the upper half plane. The rôle of the lastterm is to subtract the singularities of the first term in order to make the solution a53+ function.Explicitly, we get:gˆ(ω) = i 32 √pi2ω√ω + iε ⎡⎢⎢⎢⎣M2 sinh2 ω2 − sin2 Mω2sinh2 ω2 + sin2 Mω2+(M2+1)2ω e− iφω2pi Γ(M−i2pi ω)Γ(−M+i2pi ω)Γ2 (− iω2pi )× ∞∑n=1(−1)nnn!⎛⎝ eiφnM−iω − 2pinM−i Γ(M+iM−i n)Γ2 ( iM−i n) + e− iφnM+iω + 2pinM+i Γ(M−iM+i n)Γ2 (− iM+i n)⎞⎠⎤⎥⎥⎥⎥⎦ (3.37)This is our final expression which in general cannot be further simplified.The solution in the ξ space is the inverse Fourier transform of (3.37). Sincegˆ(ω) has a relatively simple structure of singularities in the lower half-plane, itsFourier transform can be computed using the residue theorem, which results in adouble infinite sum representation for g(ξ). The final expression (A.5) and the de-tails of the derivation are given in the appendix. This expression is very convenientfor numerical evaluation of the function g(ξ), but many quantities of interest, suchas the Wilson loop expectation value, can be calculated directly from the Fourierrepresentation.The Wilson loop is an eigenvalue average with the exponential weight. Writingthe defining equation (1.22) in terms of the endpoint variable ξ , and using (3.14)and (3.18), we getW(C) = 23/2piµ3/2 e2piµ ∫ ∞0 dξ (g(ξ)+√ξ) e−2piξ , (3.38)where as before, the integration domain is extended to infinity, allowed by thestrong coupling limit. This is a Laplace integral, and the part with g(ξ) is simplygˆ(2pii). Therefore, the Wilson loop is:W(C) = 23/2piµ3/2 e2piµ (gˆ(2pii)+ 125/2pi ) . (3.39)The last term in the brackets is the contribution of the uncorrected Wigner dis-tribution, which gives the prefactor quoted after (wrong). The Wiener-Hopf term54Figure 3.3: Scaling function g(ξ) for mass M = 0.5.gˆ(2pii) is the correction produced by the distortion of the eigenvalue distributionnear the endpoint. We study it in more detail in the next subsection.3.3.3 General structure of solutionFor moderate and small values of the mass – M of order 1 and below – g(ξ) isa featureless function whose asymptotic behavior is prescribed by the boundaryconditions of the Wiener-Hopf problem:g(ξ) ξ→0≃ B√ξ , g(ξ) ξ→∞≃ C√ξ, (3.40)where the coefficients B and C depend on the mass. The plot in fig. 3.3 shows g(ξ)at M = 1/2 for illustration.In view of (3.14), (3.18), the constant B determines the endpoint behavior ofthe eigenvalue density:ρ(x) = 2 32 (1+B)piµ32√µ −x (x→ µ) . (3.41)The constant C will be later used to compute 1/√λ corrections in the bulk of theeigenvalue distribution. We also introduceA = 25/2pi gˆ(2pii)+1, (3.42)55which determines the normalization factor in the expectation value of the Wilsonloop:W(C) = A2pi2µ3/2 e2piµ . (3.43)For pure Wigner distribution A = 1, but A is also a non-trivial function of the mass.The constants B and C can be read off from the asymptotic behavior of gˆ(ω):gˆ(ω) ω→∞≃ i 32 √piB2ω32, gˆ(ω) ω→0≃ √ipiC√ω. (3.44)The constant A is also expressed explicitly in terms of gˆ(ω). Therefore to computethese constants we do not need to perform the inverse Fourier transform back tothe ξ representation. Explicitly:A = 2piM(M2+1)2 eφsinhpiM∞∑n=1(−1)nnn!Re⎛⎝ eiφnM−i1+ inM−i Γ(M+iM−i n)Γ2 ( iM−i n)⎞⎠ (3.45)B =M2+2(M2+1) 32 ∞∑n=1(−1)nnn!Re⎛⎝e iφnM−i Γ(M+iM−i n)Γ2 ( iM−i n)⎞⎠− M2+1piarctanM (3.46)C = M2+12pi∞∑n=1(−1)nnn!Im⎛⎝e iφnM−i M−in Γ(M+iM−i n)Γ2 ( iM−i n) ⎞⎠ . (3.47)The origin of the last term in B is explained in appendix B. The first of these equa-tions solves the problem of computing the prefactor in the Wilson loop expectationvalue.For small M, we can approximateeiφnM−i Γ(M+iM−i n)Γ2 ( iM−i n) ≈ (−1)n n!2pitanpiMnM− i . (3.48)The constant A is then saturated by the n = 1 term, which develops a pole at M→ 0,while the main contributions to B and C come from terms in the sum with verylarge n ∼ 1/M. Replacing the sums by the integrals we get:A ≃ 1, B ≃ M22, C ≃ M24pi(M→ 0) . (3.49)56The constants A and B measure, in different ways, deviations from the Wignerdistribution near the endpoint. We see that these deviations vanish in the M → 0limit.Conversely, deviations from the naive strong coupling result grow with M and,as we shall see, become parametrically large at M→∞. This limit is equivalent tothe flat space limit, in which the sphere inflates to an infinite radius, correspondingto the top right hand corner of the phase diagram (fig. 3.1) approached from below.It is here that we hope to detect signs of non-trivial phase structure, making thisregime particularly interesting. It turns out that the solution indeed develops suchstructure at large M, which we investigate in detail in the next section.3.4 Decompactification limitOne of our main motivations for solving the localization matrix model at strongcoupling is to study the flat space/infinite mass limit, MR →∞. The corner ofthe phase diagram in fig. 3.1 is an accumulation point of an infinite number ofphase transitions, and even though we approach the critical point from a differentdirection, we may expect to see signatures of the non-trivial phase structure in theeigenvalue density. There are three well separated scales in the problem in thedecompactification limit: the IR cutoff scale 1/R (equal to 1 in the units we use),the mass scale M and the symmetry breaking scale µ ∼√λM. The physics behindthe phase transitions is governed by the second of these, and we expect the non-trivial structures to occur distance ∼ M away from the endpoints of the eigenvaluedistribution. But M ≪ µ and, in spite of the fact that M ≫ 1, the distances oforder M are still within the range of the scaling limit applicable near the endpoints.We thus expect that most of the interesting phenomena associated with the phasetransitions are described by the solution of the Wiener-Hopf problem. We thusneed to study the large-M limit of the solution.As M grows, the shape of the scaling function g(ξ) dramatically changes. Atfirst, a smooth profile similar to that in fig. 3.3 starts to be modulated with pe-riod approximately equal to M. At yet larger masses, the amplitude of modula-tion grows, and the scaling function develops a structure of regularly spaced peaksof diminishing amplitude (fig. 3.4). The peaks become sharper and sharper with57Figure 3.4: g(ξ) evaluated numerically for M = 10. See additional plots onpage 68.growing M and in the strict M→∞ limit morph into cusps of infinite height. Thisis precisely the phenomenon observed in [67, 68] where the flat space theory wasstudied at arbitrary coupling and the appearance of cusps was identified as a causeof the phase transitions.We now have an analytic expression for the scaling function which is validat any M, and thus can study the large-M limit rather explicitly. Moreover, thecomplicated expression (3.37) simplifies somewhat in this limit. As should beclear from the preceding discussion, there are two distinct regimes, of ξ ∼ 1 and ofξ ∼ M, which should be analyzed separately. It turns out that an additional scalearises in the UV at ξ ∼M2.3.4.1 Small ξIn the limit M→∞ and ω ∼O(1), the sums in (3.37) can be replaced by integrals,which leads to massive cancellations. This is not unexpected, since the solutionto the Wiener-Hopf problem is designed so as to subtract the singularities of thescaling function in the upper half of the complex plane. These singularities are aseries of poles (fig. 3.2), which at M →∞ collapse onto the real axis and collidewith the poles in lower half plane. In effect all the singularities happen to besubtracted, leaving behind a subleading contribution without poles, whose only58singularity is the square-root cut in the lower half-plane:gˆ(ω) ≃ M2√ipiωeωpii (ln ω2pii−1)Γ2 (1+ ω2pii) (M→∞, ω ∼ 1) . (3.50)A derivation of this result is given in appendix C.This limit describes the first peak in fig. 3.4. The Fourier transform of (3.50)has more or less the same shape, shown in fig. 3.3, as the whole scaling functionat small M. The function grows as M√ξ , reaches a maximum and then decaysas M/(2√ξ). The important difference with the small-M regime is that now thescaling function is parametrically big, proportional to M, in contradistinction tosmall M when the scaling function is just a small correction to the leading Gaussianresult. The fact that g(ξ) is the leading term now means that at large-M the fulldensity also has a peak; at smaller M the peak is diluted by the growth of the squareroot from the Gaussian approximation, with density growing monotonically awayfrom the endpoint.Notice that we cannot extract the value of the constant C defined in (3.40),(3.44) from this calculation since we tacitly assume that ω ≫ 1/M and conse-quently cannot take the limit ω → 0. To compute C, we need to consider the regimeξ ∼M⇐⇒ω ∼ 1/M, which we do in the next section.For the constants A and B defined in sec. 3.3.3, we getA ≃ 2piMe2, B ≃M (M→∞) . (3.51)These results can also be obtained by applying the formulae from appendix C di-rectly to (3.45), (3.46). We see again that the deviations from the simple Gaussianmodel are parametrically large in the decompactification limit.The regime described here completely determines the Wilson loop expectationvalue at strong coupling, including the normalization factor. The Wilson loop isdual to a fundamental string in the dual supergravity background. Consequently,from the matrix model point of view, the fundamental string probes the extremevicinity of the endpoint of the eigenvalue distribution. The features in the eigen-value density responsible for phase transitions are simply not visible to the funda-mental string probes.593.4.2 Oscillatory behaviorWhen ω ∼O(1/M), corresponding to ξ ∼O(M), the sum is dominated by termswith small n. In this case we simply take the naïve large-M limit of (3.37) (replac-ing eg. M± i by M wherever it appears). For instance, the prefactor in front of thesum in (3.37) becomesM3ω22sin Mω2e− iωM2 (3.52)where we have used the identity (3.31) for the gamma functions.Computation of the sum involves one subtlety. It turns out that we need to keepthe leading correction to the phase φ :ε ≡ pi − φM≃ 2lnMM,vanishing in the M →∞ limit. Yet this quantity leaves a finite imprint in the finalanswer. For the sum we getsum ≃ − 1M2∞∑n=1⎛⎝ e−iεnω − 2pinM + eiεnω + 2pinM ⎞⎠ = 1piM∞∑n=1Mω2pi cosεn− insinεnn2−(Mω2pi )2 (3.53)In the cos term we can set ε = 0 right away, because the sum converges. The sinterm does not contribute at first sight, but the sum converges slowly, and the limitε → 0 does not commute with summation. Indeed,limε→0∞∑n=1sinεnn= pi2, (3.54)and not zero. Taking this into account, we find thatsum = 1M2ω− e iMω22M sin Mω2. (3.55)Combining all terms together, we get for the scaling function in the large-masslimit:gˆ(ω) ≈ i3/2√pi2ω√ω + iε⎡⎢⎢⎢⎢⎣Mω e− iMω22sin Mω2−1⎤⎥⎥⎥⎥⎦ . (3.56)60Some remarks are in order here. In the limit we are considering, the poles ofthe Green’s functions G±, shown in fig. 3.2, collapse onto the real line and mergepairwise. The unprojected part of the solution (3.37) as a consequence has doublepoles in this limit. But the poles of the exact scaling function in the upper half-plane are eliminated by the + projection, and we expect the limiting scale functionto have only single poles ascending from the lower half-plane. This is exactly whathappens – the double poles get cancelled and the limiting solution has a sequenceof single poles along the real line. Their origin in the lower half-plane defines theepsilon-prescription for integrating the scaling function over frequencies.The contour of integration in the inverse Fourier transform (3.22) thus passesall the poles from above. For ξ > 0, the contour can be closed in the lower half-plane, picking up the poles along the real axis and wrapping the branch cut alongthe negative imaginary axis. The scaling function becomes a sum of two terms:g(ξ) = h0(ξ)+h1(ξ) (3.57)Let us consider the branch cut contribution first:h1(ξ) = √M2√pi ∫ ∞0 du e−uξMu3/2 (1− ueu−1) = −√ξ −√M2ζ (12,1+ ξM) , (3.58)where ζ(s,a) is the Hurwitz zeta function. Despite its appearance, the functionh1(ξ) is always positive, starts off as a constant at ξ = 0 and decays monotonicallywith ξ , asymptoting to the 1/√ξ tail at infinity.The poles giveh0(ξ) = −i∑n≠0Res[g(ω)e−iωξ , ω = 2pinM ] =√iM8∑n≠0e−2piinξ/M√n. (3.59)This function is obviously periodic with period M. In order to expose the period-icity, it is convenient to decompose ξ /M on the integer and fractional parts:{ ξM} = ξMmod1, [ ξM] = ξM−{ ξM} , (3.60)keeping in mind that h0 only depends on the fractional part. The sum can again be611 2 3 4 5 6 7 xêM-ê M , h0HxLê M(a)0 5 10 15 20 ΐM12345fHΞL M(b)Figure 3.5: (a) The scaling function g(ξ) (upper, blue curve) and the periodic func-tions h0(ξ) (lower, purple curve). At larger ξ , h0(ξ) becomes better and betterapproximation to g(ξ). (b) The eigenvalue density near the endpoint forms acomb-like structure with an infinite series of resonances on top of the leading-order square-root distribution, shown as a purple curve.expressed in terms of the Hurwitz zeta function:h0(ξ) = √M2 ζ (12 ,{ ξM}) . (3.61)Since h0(ξ) is periodic and h1(ξ) decreases at infinity, their sum asymptotesto h0(ξ) at ξ ≫ M. This is illustrated in fig. 3.5(a). The sum of h0 and h1 can be62Figure 3.6: Cusp-like structure of g(ξ) in the regime ξ ∼ O(M) for largeM. Here we compare our analytic result (3.62) with numerics (red) forM = 100.actually simplified with the help of the zeta-function identities:g(ξ) = √M2[ ξM ]∑k=01√{ ξM}+k −√ξ , (3.62)and, as we can see in fig. 3.6, it agrees well with the numeric evaluation of the sum(3.37) for M = 100. As for the scaling form of the density, we get a particularlysimple expression:f (ξ) = √M2[ ξM ]∑k=01√{ ξM}+k . (3.63)The function h0(ξ), and with it the scaling function g(ξ), blows up as M/(2√ξ)at ξ → 0. This behavior matches with the 1/√ξ tail at the upper end of the small-ξregime, which describes the first peak of the density. The change of the endpointexponent from +1/2 to −1/2 is characteristic of the infinite-volume limit, and isuniversally observed in all massive theories that can be solved on S4 by localiza-tion [67, 68, 69, 78]. But h0(ξ) is also periodic, and consequently has inversesquare-root singularities in all integer points ξ = nM. These are the resonancesthat arise due to the presence of nearly massless hypermultiplets in the spectrum.63Our analysis applies to strictly infinite coupling. Varying the coupling will causethe resonances to move, resulting in phase transitions each time a full interval istraversed and a new cusp (dis)appears in the density function.The density behaves asf (ξ) ≃ M2√ξ −nM (ξ → nM+) , (3.64)to the right of each resonance, and approaches a finite limiting value from theleft. This structure is qualitatively similar to the one previously observed at finitecoupling in the vicinity of the first phase transition [68]. But now we have ananalytic solution that describes the whole resonance structure.To move beyond the regime ξ ∼ O(M), we should recall that the poles ingˆ(ω) are really located slightly off the real axis, at ω = ± 2pinM±i . This displacementmeans the phase in the definition (3.59) of h0(ξ) acquires an imaginary compo-nent 2pii∣n∣ξ /M2, which is unimportant for ω ∼O(1/M), but for large ξ ∼O(M2)causes the peaks to decay exponentially. At these large scales the oscillations in thedensity die out and the leading contribution is the 1/√ξ tail of the function h1(ξ):h1(ξ) ≃ M4√ξ(ξ →∞) (3.65)This behavior determines the constant C defined in (3.40). Comparing (3.56) atω → 0 to (3.44) we find:C ≃ M4(M→∞) . (3.66)This result can also be obtained directly from (3.47) by methods outlined in ap-pendix C.To summarize, the overall picture of the eigenvalue density that emerges in thedecompactification limit is as follows.– Square root behavior at the extreme endpoint given byρ(ξ) = 2 32piµ32M√ξ (ξ ∼ 1). (3.67)The density reaches a peak while ξ ≪ M (corresponding to the overall peak64(a) M = 10 (b) M = 100Figure 3.7: Endpoint behavior of the eigenvalue density, showing theleading-order semi-circle solution together with a numerical evaluationof the first-order Wiener-Hopf correction for M = 10 and M = 100in the small mass solution), and decays thereafter as M/(2√ξ).– When the density is scaled to larger ξ ∼ M, the peak is not resolved anymore and becomes a cusp, thus changing the endpoint behavior of the densityfrom√ξ to 1/√ξ . Moreover, the density develops secondary cusps at theresonance points ξ = nM, and thus acquires a comb-like shape with cuspsseparated by M which are superimposed on the leading square root function.We were able to find the precise analytic form of the density in this regime:ρ(ξ) = √2Mpiµ3/2[ ξM ]∑k=01√{ ξM}+k . (3.68)– At yet larger ξ , of order M2, the amplitude of these resonances decays ex-ponentially, leaving a small 1/√ξ correction to the leading-order Wignerdistribution.– In the bulk of the eigenvalue distribution the density is given by (3.11). Theconstant β that controls the overall size of the correction is determined in thenext section.653.5 Strong-coupling expansionWe now return to the question of 1/√λ corrections. As we have shown before,the functional form of the correction to the density is fixed by the saddle-pointequations, but its overall normalization is not. The normalization constant canbe determined by matching the bulk density to the exact scaling solution near theendpoints of the distribution.Using the parametrization ξ = µ−x, the bulk solution (3.11) takes the followingform near the endpoint:ρbulk = 2 32piµ32⎛⎝√ξ + piβ4√ξ ⎞⎠ , (3.69)where we used (3.10) for the endpoint position. This has to match the asymptoticbehavior of the endpoint solution at large ξ , for which we get combining (3.14),(3.18) with (3.40):ρend = 2 32piµ32(√ξ +g(ξ))→ 2 32piµ32⎛⎝√ξ + C√ξ ⎞⎠ . (3.70)Comparing the two expressions we conclude thatβ = 4Cpi. (3.71)For the bulk eigenvalue density we thus getρbulk(x) = 8piλ (1+M2) √µ2−x2+ 4Cpiµ√µ2−x2 , (3.72)The constant C is given by an infinite sum (3.47), which at small and large Masymptotes toC ≃ M24pi(M→ 0), C ≃ M4(M→∞). (3.73)We can now determine the correction to the endpoint position, by imposing the66Figure 3.8: Fitting the curve µ(M) (3.77) to the numerical data.normalization condition on the density:∫ µ−µ dxρbulk(x) = 1, (3.74)which becomes4pi2µ2λ (M2+1) + 4Cµ = 1, (3.75)and we find:µ = √λ (1+M2)2pi−2C+O(λ−1/2). (3.76)The first strong-coupling correction to µ is thus simply related to the slope of thescaling function at large ξ .This latter relation can be checked by numerically solving the exact saddle-point equation (3.3). The method we used becomes unstable at strong couplingunless M is sufficienly small, so we restricted numerical analysis to M < 0.5, wherewe can use the small-M asymptotics (3.49) for C. We fitted the data points to:µ(M) = a√M2+1+bM2. (3.77)The results are shown in fig. 3.8, and are in perfect agreement with (3.76):67Figure 3.9: Density close to the endpoint, for M = 0.5. The first plot comparesthe endpoint solution from Wiener-Hopf method and the direct numericsolution. The second plot shows the matching condition of the Wiener-Hopf solution with the bulk solution (3.11).Analytic Values Numerical Fita 44.7214 44.7221 ± 0.0004b -0.1591 -0.1593 ± 0.0045In fig. 3.9 we compare the numerical results for the density with the Wiener-Hopfsolution in the endpoint region.The correction to the endpoint position affects the normalization of the Wilsonloop expectation value, through its exponential dependence on µ in (3.43). Takingthis correction into account, we get for the Wilson loop expectation value:W(C) ≃√ 2piAe−4piCλ34 (1+M2) 34 e√λ(1+M2), (3.78)where the constants A and C are given in (3.45), (3.47). In the decompactificationlimit, we get:W(C) ≃√ 8piMRλ− 34 e(√λ−pi)MR−2, (3.79)where we have reinstated the dependence on the radius of the four-sphere throughthe rescaling M→MR.The leading exponential term should be universal, implying that any suffi-ciently big Wilson loop in the N = 2∗ theory on R4 should obey the perimeter68law:lnW(C) = P(λ)ML (ML ≫ 1), (3.80)where L is the length of the contour C. The coefficient P(λ) governs the self-energy of an infinitely heavy quark immersed in the N = 2∗ vacuum. From (3.79)we find that at strong coupling the self-energy coefficient behaves asP(λ) = √λ2pi− 12+O( 1√λ) . (3.81)The leading order term was computed in [66] and successfully compared to the arealaw in the Pilch-Warner geometry. The second term constitutes a prediction for thefirst quantum correction to the minimal area. It should be possible to compute thiscorrection by semiclassical quantization of the string dual to the straight Wilsonline.The prefactor in (3.79) is in principle calculable by quantizing the string sus-pended on the big circle in the spherical geometry. The supergravity solution inthis case is also known [70], but unfortunately only in the five-dimensional form.To compute the Wilson loop one needs to know the string action, determined bythe ten-dimensional uplift of the solution.The prefactor in the Wilson loop expectation value (3.79) is a contour-dependentquantity. The known ten-dimensional dual of the N = 2∗ theory on R4 is of littlehelp for computing this number. However, the square-root scaling of the prefac-tor with the size of the contour may be universal and apply to any sufficiently bigWilson loop. The leading finite-size correction to (3.80) is then logarithmic withprecisely known coefficient: δfin.size lnW(C) = −(1/2) lnL.3.6 ConclusionsWe studied the strong-coupling planar limit of N = 2∗ theory compactified on S4.The supergravity approximation should be accurate in this regime, and since thesupergravity dual of N = 2∗ SYM on S4 is explicitly known1, our calculations canpotentially be compared to semiclassical string theory on the dual supergravity1An explicit analytic solution is known at infinite radius of the sphere [14], otherwise the problemreduces to a set of ODEs that can be integrated numerically [70].69background. The exponent in the Wilson loop expectation value (3.78) should thencorrespond to the area of the surface bounded by the big circle of S4. The prefactorcan be identified with the one-loop determinant due to string fluctuations aroundthe minimal surface.Interestingly, all the non-trivial features that appear at strictly infinite volume,such as consecutive phase transitions, are visible already at the first order of thestrong-coupling expansion, and it would be really interesting to explore their coun-terparts on the string theory side.As we have seen, the Wilson loop expectation value is sensitive to the imme-diate vicinity of the endpoint in the eigenvalue distribution, namely to distances oforder one in x space. In contrast, the spikes that arise in the decompactificationlimit are located at distances of order O(M). Thus Wilson loops, or semiclassi-cal strings, are not sensitive to the phase transitions, simply because they probe adifferent corner in the parameter space.Perhaps better probes are D-branes. It is known that the eigenvalue distributionas a whole can be derived from the D-brane probe analysis [77]; the question isto resolve the region distance M away from the endpoints, where the density hasnon-trivial features in the decompactification limit. We emphasize that while thedecompactification limit implies that M ≫ 1, it always remains true that M ≪ µ be-cause of the strong coupling. The D-brane probe analysis identifies the eigenvaluedistribution with a particular locus in the dual geometry. We can now pinpointexactly which parts of the eigenvalue distribution are responsible for the phasetransitions, and we can even compute the fine structure of the eigenvalue density inthis region. We can thus say that the critical behavior is associated with a particu-lar location of the 10d space-time. It would be very interesting to understand whattriggers the phase transitions in string theory.70Chapter 4The Schwinger effect fromworldline instantons4.1 IntroductionSchwinger’s famous formula [18] for what is known as the “Schwinger effect”gives the probability of the production of charged particle-antiparticle pairs by aconstant external electric field asP = 1−e−γV (4.1)where, for spin zero particles, the exponent is given byγ = ∞∑n=1(−1)(n+1)E28pi3n2e−pim2n/∣E ∣ (4.2)Here m is the mass of the particles, V is the space-time volume and E is the electricfield. We have absorbed a factor of the particle charge into E.The result (4.2) is obtained by evaluating the vacuum persistence amplitude ina theory with a charged massive scalar field exposed to an external electric field.The phase in the persistence amplitude, which normally contains the vacuum en-ergy, obtains an imaginary part. This gives a damping of the amplitude which isattributed to the production of charged particle-antiparticle pairs. As we demon-71strate in the next subsection, the problem of finding the damping rate can be posedas that of evaluating the imaginary part of the worldline path integral for the rela-tivistic particle,γ = −2I 1V ∫ ∞0 dTT ∫ [dxµ(τ)]e−∫ 10 dτ[ T4 x˙µ(τ)x˙µ(τ)+Ex1(τ)x˙2(τ)]−m2T (4.3)The integral is over periodic paths, xµ(τ + 1) = xµ(τ) and the space-time metrichas Euclidean signature. The variable T as we use it here is the inverse of whatis normally referred to as the “Schwinger proper time”. Of course, a functionalintegral such as (4.3) must be defined with care. In this chapter, we will use zeta-function regularization in order to define the formally divergent infinite productsand infinite summations which are encountered in the course of computing (4.3).Basic formulae involving zeta functions are summarized in Appendix D. In thenext subsection we give a detailed review of the derivation of this path integralformula from the usual Feynman diagram representation of the vacuum persistenceamplitude. In particular, we demonstrate that the path integral (4.3) with E = 0 andwith zeta function regularization reproduces the Feynman diagram expression forthe vacuum energy in all of its details, including its normalization.The path integral in (4.3) can be evaluated. The real part can be presented asan integral over one variable and the imaginary part can be found to coincide with(4.2). The way that it is solved is to first perform the Gaussian integration over theposition variables xµ(τ) in (4.3). In this integration, the instability of the vacuumstate of the system of charged particles when a constant electric field is applied isreflected by the subtlety that the quadratic form in the Gaussian functional integralis not positive for all values of T . This is what allows this integral of a real functionover real variables to have an imaginary part. The integral is done by first assumingthat T is in a region where the Gaussian is stable, doing the Gaussian functionalintegral over xµ(τ) and then defining the result of the integral for all values of Tby analytic continuation. The presence of values of T where the path integral wasunstable is then reflected as singularities in the remaining integration variable, T .In this case, the singularities are simple poles on the real T -axis which must bedefined carefully to take causal boundary conditions into account. The imaginarypart of the integral then comes from the sum over the residues of the poles. This72yields the infinite series quoted in (4.2) above. This is straightforward. It leads tothe exponents in the individual terms in (4.2) and, with some care in normalizingthe Gaussian functional integral involved, to the exact pre-factors in (4.2).There is another approach to computing the imaginary part of the path integral,which is less efficient, but it is often used as a starting point for computations of therate of particle production in the more general situation where the electric field isnot constant [79, 80, 81, 82]. It has also been used to discuss pair production in thecontext of AdS/CFT holography [83]. This approach is a conventional semiclas-sical evaluation of the path integral. It is generally good when the particle massis large compared to other dimensionful parameters. In our case, the parameterwhich controls the semiclassical limit is the dimensionless ratio of the electric fieldstrength to the mass squared, Em2 , which is small in the “weak field limit”. In thislimit, we treat both xµ(τ) and T as dynamical variables and solve the classicalequations of motion which follow from the worldline action,S = ∫ 10dτ [T4x˙µ(τ)2+Ex1(τ)x˙2(τ)+ m2T ] (4.4)where x˙µ ≡ ddτ xµ(τ). The classical solutions, which we denote as x0µ(τ) and T0,are a saddle point of the path integral integrand. We then compute the integral bysaddle point technique which amounts to changing integration variables asT → T0+δT (4.5)xµ(τ)→ x0µ(τ)+δxµ(τ) (4.6)and implementing perturbation theory in the fluctuations δT and δxµ(τ). Thisturns out to be an expansion in the parameter√Em2 and the expansion is valid inthe regime where this parameter is small.There is a beautiful observation, due to Affleck, Alvarez and Manton [22] thatthe classical solutions that are relevant to the Schwinger process can be interpretedas instantons. The n’th term in the summation in the Schwinger formula (4.2) canbe interpreted as a n-instanton amplitude in such a semi-classical computation of73the path integral. They showed explicitly that the first, n = 1 term in (4.2),γ1 = E28pi3 e−pim2/∣E ∣ (4.7)is obtained exactly by such a semi-classical computation where they expand abouta one-instanton solution of the classical equations of motion for xµ(τ) and T . Theexponent in (4.7) is the classical action of the instanton. The pre-factor is givenby the Gaussian integral over fluctuations about the classical solution at the lead-ing, quadratic order. It is interesting that, in the computation presented by Affleck,Alvarez and Manton, the integral is given exactly by what amounts to the lead-ing orders of an approximation. If it were an approximation, the small parameterwhich suppresses corrections would be√Em2 . However, given that, in the leadingorders they already obtained the exact result, as they noted, but did not demon-strate, higher order perturbative corrections should then cancel exactly. This wouldmean that the computation has a much larger regime where it is valid, in principlefor all values of√Em2 .The nature of the instanton is easy to understand. In Euclidean space, a Minkowskispace electric field behaves as a magnetic field. In a magnetic field, the classicalcharged particle has a cyclotron orbit. The one-instanton solution is a single cy-clotron orbit. The exponent of (4.7) is simply the classical action of the world linetheory evaluated on this orbit. The pre-factor in (4.7) is given by evaluating theGaussian integral over the fluctuations about this classical solution.In this approach, the path integral gets an imaginary part due to the fact that theinstantons in question are unstable solutions of the classical worldline theory. Theunstable fluctuation turns out to be the fluctuation of the radius of the cyclotronorbit. The Gaussian integral over the fluctuations, including the fluctuation of theradius, then produces the square root of a determinant of a matrix which has an oddnumber of negative eigenvalues, thus the factor of “i”.As well as a single instanton that leads to (4.7), there are an infinite series ofmulti-instanton classical solutions which are simply the multiple cyclotron orbits.74In the following, we shall show that all of the higher terms in (4.2),γn = (−1)n+1 E28pi3n2 e−pim2n/∣E ∣ , (4.8)with n = 2,3, . . . are produced by multi-instantons with higher wrapping number.It is easy to see (and already well known) that the exponent of the n’th term asdisplayed in (4.8) is the classical action of the n-instanton solution. What we shallshow is that the fluctuation integral produces the pre-factor of the exponential ex-actly. This has the interesting implication that the full, exact result is obtainedin the semi-classical Gaussian approximation of the worldline path integral whereone sums over all of the classical solutions. Of course, the Gaussian approximationnormally has corrections coming from expanding in the higher order non-Gaussianterms in the action, as well as corrections from an expansion about the classicalsolution of the terms which appear in the integration measure. Such terms are in-deed at least formally present in this semiclassical expansion. What we shall findhere, that the leading approximation produces the exact result, implies that the cor-rections must cancel. We shall then give a proof that this is indeed the case: allsuch corrections vanish. The proof takes advantage of an “accidental” fermionicsymmetry of the gauge-fixed action to localize the path integral on its semiclassicallimit [84] (we also present an alternative proof involving a simple scaling argumenttogether with a change of variables, originally published in [85]). This proof ex-pands the range of validity of the semiclassical computation from the weak fieldlimit to the strong field regime. Whether this can help computations in less idealproblems, for example, where the electric field is not constant, is at this point anopen question.In section 4.2, we shall perform the semiclassical computation of the path inte-gral in equation (4.3) in the n-instanton sector. We shall define the infinite productsand sums which we encounter using zeta function regularization. We show that,by careful treatment of the functional integration measure, the n’th term in theSchwinger formula (4.2), including the exact pre-factor, is obtained.In section 4.3, we examine higher order corrections beyond the leading orderin the saddle point approximation. We find a proof that all corrections beyond theintegration of quadratic fluctuations must vanish. The result is that, for computing75Figure 4.1: The Feynman diagram which must be computed to find thevacuum energy of a scalar field.the imaginary part of the vacuum persistence amplitude, the semiclassical limit ofthe worldline path integral with an external electric field is exact.The definitions and values of the relevant zeta functions are summarized inAppendix D. In Appendix F we demonstrate the semiclassical technique that weuse on a simple example. In Appendix E we give an alternative, brief perturbativeproof that all corrections to the semiclassical approximation vanish.4.1.1 Worldline path integralsIn this subsection, as a warm-up we will demonstrate that the one-loop vacuumenergy of a scalar particle is given exactly by the worldline path integral when wedefine the various infinite products and sums which occur in the latter using zetafunction regularization. The result will be equation (4.20).We begin with the usual expression for the vacuum energy density of a complexscalar field in Euclidean spaceΓ = ∫ d4 p(2pi)4 ln(p2+m2) (4.9)which is represented by the Feynman diagram in figure 4.1. Recall that a realscalar field would be the same expression with a factor of 1/2 in front. At thislevel, a complex scalar field is simply two real scalar fields which have twice asmuch vacuum energy. Here, the four-momentum is Euclidean and a space-timevolume factor has been removed so that the result of doing the integral is the energy76density. We will reorganize this integral in order to represent it as a worldline pathintegral. The singular integrals which we encounter will de defined by zeta functionregularization. To proceed, we first introduce a Schwinger parameter, T ,Γ = ∫ ∞0dTT ∫ d4 p(2pi)4 e−T(p2+m2) (4.10)We note that, in four dimensions, this integral contains an ultraviolet divergence,coming from the T ∼ 0 integration region. This divergence must be regulated inorder to obtain a sensible definition of the vacuum energy. We could regulate thisexpression by defining it as the limitln(p2+m2) = limκ→0 ddκ (p2+m2)κ = limκ→0 ddκ 1Γ[κ] ∫ dTT 1+κ e−T(p2+m2) (4.11)The appropriate quantity to study would then be ∫ dTT 1+κ e−T(p2+m2) with the expo-nent of T shifted by κ which we could always take as negative with large enoughmagnitude that the integrals to be done converge, and then define the quantity inthe region near κ = 0 by analytic continuation. In the following, we shall assumethat this regulator is implicitly there, if needed but we will stick with expression(4.10) as the writing will be slightly simpler.Now, consider the functional integral∫ [dxµ(τ)] ei∫ 10 dτ pµ(τ) ddτ xµ(τ) (4.12)where both x(τ), p(τ) have periodic boundary conditions, xµ(τ +1) = xµ(τ) andpµ(τ +1) = pµ(τ).It is very convenient to use the expansions of the integration variables in adiscrete orthonormal complete set of periodic functions,pµ(τ) = pµ + ∞∑k=1[pµk√2sin(2pikτ)+ p˜µk√2cos(2pikτ)] (4.13)77andxµ(τ) = xµ + ∞∑k=1[xµk √2sin(2pikτ)+ x˜µk √2cos(2pikτ)] (4.14)The complete set of orthonormal periodic functions is (1,√2sin(2pikτ),√2cos(2pikτ)),where the normalization is the square-integral over the interval τ ∈ [0,1]. The func-tional integration measure is then defined as the ordinary Riemann integral overeach of an infinite number of real variables,[dxµ(τ)] ≡ dxµ ∞∏k=1 dxµk dx˜µk , [d pµ(τ)] ≡ d pµ ∞∏k=1 d pµkd p˜µk (4.15)Now, with these definitions, consider∫ [dxµ(τ)]exp(i∫ 10dτ pµ(τ) ddτ xµ(τ))= ∫ dxµ ∞∏k=1 dxµk dx˜µk exp(i ∞∑k=1(2pik)[pµkx˜µk − p˜µkxµk ])=V 4∏µ=1∞∏k=1(2pi)δ((2pik)pµk) ⋅(2pi)δ((2pik) p˜µk)=V ( 1∏∞1 k)8 ∞∏k=1δ(pµk)δ( p˜µk) =V ( 1exp(−ζ ′(s)))8 ∞∏k=1δ(pµk)δ(p˜µk)=V 1(2pi)4 ∞∏k=1δ(pµk)δ(p˜µk)where we have usedlims→0∞∏1k = lims→0 exp(− dds ∞∑k=1k−s) = e−ζ ′(0)and V ≡ ∫ dxµ is the (infinite) space-time volume arising from the integral over theconstant mode xµ and the fact that it does not appear in the integrand.The identity∫ [dxµ(τ)]exp(i∫ 10dτ pµ(τ) ddτ xµ(τ)) =V 1(2pi)4 ∞∏k=1δ(pµk)δ(p˜µk) (4.16)78tells us that, if we first do the integral over xµ(τ) in the following path functionalintegral∫ d4 p(2pi)4 e−T(p2+m2) = 1V ∫ [dxµ(τ)][d pµ(τ)] e∫ 10 [ipµ(τ) ddτ xµ(τ)−T(pµ(τ)pµ(τ)+m2)](4.17)it will generate a factor of V and delta functions for all of the nonzero modes ofpµ(τ). The integrals over those nonzero modes can then be done, leaving the inte-gral over the constant mode in pµ(τ) which becomes the momentum that appearson the left-hand-side of the equation. Now, we reorganize the right-hand-side of(4.17) by doing the functional integral over pµ(τ).∫ [d pµ(τ)] e∫ 10 [ipµ(τ) ddτ xµ(τ)−T(p2µ(τ)+m2)]= ∫ d pµ ∞∏k=1 d pµkd p˜νk e∑∞k=1[(2piki)[pµk x˜µk − p˜µkxµk ]−T(p2µk+ p˜2µk)]−T p2=¿ÁÁÁÀpiT(∞∏k=1piT)2e− 14T ∑∞k=1[(2pikxµk )2+(2pikx˜µk )2] = e− 14T ∫ 10 dτ( ddτ xµ(τ))2 (4.18)where we have used the fact that the pre-factor is√piT (∏∞k=1 piT )2 = ( piT )ζ(0)+1/2 = 1.The result is then∫ d4 p(2pi)4 e−T(p2+m2) = 1V ∫ [dxµ(τ)]e∫ 10 [− 14T x˙µ(τ)2−T m2] (4.19)where the dot denotes τ-derivative and we have the path integral formula for thevacuum energy density of a complex scalar fieldΓ = 1V ∫ ∞0 dTT ∫ [dxµ(τ)] e−∫ 10 dτ[ 14T x˙µ(τ)2+T m2] , xµ(τ +1) = xµ(τ) (4.20)What we have shown is that, if one uses zeta function regularization, equation(4.20) is an identity.We can check this identity by doing the path integral directly. We begin by do-79ing the functional integral in the worldline expression (4.20) that we have derived.The integral over the constant mode of xµ(τ) gives a volume factor which cancelsthe factor of 1/V . We can then use the rules for doing Gaussian integrals to do thequadratic functional integral over nonzero modes of xµ(τ). The result isΓ = ∫ ∞0dTTe−T m2 [∞∏k=12pi ⋅2T(2pik)2 ]4 = ∫ ∞0 dTT e−T m2 [4piT ]4ζ(0)= ∫ ∞0dTTe−T m2 1(4piT)2 (4.21)Again, we have used zeta function regularization to define the infinite products.The result is identical to what is obtained by integrating (4.10) over p.Note that the variable T here is the inverse of the one that we use elsewhere(and can simply be gotten by performing the change of variable T → 1/T ).Now, consider the vacuum energy of a charged scalar particle coupled to anelectromagnetic field whose vector potential is Aµ(x). The coupling is imple-mented in the Euclidean functional integral by including the Bohm-Aharonov phasefactor, to obtainΓ = 1V ∫ ∞0 dTT ∫ 10 [dxµ(τ)] e−∫ 10 dτ[ 14T x˙µ(τ)2+T m2]+i∮ dτ x˙µ(τ)Aµ(x(τ)) (4.22)In particular, if we couple to an external constant electric field E, the gauge fieldcould be taken asAµ(x) = (0, iEx1(τ),0,0) (4.23)A physical electric field obtains a factor of i in Euclidean space. Now the pathintegral would be the Gaussian integralΓ = 1V ∫ ∞0 dTT ∫ [dxµ(τ)] e−∫ 10 [ 14T x˙µ(τ)x˙µ(τ)+Ex1(τ)x˙2(τ)+T m2] (4.24)This integral should yield the vacuum energy of a charged scalar field which iscoupled to an external electric field.804.2 Effective action from semiclassical worldline pathintegralLet us now study the case of a spinless charged particle of mass m which is subjectto a constant external electric field. Its vacuum energy is given by the worldlinepath integral (4.3). The instability of the vacuum to the production of on-shellparticle-antiparticle pairs is reflected by the fact that the vacuum energy has animaginary part. We shall compute this imaginary part in a semi-classical expansionabout a classical solution of the worldline theory.To begin, we shall first solve the classical equations of motion which are ob-tained by varying the worldline action by the dynamical variables T and xµ(τ),14m2 ∫ 10 x˙2 = 1T 2 , − T2 x¨1−Ex˙2 = 0 , − T2 x¨2+Ex˙1 = 0 , − T2 x¨3,4 = 0 (4.25)with periodic boundary conditions, xµ(τ +1) = xµ(τ). The solutions of these equa-tions arex1 = mE cos2pinτ , x2 = mE sin2pinτ , x3,4 = 0 , T = Epin (4.26)which we interpret as the n-instanton solution. Plugging these solutions into theaction (4.4) gives Scl. = pinm2E , the same expression which appears in the exponentsof the terms in (4.2).Now, we define the path integration variables as the classical solutions plusfluctuations,x1 = mE cos2pinτ +δx1 , x2 = mE sin2pinτ +δx2 , x3,4 = δx3,4 , T = Epin +δT (4.27)and we expand the action to quadratic order in the fluctuations. We obtainS = pinm2E+ 2m2(pin)3E3δT 22+ m2E(2pin)2δT ∫ dτ(cos(2pinτ)δx1+ sin(2pinτ)δx2)+ E4pin ∫ dτ [δ x˙2−4pinδx1δ x˙2]+ . . .(4.28)81To proceed, we shall use the mode expansionδxµ(τ) = xµ + ∞∑k=1[√2cos(2pikτ)akµ +√2sin(2pikτ)bkµ] (4.29)We first note that the action will not depend of the constant modes xµ . These arespace-time translation zero modes. Their integration will result in the overall factorof the space-time volume V in front of the functional integral.When we substitute (4.29) into (4.28), the action becomesS =pinm2E+ 2m2(pin)3E3δT 22+ m(2pin)22EδT (an1+bn2√2)+ 4pinE2(an1−bn2√2)2+ 4pinE2(an2+bn1√2)2+ E4pin∞∑k=1,≠n(2pik)2 [(a2kµ +b2kµ)− 2nk (ak1bk2−ak2bk1)]+ 14δT∑k(2pik)2[(aµk )2+(bµk )2]+ ∞∑k=3m2(pinE )k+1 (−δT)k (4.30)The last line of (4.30) contain terms of higher order than quadratic in the fluctua-tions. We have written them in this formula for future reference. To the leadingorder that we are studying in this section, they will be neglected.In the previous, quadratic terms in equation (4.30), we have separated the de-grees of freedom (an1,2,bn1,2) which have the same frequency as the classical so-lution and we have written them in the second line. Note that the combinationan2−bn1 does not appear in the quadratic terms in the action – this combination isa zero mode. The existence of the zero mode is due to a symmetry, the translationinvariance in τ of the action. The integration measure, as we shall define it, is alsoinvariant under translations of τ . The worldline theory is thus τ-translation invari-ant. However, the instanton solution depends on τ and it is not invariant. The resultis a zero mode in the fluctuations about the solution.The way to handle the presence of a zero mode is by using the Faddeev-Popov trick to introduce a collective variable. This technique effectively substitutesδ((an2 −bn1)/√2), accompanied by a Jacobian, into the integrand, and it multi-plies the integral by a factor of the volume of the symmetry group, ∫ 10 dτ = 1, in82this case.The introduction of a collective coordinate begins with inserting a factor of oneinto the path integral using the identity1 = 1ω ∫ 10 dtδ(χ(t))∣ ddt χ(t)∣ (4.31)where the function χ(t) should be chosen so that the integration over the zeromode becomes well-defined. Here, ω is the number of solutions of χ(t) = 0 in theinterval t ∈ [0,1]. We shall use the constraintχ(t) = ∫ 10dτ(sin(2pinτ)x1(τ − t)−cos(2pinτ)x2(τ − t)))= 1√2[([ m√2E+an1]sin(2pint)+bn1 cos(2pint))−(an2 cos(2pint)−[ m√2E +bn2]sin(2pint))](4.32)We shall later set t = 0 by translating the time variable in the path integral. Theconstraint reduces to χ(0) = 1√2[bn1−an2] which is what we need to constrain thezero mode.As a function of t, χ(t) = 0 whentan(2pint) = an2−bn1m√2E+an1+ m√2E +bn2This equation is perodic in t and it traverses 2n periods as t varies from zero to one.The fundamental domain (where it traverses one period) can be taken as − 14n < t < 14nand has length 1/2n. This fixes the constant in (4.31) as ω = 2n.The Jacobian evaluated on the constraint is1ω∣ ddtχ(t)∣χ=0 = 2pinω ∫ 10 dτ[cos(2pinτ)x1(τ)+ sin(2pinτ)x2(τ)] = ∣pimE +pi a1n+b2n√2 ∣(4.33)The net effect of this proceedure is the insertion of the delta function and mea-83sure factorδ (an2−bn1√2) [ pimE+pi a1n+b2n√2] (4.34)into the functional integral. This suppresses the integration over the zero modeand, in the leading order where we keep only the classical part of the Jacobian, itinserts the factorpimE(4.35)into the measure.We are now prepared to do the Gaussian integral. The integration of the vari-ables [δT,(an1+bn2√2) ,(an1−bn2√2) ,(an2+bn1√2)]gives the measure factor(2pi)2det− 12⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣2m2(pin)3E3m(2pin)22E 0 0m(2pin)22E 0 0 00 0 4pinE 00 0 0 4pinE⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦= ±i(2pi)2 2Em(2pin)2 14pinE(4.36)where the factor if i arises from the fact that the determinant is negative, and theplus or minus reflects the fact that there is a choice of sign when the square root istaken. (The mode with a negative eigenvalue is called a “tachyon”.)Then, we can integrate over all of the other modes. The result is the infinite84product∞∏k=1(2pi 2pinE )2 (2pik)−4 ∞∏k=1,≠n(2pi 2pinE )2 (2pik)−4⎛⎜⎜⎜⎜⎜⎝det⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣1 0 0 − nk0 1 nk 00 nk 1 0−nk 0 0 1⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦⎞⎟⎟⎟⎟⎟⎠− 12= (2pi 2pinE)4ζ(0)−2 (2pin)4 ∞∏k=1,≠n11− n2k2 = E416pi4∞∏k=1,≠n11− n2k2(4.37)In the above formula and in the following, we define infinite products using zetafunction regularization. Some of the conventions and the zeta functions that areneeded are reviewed in Appendix A.We find the identity∞∏k=1,≠n11− n2k2 = limα→n(1− α2n2 )∏∞k=1(1− α2k2 ) = limα→npiα(1− α2n2 )sinpiα= 2(−1)n+1 (4.38)Gathering measure factors (4.35), (4.36) and (4.38),• The factor of 1T in the integrandpinE• The Faddeev-Popov determinantpimE• The integral over the tachyon(±)i(2pi)2 2Em(2pin)2 14pinE85• The integral over all other modesE416pi4⋅2(−1)n+1we get the resultpimE⋅ pinE⋅(±)i(2pi)2 2Em(2pin)2 14pinE ⋅ E416pi4 ⋅2(−1)n+1 = ±i E216pi3n2 (−1)n+1(4.39)With the appropriate choice of sign, and the factor of 2 from the formula (4.3),we can see that we obtain, as the pre-factor of the exponential of the classicalaction, the factor E28pi3n2 (−1)n+1 which matches the pre-factors of the exponential ineach term in the summation (4.2) exactly. The semiclassical integration has givenus the exact result for the imaginary part of the integral in the n-instanton sector,γn = (−1)n+1 E28pi3n2 e−pim2n/∣E ∣. Summation of the instanton number results in the sumover n which appears in the Schwinger formula.It is interesting that we have produced the imaginary part of the functional in-tegral exactly at this order of what is putatively an approximate computation. Thismeans that all of the higher order corrections to this approximation must cancel.We shall explore this issue in the next section.4.3 No more correctionsNow let us examine the corrections to the saddle point approximation which weperformed in the previous section. Corrections arise from the expansion of theintegrand about the saddle-point. If we expand the non-Gaussian parts of the in-tegrand in a power series in the fluctuations, we can use the functional version ofWick’s theorem to compute the corrections. Since we have already obtained the ex-act result in the next-to-leading order of this expansion, we expect that the higherorder corrections must find a way to vanish. In this Section, we shall prove thatthey indeed vanish. (In subsection 4.3.1 we give an alternative formulation of theproof, avoiding the use of fermionic variables).86Recall that our “gauge-fixing” procedure amounted to inserting the identity1 = 1ω ∫ 10 dtδ (∫ 10 dτ [x˙(n)0µ (τ)xµ(τ + t)])∣ ddt ∫ 10 dτ x˙µ(τ)δxµ(τ + t)∣ (4.40)into the path integral. Upon using translation invariance of the integrand and mea-sure, we can see that this is equivalent to inserting the gauge fixing factor∫ [dbdcdc¯] e∫ 10 dτ[2piiωbx˙(n)0µ (τ)δxµ(τ)+c¯x˙(n)0µ (τ)x˙µ(τ)c] (4.41)into the path integral. Here, the τ-independent variables c and c¯ are anti-commutingFaddeev-Popov ghosts and b is a Lagrange multiplier.We add the exponent in the integrand of equation (4.41) to the action S to getthe “gauge fixed action” Sgf. The worldline path integral (4.24) is nowiΓ = −2 ∞∑n=11V ∫ [d(δxµ(τ))d(δT)dbdcdc¯] 1T (n)0 +δT e−Sgf[x(n)0µ +δxµ ,T (n)0 +δT,b,c,c¯](4.42)where, because it turns out that each instanton sector has an odd number of tachyons,the Euclidean partition function is purely imaginary in all of the instanton sectors(so we have removed the symbol I). When it is expanded about the n-instanton so-lution, the action contains classical, quadratic and higher order (interaction) terms,Sgf = S(n)classical+S(n)quad+S(n)int (4.43)87respectively, andS(n)classical = ∫ 10 dτ⎡⎢⎢⎢⎢⎣T (n)04x˙(n)0µ (τ)2+Ex(n)01 (τ)x˙(n)02 (τ)+ m2T (n)0⎤⎥⎥⎥⎥⎦ = pinm2E(4.44)S(n)quad = ∫ 10 dτ⎡⎢⎢⎢⎢⎣δT2 x˙(n)0µ δ x˙µ +T (n)04δ x˙2µ +Eδx1δ x˙2+ m2δT 2T (n)30 −2piiωbx˙(n)0µ δxµ − c¯cx˙(n)0µ x˙(n)0µ⎤⎥⎥⎥⎥⎦(4.45)S(n)int = ∫ 10 dτ [14δT δ x˙µ(τ)δ x˙µ(τ)− x˙(n)0µ (τ) c¯cδ x˙µ(τ)]+ ∞∑k=3 m2T (n)0(−δT)kT (n)k0(4.46)It is in the remaining integral that we want to show that the interaction terms, Sint,and the term in the measure, 1/(T (n)0 +δT), can be replaced by zero and 1/T (n)0 ,respectively.For this purpose, we define the fermionic transformation∆c¯ = 12δT , ∆(δxµ(τ)) = cx(n)0µ (τ) , ∆c = 0 , ∆b = 0 , ∆(δT) = 0 (4.47)We note that∆2(anything) = 0 , ∫ [d(δxµ(τ))d(δT)dbdcdc¯] ∆(anything) = 0 (4.48)and the three parts of the action are invariant separately,∆S(n)classical = 0 , ∆S(n)quad = 0 ,∆S(n)int = 0. (4.49)Moreover, the interaction terms in the action and in the measure can be seen to be88exact, that is, they areS(n)int = ∆ψ (4.50)ψ = c¯2 ∫ 10 dτδ x˙µ(τ)2−2c¯ ∞∑k=3 m2T (n)0(−δT)k−1T (n)k0 (4.51)1T (n)0 +βδT = 1T (n)0 +∆χ (4.52)χ = c¯ ∞∑k=1(−β)kδT k−1T (n)k0 (4.53)Now, considerI(β ,λ) ≡ ∫ [d(δxµ)d(δT)dbdcdc¯]⎛⎝ 1T (n)0 +βδT⎞⎠ e−S(n)classical−S(n)quad−λS(n)intWe are interested in this integral when the parameters β = 1 and λ = 1. However,we can easily show that it is independent of both β and λ . Consider∂∂λI(β ,λ) = ddλ ∫ [d(δxµ)d(δT)dbdcdc¯]⎛⎝ 1T (n)0 +∆χ⎞⎠ e−S(n)classical−S(n)quad−λ∆ψ= −∫ [d(δxµ)d(δT)dbdcdc¯]⎧⎪⎪⎨⎪⎪⎩⎛⎝ 1T (n)0 +∆χ⎞⎠⎫⎪⎪⎬⎪⎪⎭∆ψ e−S(n)classical−S(n)quad−λ∆(ψ)= −∫ [d(δxµ)d(δT)dbdcdc¯] ∆⎧⎪⎪⎨⎪⎪⎩⎛⎝ 1T (n)0 +∆χ⎞⎠ψe−S(n)classical−S(n)quad−λ∆ψ⎫⎪⎪⎬⎪⎪⎭ = 0Similarly,∂∂βI(β ,λ) = ∂∂β ∫ [d(δxµ)d(δT)dbdcdc¯]⎛⎝ 1T (n)0 +∆χ⎞⎠ e−S(n)classical−S(n)quad−λS(n)int= ∫ [d(δxµ)d(δT)dbdcdc¯] ∆( ddβ χ) e−S(n)classical−S(n)quad−λS(n)int= ∫ [d(δxµ)d(δT)dbdcdc¯] ∆{( ddβ χ) e−S(n)classical−S(n)quad−λS(n)int } = 089and the integralI(β ,λ) = ∫ [d(δxµ)d(δT)dbdcdc¯]⎛⎝ 1T (n)0 +βδT⎞⎠e−S(n)classical−S(n)quad−λS(n)intis independent of the parameters λ and β . Both of these parameters can then bedeformed to zero, yieldingiγ = −2 ∞∑n=11V ∫ [d(δxµ(τ))d(δT)dbdcdc¯] 1T (n)0 e−S(n)classical−S(n)quad (4.54)which, in each instanton sector, contains the classical solution plus quadratic fluc-tuations only. The remaining integral was performed in section 4.2, yielding theknown Schwinger formula (4.2).4.3.1 Another proofWe consider the action (4.30) and we make the change of variablesxµ(τ) = x˜µ(τ˜) , τ˜ = τβ , x˜µ(τ˜) = x˜µ(τ˜ +β) (4.55)T = T˜ /√β (4.56)The path integral measure [dxµ] is invariant under this change of variables. 1 Thescaling of T cancels in the measure of the integral. The worldline action becomes1To demonstrate this, we can show that the following Gaussian integral is independent of β :∫ [dxµ ]e−∫ β0 dτ T4 β x˙2µ , xµ(τ +β) = xµ(τ)Using the mode expansion (4.59) and [dxµ ] = dxµ∏∞n=1 danµdbnµ ,∫ [dxµ ]e−∫ β0 dτ T4 β x˙2µ = ∫ dxµVδD⎛⎝ xµ√β ⎞⎠ ∞∏n=1danµdbnµ exp(− T4β∞∑n=1(2pin)2(a2nµ +b2nµ))=VβD/2 ∞∏n=1[ 4piβ(2pin)2T ]D =V [ T4pi]D/2where we have gauge fixed by inserting 1 = ∫ dXµδD (Xµ − 1β ∫ β0 dτxµ(τ)). This results in thefactor VδD( xµ√β) with V the space-time volume. We have also used zeta function regularization todefine the infinite product. The result does not depend on β . This is so in any dimension D.90(dropping the tildes)S = ∫ β0dτ [√β T4x˙µ(τ)2+Ex1(τ)x˙2(τ)]+√β m2T(4.57)The path integral cannot depend on the parameter β . Moreover the limit whereβ is large is the semiclassical limit. In the following we shall take this limit withsome care to show that it indeed projects the full path integral to the semiclassicalone which we computed in the previous section, where we only kept the classicaland Gaussian terms in the action and the classical terms in the integration measure.Again, we shall expand the integration variables about the classical solution,x1(τ) = mE cos 2pinτβ +δx1(τ) , x2(τ) = mE sin 2pinτβ +δx2(τ)x3,4(τ) = δx3,4(τ) , T =√β Epin +δT (4.58)with the fluctuations expanded asδxµ(τ) = xµ√β+ ∞∑k=1[√2βcos2pikτβakµ +√ 2β sin 2pikτβ bkµ] (4.59)The measure in the path integral is nowdδT√β Epin +δT Vδ ⎛⎝ xµ√β ⎞⎠δ ⎛⎝bn1−an2√2β ⎞⎠⎡⎢⎢⎢⎣pimE + pi√β (an1+bn2√2 )⎤⎥⎥⎥⎦4∏µ=1dxµ∞∏k=1dakµdbkµ(4.60)91The action becomesS =pinm2E+ 2m2(pin)3βE3δT 22+ m(2pin)22βEδT (an1+bn2√2)+ 4pinE2β(an1−bn2√2)2+ 4pinE2β(an2+bn1√2)2+ E4pinβ∞∑k=1,≠n(2pik)2 [(a2kµ +b2kµ)− 2nk (ak1bk2−ak2bk1)]+ 14β32δT∞∑k=1(2pik)2[(aµk )2+(bµk )2]+∞∑k=3m21βk2(pinE)k+1 (−δT)k (4.61)Now, in order to make the quadratic terms β -independent, we rescaleδxµ(τ)→√βδxµ(τ) (4.62)The Jacobian for this transformation is (√β)4+8ζ(0) =1 where we have used ζ(0)=−1/2. The integration measure becomesdδT√β Epin +δT Vδ (xµ)δ (bn1−an2√2 )[pimE +pi (an1+bn2√2 )]4∏µ=1dxµ∞∏k=1 dakµdbkµand the action isS =pinm2E+ 2m2(pin)3βE3δT 22+ m(2pin)22E√βδT (an1+bn2√2)+ 4pinE2(an1−bn2√2)2+ 4pinE2(an2+bn1√2)2+ E4pin∞∑k=1,≠n(2pik)2 [(a2kµ +b2kµ)− 2nk (ak1bk2−ak2bk1)]+ 14β12δT∞∑k=1(2pik)2[(aµk )2+(bµk )2]+∞∑k=3m21βk2(pinE)k+1 (−δT)k (4.63)As it stands, we cannot directly set β to infinity; in this limit the measure diverges,as does the integral over (an1+bn2√2) and δT . Therefore we further rescale the single92mode v,v→√β v, where v ≡ (an1+bn2√2) . (4.64)This modifies the measure todδTEpin + δT√β Vδ (xµ)δ (bn1−an2√2)[pimE+√βpi (an1+bn2√2)] 4∏µ=1dxµ∞∏k=1 dakµdbkµ .(4.65)Now consider the integral over the mode v, which after the above rescaling be-comes. . .∫ dv(pimE +√β piv)e− 2(pin)2mE δT(v+√β E2mv2) . . . . (4.66)The ellipsis stands for the rest of the path integral. In terms of the variableξ ≡ v+√βE2mv2 (4.67)this is justpimE ∫ dξ e− 2(pin)2mE ξ ⋅δT , (4.68)which demonstrates that we can simply drop the terms proportional to β+ 12 in themeasure and action. Equation (4.67) is the “Nicolai map” that reduces this factorof the path integral to Gaussian form.The remaining corrections to the semiclassical approximation, in both the mea-sure and the action, are suppressed by powers of√β . Now, we remember that theintegral is independent of β . The original integral that we computed was for thecase β = 1. Assuming smooth behavior in β , we can set the original integral equalto the limit of the above as β →∞. In that limit, the interaction terms in the actionand in the measure go to zero and the integral is reduced to the Gaussian one whichwe have already computed in the previous section where we found that it gives theexact result.934.4 DiscussionIn conclusion, we note that there are circumstances where the worldline path inte-gral in the presence of more general, non-constant electric fields is thought to beexact [86, 87]. Although we shall not do so here, it would be very interesting tounderstand whether our results could be extended to those cases.One generalization which our results can be considered a preparation for is theinclusion of dynamical gauge fields. That could be done by including the Wilsonloop in the word-line path integral,Γ = 1V ∫ ∞0 dTT ∫ 10 [dxµ(τ)] e−∫ 10 dτ[ 14T x˙µ(τ)2+ 12 Fµνxµ(τ)x˙ν(τ)+T m2] ⟨ei∮ dτ x˙µ(τ)Aµ(x(τ))⟩(4.69)where the bracket is the expectation value of the operator in the relevant quan-tum field theory and we have separated a constant background field Fµν fromthe fluctuating gauge field of the quantum field theory. The expectation value,⟨ei∮ dτ x˙µ(τ)Aµ(x(τ))⟩ is a functional of the trajectory xµ(τ). A semi-classical ap-proximation to the amplitude begins with seeking a solution of the “classical” equa-tion of motion, which now must be derived from the action including the Wilsonloop. The latter provides a potential whose derivative is a force term which appearsin the equation of motion of the particle− 12Tx¨µ(τ)+Fµν x˙ν(τ) = δδxµln⟨ei∮ dτ x˙µ(τ)Aµ(x(τ))⟩By symmetry, in a Euclidean rotation invariant field theory, due to the symmetryof a circle under rotations about its centre,δδxµ(τ) ln⟨ei∮ dτ x˙µ(τ)Aµ(x(τ))⟩∣xµ=circle = 0In an external electric or magnetic field, knowing that the circle trajectory is still aclassical solution, and the understanding that in the absence of gauge field fluctua-tions, the semi-classical expansion beginning with the circle trajectory leads to thecorrect result for the Schwinger formula in the n-instanton section provides a start-ing point for studying corrections from quantum fluctuations of the gauge fields.94This idea was first exploited by Affleck, Alvarez and Manton [22] to compute theleading correction from photon exchange and it has recently been used to study thestrong coupling limit of the Schwinger formula and the behaviour of heavy quarksin electric fields in the context of AdS/CFT holography [83, 88].95Chapter 5String pair production in abackground field from worldsheetinstantons5.1 IntroductionWe recall Schwinger’s formula (4.2), which we generalize here to D space dimen-sions, for the rate of vacuum decay, per unit volume per unit time, by the productionof charged particle-anti-particle pairs in a constant electric field:Γparticle = (2J+1)pi ∞∑k=1(−1)(k+1)(2J+1) [ E4pi2 k]D+12e− pim2E k. (5.1)E is the strength of the electric field and J and m are the spin and mass of theparticle. We have absorbed the electric charge into the field E. This is one of thefew non-perturbative formulae describing a quantum field theory process whichcan be obtained exactly, normally by computing the imaginary part of the vacuumenergy, and thereby the vacuum decay rate, when a constant electric field is present.There exists a formula analogous to equation (5.1) for the pair production ofelectrically charged strings by a constant electric field. In that case, the chargesreside on the endpoints of open strings which are in turn confined to travel on D-96branes. When the internal electric field on an infinite, flat D-brane has strength E,and assuming that the other end of the string goes to a parallel D-brane with noelectric field, the formula for the rate of vacuum decay due to string pair creation,per unit volume per unit time is [89, 90]Γstring =∑SEE pi ∞∑k=1(−1)k+1 [ E4pi2k]D+12e− pim2SE k−2piα ′Ek (5.2)whereE = arctanh2piα ′E2piα ′ ∼ E [1+O(2piα ′E)2] (5.3)The sum over S is the sum over the particle states in the string spectrum. The mSare the masses of the particles and the multiplicity of the states with each mass isanalogous to the (2J+1) factor in the Schwinger formula (5.1).The string formula (5.2) reduces to the particle formula (5.1), summed over theparticles in the string spectrum, in the limit where the electric field is much smallerthan the string tension, 2piα ′E << 1 so that E ≈ E. Away from that limit, the stringformula (5.2) is not simply identical to the particle formula (5.1) summed overthe particle states in the string spectrum. The main (but not only) difference is thereplacement of the electric field E by the parameter E , which can be thought of as atype of screening. Moreover, since arctanh2piα ′E becomes complex if 2piα ′E > 1,there is an upper critical electric field,Ecrit. ≡ 12piα ′ (5.4)which agrees with other derivations of the upper critical electric field, being simplythe value of the electric field which balances the string tension on flat space. Itcan already be seen to be the singular point in the Born-Infeld action1 which is1 The leading terms in an expansion in derivatives of Fµν of the disc amplitude is the Born-Infeldeffective actionSDBI = 1(2pi)Dα ′(D+1)/2gs ∫ dD+1ξ√det(gµν(ξ)+2piα ′Fµν(ξ))integrated over the D-brane world-volume. For a flat brane, gµν = δµν and constant electric field,where the non-zero components of Fµν are F01 = −F10 = E, the Born-Infeld action does not have97contained in the disc amplitude for the string sigma model [91, 92, 93]. Equation(5.2) arises from the next order in the string loop expansion, the cylinder amplitude.There is thus something that is intrinsically stringy about the Schwinger pro-cess for strings. One might speculate that, since we are discussing a transient stateof the string theory – it is a state which is decaying – it is not an on-shell solutionof string theory. In this sense, the interesting formula (5.2) could well be a simpleprobe of off-shell string theory.The weak field limit of equation (5.2) was first discussed by Burgess [94].Bachas and Porrati [89] derived the full expression by finding an operator solutionof the string sigma model with an electric field. The solution was confirmed usingthe boundary state technique [95]. In this chapter, we shall discuss how it canalso be obtained by integrating the functional integral for the bosonic string sigmamodel which describes the appropriate configuration of open strings, the cylinderamplitudeΓstring = 2V I∫ ∞0 dT2T ∫ [dXµ(σ ,τ)] [ghost] e−S[X ,T ] (5.5)where the Polyakov action in the conformal gauge isS = 14piα ′ ∫ 10 dτ∫ 10 dσ [T X˙µ(σ ,τ)2+ 1T X ′µ(σ ,τ)2]− E2 ∫ 10 dτ [X0(0,τ)X˙1(0,τ)−X1(0,τ)X˙0(0,τ)](5.6)and the metrics of both spacetime and the string worldsheet are Euclidean. Asusual, we denote x˙ ≡ ∂∂τ x and x′ ≡ ∂∂σ x. Here, T is the modular parameter of thecylinder. The factor of 1/2 in the measure reflects the symmetry under time rever-sal on the annular worldsheet. (For the charged scalar particle treated in [85, 84]this factor is absent as the scalar field is complex). The ghost determinant is (seean imaginary part, as long as the electric field is less than the critical field E ≤ 12piα ′ . In stringperturbation theory about flat space, and with E ≤ 12piα ′ , the imaginary part of the vacuum energyfirst appears in the cylinder amplitude. As we see in equations (5.2),(5.4), the upper critical electricfield also appears in the cylinder amplitude.98subsection 5.2.2)[ghost] = det[− 1T∂ 2σ −T∂ 2τ ] (5.7)In appendix G, we will review how the imaginary part of the cylinder ampli-tude can be computed, and equation (5.2) obtained directly by first performing theGaussian integral over the embedding coordinates of the string and then finding theimaginary contributions of some poles in the modular parameter integral. We canconsider this a confirmation of equation (5.2) which, as we have discussed above,was found by other techniques. One interesting point is that, when zeta-functionregularization is used in order to define the various infinite products and summa-tions which are encountered in taking the functional integral, the result turns out toreproduce (5.2) in every detail, including the overall normalization.Here, we wish to emphasize an alternative approach which, at the outset, ap-pears less efficient. It is a true semiclassical computation of the partition functionwhere we begin with a classical instanton solution of the equations of motion whichare obtained by treating both the embedding coordinates and the modular param-eter T as dynamical variables. The action evaluated on such classical instantonshas already been seen to produce the large m2S limit of the exponent in equation(5.2) [96]. We will then perform a detailed analysis of the fluctuations about theclassical solution, using the Gelfand-Yaglom approach to computing functional de-terminants. We will find that, with zeta-function regularization, we produce the fullexpression in equation (5.2), complete with the prefactors. This suggests that thissemi-classical calculation is giving us the exact result. We then fashion a proofthat the functional integral in an instanton sector is indeed given exactly by thesemi-classical, WKB limit. This proceeds by identifying an interesting nilpotentfermionic transformation of the dynamical variables which is a symmetry of thefunctional integral. This symmetry is fermionic in that it uses the Fadeev-Popovghost variables which appear due to a certain gauge fixing, but it differs from theusual BRST supersymmetry. Then we show that the interaction terms in the actionin the multi-instanton sector, as well as being closed forms are also exact forms andcan therefore be deformed to zero. The WKB approximation then gives the exactresult.99Although our calculation does not provide any new information beyond equa-tion (5.2) which is already known, we consider it worth presenting nonetheless. Itis one of the few explicit examples of localization of a functional integral and themechanism for this localization is interesting and new. It becomes one of a shortlist of instanton computations that can be done exactly and the sum over all instan-ton sectors indeed reproduces (5.2), which confirms its validity from a fourth pointof view.Also, this semiclassical approach can be a starting point for other interest-ing calculations where the other approaches do not work, for example, where thespacetime of the D-brane world-volume is curved or where the electromagneticfield is not constant and the integral over embedding coordinates is not Gaussian.Then the perturbative approach which we espouse would be the conventional start-ing point. Although we shall not explore these issues here, it could be that our result– the knowledge that the flat space, constant field limit is WKB exact – would beof value in understanding corrections to that limit when spacetime is not flat andgauge fields are not constant or where the gauge fields are dynamical.2The rest of this chapter is structured as follows. In Section 5.2, we performa semi-classical computation of the cylinder amplitude in the bosonic open stringsigma model. We discuss the instanton solution and we consider the fluctuationsof the dynamical variables of the sigma model about the classical solution in theGaussian approximation. We find that these, subsequently summed over all instan-ton numbers, gives the exact result, the formula in equation (5.2). In Section 5.3we shall find a fermionic symmetry of the theory in the multi-instanton sector. Wethen use this fermionic symmetry to demonstrate that all perturbative correctionsto the WKB limit indeed cancel so that the WKB approximation is exact.In Appendix F we examine, as a toy model, a simple integral which has featuressimilar to the string functional integral that we evaluate, and which illustrates thetechnique of evaluating the instanton amplitude, including exactness of the WKBapproximation. The supersymmetry that is identified and which we use there is aclose analog of the one that we found in the previous chapter for the particle in an2Some exact results for pair-production in non-constant background fields have been obtainedin the case of QED; see [87] and references therein. For the QED analog of the present analysis,including of our localization calculation, see [85].100electric field [85, 84] and the string in an electric field in this chapter. In appendixG we evaluate the imaginary part of the cylinder amplitude (5.5) by first integratingover the string embedding coordinates and subsequently picking up the imaginarycontributions of poles in the integral over the modular parameter T . In appendixH we carry out the computation of section 5.2 using an explicit mode expansioninstead of the Gelfand-Yaglom method. In subsequent appendices, we collect someproperties of the Dedekind eta-, Jacobi theta and Riemann zeta-functions and theirtransformations which we make use of in the computations in section Semiclassical evaluation of the cylinder amplitudeNow let us consider the case of a bosonic string in an electric field. The scenariowe are interested in has an open string that is suspended between two parallel D-branes which both have flat geometry. We turn on a constant U(1) electric field inone of the D-branes. We are interested in the amplitude for the creation of pairs ofcharged states of the string by the same tunneling process as the Schwinger processfor particles. We will discuss a semiclassical computation which takes into accountworldsheet instantons.The instanton solution of the open string theory has already been found andshown to produce the classical limit of the amplitude [96]. In this section, we shallexpand on that calculation. In particular, we will include fluctuations about theclassical instanton.To evaluate the amplitude for the Schwinger process for charged strings inan electric field, we shall look for an imaginary part of the cylinder amplitude.We begin with the amplitude in the conformal gauge appearing in equations (5.5),(5.6) and (5.2.2). For the coordinates which are affected by the electric field, it isconvenient to use the complex combinationz(σ ,τ) = 1√2[X0(σ ,τ)+ iX1(σ ,τ)] (5.8)101In this notation, the action becomesS = 12piα ′ ∫ 10 dτ∫ 10 dσ [T ∣z˙(σ ,τ)∣2+ 1T ∣z′(σ ,τ)∣2]+ iE∫ 10 dτ z¯(0,τ)z˙(0,τ)+ D∑a=214piα ′ ∫ 10 dτ∫ 10 dσ [T x˙a(σ ,τ)2+ 1T x′a(σ ,τ)2]+ 25∑A=D+114piα ′ ∫ 10 dτ∫ 10 dσ [T x˙A(σ ,τ)2+ 1T x′A(σ ,τ)2](5.9)where we label the coordinates which have Neumann boundary conditions as xa(σ ,τ),with a = 0, ....,D and those which have Dirichlet boundary conditions as xA(σ ,τ)with A = D+ 1, ...,25. The D-brane is flat and infinite, filling the spacetime co-ordinates x0, ...,xD. The boundary conditions are periodic in worldsheet time,Xµ(σ ,τ +1) = Xµ(σ ,τ) andz′(τ,σ = 1) = 0 , z′(τ,σ = 0) = 2piα ′iET z˙(τ,σ = 0) (5.10)x′a(τ,σ = 1) = 0 , x′a(τ,σ = 0) = 0 , a = 2, ...,D (5.11)xA(τ,σ = 1) = dA , xA(τ,σ = 0) = 0 , A =D+1, ...,25 (5.12)As usual, the presence of the electric field, which will not appear in the equationsof motion, is in the boundary condition (5.10).5.2.1 Worldsheet instantonsWe will treat the string coordinates z(σ ,τ),xa(σ ,τ),xA(σ ,τ) and the modular pa-rameter T as dynamical variables. We will begin by finding the saddle points ofthe integrand in the functional integral by solving the classical equations of mo-tion. Those equations are obtained by applying the variational principle to theaction (5.9). The equations arez′′(σ ,τ)+T 2z¨(σ ,τ) = 0 (5.13)x′′(σ ,τ)+T 2x¨(σ ,τ) = 0 (5.14)∫ dτdσ [T 2(∣z˙(σ ,τ)∣2+ 12 x˙(σ ,τ)2)−(∣z′(σ ,τ)∣2+ 12x′(σ ,τ)2)] = 0 (5.15)102Figure 5.1: The worldsheet instanton configuration in (5.16). The parallelsheets are D-branes, with the lower one having a uniform worldvolumeelectric field turned on.The solutions must also obey the boundary conditions (5.10)-(5.12). With theseboundary conditions, the solutions of the above equations arez0(σ ,τ) = 1√2∣d⃗∣ e2piikτ cosh(2piα ′E(σ −1))2piα ′E , k = 1,2, . . . (5.16a)x0a(σ ,τ) = 0 , x0A(σ ,τ) = dAσ , T0 = 2piα ′E2pik (5.16b)where 2piα ′E = arctanh2piα ′E. The solution is a worldsheet with cylindrical topol-ogy which intersects the two parallel D-branes between which it is suspended oncircles. The positive integer k is the instanton number. It is the number of timesthat the embedding wraps the cylinder.The circle which the endpoint of the open string traces on the D-brane withthe electric field can be thought of as a cyclotron orbit. In Euclidean space, anelectric field behaves as a magnetic field and the charged particle residing at theend of the string follows a cyclotron orbit. The radius of the circle is related tothe magnitude of the electric field and the separation of the D-branes. Since theminimum (classical) mass of the classical string state is given by the string tension103times the string length,m˜0 =¿ÁÁÁÀ( ∣d⃗∣2piα ′)2− 1α ′ ≈ ∣d⃗∣2piα ′the radius of the orbit isradiusE = m0 cosh(2piα ′E)E = 1√1− E2(2piα ′)2m0arctanh(2piα ′E)/2piα ′ (5.17)which is equal to the cyclotron radius of a relativistic particle, m0E when E << 2piα ′but gets very large as E approaches the string scale 2piα ′, going to infinity at thecritical field. The string endpoint on the D-brane with no electric field also getsdragged in a circle which has a smaller radiusradius0 =m0 1E = m0arctanh(2piα ′E)/2piα ′ =¿ÁÁÀ1− E2(2piα ′)2 ⋅ radiusEThe classical action in this k-instanton sector is given bySclassical = pikE d⃗2(2piα ′)2 = pim20E k (5.18)which matches the first, dominant term in the exponent in the string amplitude inequation (5.2). This formula is known from previous work [96]. It also approachesthe classical action for the cyclotron orbit of a relativistic particle, pim20E k, whenE << 12piα ′ . When E is of order the string scale, the result is much smaller than thatfor a particle, going to zero at the critical E → 12piα ′ .For the remaining sections we defineε ≡ 2α ′E (5.19)to bring our formulae in line with the notation of e.g. [89, 97, 96].1045.2.2 Fluctuations about the instantonWe now expand the dynamical variables about the classical solution obeying theseboundary conditions asz(σ ,τ) = z0(σ ,τ)+δ z(σ ,τ), xa(σ ,τ) = δxa(σ ,τ),xA(σ ,τ) = dAσ +δxA(σ ,τ), T = T0+δT, (5.20)where the boundary conditions for the fluctuations areδ z′(τ,σ = 1) = 0 , δ z′(τ,σ = 0) = 2piα ′iET δ z˙(τ,σ = 0) (5.21a)δx′a(τ,σ = 1) = 0 , δx′a(τ,σ = 0) = 0 , a = 2, ...,D (5.21b)δxA(τ,σ = 1) = 0 , δxA(τ,σ = 0) = 0 , A =D+1, ...,25 (5.21c)The expansion of the action to quadratic order in the fluctuations isS = km20 2piα ′ε + 12piα ′ δT 2T 30 ∫ 10 dσdτ [∣z′0∣2+ 12x′20A]+ iE∫ 10 dτδ z¯δ z˙∣σ=0+ 12piα ′ δTT 20 ∫ 10 dσdτ [T 20 ( ˙¯z0δ z˙+δ ˙¯zz˙0)−(z¯′0δ z′+δ z¯′z′0)]+ 12piα ′ ∫ 10 dσdτ [T0∣δ z˙∣2+ 1T0 ∣δ z′∣2+ T02 δ x˙2+ 12T0 δx′2]+ . . . (5.22)where T0, z0, x0A are the classical solutions given in (5.16a,5.16b), and we definedδ x⃗≡ (δ x⃗aδ x⃗A). Terms of cubic and higher order in fluctuations will be studied in section5.3. This stringy fluctuation path integral has a similar structure to that of theworldline fluctuation integral mediating the Schwinger effect in scalar QED [85].Accordingly, the mechanism by which it reduces to its semiclassical approximationwill largely parallel that of [85], up to some technical complications.Structure of fluctuation integralLet us set up the evaluation of the above fluctuation integral with care, as it willstreamline the calculation considerably. Ignoring for the moment the second line of(5.22), we have a quadratic form in each of δxa, δxA, δ z, with identical fluctuation105operatorLˆ = 12piα ′T0 (−∂ 2σ −T 20 ∂ 2τ ) (5.23)but different boundary conditions. The τ-dependence is trivially diagonalized byFourier transforming,δ z(σ ,τ) = ∞∑n=−∞e2piinτδ zn(σ), δxi(σ ,τ) = ∞∑n=−∞e2piinτδxi,n(σ) (5.24)thereby reducing the 2-d spectral problem to 1-d, with Robin, Neumann and Dirich-let boundary conditions for δ z,δxa, and δxA respectively. Gaussian integrationthen generates a product over n of functional determinants for the 1-d operatorLˆn = 12piα ′T0 (−∂ 2σ +Ω2n) , Ωn ≡ 2pinT0. (5.25)Both the determinants themselves and their product over n are formally infinite andwill require regularization.At this stage we note a couple of complications. Firstly, we have ignored thecoupling between δT and δ z in the second line of (5.22). δT couples to an infinitenumber of n = k modes of Rδ z.3 Second is the issue of zero modes. Both z0 and z˙0are non-constant eigenfunctions of Lˆk with zero eigenvalue4.Let us label the normalized eigenmodes of Lˆk with Robin boundary conditions,and their respective Fourier coefficients, as followsZero modes ∶ zˆ0(σ) ≡ z0(0,σ)∥z0∥ , vˆ˙z0(σ) ≡ z˙0(0,σ)∥z˙0∥ , zNon-zero modes ∶ yi(σ) (i > 0), yi (5.26)For later convenience we introduce the following notation for the inner product and3Unlike the analogous worldline path integral for scalar QED [85], where δT coupled only to thesingle mode z0.4 There is also a constant zero mode for the 0, . . . ,(25-D) directions, which generates a factor ofthe brane worldvolume.106norm, respectively, on the Hilbert space L2 ([0,1]× [0,1]):⟨z∣w⟩ ≡ ∫ 10dτ∫ 10dσ [z¯(σ ,τ)w(σ ,τ)+ w¯(σ ,τ)z(σ ,τ)] (5.27)∥z(σ ,τ)∥ ≡ √⟨z∣z⟩ (5.28)Similarly, for the real coordinates Xµ (µ = 2, . . . ,25) we define⟨X ∣Y ⟩ ≡ ∫ 10dτ∫ 10dσ X⃗(σ ,τ) ⋅Y⃗(σ ,τ) (5.29)∥X(σ ,τ)∥ ≡ √⟨X ∣X⟩. (5.30)ˆ˙z0(σ) is a genuine zero mode of the quadratic fluctuation action (5.22), i.e. theaction is independent of z. This is a familiar consequence of expanding a τ-translationally invariant action about a τ-dependent solution, z0(σ ,τ). This willhave to be gauge-fixed and the Faddeev-Popov jacobian JFP accounted for. We dothis in subsection 5.2.2. On the other hand, z0(σ ,τ) couples linearly to δT , andtherefore is not a genuine zero mode of the action.In fact these comments lead to a key simplifying observation regarding thecombined δ zk(σ), δT quadratic form. This part of the fluctuation action (5.22)can be written, schematically, asSδ z,δT ∝ 12aδT 2+δT(bv+ c⃗ ⋅ y⃗)+ 12 y⃗T D y⃗ (5.31)where the values of the constants a, b, c⃗ and matrix D can be read off from (5.22).Now this quadratic form is not positive definite, but must be defined by analyticcontinuation, v→ ±iv. It is in this manner that the path integral obtains an imagi-nary part, and thus a non-zero tunneling probability. The advertised simplificationis that all dependence on both c⃗ and a cancels out once we integrate out δT , v, andy⃗:∫ dδT dv N∏i=1 dyie−Sδ z,δT = ±√−1(2pi)N+1bdetD(5.32)We have left implicit the limit N →∞. The upshot is that we can as well make thereplacement δ z(σ ,τ)→ zˆ0(σ ,τ)v in the second line of (5.22), to determine thetachyonic contribution denoted b in (5.32), and we simply omit the zero eigenvalue107in our evaluation of detLˆk.The semiclassical approximation to (5.5) can then be summarized by the fol-lowing expression:Γstring ≃ 2V I V2T0 (∏√2pi)e−Sclassical [ghost]∣T0⎡⎢⎢⎢⎢⎣(det′RLˆ0)(det′RLˆk) ∏n≠0,k detRLˆn⎤⎥⎥⎥⎥⎦−1[det′N Lˆ0∏n≠0detN Lˆn]−D−12 [ ∞∏n=−∞detDLˆn]D−252 ⋅JclassicalFP ⋅(tachyon) (5.33)Here JclFP is the leading-order (in fluctuations) part of the Faddeev-Popov deter-minant. By “tachyon” we mean the imaginary contribution of the δT ,δ z form(this is described above and indicated by ±i/b in equation (5.32)). The expression(∏√2pi) indicates the Gaussian integration normalization, and will be carefullyaccounted for later. The operator Lˆn is defined in (5.25). The subscripts R(obin),N(eumann) and D(irichlet) refer to the relevant boundary conditions:R ∶ δ z′n(σ = 1) = 0 , δ z′n(σ = 0)+2piα ′EΩnT0δ zn(σ = 0) = 0 (5.34a)N ∶ δx′a(σ = 1) = 0 , δx′a(σ = 0) = 0 , a = 2, ...,D (5.34b)D ∶ δxA(σ = 1) = 0 , δxA(σ = 0) = 0 , A =D+1, ...,25 (5.34c)We now proceed to determine each of the above components. Evaluation of thefunctional determinants is straightforward if one knows the eigenvalues. For theNeumann/Dirichlet cases this is not a problem, but for Robin boundary conditionsthe eigenvalues are not explicitly known, being determined by a transcendentalequation. One option then is to expand the fluctuations δ zn in modes satisfyingincorrect boundary conditions, say δ z′ = 0. The quadratic form will not be diagonalin this basis, so that more work is required to evaluate its determinant, but we showin appendix H that this can be done in detail and it obtains the same result as theone which we derive below.A more elegant approach which does not rely on knowledge of the eigenvalues,and which will generalize readily to more complicated setups (e.g. non-constantbackground fields), is the method of Gelfand-Yaglom. We employ this method insubsection 5.2.3 to evaluate all determinants in (5.33).108Zero modeSince Im[δ z] does not couple to δT , in this case the zero eigenvalue of Lˆk cor-responds to a genuine zero-mode of the quadratic action. This is expected, andresults from proper-time-translation invariance and the fact that the instanton de-pends on the world-sheet time. Denoting the gauge transformation parameter by tas followszt(σ ,τ) = z(σ ,τ + t) (5.35)one has that z˙0,t(τ) is a zero mode of the quadratic action:δ 2Lδ zδ z¯∣z0,tddtz0,t = ddt (δLδ z¯ )∣z0,t = 0 (5.36)We gauge fix by introducing a collective coordinate. The Faddeev-Popov trickbegins by introducing unity into the path integral in the form1 = 1ω ∫ 10 dt δ (g(t)) ⋅ ddt g(t) (5.37)The gauge-fixing function g(t) is chosen so as to render the integration over thezero mode well-defined. The “Gribov” factor ω is the number solutions of g(t) = 0in the interval 0 < t < 1. We chooseg(t) = 1∥z˙0∥ ⟨zt ∣z˙0⟩ , (5.38)for whichω = 1/2k. (5.39)The classical solution z0(σ ,τ) was defined in (5.16). Then the time translationsymmetry of the path integral is used to translate the argument t to zero. Thisprocedure then amounts to insertingδ(z) g˙(0)2k(5.40)109whereg˙(0) ≡ ddtg(t)∣t=0 = 1∥z˙0∥ ⟨z˙∣z˙0⟩ (5.41)= ∥z˙0∥+ 1∥z˙0∥ ⟨δ z˙∣z˙0⟩ (5.42)This allows the integration over z to be done using the delta function, and leaves aFaddeev-Popov jacobian whose classical contribution is the first, constant termabove. The second term is a correction which we ignore in the present semi-classical computation. The net result then is the insertion ofJclassicalFP = pi∥z0∥ (5.43)GhostsFor completeness we briefly review here the contribution of reparametrization ghoststo the path integral. These arise from gauge-fixing of the Diffeomorphism ⊗ Weylsymmetry on the worldsheet, as explained in detail in e.g. [98]. An arbitrary,infinitesimal such symmetry variation is given byδgab = 2ωgab−∇aδσb−∇bδσa (5.44)where (σ0,σ1) ≡ (σ ,τ), gab is the worldsheet metric, and ω parametrizes an in-finitesimal Weyl transformation. The Faddeev-Popov procedure applied to thisinvariance leads to the following ghost action in conformal gauge:5Sgh = 12 ∫ d2z(b∂z¯c+ b˜∂zc˜) (5.45)≡ 12 ∫ 1/T0 dτ∫ 10 dσ {b(∂σ + i∂τ)c+ b˜(∂σ − i∂τ)c˜} (5.46)5 In this subsection we use the following notation:z = σ + iτ∂z,∂z¯ = 12(∂σ ∓ i∂τ)d2z = 2dσdτ110The integration limit 1/T follows from our definition of T in (5.9) (usually themodular parameter is taken as the reciprocal of this). b, b˜, c and c˜ are ghost fieldswhose boundary conditionsc = c˜, b = b˜, on boundary, (5.47)are inherited from the worldsheet reparametrizations. These conditions are easilyimplemented using the so-called doubling trick. Define B(σ ,τ), C(σ ,τ) on thedoubled domain σ ∈ [−1,1] byC(0 < σ < 1) ≡ c(σ), C(−1 < σ < 0) ≡ c˜(−σ) (5.48)B(0 < σ < 1) ≡ b(σ), B(−1 < σ < 0) ≡ b˜(−σ) (5.49)with periodic boundary conditions on B, C. The ghost action is thenSgh = ∫ 1/T0dτ∫ 1−1dσ B(σ ,τ)∂z¯C(σ ,τ) (5.50)and the expansionB(σ ,τ) = ∞∑m=−∞∞∑n=−∞e2piinτT+piimσBmn , C(σ ,τ) = ∞∑m=−∞ ∞∑n=−∞e2piinτT+piimσCmn(5.51)leads to (we exclude the simultaneous zero mode m = n = 0)[ghost] = (∏n≠0(2pin))∏n ∏m≠0[2pin− ipim/T ] (5.52)= η2(i/2T) (5.53)As usual we have used ζ -regularization - see appendix D. For the semiclassicalapproximation we approximate T ≃ T0 to get[ghost]∣T0 = η2(ik/ε). (5.54)111TachyonAs described in section 5.2.2, the variables δT and v couple in the followingquadratic form (δT, v)⎛⎝ a b/2b/2 0 ⎞⎠(δTv ) (5.55)and we can ignore the coupling of δT to the other modes. b is simply the coefficientof v ⋅δT obtained by substituting δ z(σ ,τ)→ zˆ0(σ ,τ)v in (5.22), and we find(tachyon) = ±ipib= ±iα ′k2∥z0∥R2(5.56)where R ≡ dpiε .5.2.3 Determinants à la Gelfand-Yaglom from contour integrationThe technique of evaluating functional determinants by relating them to the solu-tion of an initial value problem is originally due to Gelfand and Yaglom [99]. Ithas since been extended in various directions, including more general boundaryconditions, operators with zero modes and partial differential operators, [100, 101,102, 103, 104, 105]. See [106] for a review. Here we briefly review an elegantand simple derivation based on contour integration [102], for ordinary differentialoperators with quite general boundary conditions.Given an ordinary differential operator Lˆ and boundary conditions on [0,1] ⊂R we can associate to the spectral problem Lˆψi(x) = λiψi(x) a generalized zetafunctionζ(s) ≡ ∞∑i=1λ−si . (5.57)For simplicity we will assume positive eigenvalues, λi > 0. The above sum is onlyconvergent for sufficiently large s; however, ζ(s) can be analytically continuedto give a meromorphic function on C. The determinant of Lˆ, which nominallyis given by the divergent infinite product ∏iλi>0, is then defined in zeta functionregularization asdetLˆ ≡ exp(−ζ ′(0)) (5.58)Now suppose we have a function U(λ) with zeros at precisely the eigenvalues112λ = λi of Lˆ (without necessarily knowing the values of these eigenvalues). We willcome to the problem of constructing such a function presently. Then the logarith-mic derivative of U(λ) has poles of unit residue at each λi, and we therefore havethe following contour integral representation of the sum (5.57):ζ(s) = 12pii ∫γdλ λ−s ddλ lnU(λ) (5.59)The contour γ encloses all poles λi ∈R+ in a counterclockwise sense. Deforming itto the negative real axis we then obtainζ(s) = sinpispi ∫ ∞0 dλ λ−s ddλ lnU(−λ) (5.60)Clearly the above expressions are not convergent for all s, (and in particular not fors = 0). The usual way of handling this is to compute the ratio of determinants (withrespect to the same boundary conditions) of Lˆ and some simpler normalizing op-erator Lˆ0 whose spectrum is known. This improves the convergence; in particular,if the coefficient of the derivative of highest degree in Lˆ0 is the same as for Lˆ, theratio of their determinants is finite. Since the large-λ asymptotics of U and U0 arethe same, the representationζ1(s)−ζ2(s) = sinpispi ∫ ∞0 dλ λ−s ddλ ln U1(−λ)U2(−λ) (5.61)is valid around s = 0, and one has detLˆ/detLˆ0 = exp{−[ζ ′(0)−ζ ′1(0)]}.For the fluctuation problem in this chapter, the main difficulty stems from theRobin boundary conditions. Even for the “massless” operator −∂ 2 we do not knowthe spectrum explicitly. Therefore we will study the “absolute” determinant di-rectly, performing an explicit analytic continuation.The idea then is to improve the large-λ behavior. By splitting the integrationrange and subtracting off the leading asymptotic formU(−λ) λ→∞∼ U∞(−λ) (5.62)113of U we have [105, 104]ζ(s) = ζ f in(s)+ζ∞(s) (5.63)where the finite and asymptotic contributions are respectivelyζ f in(s) = sinpispi (∫ 10 dλ λ−s ddλ lnU(−λ)+∫ ∞1 dλ λ−s ddλ ln U(−λ)U∞(−λ))ζ∞(s) = sinpispi ∫ ∞1 dλ λ−s ddλ lnU∞(−λ) (5.64)Since U∞ is a much simpler function, it will be possible to evaluate the integral(5.64) explicitly (assuming large s), and the resulting meromorphic expressiontaken to define the analytic continuation of ζ∞ by allowing s ∈ C. Meanwhileζ f in(s) is now well-defined at s = 0, and we haveζ ′f in(0) = − ln[ U(0)U∞(−1)] (5.65)In conclusion,detLˆ = U(0)U−1∞ (−1)e−ζ ′∞(0) (5.66)At this stage the problem boils down to constructing a function U(λ) with thedesired arrangement of zeros. To this end, consider again the eigenvalue equation(Lˆ−λ)uλ (σ) = 0 (5.67)with associated boundary conditions expressed in the general formM(uλ (0)u′λ (0))+N(uλ (1)u′λ (1)) = (00), (5.68)for some constant matrices M, N. Suppose for now, however, we instead imposesome arbitrary initial (σ = 0) conditions on (5.67), leaving the right-hand boundarycondition free. This uniquely fixes two independent solutions6 u(1,2)λ (σ) of (5.67),from which we can construct a general solution uλ (σ) = αu(1)λ (σ)+βu(2)λ (σ).Such a solution evaluated at the boundary, σ = 1, defines a function of λ . It is then6assuming Lˆ is of second order114straightforward to show that the condition for the existence of a solution uλ (σ)satisfying the boundary conditions (5.68) is [102]det(M+NHλ (1)H−1λ (0)) = 0 (5.69)where Hλ (σ) is the fundamental matrix defined byHλ (σ) ≡ ⎛⎝u(1)λ (σ) u(2)λ (σ)u′(1)λ (σ) u′(2)λ (σ)⎞⎠ . (5.70)It proves convenient to impose the particular initial conditionHλ (0) = ⎛⎝1 00 1⎞⎠ (5.71)Then the function we are after is given byU(λ) = det(M+NHλ (1)) (5.72)Exclusion of zero eigenvalueWhen Lˆ has a zero mode, the usual approach is to introduce an ad hoc regulatorand divide out the pseudo-zero-eigenvalue by hand. A more rigorous, regulator-independent approach detailed in [102, 103] involves a slight modification to theabove zeta-function computation. We briefly summarize their results, as pertainingto the calculation in this chapter.The required determinant det′Lˆ with zero eigenvalue excluded follows from(5.58,5.59) if we modify ζ(s) appropriately. Namely, we replace U(λ) in (5.59)by a function f (λ) with the same positive zeros, but which is non-zero at the origin.This allows for γ to be deformed to R− as before, by eliminating the singularity at0. Such a function is given byf (λ) ≡ − 1λdet(M+NHλ (1)) (5.73)115Furthermore, if we defineuλ (x) ≡ −[m12+n11u(2)(1)+n12u′(2)(1)]u(1)(x)+[m11+n11u(1)(1)+n12u′(1)(1)]u(2)(x) (5.74)with u(1,2)(x) as before, then we havef (λ) = −B∫ dxu0(x)∗uλ (x) (5.75)where the constant B is given byB = n12u′0(1)∗ if n12 ≠ 0; B = −n22u0(1)∗ if n22 ≠ 0. (5.76)for the general case of “separable” boundary conditions, i.e. detM = detN = 0.Equation (5.66) is thus replaced withdet′Lˆ = −B∣y(σ)∣2 e−ζ ′∞(0)U∞(−1) (5.77)where, in the notation of [102], y1(σ) ≡ limλ→0 uλ (0).In subsection 5.2.3 we evaluate several determinants with excluded zero modesusing a regulator, but it is satisfying to note that the results are reproduced in eachcase by the formula (5.77).Evaluation of the Functional DeterminantsIt is now a simple matter to apply the algorithm of section 5.2.3 to the evaluationof detµ Lˆn, for (µ = R,N,D). We first normalize the “kinetic” term to unity, as iscustomary. That is, we will compute the determinants of the operatorLn = −∂ 2σ −Ω2n, Ωn ≡ 2pinT0 (5.78)116This normalization introduces a factor ofN into the integral form of the zeta func-tion as follows:ζ(s)→ ζ˜(s) = sinpispi ∫ ∞0 dλ (Nλ)−s∂λ lnU(−λ) (5.79)Clearly ζ˜ ′f in(0) = ζ ′f in(0), but the asymptotic piece gives a non-trivial contribution:ζ˜ ′∞(0) = dds (N −sζ∞(s))∣s=0 (5.80)= (− logN )ζ∞(0)+ζ ′∞(0) (5.81)Consequently, the relation between the determinants of Lˆ and L isdetµ Lˆn =N ζ (µ)∞ (0)detµLn, µ = R,N,D (5.82)However, the (Riemann-zeta-regularized) product over all n ∈ Z erases this depen-dence on N , since ∏∞n=−∞(const) = 1, and ζ∞(s) does not depend on n. The onlysuch contribution then is from the excluded pseudo-zero-eigenvalues, so effectivelyeach “prime” on a determinant is accompanied by N −1.For convenience, we will absorb the conventional powers of√2pi accompany-ing each Gaussian integration (and hence eigenvalue) into N , definingN ≡ 12piα ′T012pi= k2pi2α ′ε . (5.83)With these notations, we write (5.33) asΓstring ≃ I 2V V2T0 e−Sclassical [ghost]∣T0⎡⎢⎢⎢⎢⎣N −2 (det′RL0)(det′RLk) ∏n≠0,k detRLn⎤⎥⎥⎥⎥⎦−1[det′NL0N ∏n≠0detNLn]−D−12 [ ∞∏n=−∞detDLn]D−252 ⋅JclFP ⋅(tachyon) (5.84)Without further ado, let us compute the determinants. The fundamental matrix117(5.70) satisfying (5.71) isHλ (σ) = ⎛⎜⎜⎝ cos(σ√λ −Ω2n) sin(σ√λ−Ω2n)√λ−Ω2n−√λ −Ω2n sin(σ√λ −Ω2n) cos(σ√λ −Ω2n)⎞⎟⎟⎠ (5.85)• Robin boundary conditionsIn the notation of (5.68) we haveM = ⎛⎝2piα ′EΩn 10 0⎞⎠ , N = ⎛⎝0 00 1⎞⎠ (5.86)ThusU(−λ) = 2piα ′EΩn cosh√λ +Ω2−√λ +Ω2 sinh√λ +Ω2 , (5.87)U∞(−λ) = −12√λe√λζ∞(s) = sinpis2pi ⎛⎝1s + 1s− 12 ⎞⎠ , lims→0ζ ′∞(0) = −1 (5.88)Therefore recalling (5.66)detL = U(0)U−1∞ (−1)e−ζ ′∞(0) (5.89)we havedetRLn = 2Ωn sinhΩn [1−2piα ′E cothΩn] (5.90)Using the infinite product formulae derived in appendix D, we obtain∏n≠0,k detRLn = [(−1)k+1e−pikεcosh2piε]−1 η2(ik/ε)2piε sinhpiε2k2ε2(5.91)To handle the zero modes, we choose to modify the operators with a regulatorδ ≪ 1, leaving the boundary conditions unchanged:n = k: Take L→ L˜ = −∂ 2σ −Ω2(1+δ)2 (5.92)118Note that z˙0 (z0) is still an eigenfunction, but with eigenvalue λ0 = 2δ(piε)2,and the determinant is7det′RLk = limδ→0 detRL˜kλ˜0 = 2piε + sinh2piε2piε sechpiε = 4∣∣z0∣∣2R20 coshpiε(5.93)n = 0: The boundary conditions reduce to pure Neumann (A0 = 0). TakingL˜0 = −∂ 2σ +δ 2, the zero mode y0(σ) = 1 is unchanged but eigenvalue shiftsto δ 2, givingdet′RL0 = 2 (5.94)Therefore the “Robin” contribution [N −2 (det′RL0)(det′RLk)∏n≠0,k detRLn]−1isRobin =N 2 [(−1)k+1e−pikεcosh2piε] (2piε)sinhpiεη2(ik/ε) ( ε2k)2 R2 coshpiε2∣∣z0∣∣2 (5.95)• Neumann boundary conditionsThe result follows from (5.90,5.94) and the infinite product formulae in ap-pendix D by setting A = 0:Neumann = [N −1det′NL0∏n≠0detNLn]−D−12(5.96)= N D−12 [2∏n≠02Ωn sinhΩn]−D−12(5.97)= [ ε8pi2α ′k]D−12η1−D(ik/ε) (5.98)• Dirichlet boundary conditions7 Alternatively, the result (5.77) givesdet′RLk = −2B∣y(σ)∣2,with y(σ) = tanh(piε)sinh(Ωkσ)−cosh(Ωkσ) and B = coshpiε , agreeing with (5.93).119We have M = ⎛⎝1 00 0⎞⎠ and N = ⎛⎝0 01 0⎞⎠, thereforeU(−λ) = sinh√λ +Ω2n√λ +Ω2n , U∞(−λ) = e√λ2√λζ∞(s) = sinpis2pi ⎛⎝ 1s− 12 − 1s⎞⎠ , lims→0ζ ′∞(0) = −1 (5.99)anddetDLn = 2sinhΩnΩn . (5.100)The net contribution is thusDirichlet = ∞∏n=−∞(detDLn)D−252 = [η(ik/ε)]D−25 (5.101)5.2.4 ResultGathering all factors in (5.33), we find that the semi-classical approximation to theannular string partition function with action (5.9) yieldsΓsemicl. = tanhpiεε(−1)k+1e−pikεη24(ik/ε) [ ε8pi2α ′k]D+12e−2piα ′km20/ε , (5.102)where we recall that m0 = d2piα ′ and ε = 1pi arctanh(2piα ′E). This is identical tothe result (equation (25)) of [89]. To see this, it is useful to recall the mass-shellrelation of the open string,m2S = d2(2piα ′)2 + 1α ′ (N −1) (5.103)where N is the level number, from which one obtains∑Se−(2piα ′)km2S/ε = e−(2piα ′)km20/εη−24(ik/ε). (5.104)120(Note that in contrast to our setup, theirs allows for both string endpoints to becharged, but is specialized to a spacetime-filling D-brane, D = 25.)1215.3 Exactness of semiclassical approximation: Proof bylocalizationIt is remarkable that our semi-classical computation has produced the exact am-plitude for pair production. It implies that all higher-order corrections must find away to cancel. In this section we will study the full path integral and construct aproof that it localizes onto its semi-classical approximation.There is an analogous localization of the worldline path integral of scalar QEDin a constant electric field, which we addressed in [85] (and which inspired thepresent investigation). While the underlying mechanism by which the two pathintegrals localize is essentially the same, the 2d worldsheet with boundary doesintroduce some complications relative to the particle worldline, and in particularthe manipulations of this section become significantly more cumbersome.The cleanest approach proceeds by identifying a fermionic symmetry of thegauge-fixed action, mixing ghost and bosonic variables (but distinct from the usualBRST symmetry). Recall that the quadratic action had a zero-mode associated withproper time translation invariance. The gauge-fixing factor introduced in equation(5.37) can be represented as follows1ω ∫ 10 dt δ (g(t)) ddt g(t) = 1ω ∫ dt∫ [dBdcdc¯] e−[2piiB⋅g(t)+c¯c⋅ ddt g(t)], (5.105)where c and c¯ are constant, anti-commuting Faddeev-Popov ghosts, and B is aLagrange multiplier8. Thus after using the gauge-invariance of the path integral to8 The gauge-fixed action Sg f , which now includes the exponent in (5.105), enjoys as usual aBRST symmetryδˆ Sg f = 0, (5.106)whereδˆ t = −c, δˆ c¯ = 2piiB, δˆ c = δˆ B = δˆ (δT) = 0. (5.107)Recall that t parametrizes the gauge transformation, δ zt(σ ,τ) ≡ δ z(σ ,τ + t). The gauge-fixing partintroduced in (5.105) can be written as the following BRST exact expression:δˆ [c¯g(t)] .122translate t to zero, we can write the full path integral as followsΓ= 2ωVI∞∑k=1∫ [d(δxµ)]d(δT)dBdcdc¯η2( i/2T (k)0 +δT )T (k)0 +δT e−S(X(k)0 +δX ,T (k)0 +δT)−[2piiB⋅g(0)+c¯c⋅g˙(0)](5.108)In order to construct the sought-after transformation, let us specify our gauge-fixing function. For present purposes, a judicious choice will beg(t) = 2pikκ[⟨zt ∣z˙0⟩− 1(2pik)2 1T0T ⟨z′t ∣z˙′0⟩] (5.109)where zt ≡ z(σ ,τ + t) and κ is defined asκ ≡ 1∥z0∥ (∥z˙0∥2− 1T 20 ∥z′0∥2) = (2pikR)2∥z0∥ , R ≡ d/piε. (5.110)This differs from our earlier choice (5.38) by the addition of the second term, butit is normalized such that the leading order Jacobian JclassicalFP is the same as be-fore. The Gribov factor is still ω = 2k. We will argue that the fluctuation part canultimately be set to zero. The constraint function becomesg(0) = 2pikκ[⟨δ z∣z˙0⟩− 1(2pik)2 ⟨δ z′∣z˙′0⟩T0T ]= 2pikκ[ z∥z˙0∥ (∥z˙0∥2− ∥z′0∥2T0T )]− f (δT,{yi})= z[1− κ4piα ′ δT h(δT)]− f (δT,{yi}) (5.111)where the function f depends on all modes except v and z, while h depends onlyon δT . Our symmetry argument will eliminate the second term in square brackets,so that the constraint will reduce to δ(z− f ) with unit coefficient in front of z.123Similarly, the Faddeev-Popov jacobian becomes9JFP(z,T) ≡ g˙(0) = 2pikκ [⟨z˙∣z˙0⟩− ⟨z′∣z′0⟩T0T ] (5.112a)= ∥z˙0∥+ . . . (5.112b)Expanding the full gauge-fixed action Sg f about the k’th instanton, we haveS(k)g f = S(k)classical+S(k)quad+S(k)int , (5.113)S(k)classical is given by (5.18), and using (5.22),(5.105),(5.111) and (5.112), we have10S(k)quad = 14piα ′ [T0∥δ X˙∥2+ 1T0 ∥δX ′∥2]+ δT2piα ′ [⟨z˙0∣δ z˙⟩− 1T 20 ⟨z′0∣δ z′⟩]+ iE∫ 10 dτ δ z¯δ z˙∣σ=0+12aδT 2+2piiB(z− f )+ c¯cJclassicalFPS(k)int = δT4piα ′ [∥δ X˙∥2− 1T 20 ∥δX ′∥2]− 14piα ′∞∑j=2(−δT) jT j+10 [2⟨X ′0∣δX ′⟩+∥δX ′∥2] (5.114)− κ4piα ′ 2piiBzδT ⋅h(δT)+ c¯c[JFP(z,T)−∥z˙0∥] (5.115)(Although f is not quadratic, we include it in Squad for convenience since the finalresult will be independent of f ). Similarly, we expand the measure around T =T (k)0 :1Tη2 [i/2T ] = 1T (k)0 η2 [i/2T (k)0 ]e−F(k)(δT) = 1T (k)0 η2 [i/2T (k)0 ](1+O(δT))(5.116)9Here we exclude the Gribov factor 1/ω from our definition of JFP, unlike in the previous section.10 Recall that uppercase Xµ stands for all spacetime components, µ = 0, . . . ,25. Lowercase xµ hasµ = 2, . . . ,25, i.e. ∥δ X˙∥2 = ∥δ z˙∥2+∥δ x˙∥2124where the fluctuation factor F(k), given byF(k)(δT) = log⎡⎢⎢⎢⎢⎣ η2(i/2T)(1+δT /T (k)0 )η2(i/2T (k)0 )⎤⎥⎥⎥⎥⎦ (5.117)= ∞∑j=1F(k)j δTj, (5.118)contains all corrections to the measure, including the reparametrization ghost con-tribution. The Taylor expansion starts at δT 1, and is well-defined since η(x) is aholomorphic function in the upper half-plane. The amplitude is thereforeΓ(λ) ≡ 2ωVI∞∑k=1e−S(k)classical∫ ∞−∞ d(δT)∫ [d(δxµ)]dBdcdc¯η2 [i/2T (k)0 ]T (k)0 e−S(k)quad−λ(S(k)int +F(k)(δT)) (5.119)evaluated at λ = 1. Moreover, the semiclassical approximation to Γ that we com-puted in section 5.2 is given by Γ(0).In what follows, we will demonstrate that ∂∂λ Γ(λ) = 0 and therefore λ canbe deformed to zero without altering the value of Γ. From now on we drop thesuperscript (k).Define the nilpotent fermionic transformation ∆ by∆c¯ = κ4piα ′ δT, ∆δ z(σ ,τ) = −12(2pik)z0(σ ,τ) ⋅c, ∆(other) = 0. (5.120)with κ given by (5.110). It leaves the quadratic action invariant:∆S(k)quad = 0. (5.121)We will show that Sint is ∆-exact (and therefore in particular it is ∆-closed: ∆Sint =0). In a given (k-)instanton sector, all bosonic interaction terms are contained in125the combinationT ≡ S(T,X)−S(T (k)0 ,X)+F(k)(δT) (5.122)= κ4piα ′ δT ⋅ξ [T,X] (5.123)where we have definedξ ≡ 1κ(∥X˙∥2− 1T0T∥X ′∥2)+ 4piα ′κF(k)(δT)/δT (5.124)We showed earlier that in the absence of interaction terms, the path integral doesnot depend on the δT 2 term or the δT ⋅yi cross terms. Therefore the semi-classicalapproximation corresponds, in the bosonic sector, to replacing ξ → 2v.It is now easy to see that T and c¯cg˙(0) are generated by ∆ as follows:∆(c¯ξ) = T + c¯cJFP, (5.125)while the correction to the “constraint” term can be written∆(−c¯ ⋅2piiBzh(δT)) = − κ4piα ′ 2piiBzδT ⋅h(δT). (5.126)We have thus written a part of the action, including Sint, as a ∆-exact quantity.However, we cannot simply set this to zero, as all v and c¯c dependence would beeliminated, i.e. we would get Γ ∼ 0 ⋅∞. Before localizing we must separate outthe quadratic part (c¯cJclassicalFP + κ4piα ′ δT (v+ c⃗ ⋅ y⃗)) of (5.125) by substracting off itspreimage under ∆ on the left hand side, namelyξ0 ≡ 2κ (⟨δ X˙ ∣X˙cl⟩− 1T 20 ⟨δX ′∣X ′cl⟩) ; ∆ξ0 = c¯cJclassicalFP + κ4piα ′ δT (v+ c⃗ ⋅ y⃗) .(5.127)(The quantities c⃗ and y⃗ were introduced in equation (5.31)). Finally, we haveSint(δT,δX)+F(k)(δT) = (T − κ4piα ′ δTξ0)+ c¯c(JFP−JclFP)+2piiBz κ4piα ′ δT ⋅h(δT)= ∆ψ, (5.128)126whereψ ≡ c¯(ξ −ξ0−2piiBzh(δT)) , (5.129)and ξ , ξ0 and h(δT) were defined in (5.124), (5.127) and (5.111) respectively.Consequently,∂∂λΓ(λ) = 2ωVI∞∑k=1∫ [d(δXµ)]d(δT)dBdcdc¯ η2 [i/2T (k)0 ]T (k)0 (∆ψ)e−S(k)classical−S(k)quad−λ∆ψ= 2ωVI∞∑k=1∫ [d(δXµ)]d(δT)dBdcdc¯ ∆⎧⎪⎪⎪⎨⎪⎪⎪⎩η2 [i/2T (k)0 ]T (k)0 ψ e−S(k)classical−S(k)quad−λ∆ψ⎫⎪⎪⎪⎬⎪⎪⎪⎭= 0, (5.130)where the last equality follows from the fact that we are integrating a total deriva-tive. We have thus proven that the full, interacting path integral (5.5) is givenexactly by its semi-classical approximation.5.4 DiscussionIn this chapter we have studied the open bosonic string where the D-brane on whichthe string ends contains a constant electric field. We have examined the ampli-tude for string pair creation by tunneling, the analog of the Schwinger effect forcharged particle-antiparticle pairs in an electric field. The string theory tunnelingprocess is mediated by instantons of the string sigma model which computes theopen string annulus amplitude. We have analyzed fluctuations about the classicalmulti-instanton solutions and we found that integrating the Gaussian fluctuationsand summing over all possible multi-instanton configurations obtains the knownformula for the amplitude. This can be regarded as another confirmation of thatformula. We call the approximation which retains the classical instanton actionand the determinants due to the quadratic fluctuations the WKB limit and our firstresult suggests that the WKB limit is exact. We have then fashioned a localizationargument to prove that it is indeed exact. The cohomology used for the localiza-tion of the functional integral utilizes the Fadeev-Popov ghosts which arise fromthe introduction of a particular collective coordinate, but it differs in form from the127usual BRST cohomology.The fermionic symmetry which we find could be useful in computing correla-tion functions, for example, even in integrals which are not WKB exact, it could beused for re-organizing the integral to a more convenient form. We leave a detailedstudy of whether this can indeed be used to simplify perturbative computationsof more complicated scenarios such as those with non-constant electric fields, tofuture study. The problem of string pair production in non-homogeneous back-ground fields has, to our knowledge, only been addressed in a few works to date[107, 108, 109] and demands further exploration.Finally let us point out some appealing connections with the literature. Undera T-duality transformation in the E⃗ direction, our Dp-brane setup is mapped toa pair of D(p− 1)-branes with constant relative velocity (but zero extension) inthe E⃗ direction. This establishes a connection to questions of D-brane dynamics,see for example [110, 111, 112]. From this perspective, pair production occursdue to time dependence of the brane separation and, consequently, the open stringspectrum. The critical electric field (5.4) manifests itself as a limiting velocity forthe relativistic mechanics of D-branes, namely the speed of light.Our work here may serve as a useful toy model for related calculations incurved space, in particular in the holographic context, for example involving the in-stanton fluctuation prefactor in the holographic Schwinger effect [83], and perhapseven the meson decay process analyzed in [113]11. Another intriguing applica-tion is to pomeron physics. In [114], high energy, inelastic scattering of dipolesin holographic QCD is studied and is found to be well modeled in a certain “softpomeron” regime by D0-brane scattering in flat space. This is in turn related – viathe T-duality just discussed – to Schwinger pair production mediated by worldsheetinstantons.Lastly, it could be interesting to consider a hybrid particle-string version of themodel (5.6), wherein one includes a particle-like kinetic term on the boundary inaddition to the gauge field coupling and string bulk terms. On the one hand, such amodel provides a kind of interpolation between the string and particle models con-11 In the latter, mesons at finite temperature are modeled by strings connected to a D7 “flavor”brane outside of a black hole in AdS5 ×S5. Dissociation of the mesons is mediated by worldsheetinstantons, which allow for leaking of mesons into the black hole.128sidered in this chapter and the previous one (see appendix A of [115] where sucha calculation is performed explicitly for the string disk worldsheet). On the otherhand, it was shown in [116] by integrating out the free bulk degrees of freedom ofthe worldsheet that such a setup is equivalent to the dissipative quantum mechanicsof Caldeira and Leggett [117, 118]. This permits one to study, for example, pairnucleation of particles in a dissipative setting [119].129Chapter 6ConclusionLet us briefly review what we have accomplished in this thesis. For more detaileddiscussion of the issues and of future research directions, see the concluding dis-cussions at the end of each of the chapters 2, 3, 4 and 5.In the first part of the thesis – chapters 2 and 3 – we used the results of su-persymmetric localization to study aspects of N = 4 and N = 2∗ supersymmetricYang-Mills theories relevant to gauge/gravity duality (holography).In chapter 2, we focused on the 12 -BPS circular Wilson loop of N = 4 SU(N)SYM in the totally antisymmetric representation of the gauge group. Specifically,we set up a systematic 1/N expansion for the expectation value of the Wilsonloop, for the regime where k, the rank of the representation, is of order O(N).At strong coupling, this regime is captured holographically by the physics of aprobe D5-brane embedded in the dual gravity theory on AdS5 ×S5. Furthermore,we computed explicitly the first 1/N correction to the Wilson loop expectationvalue, providing a precise prediction for holography1. The interesting problem ofreproducing this prediction on the gravity side of the duality remains open.In chapter 3 we performed a detailed analysis of the eigenvalue density (or“master field”) and the expectation value of the large circular fundamental Wilsonloop in planar N = 2∗ SYM on S4 at strong coupling. Our analysis gave a detailedpicture of the critical behaviour of the theory at strong coupling. In particularwe found that the sequence of phase transitions that occur as the coupling is var-1Our prediction has since been corroborated independently in the work [120].130ied should be visible in the (known) holographically dual theory, in the regime inwhich semiclassical string quantization is valid. We also generated a quantitativeprediction for the first quantum correction to the minimal area of the fundamentalstring which is holographically dual to the Wilson loop.The second half of the thesis involved a study of the Schwinger effect in scalarquantum electrodynamics and bosonic string theory, with constant backgroundelectric fields. We were interested in the semiclassical treatment of the path in-tegral which, although a priori not exact, provides a useful framework for tack-ling challenging problems involving, for example, non-constant and/or dynamicalgauge fields, as well as nontrivial background geometries.In chapter 4 we analysed the semiclassical expansion of the worldline pathintegral for a charged particle in a constant background electric field. We showedin detail how it leads to the exact Schwinger formula for the rate of spontaneouscharged particle pair production, and constructed a proof based on localization thatall corrections to the semiclassical approximation vanish.Chapter 5 generalized this analysis to string theory. Specifically, we consideredparallel D-branes, one of them having a constant worldvolume U(1) gauge field.We evaluated the (Euclidean) Polyakov path integral corresponding to the one-loop open string diagram, namely the cylinder, in the semiclassical approximation.After finding the classical “worldsheet instanton” solutions that mediate the pair-production process we computed the quadratic fluctuation contribution and foundthat these gave the exact production rate. Thereafter we constructed a proof basedon localization that the path integral localizes onto the worldsheet instantons andthe semiclassical approximation is, remarkably, once more exact.131Bibliography[1] J. Austen, Pride and prejudice. 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Vol. 1: An introduction to the bosonic string.Cambridge University Press, 2007. → pages 153142Appendix AExact expression for g(ξ)Here, we are going to derive a sum representation for g(ξ), which is useful forstudying the function numerically.The Fourier integral (3.22) can be solved by applying the residue theorem,where we sum over the remaining poles of gˆ(ω) in the lower half plane. Notice thatwe also have a branch cut in the lower complex half-plane, due to the 1/√ω + iεterm in our solution (3.37). Our strategy is to inverse Fourier transform the branchcut and the part with poles separately. In the coordinate space, the final expressionis then the convolution of these terms.Let us factorize gˆ(ω) ≡ aˆ(ω)bˆ(ω), with aˆ(ω) ≡ √ipi√ω + iε . In the coordinatespace, the latter becomes:a(ξ) = 1√ξθ(ξ), (A.1)where θ(ξ) is the Heaviside step function.Regarding bˆ(ω), the only poles in the lower half plane are due toΓ(−(M+ i)ω2pi)Γ((M− i)ω2pi) ,which comes from the expression for G+(ω), (3.29). These poles are the complexconjugate of (3.32), i.e.ω¯n = 2pi (Mn− i∣n∣)M2+1 , n = ±1,±2, . . . (A.2)143Applying the residue theorem to bˆ(ω), we obtain:b(ξ) = −i ∞∑n=−∞Res(bˆ(ω)e−iωξ ,ω¯n)θ(ξ). (A.3)Hence,g(ξ) = ∫ ∞−∞ dη a(η)b(ξ −η)= ∫ ξ0dη1√ξ∞∑n=−∞Res(bˆ(ω),ω¯n)e−iω¯n(ξ−η)As the sum is convergent, the integral and the sum commute. The integral gives:∫ ξ0dη1√ηe−iω¯n(ξ−η) = e−iω¯nξ√ ipiω¯nerf(√−iω¯nξ) (A.4)The residue term for n = 1,2, . . . is explicitly:Res(bˆ(ω),ω¯n) = −(M2+1)(−1)n2pin sinh2 ( pinM+i)sinh(pi(1+iM)nM+i )− (M2+1)2 (−1)n ipiM+ i Γ((M+i)nM−i )n!Γ( inM−i)2 einφM−iA( 2pinM+ i)where A(ω) is the sum in (3.37). For negative values of n, the residue is thenegative complex conjugate of the expression above. Hence we need only theimaginary part of the positive n sum. The final expression for g(ξ) is thus:g(ξ) = 2 ∞∑n=1I⎡⎢⎢⎢⎣Res(bˆ(ω),ω¯n)e−iω¯nξ√ipiω¯nerf(√−iω¯nξ)⎤⎥⎥⎥⎦ (A.5)144Appendix BAnomalous contribution to BIn computing B as defined in (3.44), we need to take the ω →∞ limit of the scalingfunction (3.37). The naive limit gives the first two terms in (3.46), but (3.37) con-tains an infinite sum and one has to be careful and do the summation first, beforetaking ω →∞. It turns out that the summation and taking the limit do not com-mute, and B receives an anomalous contribution. To isolate this contribution wecan divide the sum into two parts, from 1 to N0 and from N0 to infinity for someN0 ≫ 1. The anomalous contribution can only come from the second part:δBanom = limω→∞ i(M2+1)ω2pi ∞∑n=N0 1n ⎛⎝ 1ω − 2pinM−i − 1ω + 2pinM+i ⎞⎠ (B.1)Here we used that n ≥ N0 ≫ 1 to simplify the summand. This expression can alsobe written asδBanom = i(M2+1) limω→∞ ∞∑n=N0 [ 1(M− i)ω −2pin + 1(M+ i)ω +2pin] . (B.2)The naive ω →∞ limit would give zero, but we need to first sum and then take thelimit, and this gives a finite result:δBanom = i(M2+1)2pi limω→∞ ln 2piN0−(M− i)ω2piN0+(M+ i)ω = −M2+1pi arctanM. (B.3)145Appendix CLarge M limit of scaling functionHere we give the details on the derivation of the limiting expression (3.50) for thescaling function from the exact one (3.37), in the case when M →∞ and ω staysfinite. We assume that ω is real throughout the derivation.We start by examining the infinite sums appearing in (3.37):A± = ∞∑n=1a± ( nM±i)ω ± 2pinM±i , (C.1)wherea±(x) = e∓i[φ−(M±i)pi]x(M± i)2x2 Γ((M∓ i)x)Γ((M± i)x)Γ2 (∓ix) . (C.2)In terms of these sums,gˆ(ω) = i 32 √pi2ω32⎡⎢⎢⎢⎣M2 sinh2 ω2 − sin2 Mω2sinh2 ω2 + sin2 Mω2+(M2+1)2ω e− iφω2pi Γ(M−i2pi ω)Γ(−M+i2pi ω)Γ2 (− iω2pi ) (A−+A+)⎤⎥⎥⎥⎦ . (C.3)Since a±(x) has a finite limiting value at zero:a±(0) = − 1M2+1 , (C.4)the sums (C.1) appear linearly divergent if M is sent to infinity independently in146each term. The main contribution consequently comes from very large n ∼ M,because then a±(x) become slowly varying functions of their argument, namelya±(x) M→∞≃ e∓2ix(ln(∓ix)−1)M2x2Γ2 (∓ix) , (C.5)assuming x > 0. We are not going to use these approximate expressions, becauseof the necessity to keep the next-to-leading order accuracy. It will suffice to knowthat the exact a+(z) is an analytic function in the upper half plane, decreases as 1/zin its domain of analyticity, and satisfies the following functional identity:(M+ i)a+(−x) = (M− i)a−(x) sinpi(M+ i)xsinpi(M− i)x e2pix. (C.6)Normally, the sum of f (n/M), where M is a big paramater, is well approxi-mated by the intergal with the help of the Euler-Maclaurin formula, but here weneed to be more careful. The summands in A± have poles at 2pin/(M± i) =ω thatcollapse onto the contour of integration in the M →∞ limit. In the vicinity of thepoles the summand is not a slowly varying function of n/M, and the summationhas to be performed exactly. As a result a more general formula applies:A± ≃ (M±i)⨏ ∞0dxa±(x)ω ±2pix − a±(0)2ω +M± i2 θ(∓ω)a±(∓ ω2pi )cot (M± i)ω2 . (C.7)The second term is the Euler-Maclaurin correction, which we need to make thisformula correct throughout the next-to-leading order. The last term is the result ofsummation around n ∼ Mω/2pi . We excluded this region from the integral by theprincipal-value prescription, to avoid double-counting. This formula can be viewedas a contour-deformation prescription that takes into account the discreteness of thesum in the residue term. The formula can be brought to the formA± ≃ (M± i)∫ ∞0dxa±(x)ω ±2pix+ iε − a±(0)2ω + M± i2 θ(∓ω)a±(∓ ω2pi ) ei(M±i)ω2sin M±i2 ω ,(C.8)that facilitates rotation of the contour of integration into the domain of analyticityof the integrand.147Substitution of (C.8) into (C.3) leads to massive cancellations. Let us firstconcentrate on the integral terms. Using the identity (C.6) we can transform theirsum as(M+ i)∫ ∞0dxa+(x)ω +2pix+ iε +(M− i)∫ ∞0 dxa−(x)ω −2pix+ iε= (M+ i)∫ ∞−∞ dxa+(x)ω +2pix+ iε+(M− i)∫ ∞0dxa−(x)ω −2pix+ iε (1− sin(piMx+ ix)sin(piMx− ix) e2pix) . (C.9)We are going to argue that both terms are negligible in the large-M limit. The firstintegral actually vanishes identically, which follows from the contour argumentsince the intergand has no singularities in the upper half plane. The second integralcontains a rapidly oscillating function that depends on the slow variable x and thefast variable piMx. The dependence on the fast variable is periodic, and integrationover x goes through many periods of oscillations before the slow dependence onx can substantially alter the integrand. In any integral of this type, the integrandF(piMx,x) can be replaced by its average:∫ dxF(piMx,x) ≃ ∫ dx ⟨F(Ω,x)⟩ , ⟨F(Ω,x)⟩ ≡ ∫ 2pi0dΩ2piF(Ω,x).(C.10)It is easy to show that ⟨1− sin(Ω+ ix)sin(Ω− ix) e2pix⟩ = 0for x > 0. We can therefore drop the last integral in (C.9) too.A relatively long but straightforward calculation shows that the residue term in(C.8) cancels with the first term in the square brackets in (C.3) with the requisite,O(M0) accuracy. Thus only the Euler-Maclaurin term contributes and we are leftwithgˆ(ω) = i 32 √piM22ω32e− iφω2pi Γ(M−i2pi ω)Γ(−M+i2pi ω)Γ2 (− iω2pi ) . (C.11)Using the basic gamma-function identity (3.31), and the Stirling formula we get at148large M:gˆ(ω) ≃ i 32 √piM4√ωeωpii (ln ω2pii−1)Γ2 (1+ ω2pii) e− iMω2 + ∣ω ∣2sin(Mω2 + i∣ω ∣2 ) . (C.12)The last factor still depends on M through the periodic dependence on the fastvariable Mω/2. The Fourier transform back to the ξ -representation will effectivelyaverage over rapid oscillations, so the last factor can be replaced with⟨ e−iΩ+ ∣ω ∣2sin(Ω+ i∣ω ∣2 )⟩ =2i, (C.13)which gives the final result (3.50) quoted in the main text.149Appendix DInfinite products and specialfunctionsD.1 Infinite products and zeta-function regularizationWe use zeta-function regularization to define all divergent summations and prod-ucts in this thesis. The Riemann zeta-function is given by the infinite sumζ(s) = ∞∑n=1n−s,defined as a function of a complex variable s where the real part of s should be largeenough so that the sum converges. The function is then analytically continued tothe entire complex plane where it is a meromorphic function on the whole complexs-plane, which is holomorphic everywhere except for a simple pole at s = 1, withresidue 1. For completeness we give here explicit computations of some of theinfinite products occurring throughout the paper. The values of the zeta-functionof interest to us areζ(0) = −12, ζ(−1) = − 112, ζ ′(0) = −12ln(2pi)150Examples of infinite sums and products which we encounter in our computationsare∞∑m=11 ≡ lims→0ζ(s) = −12 (D.1)∞∑m=1m ≡ lims→−1ζ(s) = − 112 (D.2)∞∏k=1α = lims→0αζ(s) = α− 12 (D.3)∞∏n=1(2pin) ≡ lims→0(2pi)ζ(s) ⋅ lims¯→0 e− dds¯ ζ(s¯) = 1 (D.4)∏n>0Ωn ≡ ∏n>0(2pin) ε2k =√2kε(D.5)Using the above regularizations, and the product formula∞∏n=1[1+ a2n2] = sinhpiapia(D.6)we find∞∏n=−∞[(pin)2a2+(pim)2b2] = (pimb)2∏n>0[(pin)2a2+(pim)2b2]2= (pimb)2∏n>0(pina)4∏n>0[1+(pimbpina )2]2= (pimb)2(2a)2 sinh2(pimb/a)(pimb/a)2= 4sinh2(pimb/a)By (D.8) and (D.3) we further have that∏m≥12sinh(pimb/a) = epib/a∑∞m=1 m∏m≥1(1−e−2pimba)= e− 112pib/a ∞∏m=1(1−e−2pimb/a)= η(ib/a) =√abη(ia/b).151In section (5.2.3) and appendix (H) one encounters the following infinite product:∏n≠0,k1det(I −E (n)) = 1∏n≠0,k [1− tanhpiε cothpiεn/k]= limκ→k [1− tanhpiε cothpiεk/κ]∏∞n=1 [1− tanh2piε coth2piεn/κ]= limκ→k[1− tanhpiε cothpiεk/κ] κε tanhpiε e−piκεsin(piκ) ⋅⋅ ∞∏n=1[1−e−2piκn/ε]2(1−e−2piκ(n+iε)/ε)(1−e−2piκ(n−iε)/ε)= (−1)k+1e−pikεcosh2piε(D.7)In the third line we performed a modular transformation as in equation (G.17), andfor the last step observed that the remaining infinite product is regular for all κ > 0.D.2 The Dedekind eta and Jacobi theta functionsThe Dedekind eta function is defined byη(τ) = epiiτ/12 ∞∏k=1(1−e2piikτ) (D.8)It has the property thatη(−1/τ) =√−iτ η(τ) (D.9)We shall also make use of the Jacobi theta functionΘ11(ν ∣τ) = −2ei piτ4 sinpiν ∞∏m=1(1−e2piiτm)(1−e2pii(mτ+ν))(1−e2pii(mτ−ν)) (D.10)This theta function has the modular transformationsΘ11(ν ∣τ +1) = ei pi4Θ11(ν ∣τ) (D.11)Θ11(ν/τ,−1/τ) = −i(−iτ) 12 epiiν2/τΘ11(ν ∣τ) (D.12)152The definition, further properties and uses of the eta- and theta-functions can befound in string theory textbooks. Here, we use the notational conventions ofPolchinski [121].153Appendix EProof without scaling:order-by-order cancellationsAs an alternative to the supersymmetric and scaling arguments of section 4.3 wenow present a perturbative proof of cancellations of all corrections. First note thatall higher-order terms in the action, which we collectively denote Sint , as well ascorrections to the factor 1T ≈ 1T0 in the measure, are proportional to δT p, (p ∈ Z+).If we first separate out the factor e−(pinv)2δT , and then Taylor expand the remaininge−S˜int , the path integral becomes a sum of expectation values of monomials in δT .Focusing on the v, δT part∫ dvdδT (1+ Emv)e− A2 δT 2−BvδT e−(pinv)2δT δT p≡ ⟨(1+ Emv)e−(pinv)2δTδT p⟩ (E.1)= ∞∑k=0(−1)k(pin)2kk!(⟨v2kδT k+p⟩+ Em⟨v2k+1δT k+p⟩) (E.2)154Performing the δT integration (and analytically continuing v→ iv as in section 4.2)one obtainsi√2piA ∫ dv[ ∞∑k=p (−1)k(pin)2kBk+pk! (2k)!(k− p)!(iv)k−p+ ∞∑k=p−1(−1)k(pin)2kBk+pk! (2k+1)!(k− p+1)! Em(iv)k−p+1]e−(Bv)2/2A (E.3)Now since B = 2(pin)2mE , there is an exact term-by-term cancellation between thesetwo sums when p > 0. For p = 0 the only difference is that the first term in theright-hand sum is absent, and therefore the first term in the left-hand sum is notcancelled. This term gives precisely the leading, semi-classical contribution to thepath integral.155Appendix FA toy modelIn this appendix, we shall test the saddle point approximation for the case of anintegral which is similar to, but is much simpler than the worldline and worldsheetpath integrals that we studied in chapters 4 and 5. Consider the integralZ = I∫ ∞01T ∫ d2ze−(T−T0)z¯z−M2/T (F.1)The integral measure and the integrand are real and positive, so the reader mightwonder where it gets an imaginary part. To understand this, notice that, when T0is real and positive, the coefficient of the quadratic form z¯z in the integrand is notpositive for all values of T , if T < T0 it is negative and the z-integration woulddiverge. To define the integral, we first assume that T0 is negative, perform theintegration which is then well-defined, and obtain a function of T0 which can beanalytically continued to the complex T0 plane. It is this analytic continuationwhich produces an imaginary part for the integral when T0 is positive.Explicitly, we first do the Gaussian integral over z to getZ = I∫ ∞01TpiT −T0− iε e−M2/T = pi2T0 e−M2/T0 (F.2)Then, if T0 is real and positive, we define the distribution in the integrand by takingT0 to the real axis from the upper half-plane by replacing piT−T0 with piT−T0−iε . Wecan then use the identity I 1T−T0−iε = piδ(T −T0). This allows us to find the exact156imaginary partZ = pi2T0e−M2/T0 (F.3)Alternatively, we could consider an approximate evaluation of the integral by sad-dle point technique. Such an approximation should be accurate when the “action”S = (T −T0)z¯z+M2/T (F.4)is large, that is then M2T0>> 1. To implement the saddle-point technique, we considerthe classical equations of motion,z¯z−M2/T 2 = 0 , (T −T0)z = 0 (F.5)and we find the classical solutions z = MT0 , T = T0 and we expand the integrationvariables as the classical solutions plus “fluctuations”,z = MT+δ z , T = T0+δT (F.6)Before we proceed, we notice that the action has a flat direction, it is invariantunder the phase transformation z→ e−iθ z, z¯→ eiθ z¯. Such a symmetry will lead toa zero mode in the fluctuations about the classical solution. We must take care ofthis symmetry, and degeneracy of the solution by gauge fixing. Most convenient isthe Fadeev-Popov trick of inserting unity into the integration measure in equation(F.1) in the following form,1 = 12 ∫ 2pi0 dθδ ( 12i(eiθ z¯−e−iθ z))∣ ddθ ( 12i(eiθ z¯−e−iθ z))∣ (F.7)The 12 which appears in front of the right-hand-side arises from a Gribov copy –there are two solutions of the equation eiθ z¯− e−iθ z = 0 in the interval θ ∈ [0,2pi).One then removes θ by a symmetry transformation. The upshot is the insertion ofthe following into the integral in equation (F.1),piδ(Iz)∣Rz∣ = ∫ dc¯dcdb e−piRzc¯c−2piibIz157where c and c¯ are Fadeev-Popov ghosts. It now takes the formZ = I∫ [dT d2zdc¯dcdb] 1T e−(T−T0)z¯z−M2/T−piRzc¯c−2piibIz (F.8)Now, we expand the action about the classical solution (with the classical part ofthe ghosts vanishing). It becomesS = S0+Squad+SintS0 = M2T0 (F.9)Squad = MT0 δT(δ z+δ z¯)+ M2T 30 δT 2+ MT0pi c¯c+2piiIzb (F.10)Sint = δTδ z¯δ z+piRδ zc¯c+ ∞∑k=3M2T k+10 (−δT)k (F.11)The classical part of the action S0 = M2T0 matches the exponent in equation (F.3).The corrections to this classical limit begin with dropping all terms of order higherthan quadratic in the fluctuations, that is, dropping Sint, and doing the remainingGaussian integral over the fluctuations. The resulting integral is complex, due tothe fact that the determinant of bosonic quadratic from is negativedet⎡⎢⎢⎢⎢⎣0 MT0MT02 M2T 30⎤⎥⎥⎥⎥⎦ = −(MT0 )2(F.12)and the gaussian integration contributes the inverse of the square root of this deter-minant. This produces a factor of ipi T0M in the measure, where we have chosen anappropriate sign for i =√−1. The fermionic quadratic form contributes pi MT0 so thenet factor from integrating the fluctuations is ipi2. The remainder of the integrandis evaluated at the classical solution. The result of the WKB approximation to theintegral in this “instanton” sector is then purely imaginaryZ = I{ipi2T0e−M2/T0}and it appears to get the imaginary part of the integral, compare with (F.3), exactly.158However, this should be an approximation. There are higher order than quadraticterms in the action, Sint as well as the factor 1T in the measure, and there should becorrections to this result. Apparently, if the two computations are to match, suchcorrections must cancel.To see how this can happen, we observe that the action in equations (F.9)-(F.11)has a fermionic symmetry under the transformations∆c¯ = −δT , ∆δ z = ∆δ z¯ = pi2c , (F.13)∆c = ∆δT = ∆b = 0 (F.14)This transformation is nilpotent , ∆2(anything) = 0 and also∫ dc¯c∆( anything even in c, c¯) = 0since the integral of any quantity which is odd in the Grassman numbers mustvanish.It is easy to check that ∆S = 0. As well, the measure factor, 1T = 1T0 11+δT/T0 , isinvariant∆[ 1T011+δT /T0 ] = 0Moreover, the “interaction” terms in the action can be written as transforma-tions of simple functions,Sint = δTδ z¯δ z+piδ zc¯c+ ∞∑k=3M2T k+10 (−δT)k = ∆(−δ z¯δ zc¯−∞∑k=3M2T k+10 (−δT)k−1c¯)(F.15)(F.16)In addition the measure factor1T011+ δTT0 = 1T0 +∆[∞∑k=1 c¯(−δT)k−1T k0]159Then,Z = ∫ [dT d2zdc¯dcdb] 1T0 11+ δTT0 e−S0−Squad−Sint= ∫ [dT d2zdc¯dcdb][ 1T0 e−S0−Squad +∆{...}] = ∫ [dT d2zdc¯dcdb] 1T0 e−S0−Squad(F.17)which tells us that the saddle point approximation of the integral is equal to theexact result in the “instanton sector”.160Appendix GGaussian integral computationWe can evaluate the partition function with action (5.9) and open string boundaryconditions in two different ways. In section 5.2 we considered the semi-classicallimit and integrated both T and the coordinates using a saddlepoint technique. Inthis appendix we shall do the Gaussian functional integral over the string embed-ding coordinates (z,xa,xA) and then the integral over the modular parameter of thecylinder, T . By picking up the residues at an infinite sequence of poles we shallreproduce the formula (5.102) for the imaginary part of the vacuum energy [89].For this computation, we introduce the mode expansionz(σ ,τ) = ∞∑n=−∞∞∑m=1e2piinτ√2cosmpiσ znm+ ∞∑n=−∞e2piinτ zn0 (G.1)xa(σ ,τ) = ∞∑n=−∞∞∑m=1e2piinτ√2cosmpiσ xmna+ ∞∑n=−∞e2piinτ xan0 (G.2)xA(σ ,τ) = σ d⃗+ ∞∑n=−∞∞∑m=1e2piinτ√2sinmpiσ xAnm (G.3)where we have used Neumann boundary conditions for xa and Dirichlet boundaryconditions for xA. We will also use Neumann boundary conditions for the normalmodes in which we expand z(σ ,τ). These are not the correct boundary conditionsfor z. The result of using these boundary conditions is that the quadratic formwhich we will obtain is not diagonal.161When we plug the mode expansions into the the action, it becomesS = d⃗24piα ′T+ ∞∑m=0∞∑n=−∞[(2pin)2T 2+(pim)2]4piα ′T ∣xanm∣2+ ∞∑m=1∞∑n=−∞[(2pin)2T 2+(pim)2]4piα ′T ∣xAnm∣2+ ∞∑m=0∞∑n=−∞[(2pin)2T 2+(pim)2]2piα ′T ∣znm∣2− ∞∑n=−∞(2pinE) ∣zn0+ ∞∑m=1√2znm∣2(G.4)The path integral measure is∫ ∞0dT2T ∫ [dXµ(σ ,τ)] [ghost] = V ∫ ∞0 dT2T ∞∏n=−∞ ∞∏m=1dznmdz¯nmdxanmdxAnm⋅⋅ ∞∏n=−∞,n≠0dzn0dz¯n0dxan0 det′ [− 1T ∂ 2σ −T∂ 2τ ](G.5)where we have included the ghost determinant and the prime on the ghost determi-nant indicates that simultaneous zero mode of ∂τ and ∂σ is omitted. The factor ofthe D-brane worldvolume V is from the integral over similar zero modes of xa andz, namely xa00 and z00.The quadratic form in the last line of the action in equation (G.4) is a non-diagonal matrix. This is not surprising, as we have expanded in modes whichdo not obey the correct boundary condition. To proceed, we will have to findthe eigenvalues of that matrix. We can make this slightly easier to deal with byrescaling the modes byznm →√ 2piα ′T(2pin)2T 2+(pim)2 znm , z¯nm →√2piα ′T(2pin)2T 2+(pim)2 z¯nm (G.6)162The resulting action isS = d⃗24piα ′T + ∞∑m=0∑n≠0 z¯nm [δmm′ −Emm′]zm′n+∞∑m=1 ∣zm0∣2+ ∞∑m=0∞∑n=−∞[(2pin)2T 2+(pim)2]4piα ′T ∣xanm∣2+ ∞∑m=1∞∑n=−∞[(2pin)2T 2+(pim)2]4piα ′T ∣xAnm∣2(G.7)where we define the matrix⎡⎢⎢⎢⎢⎣E00(n) E0m′(n)Em0(n) Emm′(n)⎤⎥⎥⎥⎥⎦ = 2pinE⎡⎢⎢⎢⎢⎣ g(n,0)√2g(n,0)g(n,m′)√2g(n,m)g(n,0) 2√g(n,m)g(n,m′)⎤⎥⎥⎥⎥⎦ , (G.8)g(n,m) = 2piα ′T(2pin)2T 2+(pim)2 . (G.9)Taking into account the Jacobian in the measure resulting from this rescaling, andthen doing the Gaussian integrals over the coordinates will result in the appearanceof the determinants in the integrand:Z =V ∫ ∞0dT2Te− d⃗24piα′T ∞∏n=−∞∞∏m=1(2pi)12⎡⎢⎢⎢⎣[(2pin)2T 2+(pim)2]2piα ′T⎤⎥⎥⎥⎦−12 ⋅⋅∏n≠0(2pi)D−12 [(2pin)2T2piα ′ ]−D+12 ⋅ 12pi∏n≠0[det[δmm′ −Emm′]]−1 (G.10)where the infinite products∞∏n=−∞∞∏m=1(2pi)12⎡⎢⎢⎢⎣[(2pin)2T 2+(pim)2]2piα ′T⎤⎥⎥⎥⎦−12which appear in the first line are the determinants arising from integrating xanm,a factor from scaling znm, z¯nm and the ghost determinant. The determinant in thesecond line is from the integral over znm and z¯nm. When infinite products diverge,we use ζ -function regularization to define them (see appendix D. For example, infinding the determinant in the second line of (G.10), we encounter a product over163all of the modes of the factor 2pi ,∏mn(2pi) = (∏m≥12pi)(∏n≠0∏m≥02pi) = (2pi)ζ(0)(∏n≠0(2pi) ⋅(2pi)ζ(0)) = 1√2pi (∏n≠0√2pi)= 1√2pi(∏n≥12pi) = 1√2pi (2pi)ζ(0) = 12pi , (G.11)which is the factor in front of the determinant.The infinite products in (G.10) can be put in a more convenient form. Usingthe results of appendix D we have that the infinite product in the first line of (G.10)reduces to the usual modular form∞∏m=1∞∏n=−∞⎡⎢⎢⎢⎣[(2pin)2T 2+(pim)2]2piα ′T⎤⎥⎥⎥⎦−12 = [e−pi/12T ∞∏m=1[1−e−pim/T ]2]−12 = η−24(i/2T)(G.12)where η(τ) is the Dedekind eta-function (see Appendix A). In addition, usingzeta-function regularization,∞∏n=−∞,n≠0(2pi)D−12 [(2pin)2T2piα ′ ]−D+12 = 2pi ∞∏n=1[(2pin)2T4pi2α ′ ]−(D+1) = 2pi [ T4pi2α ′ ]D+12.We are left with the last determinant in the second line of (G.10). Observe that thematrix E defined in (G.8) is the outer product of a vector and its transpose:E = tanh(piε)coth(2pinT)vmvm′ , (G.13)where the normalized vector v⃗ is given byvm =√ nα ′ tanh(2pinT)(√2)1−δm0√g(n,m). (G.14)It therefore has only one non-zero eigenvalue, namely2piα ′E coth(2pinT) = tanh(piε)coth(2pinT)Inserting this result into the partition function, and after transforming the integra-164tion variable T → 1/2T , we find the expressionZ =V ∫ ∞0dTTe−T d⃗2/2piα ′η−24(iT)[8pi2α ′T ]−D+12∏∞n=1 [1−(2piα ′E)2 coth2(pin/T)] (G.15)When E → 0 this expression approaches the usual cylinder amplitude of the openbosonic string suspended between two D-branes. Moreover, the integrand now haspoles at discrete values of T ,Tk = kε (k ∈Z+), tanhpiε ≡ 2piα ′E (G.16)To proceed, it is necessary to find the residues of the poles. For this purpose, itis convenient to perform a modular transformation of the infinite product. In orderto do this, it is convenient to first write the infinite product as a product representa-tion of Jacobi theta functions1 and then to use the known modular transformationproperty of the theta function. For this purpose, we use the following sequence ofmanipulations,∞∏n=11[1− tanh2piε coth2(pin/T)] = ∞∏n=1 cosh2piε sinh2(pin/T)[sinh(pin/T −piε)sinh(pin/T +piε)]= 1coshpiε∞∏n=1[1−e−2pin/T ]3[1−e−2pin/T ][1−e−2(pin/T−piε)][1−e−2(pin/T+piε)] = 2icothpiε η3(i/T)Θ11(iε ∣i/T)Using the modular transformation of the theta- and eta-functions, we find∞∏n=11[1−(2piα ′E)2 coth2(pin/T)] = 2T tanhpiε ⋅e−piε2T η3(iT)Θ11(−εT ∣iT)= tanhpiε Te−piε2Tsin(piεT) ∞∏n=1 [1−e−2piT n]2(1−e−2Tpi(n+iε))(1−e−2Tpi(n−iε)) (G.17)Now, the factors in the infinite product are regular for nonzero real values of T .The poles on the real T -axis originate from the factor of the inverse of sin(piεT)which is outside of the infinite product. The poles occur at T = k/ε and, at each1See Appendix A for a definition of the relevant theta function and its modular transformationproperty.165pole, the residue is (−1)kke−pikε tanhpiεpiε2Note that the remaining infinite product is simply equal to one at the position ofthe pole. The imaginary part of the partition function is then given by a sum over(half-) residues at the poles,Z =R(Z)+V 122pii∞∑k=112Tk(−1)kke−pikε tanhpiεpiε2e−kd⃗2/2piα ′εη24(ik/ε) [8pi2α ′k/ε]−D+12(G.18)so that the rate of pair production isΓ = 2VI(Z) = tanhpiεε∞∑k=1(−1)k+1e−pikM2 2α′ε −kpiεη24(ik/ε) [ε/2α ′4pi2k ]D+12. (G.19)166Appendix HFluctuation prefactor fromexplicit mode expansionIn section 5.2 we evaluated the quadratic fluctuation prefactor using the Gelfand-Yaglom approach for functional determinants. In this appendix we present, forcompleteness, a “brute-force” calculation of the same fluctuation integral using anexplicit mode expansion. Since the eigenvalues of the µ = 0,1 fluctuation operatorare determined by a transcendental equation, and therefore not known explicitly,we use modes for δ z which do not obey the correct boundary condition, but wouldbe appropriate to the same problem with Neumann boundary conditions. Thisyields a non-diagonal quadratic form. With a little work, and some cavalier manip-ulations of infinite matrices, we are able to find its determinant, as well as extractthe zero- and tachyonic modes, to obtain finally the result (5.2).Returning therefore to equation (5.22), we now expand the fluctuations inmodes asδ z(σ ,τ) = ∞∑n=−∞∞∑m=1e2piinτ√2cospimσ δ znm+ ∞∑n=−∞e2piinτ δ zn0 (H.1)δxa(σ ,τ) = ∞∑n=−∞∞∑m=1e2piinτ√2cospimσδxanm+ ∞∑n=−∞e2piinτδxan0 (H.2)δxA(σ ,τ) = ∞∑n=−∞∞∑m=1e2piinτ√2sinpimσδxAnm (H.3)167Even though δ z has the wrong boundary conditions, the linear terms in δ z vanishdue to the fact that the classical solution obeys the equation of motion and it hasthe correct boundary conditions. Using this mode expansion and the equations ofmotion the action becomesS = km20 2piα ′ε +aδT 2+ d2piα ′T 20 sinh(piε)δT∞∑m=0qm(δ zkm+δ z¯km)√2+ ∞∑m=0∞∑n=−∞∣δ znm∣22piα ′T0 [T 20 (2pin)2+(pim)2]− ∞∑n=−∞2pinE ∣δ zn0+ ∞∑m=1√2δ znm∣2+ ∞∑m=0∞∑n=−∞∣δxanm∣24piα ′T0 [T 20 (2pin)2+(pim)2]+ ∞∑m=1∞∑n=−∞∣δxAnm∣24piα ′T0 [T 20 (2pin)2+(pim)2]+ . . . (H.4)whereqm ≡ (√2)1−δm0 (piε)2−pi2m2(piε)2+pi2m2 , a ≡ d22piα ′ 14T 30 (1+ cosh(piε)sinh(piε)piε )(H.5)Transverse fluctuations and ghostsPerforming the Gaussian integral over the coordinates δxa,A yields the followingfactors in the path integral measure, for each of the modes with m ≠ 0∞∏n=−∞∞∏m=1[ 2piα′T0(2pin)2T 20 +(pim)2 ]12 = η−24(i/2T0) (H.6)where η(z) is the Dedekind eta-function. For the modes with m = 0∏n≠0(2pi)D−12 [ 2piα′(2pin)2T0 ]D−12 =∏n>0[ 4pi2α ′(2pin)2T0 ]D−1= [ 4pi2α ′(2pi)2T0 ](D−1)ζ(0)e2(D−1)ζ ′(0) = [ T04pi2α ′ ]D−12. (H.7)168where we have used zeta-function regularization to define the formally divergentinfinite product (see appendix D). Evaluating the ghost determinant (see discussionof reparametrization ghosts in section 5.2) yields the factordet[− 1T0∂ 2σ −T0∂ 2τ ] = ∞∏n=−∞∞∏m=1(2pin)2T 20 +(pim)22piα ′T0 = η+2(i/2T0) (H.8)Lightcone-coordinate fluctuationsConsider the quadratic form containing the variables δ znm. In order to evaluate itsdeterminant, we shall have to find its eigenvalues. In order to define its eigenvaluesit is convenient to rescale all δ znm where either m or n is nonzero asδ znm →√g(n,m)δ znm, with g(n,m) = 2piα ′T0T 20 (2pin)2+(pim)2 (H.9)This rescaling results in a Jacobian in the measure, (where the second productarises from the m = 0 modes)∞∏n=−∞∞∏m=1[ 2piα′T0T 20 (2pin)2+(pim)2 ]∞∏n=1[ 2piα′T0T 20 (2pin)2 ]2= { ∞∏n=−∞∞∏m=1[ 2piα′T0T 20 (2pin)2+(pim)2 ]}[ 2piα′T0T 20 (2pi)2 ]2ζ(0)e4ζ′(0)= 2piη2(i/2T0) T04pi2α ′ (H.10)This leaves the following quadratic form in the actionSquadratic = aδT 2+ d2piα ′T 20 sinh(piε)δT∞∑m=0qm√g(k,m)(δ zkm+δ z¯km)√2+ ∞∑m=0∑n≠0δ z¯nm [δmm′ −Emm′(n)]δ znm′ +∞∑m=1 ∣δ z0m∣2+ . . . (H.11)169where⎡⎢⎢⎢⎢⎣E00(n) E0m′(n)Em0(n) Emm′(n)⎤⎥⎥⎥⎥⎦ =2pinE⎡⎢⎢⎢⎢⎣ g(n,0)√2g(n,0)g(n,m′)√2g(n,m)g(n,0) 2√g(n,m)g(n,m′)⎤⎥⎥⎥⎥⎦ , m,m′ = 0,1,2, ...(H.12)= tanh(piε)coth(piεn/k)vmvm′ (H.13)This is just the matrix in (G.8), evaluated at T = T0. Consequently the quadraticform (I −E (n)) has eigenvalues 1 and 1− tanh(piε)coth(piεn/k). When n = k thelatter is zero. The integration of the modes δ znm with n = k excluded produces thefactor∏n≠0,k1det(I −E (n)) = 1∏n≠0,k [1− tanh(piε)coth(piεn/k)]= (−1)k+1e−pikεcosh2piε(H.14)which is derived in appendix D, equation (D.7).Tachyon: real n = k modes coupled to δTNow consider n = k, for which (I−E ) has a zero eigenvalue. Note that δT couplesonly to the real part of δ zk. More precisely, we have that δ zkm+δ z¯km√2 and δT arecoupled in the following quadratic form:M = ⎛⎝ a J⃗⊺J⃗ I −Ek ⎞⎠ (H.15)where the δT , δ z cross-term isJm = d sinh(piε)4piα ′T 20 qm√g(k,m) (H.16)and qm is given by (H.5). We found the spectrum of the submatrix (I−E ) above. Inthe gaussian integration over the quadratic form M, we will find that although δTcouples to all eigenmodes of (I −E ), only its coupling to the would-be zero mode170v contributes to the Gaussian integral. We demonstrate this as follows. (I −E ) isdiagonalized by its matrix of orthonormal eigenvectorsΛ ≡ S⊺(I −E )S= diag[0,1,1,1 . . .] , (H.17)S = (v⃗, u⃗1, u⃗2, . . .) (H.18)v⃗ is defined in (G.14), and we will not need the precise form of the u⃗’s. Under thischange of variables, whose jacobian is 1, M becomes⎛⎝ 1 S⊺ ⎞⎠M⎛⎝ 1 S ⎞⎠ =⎛⎜⎜⎜⎜⎜⎜⎜⎜⎝a J⃗ ⋅ v⃗ J⃗ ⋅ u⃗1 J⃗ ⋅ u⃗2 . . .J⃗ ⋅ v⃗ 0J⃗ ⋅ u⃗1 λ1J⃗ ⋅ u⃗2 λ2⋮ ⋱⎞⎟⎟⎟⎟⎟⎟⎟⎟⎠(H.19)We denote our new integration variables ym′ ≡ (S⊺)m′m δ zkm+δ z¯km√2 , and for the tachy-onic mode, y0 ≡ v = ∑∞m=0 vm δ zkm+δ z¯km√2 . We can now write the quadratic form asfollows(δT, y⃗⊺)M⎛⎝ δTy⃗ ⎞⎠ = y⃗⊺Λy⃗+2δT ∞∑i=1 yiJ⃗ ⋅ u⃗i+(δT,v+)⎛⎝ a J⃗ ⋅ v⃗J⃗ ⋅ v⃗ 0 ⎞⎠⎛⎝ δTv+ ⎞⎠(H.20)The first two terms on the RHS are independent of y0. Thus completing the squarein yi (i > 0) to eliminate the second term only has the effect of modifying the co-efficient a of δT 2 appearing in the last term. But due to the form of the latter, thedeterminant obtained after integrating out δT , y0 is independent of a.171Therefore the total contribution from integrating out Re[δ zkm] and δT isdet′ (I −E (k))−1/2 det⎡⎢⎢⎢⎢⎣⎛⎝ a J⃗ ⋅ v⃗J⃗ ⋅ v⃗ 0 ⎞⎠⎤⎥⎥⎥⎥⎦−1/2 = (∏′1)det⎡⎢⎢⎢⎢⎣⎛⎝ a J⃗ ⋅ v⃗J⃗ ⋅ v⃗ 0 ⎞⎠⎤⎥⎥⎥⎥⎦−1/2= ±i ∣J⃗ ⋅ v⃗∣−1 (H.21)= ±i√4piα ′T 30 sinh(2piε)d2piε(H.22)where the prime means we exclude the zero mode. The square root of a negativedeterminant gives rise to a factor i. This “tachyonic” mode corresponds to fluctua-tions in the radius of the instanton, with respect to which it is unstable.Imaginary n = k modes & zero modeReturning to (H.11), we note that Im[δ z] does not couple to δT , so in this case thezero eigenvalue of (I−E (k)) corresponds to a genuine zero-mode of the quadraticaction:z ≡ ∞∑m=0vmδ zkm−δ z¯km√2i(H.23)The gauge-fixing procedure is described in subsection 5.2.2. The net result is theremoval of the zero eigenvalue, and the introduction of a compensating Faddeev-Popov jacobian,∬ dσdτ [ ˙¯z(σ ,τ) ˙ˆzcl(σ ,τ)+c.c.] = ∥z˙cl∥+∬ dσdτ [δ ˙¯z ˙ˆzcl(σ ,τ)+c.c.] (H.24)The first term evaluates to2pikd√2piε√2piε + sinh(2piε)4piε,while the latter term, which is the projection of δ z onto the tachyonic mode zcl , isa subleading correction and is to be dropped in the semiclassical approximation.It is important here to note the following subtlety, that the rescaling of δ znm(equation (H.9)) does not commute with the introduction of our collective coordi-nate. To account for this, we regulate the determinant, divide out by the (putative)172zero eigenvalue, and then take the limit of the regulator going to zero.We want the determinant, with zero-eigenvalue excluded, of the operatorLˆ ≡ 12piα ′T0 (−∂ 2σ +(piε)2)with the appropriate boundary conditions. A possible regularization isLˆ(δ) = 12piα ′T0 (−∂ 2σ +(piε(1+δ))2)Then the zero eigenvalue gets shifted toλ (δ)0 = 2pikεα ′ δ +O(δ 2)and g(n,m) (equation (H.9)) gets modified in accordingly. Now rescale as before,δ zkm →√g(δ)km δ zkm. The corresponding jacobian Jg has an O(δ) correction whichwill not be important. The resulting quadratic form is now (I −E (δ)(k)), whereE (δ) is defined as in (H.12) but with g(n,m) replaced everywhere by g(δ)(n,m).This has the same structure as before, with only the pseudo-zero eigenvalue λ˜0modified toλ˜ (δ)0 = 1−2pikE [g(δ)k0 +2 ∞∑m=1g(δ)km ] = 1− tanh(piε)coth(piε(1+δ))(1+δ)= (1+2piε csch(2piε)) ⋅δ +O(δ 2) (H.25)The determinant is then(det′Lˆ)−1/2 = limδ→0¿ÁÁÀ λ (δ)0detLˆ(δ) = limδ→0J(δ)g √λ (δ)0 /λ˜ (δ)0 = Jg [2pikεα ′ sinh(2piε)2piε + sinh(2piε)]1/2(H.26)The factor Jg was already accounted in (H.10), so the net contribution obtainedhere for the n = k imaginary modes (including the Gribov factor ω−1 = 1/2k) is[2pikεα ′ sinh(2piε)2piε + sinh(2piε)]1/2 ⋅ 12k ⋅ 2pikdpiε√2piε + sinh(2piε)4piε= d ksinh(piε)ε√kα ′ tanh(piε)(H.27)173Final ResultGathering all of the factors, including the product ∏nm(2pi) evaluated in (G.11),we obtain for the tunneling amplitude1Γstring = 2V I(Z) = ±i tanhpiεε ∞∑k=1 (−1)k+1e− 2piα′kM2ε −kpiεη24(ik/ε) [ε/2α ′4pi2k ]D+12(H.29)in agreement with the Gelfand-Yaglom calculation of section 5.2.1 In detail:2VI(Z) = 2 ∞∑k=1e−kd22piα′ε ⋅ 12T0dcurlymeasure⋅η−24(i/2T0)´udcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymod¸udcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymod¶δxm>0a/A⋅[ T04pi2α ′ ]D−12´udcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymod¸udcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymod¶δxm=0a/A⋅η+2(i/2T0)´udcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymod¸udcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymod¶ghosts⋅ 2piη2(i/2T0) T04pi2α ′´udcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymod¸udcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymod¶δ z rescaling⋅(−1)k+1e−pikεcosh2piε´udcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymod¸udcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymod¶δ zn≠k⋅(±i)¿ÁÁÀ4piα ′T 30 sinh(2piε)d2piε´udcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymod¸udcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymod¶tachyon⋅ d k sinh(piε)ε√kα ′ tanh(piε)´udcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymod¸udcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymod¶JFP⋅ 12pi(H.28)174


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