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Emergent geometry through holomorphic matrix models Pietromonaco, Stephen 2017

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Emergent Geometry Through HolomorphicMatrix ModelsbyStephen PietromonacoB.Sc., The University of New Mexico, 2014A THESIS SUBMITTED IN PARTIAL FULFILLMENT OFTHE REQUIREMENTS FOR THE DEGREE OFMASTER OF SCIENCEinThe Faculty of Graduate and Postdoctoral Studies(Physics)THE UNIVERSITY OF BRITISH COLUMBIA(Vancouver)August 2017c© Stephen PietromonacoAbstractOver the years, deep insights into string theory and supersymmetric gauge theorieshave come from studying geometry emerging from matrix models. In this the-sis, I study the N = 1∗ and N = 2∗ theories from which an elliptic curve withmodular parameter τ is known to emerge, alongside an elliptic function called thegeneralized resolvent into which the physics is encoded. This is indicative of thecommon origin of the two theories in N = 4 SYM. The N = 1∗ Dijkgraaf-Vafamatrix model is intrinsically holomorphic with parameter space corresponding tothe upper-half plane H. The Dijkgraaf-Vafa matrix model ’t Hooft coupling S(τ)has been previously shown to be holomorphic on H and quasi-modular with re-spect to SL(2,Z). The allowed N = 2∗ coupling is constrained to a Hermitianslice through the enlarged moduli space of the holomorphic N = 1∗ model.After explicitly constructing the map from the elliptic curve to the eigenvalueplane, I argue that theN = 1∗ coupling S(τ) encodes data reminiscent ofN = 2∗.A collection of extrema (saddle-points) of S(τ) behave curiously like the quantumcritical points ofN = 2∗ theory. For the first critical point, the match is exact. Thiscollection of points lie on the line of degeneration which behaves in a sense, like aboundary at infinityI also show explicitly that the emergent elliptic curve along with the generalizedresolvent allow one to recover exact eigenvalue densities. At weak coupling, mymethod reproduces the inverse square root of N = 2∗ as well as the Wigner semi-circle in N = 1∗. At strong coupling in N = 1∗, I provide encouraging evidenceof the parabolic density arising in the neighborhood of the line of degeneration.To my knowledge, the parabolic density has only been observed asymptotically. Itis interesting to see evidence that it may be exactly encoded in the other form ofemergent geometry: the elliptic curve with the generalized resolvent.iiLay AbstractThe main novelty of holomorphic matrix models is the enlarged moduli space theyenjoy. A particular instance I focus on is the N = 1∗ matrix model where analgebraic curve emerges. I uncover connections to N = 2∗ theory which wouldbe invisible without fully exploiting the enlarged moduli space. I also show thateigenvalue densities are encoded into the algebraic curve. Using this, I recover thefamous Wigner semi-circle and have encouraging evidence of a parabolic eigen-value density encoded into the bulk of the moduli space. This parabolic densityis of fundamental importance in other (seemingly unrelated) theories of emergentgeometry.iiiPrefaceThis thesis employs a combination of analytical, numerical, and computationaltechniques. I use so-called holomorphic matrix models to conjecture extensionsof previously established results about certain quantum field theories. I used Math-ematica to construct the necessary algorithms. The elliptic and modular functionsof interest are sufficiently chaotic at the extreme parameter values needed. There-fore, basic numerical procedures had to be employed. To analyze the resulting data,I used standard techniques in curve fitting.ivTable of ContentsAbstract . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . iiLay Abstract . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . iiiPreface . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ivTable of Contents . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . vAcknowledgements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . vii1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12 Introduction to Holomorphic Matrix Models . . . . . . . . . . . . . 82.1 Definition of the Model at Finite N . . . . . . . . . . . . . . . . 82.2 The ’t Hooft Limit and the Emergence of a Hyperelliptic Curve . 102.3 Physical Discussion of Classical and ’t Hooft Limits . . . . . . . 142.4 Summary and Outlook . . . . . . . . . . . . . . . . . . . . . . . 153 Introduction to the N = 1∗ and N = 2∗ Theories . . . . . . . . . . 173.1 N = 4 Super Yang-Mills Theory (SYM) in Four Dimensions . . 173.2 The N = 1∗ Theory and the Holomorphic Matrix Model . . . . . 193.3 The N = 2∗ Theory and its Critical Points . . . . . . . . . . . . 244 Embedding the Elliptic Curve into the Eigenvalue Plane . . . . . . 294.1 The Construction . . . . . . . . . . . . . . . . . . . . . . . . . . 304.2 A Summary of the Method . . . . . . . . . . . . . . . . . . . . . 334.3 Schematic Description of the Moduli Space of the Theory . . . . 355 The Quasi-Modular Coupling S(τ) . . . . . . . . . . . . . . . . . . 395.1 The ’t Hooft Coupling of N = 1∗ . . . . . . . . . . . . . . . . . 40vTable of Contents5.2 A Relationship between the Gauge Theory and the Matrix Model 445.3 Weak Coupling Expansion of S(τ) . . . . . . . . . . . . . . . . 465.4 Surprising Connections to N = 2∗ Theory . . . . . . . . . . . . 476 Eigenvalue Densities . . . . . . . . . . . . . . . . . . . . . . . . . . . 546.1 The General Method . . . . . . . . . . . . . . . . . . . . . . . . 556.2 N = 2∗ at Weak Coupling: Shenker’s Inverse Square Root . . . . 586.3 N = 1∗ Eigenvalue Densities . . . . . . . . . . . . . . . . . . . 587 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 66viAcknowledgementsFirst and foremost, I am deeply indebted to my advisor, Dr. Joanna Karczmarek.Without her tremendous technical assistance, intuition, support, and patience, thisproject could not have been completed. I am also very grateful to Jamie Gordon,Malik Barrett, Javier Gonza´lez Anaya, Elliot Cheung, and Jake Bian for enlighten-ing conversations and suggestions. Finally, I am thankful to Konstantin Zarembofor gracious and helpful correspondence.viiChapter 1IntroductionThe notion of “emergent geometry” in theoretical physics is the idea that the geom-etry underlying a theory is not provided as input, but rather “emerges” from some-thing more fundamental. This has the manifest benefit of making the theory lessarbitrary. For example, one usually specifies a quantum field theory by choosing aspacetime manifoldM , inventing an action depending on structures defined onM ,and then attempting to extract information from Feynman’s path integral prescrip-tion, the hope being that one can compute algebraic information like correlationfunctions. Therefore, we have data in the form of correlation functions emergingfrom our choice of manifold M . The obvious problem with this program is the ar-bitrary choice we must make for a spacetime geometry. One can hope to turn thisprogram around, and find geometrical structures emerging from more fundamentalinput data.Two questions immediately arise. First, what should we take as the initial inputdata? This should be something concrete, and physically motivated. Second, whatsort of geometry will emerge, and what information will it provide? Let us beginby tackling the first question. One of the most basic, algebraic objects motivatedby physics is a matrix model. This is the simplest possible quantum field theory:namely, a gauge theory defined on a 0-dimensional manifold. A particular matrixmodel is determined by a choice of polynomial superpotential W (x) and a certaincollection Γ of N ×N matrices Φ over which to integrate. The partition functionis then given byZN (Γ, gs) := 1C∫ΓdΦe− 1gsTrW (Φ), (1.1)where C is a normalization constant and gs is the coupling, playing an analogousrole to ~ in quantum mechanics. Notice that in a general gauge theory, the actionrequires integrating the Lagrangian density over the manifold, but here, with a zero-dimensional spacetime, this integral is replaced by the trace. This is a concreteand computable quantity, certainly in comparison to richer quantum field theories.1Chapter 1. IntroductionHowever, such a model appears to be completely devoid of any elegant geometryat first sight. One profoundly important insight has been that by taking the ’t Hooftlimit of certain matrix models, one can extract quite rich geometrical structures.To tackle the second question, we ask what kinds of geometries do these modelsencode? The first example of geometry emerging from matrix models is througheigenvalue densities. This originated with Eugene Wigner, who considered Her-mitian matrix models with Gaussian actions [20]. Hermitian matrix models arematrix models where the eigenvalues distribute themselves on the real axis. In suchmodels with Gaussian actions, one can study the statistical distribution of the eigen-values in the ’t Hooft limit. Wigner noticed that in this limit, the eigenvalues aredistributed continuously over a path, called a cut. The density of this distribution isa semi-circle. This is Wigner’s famous semi-circular distribution and is the first in-stance of geometry emerging from matrix models. It is rather remarkable that sucha rigid shape can emerge from a path integral over matrices. In this thesis, I willstudy models which coincide with Wigner’s Gaussian model in certain limits, andI exactly recover the semi-circular distributions. In a related fashion, I will also ex-actly recover the inverse square-root of Shenker and Douglas [8]. Recently, peoplehave been interested in matrix models in which a parabolic eigenvalue distributionemerges. This density indicates a multi-matrix model in a commuting phase, and tomy knowledge, has only been observed numerically, or by employing approxima-tions. I will provide partial evidence in support of an analytical method to recoverthe parabolic density.The second form of emergent geometry is the algebraic varieties which arisefrom the equations of motion of certain matrix models. In [3] it was realizedthat one-matrix models with a polynomial superpotential determined hyperellipticcurves. This was studied in detail in [15], which motivated our following chapter.Also in [3] it was shown that these matrix models play a crucial role in the geomet-ric transitions in topological string theories on Calabi-Yau manifolds. In this thesisI will be concerned with the well-known solutions to N = 1∗ and N = 2∗ theo-ries in which an elliptic curve emerges. It is quite astounding that something likean algebraic variety can emerge from something seemingly so structureless like amatrix model.There is also a third form of emergent geometry from matrix models: the clas-sical equations of motion of a multi-matrix model may define a non-commutative(fuzzy) geometry. This is usually referred to as a non-commutative backgroundgeometry. For example, one might have a three-matrix model involving a path inte-2Chapter 1. Introductiongral over three matrices Φ1, Φ2, and Φ3 in which the classical equations of motionconstrain the matrices to satisfy[Φ1,Φ2] = iΦ3, (1.2)in addition to the two cyclic permutations [16]. In this case, the background geom-etry is a non-commutative two-sphere.I have reviewed three manifestations of geometry encoded into matrix models:the distributions of eigenvalues, the semi-classical algebraic curves, and the non-commutative background geometries. A connection was recently established be-tween two of these [1, 2, 16, 21] studying multi-matrix models which enjoy a com-muting phase given by a parabolic eigenvalue density, as well as a non-commutingphase. In the low-temperature limit, as one passes to the non-commuting phase,the non-commutative sphere emerges. The conclusion is that a geometry emergesas a system “cools”; perhaps the geometry in our universe emerged post-Big Bangin an analogous (albeit, much more complex) way. So these particular models linkemergent eigenvalue densities and emergent non-commutative geometries.One aim of my thesis is to describe how this story relates to an emergent alge-braic curve. I will attempt to recover eigenvalue densities from this curve, whichwill connect all three forms of emergent geometry discussed above. As mentioned,I will study models which are intimately related to an elliptic curve. The physicswill determine a modular parameter τ , as well as a particular elliptic function on thecurve. From these two bits of physically determined data, I will provide an exactalgorithm for recovering eigenvalue densities, which to my knowledge, is not ex-plicit in the literature. This method will recover not only Wigner’s semi-circle andShenker’s inverse square-root, but also a hint of the parabolic eigenvalue density.So it appears that in certain cases, the eigenvalue densities are actually encoded intoan emergent algebraic variety.The discussion will rely on a generalization of an Hermitian matrix model, al-lowing for much richer structure. These are known as holomorphic matrix modelsand I introduce them in the next chapter. In broad brush, these models allow foreigenvalues to be supported on more general paths in the complex plane, whereasHermitian models have eigenvalues constrained to the real axis. The emergent al-gebraic curves are varieties over C, not R, so this is very natural. Holomorphicmatrix models explore in full the relevant portion of the moduli spaces of thesecurves. Their Hermitian siblings, merely explore a “slice” through the relevantmoduli space. Having access to a larger region within a moduli space means that3Chapter 1. Introductionone can potentially avoid critical points. In a Hermitian matrix model, there existsthe possibility of complete termination upon colliding with a critical point. How-ever, if it’s possible to embed the theory into a holomorphic model, one can avoidthe critical point and explore the bulk of the moduli space. I will attempt a ma-neuver of this sort in this thesis. In particular, the N = 2∗ theory is well-knownto have an infinite sequence of critical points accumulating at strong coupling. Itis also well-known that at weak coupling an emergent elliptic curve offers an exactsolution of the theory, but beyond the first critical point the Hermitian slice degen-erates completely. I hope to argue that by employing holomorphic matrix models,we may by-pass this first critical point, continuing to a region of the moduli spacewhich may shed light on strong coupling behavior. More generally, the fundamentalquestion I will consider is the following.Given a Hermitian matrix model in the ’t Hooft limit whose weak coupling regionembeds naturally along the Hermitian slice up to the first critical point, can thegeometrical solution using an elliptic curve detect the strong coupling region?In Chapters 5 and 6, I answer partially in the affirmative for the N = 2∗ theory aswell as the Hermitian model studied in [1, 2, 16, 21], respectively.Earlier I indicated that the program was to choose a matrix model as input, fromwhich richer structure may emerge. This is a good general philosophy, however thematrix models considered in my thesis actually arise from attempts to answer diffi-cult questions in quantum field theory. In quantum field theories, computing phys-ical observables like the partition function, or correlation functions, requires com-puting Feynman’s path integral. In general, the path integral is understood only as aschematic or heuristic device. In certain special cases however, the infinite dimen-sional path integral “localizes” to something manageable. For example, it mightlocalize to an ordinary integral over a concrete space, or a matrix model, or perhapseven a sum. Compared to an infinite dimensional path integral, a matrix model isa relatively computable quantity. It is in this sense that an impenetrable observablein a quantum field theory may be computed by considering a matrix model. Morespecifically, there will be a certain class of observables called a subsector whichcan be exactly computed. Using matrix models to compute these observables willnot be the main focus of my thesis. Rather, I will be studying the matrix modelsthemselves. Nevertheless, it is important to understand where the objects one isstudying came from. I provide a very brief outline of this.4Chapter 1. IntroductionOne of the most fundamental quantum field theories isN = 4 super Yang-Mills(SYM) in four dimensions. This theory is superconformally invariant, even quan-tum mechanically, and it enjoys the mysterious S-duality, which seems to identifyeach physically distinct parameter value with an elliptic curve. This theory has twounique massive deformations preservingN = 1 andN = 2 supersymmetry: theseresulting theories are known as N = 1∗ and N = 2∗, respectively.The N = 1∗ TheoryIn general, N = 1 supersymmetric gauge theories onM = R4 contain a “holomor-phic subsector.” This is a special class of physical observables which are constrainedto depend holomorphicaly on the parameters of the theory. More specifically whatthis means is that given such a theory, one can compute an effective superpoten-tial Weff, which is a holomorphic function of the parameters. In turn, the vacuumexpectation values of the special observables are given by differentiating Weff withrespect to these parameters. All observables that can be obtained in this mannerdefine the holomorphic subsector. The question immediately becomes, how doesone compute Weff? Dijkgraaf and Vafa [5] have proposed a solution that works inparticular for the N = 1∗ theory.The N = 1∗ theory enjoys an extremely rich classical vacuum structure. Infact, each classical vacuum will be given by a non-commutative background ge-ometry of the form described earlier. The vacua are discrete, and depend on theamount of symmetry broken in the original gauge group. There will be a differenteffective superpotential for each different vacuum. The Dijkgraaf-Vafa prescriptionis that one should consider a holomorphic matrix model expanded around the rele-vant vacuum. They conjecture that the ’t Hooft limit of this matrix model computesobservables in the field theory for finiteN in the holomorphic subsector. In particu-lar, one can exactly compute the effective superpotentialWeff in this vacuum whichwill depend on a modular parameter τ . The holomorphic observables will arise bydifferentiating Weff with respect to τ .This conjecture is in the same spirit described above: an impenetrable compu-tation in quantum field theory localizes to something manageable and computable.In this case, the localization refers to considering a matrix model as a fluctuationabout a particular classical vacua.5Chapter 1. IntroductionThe N = 2∗ TheoryOne of the landmark achievements recently in theoretical physics, is the realizationby Pestun [17] that certain observables inN = 2 supersymmetric gauge theories areexactly computable on a spacetime M = S4. This remarkable method is known aslocalization. Roughly speaking, the field content of N = 2 SYM contains a scalarfield Φ transforming in the adjoint representation of a gauge group, whose vacuumexpectation value (VEV) is〈Φ〉 = diag(a1, . . . , aN ). (1.3)The eigenvalues {aI} of the adjoint scaler field VEV are called Coulomb mod-uli as they parameterize the Coulomb branch moduli space. Pestun’s localizationreduces the computation of certain observables to finite-dimensional matrix inte-grals over the Coulomb moduli. For instance, the partition function ofN = 2 superYang-Mills compactified onS4, localizes to a finite-dimensional matrix model. Thepartition function of N = 2∗ on S4 also localizes to a matrix model. I will studythis matrix model by letting the radius of S4 go to infinity (the decompactificationlimit) and simultaneously taking the ’t Hooft limit. It is precisely in such a scenariowhere S4 can be approximately identified with R4 and we may see connections tothe emergent geometry of N = 1∗.I make one final informal remark about the mathematical analog to localizationin physics. Ideally, one would like to be able to integrate any differential form overany smooth manifold M . This is a non-trivial problem given that in general, ac-tually evaluating an integral is merely inspired guesswork. If however, one has agroup action on the manifold, there is a special class of forms called equivariantforms whose integral over M reduces to a sum over indices at the fixed points ofthe group action. This is known as Atiyah-Bott localization in mathematics. Butnotice how analogous this is to our physical problem in quantum field theory: ob-servables require computing a path integral, even whose definition, much less exactvalue eludes us. However, in certain theories, we have a special class of observables(a subsector) which may be computed exactly in an easy way using some “localiza-tion” method. In N = 2∗ this is the localization of Pestun, while in N = 1∗, thisis the Dijkgraaf-Vafa conjecture using holomorphic matrix models.Again, the quantum field theory questions will not be the main focus of my thesis,but rather the matrix models themselves. However, I hope that it is enlightening tounderstand why I am studying these matrix models, and where they came from.6Chapter 1. IntroductionMoreover, certain results can be interpreted as a “shadow” of some structure in theoriginal theory before deformation or localization. For instance, the elliptic curvesemerging from the mass-deformed matrix models seem to be an artifact of theirorigin in N = 4 SYM. Finally, and most importantly, perhaps some matrix modelresult might help in understanding quantum field theory questions in future work.7Chapter 2Introduction to HolomorphicMatrix ModelsThis chapter is modeled on Calin Lazaroiu’s excellent paper [15] and contains nooriginal results. The goal is to introduce the foundations of holomorphic matrixmodels which will pervade the later chapters.2.1 Definition of the Model at Finite NI begin by considering MatN (C), the space of N × N matrices with complex en-tries. This is an N2-dimensional manifold identified with CN2 . Let the subsetD ⊂ MatN (C) be defined to contain all matrices which are diagonalizable. In otherwords, for all Φ ∈ D, there exists a general linear transformation S ∈ GLN (C) suchthat Φ becomes diagonal upon conjugation by SSΦS−1 = diag(λ1, . . . , λN ), (2.1)where the λI are the eigenvalues of Φ. The subset D constitutes an open submani-fold of MatN (C) = CN2 . One can define the spectrum σ(Φ) of a matrix Φ ∈ D tobe simply the set of eigenvaluesσ(Φ) := {λ1, . . . , λN}. (2.2)Let Γ denote a connected, non-compact, boundary-less submanifold ofD, which ishalf-dimensional, i.e.,dimRΓ = dimCD = N2. (2.3)In other words, the real dimension of Γ is precisely the complex dimension ofD; itis in this sense that Γ is a half-dimensional “real slice” through D.82.1. Definition of the Model at Finite NA polynomial superpotential is defined by the following degree-(n + 1) poly-nomial (n ≥ 2) with complex coefficientsW (x) =n+1∑I=0uIxI . (2.4)We will soon see that for my purposes, only the critical points of W (x) will berelevant and thus, all polynomial superpotentials which differ by a constant willbe identified. With these basic notions in hand, I can now give a definition of aholomorphic matrix model.Definition Given a choice of superpotentialW (x), a choice of Γ ⊂ D as describedabove, and a choice of N , a holomorphic matrix model is defined by the partitionfunctionZN (Γ, gs) := 1C∫ΓdΦe− 1gsTrW (Φ), (2.5)where C is a normalization constant, dΦ = ∧I,JΦI,J is the familiar matrix integra-tion measure, and gs is a coupling constant.A holomorphic matrix model is a path integral over a certain half-dimensional “con-tour” within the space of complex matrices. However, it is much nicer to work withthe eigenvalue representation of a matrix model. Therefore we want to translate theabove definition into an integral over the eigenvalues of the matrix. Let C be calledthe eigenvalue plane and let γ : R → C be an open, immersed path in this plane,without self-intersections. We can use this path γ to define the subspace of DΓ(γ) := {Φ ∈ D∣∣σ(Φ) ⊂ γ}, (2.6)where I slightly abuse notation and identify the path γ with its image in C. Noticethat since γ ⊂ C is half-dimensional (i.e., the real dimension of γ coincides withthe complex dimension of C), it follows that Γ(γ) is in turn also half-dimensionalinside ofD. Lazaroiu [15] proves that the original partition function (2.5) is actuallygauge invariant, with the gauge orbits being given by the complex homogeneousspace H := GLN (C)/(C∗)N . Denote by hvol(H) the volume of the space H .Using this, we can choose the normalization C to cancel all unwanted factorsC = (−1)N2(N−1)/2 1N !hvol(H). (2.7)92.2. The ’t Hooft Limit and the Emergence of a Hyperelliptic CurveWith this normalization, we can give a very nice expression for the eigenvalue rep-resentation of a holomorphic matrix modelZN (γ, gs) =∫γdλ1 · · ·∫γdλN∏I 6=J(λI − λJ)2e−1gs∑NJ=1W (λJ ), (2.8)where we simply define ZN (γ, gs) := ZN (Γ(γ), gs). When changing variablesto the eigenvalue representation, it is the Jacobian which gives rise to the Vander-monde determinant ∆(λ) :=∏(λI − λJ).At this stage, it is obvious what a holomorphic matrix model is: we must providethe data of a polynomial superpotential W (x) as well as a suitable path γ, andthe model will be defined as a path integral over all matrices whose eigenvalueslie entirely within γ. If we choose γ to be the real axis, we recover the familiarHermitian matrix models. Similarly, choosing γ to be the imaginary axis or theunit circle, gives rise to an anti-Hermitian matrix model or unitary matrix model,respectively. In this sense, holomorphic matrix models generalize these modelsby allowing the eigenvalues to be supported on more general paths in the complexplane.Section 2 of Lazaroiu’s paper deals with convergence issues. I will not providethe details here, as they will not constrain us greatly. Briefly, depending on the poly-nomial W (x), the path γ needs to asymptote at infinity to certain specific sectorsof the complex plane. For example, if we choose W (x) to have odd degree, thenchoosing γ to be the real axis will not work. This is an illustration of the familiar factthat Hermitian matrix models are not well-defined for polynomial superpotentialsof odd degree, since the real part of the potential cannot be bounded from belowalong the real axis. In Section 2 of his paper [15], Lazaroiu provides the constraintson γ from W (x).2.2 The ’t Hooft Limit and the Emergence of aHyperelliptic CurveLet γ be a path in C as described in the previous section. Following Lazariou, let sdenote a length coordinate on the path. Thus, λ(s) shall be the parameterization ofthe path, and for notational ease, let λI = λ(sI). Since the matrix model eigenval-ues are constrained to live in γ, we want to introduce an eigenvalue density along102.2. The ’t Hooft Limit and the Emergence of a Hyperelliptic Curvethe path. For finite N , this is simply a sum of delta functionsρ(s) :=1NN∑I=1δ(s− sI), (2.9)with normalization condition ∫ ∞−∞dsρ(s) = 1. (2.10)Note that the range of the integral is from −∞ to∞. Following the usual matrixmodel constructions, define the resolventω(x) :=1NN∑I=01x− λI . (2.11)In the eigenvalue representation of the partition function (2.8), we integrated overthe eigenvalues λI . Now, make a change of variables λI = λ(sI), and rewrite thepartition function in terms of the sI .ZN (γ, gs) =∫γds1 · · ·∫γdsNN∏I=1λ˙(sI)∏I 6=J(λ(sI)−λ(sJ))2e− 1gs∑NI=1W (λ(sI)).(2.12)It is illustrative to introduce an effective action Seff,Seff(s1, . . . , sN ) :=N∑J=1W (λJ)− 2gs∑I 6=Jlog(λI − λJ)− gsN∑I=1log λ˙I , (2.13)such that the total partition function can now be written asZN (γ, gs) =∫γds1 · · ·∫γdsN e− 1gsSeff . (2.14)One can then vary the effective action with respect to parameter λI = λ(sI) whichgives the equation of motionW ′(λI)− 2gs∑J 6=I1λI − λJ − gsλ¨Iλ˙2I= 0. (2.15)112.2. The ’t Hooft Limit and the Emergence of a Hyperelliptic CurveWe see here that the equation of motion depends onW ′, notW , further emphasizingthat superpotentials differing by a constant should be physically identified. Giventhe definition of the resolvent, one can show that the equation of motion above isequivalent toω(x)2 +1gsNW ′(x)ω(x) +14(gsN)2f(x) +1Nω′(x) +1N2N∑I=1λ¨Iλ˙2I= 0, (2.16)where the polynomial f(x) is defined asf(x) := 4gsN∑I=1W ′(x)−W ′(λI)x− λI . (2.17)Notice that the equation of motion (2.16), as written, is a differential equation forthe resolvent ω(x). If we letN →∞, it seems like we can discard the term with thederivative, leaving simply an algebraic equation. The downside to this appears to bethat we will lose almost every term above, leaving the trivial condition ω(x)2 = 0.However, we can take the large-N limit in specifically such a way as to keep mostof the relevant terms above. Let N → ∞, and simultaneously let gs → 0, whilekeeping fixed the ’t Hooft coupling which is defined asS = gsN. (2.18)This is known as the ’t Hooft limit and it is one of the most important limits in agauge theory. It is precisely the limit in which many gauge theories are conjecturedto find intimate ties to string theory.Taking the ’t Hooft limit of (2.16), the last two terms drop out entirely, leavingthe purely algebraic equationω(x)2 +1SW ′(x)ω(x) +14S2f(x) = 0. (2.19)There is one subtlety here (see Lazariou for details). In taking the ’t Hooft limit,we replace averaged quantities with their expansions in 1N . For example, we have〈ρ(s)〉 = ρ0(s)+O( 1N ). Thus, when defining the resolvent and f(x) in the ’t Hooftlimit, it is done with respect to the eigenvalue density ρ0(s), as follows,ω(x) =∫ ∞−∞dsρ0(s)x− λ(s) , (2.20)122.2. The ’t Hooft Limit and the Emergence of a Hyperelliptic Curvef(x) = 4S∫ ∞−∞dsρ0(s)W ′(x)−W ′(λ(s))x− λ(s) . (2.21)There’s a constraint on f(x) coming from the normalization of the eigenvalue den-sity. One can see that the leading coefficient must be 4S(n + 1)un+1, where werecall that un+1 is the leading coefficient of W (x), [15].A strong word of warning is due at this point. The eigenvalue density ρ0(s) isactually complex-valued. This will become important later in the thesis. The detailscan all be found in Lazaroiu [15], noting that he renames the large-N quantitiesω0(x) and f0(x).Defining the functiony(x) = 2Sω(x) +W ′(x), (2.22)one can show that the equation of motion in the ’t Hooft limit (2.19) is equivalenttoy2 −W ′(x)2 + f(x) = 0. (2.23)Definition A smooth hyperelliptic curve of genus g is an algebraic curve over Cgiven by y2−F (x) = 0 where F is a polynomial of even degree 2n = 2(g+1) ≥ 4with 2n distinct roots and complex coefficients. For our purposes, a maximallysingluar hyperelliptic curve is the singular curve of the form y2 −H(x)2 = 0 fora polynomial H of degree n ≥ 2 with distinct roots.Since W (x) is a degree-(n + 1) polynomial, W ′(x)2 is a degree-2n polynomial.The polynomial f(x) is of degree-n, so we see that the algebraic equation (2.23)is exactly that of a hyperelliptic curve! Thus, in the ’t Hooft limit, the equation ofmotion of the matrix model gives rise to a hyperelliptic curve, which is thought todouble-cover the eigenvalue plane compactified to P1, branched over the 2n zerosofW ′(x)2−f(x). This is an example of an emergent algebraic curve, as mentionedin the Introduction. In algebraic geometry, hyperelliptic curves have a number ofmoduli determining the complex structure. If we choose to consider a Hermitianmatrix model, we would be constraining ourselves to a “slice” through the modulispace of hyperelliptic curves. It is the holomorphic models which explore the fullmoduli space of emergent structures.132.3. Physical Discussion of Classical and ’t Hooft Limits2.3 Physical Discussion of Classical and ’t Hooft LimitsThe previous section developed the structure of the theory, but I have yet to provideany physical interpretation or intuition. In this section, I describe the physics of aholomorphic matrix model in the ’t Hooft limit [3, 4], which was introduced above.But first, I discuss the classical limit gs = 0. This is analogous to taking the ~ = 0limit in quantum mechanics. Mathematically, this limit corresponds to a maximallysingular hyperelliptic curve.The Classical LimitAs mentioned, the classical limit of the theory corresponds to fixing gs = 0. Intaking the ’t Hooft limit, we let gs → 0, but this is fundamentally different. If onehas already taken the ’t Hooft limit, the classical limit can be recovered by requiringS = 0. In this limit, we see from (2.13) that the effective action is given bySeff =N∑J=1W (λJ), (2.24)which is simply the action for N non-interacting eigenvalues, under the influenceof a polynomial potential. In addition, from (2.15) the equation of motion reducestoW ′(λI) = 0, (2.25)which is the statement that the eigenvalues live precisely at the extrema of the poly-nomial potential.1 Most importantly, we see from (2.17) that the polynomial f(x)vanishes when gs = 0. That is to say, the holomorphic matrix model in its classicallimit is described by a maximally singular hyperellpitic curve [3],y2 = W ′(x)2. (2.26)The singularities of the hyper elliptic curve coincide with the extrema ofW (x). Thephysical interpretation is that we’ve distributed the N eigenvalues into the extremaof the polynomial, and since there are no interaction terms, they simply reside at1For n ≥ 1 extrema, one must specify filling fractions which are n rational numbers satisfyingq1 + · · ·+ qn = 1 which specify the fraction of eigenvalues in each extrema.14The ’t Hooft Limitthe extrema. As noted in [3], working in the holomorphic setting, one does notdistinguish between minima and maxima; all extrema are saddle points.The ’t Hooft LimitTurning on the coupling gs, the effective action retains its form in (2.13). The firstlogarithmic term in the effective action acts to repel the eigenvalues from one an-other. This provides the impetus for the eigenvalues to depart from their positionsat the extrema of the polynomial. Thus, the two physical phenomena present are anattraction of each eigenvalue to the extrema ofW (x), as well as a mutual repulsionof the eigenvalues. As we have seen, in the ’t Hooft limit the only free physicalparameter is the coupling S = gsN . It is clear from the equations that S = 0recovers the classical limit, and singular algebraic curve. If S 6= 0, the eigenvalueswill spread out into a continuous cut. Recalling that the eigenvalues are constrainedto live in the path γ these cuts are completely supported within γ. These cuts areprecisely the branch cuts of the hyperelliptic curve (2.23) and the endpoints corre-spond to the branch points. Let C1, . . . , Cn denote the n cuts. Important quantitiesare the periods of the Riemann surface given as certain contour integrals around thecuts,Si =12pii∮Ciy(x)dx, (2.27)where S = S1 + · · · + Sn. Note that these periods are invariants of a Riemannsurface with a given complex structure. They are holomorphic functions of thecomplex moduli of the surface.22.4 Summary and OutlookThe most fundamental properties of holomorphic matrix models are that they haveaccess to a larger moduli space than their Hermitian counterparts, and the physicalcouplings (like Si) emerge as holomorphic functions of these complex moduli. Inthe remainder of this thesis I study theN = 1∗ matrix model, which is intrinsicallyholomorphic. Though it is of a slightly more complicated form than the modelsintroduced here, it shares these same fundamental properties and in the coming2For the remainder of my thesis, the matrix models will have only one cut, in which case S1 = S.152.4. Summary and Outlookchapters, I attempt to find interesting phenomena encoded into theN = 1∗ modulispace as well as possible connections to N = 2∗.16Chapter 3Introduction to the N = 1∗ andN = 2∗ TheoriesThe N = 1∗ and N = 2∗ theories are quantum field theories defined as massivedeformations of N = 4 super Yang-Mills (SYM) in four dimensions. I begin thischapter by briefly describing this theory and defining the massive deformations.The remaining two sections will be devoted to studying the N = 1∗ and N =2∗ theories, respectively, in detail. The exact solutions of both models requiresconsidering certain matrix models. In both cases, the equations of motion of thematrix models determine a particular elliptic curve with modular parameter τ , aswell as a certain elliptic function on the curve. The physical parameters of thematrix model are encoded into τ as essentially a change of variables. The similarityof the two solutions indicates that perhaps through the philosophy of holomorphicmodels, an enlarged moduli space may uncover connections between the theories.Such connections will occupy later chapters. We will see in the next section thatN = 4 SYM is intrinsically connected to an elliptic curve. The fact that both theN = 1∗ and N = 2∗ theories are solved in the same way, using an elliptic curve,can probably be traced back to their common origin in N = 4 SYM. As pointedout in [6], the fact that the matrix model can uncover this fact is remarkable.3.1 N = 4 Super Yang-Mills Theory (SYM) in FourDimensionsLet the symmetry group of the theory be a compact Lie group G, and let M bea four-dimensional spacetime manifold. Take all fields to transform in the adjointrepresentation of G. In addition, let gYM denote the Yang-Mills coupling of thegauge theory. This is not to be confused with the coupling gs of a matrix model tocome.N = 4 SYM in four dimensions actually arises as a dimensional reduction of173.1. N = 4 Super Yang-Mills Theory (SYM) in Four Dimensionsan even simpler theory: N = 1 SYM in ten dimensions. This latter theory containsonly vector multiplets (AM , ψ), where AM is a ten-dimensional gauge field withM a ten-dimensional index, and ψ is a 16 component Majorana-Weyl spinor. Wecan dimensionally reduce this theory to four dimensions by regarding the latter sixcomponents of the gauge field as scalar fields which we call φ4, . . . , φ9, while thefirst four components transform as a four-dimensional gauge field Aµ. In addition,there are right and left moving fermions. The decomposition of the field contentunder this dimensional reduction is(AM , ψ) −→ (Aµ, φ4, . . . , φ9, ψL, ψR, χL, χR). (3.1)The resulting theory is known as N = 4 SYM in four dimensions. This theoryis superconformally invariant, even quantum mechanically. This is surprising be-cause, in general, Yang-Mills theories in four dimensions are classically conformal,but this symmetry is broken by quantum anomalies.Yang-Mills theories on four dimensional spacetimes are well known to have richstructure. Specifically in four dimensions, one can add the following topologicalterm to the Lagrangian of the theory,θ8pi2∫MTrF ∧ F, (3.2)where F is the curvature two-form associated to the gauge connectionA, and θ is acoupling called the “theta angle.” This quantity depends only on the topology of theprincipal G-bundle underlying the gauge theory, and is special to four-dimensionalmanifolds since TrF ∧ F is a 4-form. We can package the Yang-Mills couplinggYM and the theta angle into the complexified gauge coupling,τ0 =θ2pi+4piig2YM. (3.3)It’s crucial to note that τ0 is the coupling of the gauge theory. We will soon see a pa-rameter τ arising from a matrix model, and the two are not to be identified a priori.The fact that they are related is a very deep connection made by Dijkgraaf and Vafa,which we will come to later. The conjectured S-duality asserts that N = 4 SYMin four dimensions is invariant under SL(2,Z) transformations of τ0. These trans-formations are generated by τ0 → τ0 + 1 and τ0 → − 1τ0 . Thus, four-dimensionalN = 4 SYM appears to be intrinsically connected to elliptic curves: (3.3) essen-tially coordinatizes the moduli space of elliptic curves M1,1 with N = 4 SYMmoduli.18The Mass Deformations of N = 4 SYMThe Mass Deformations of N = 4 SYMOne often hears about re-writing the field content of some theory in the languageof another theory. This does not refer to any sort of deformation or dimensionalreduction. Rather, it’s simply a repackaging of the field content of one theory, intoa multiplet transforming properly under action by the symmetry group of anothertheory. If we re-write the field content of N = 4 SYM in N = 1 language, we geta single N = 1 vector multiplet (Aµ, λ) and three chiral multiplets Φi = (Zi, ψi),for i = 1, 2, 3. Here, Zi are three complex scalar fields, λ and ψi are fermions,and Aµ is the four-dimensional gauge field. The chiral multiplets interact via acubic classical superpotential Φ1[Φ2,Φ3]. Notice that there are still 6 real scalarfields, one gauge field, and four fermions. We have not gained or lost any data,we have only repackaged it. We can just as easily rewrite the field content ofN = 4 SYM in N = 2 language. The data is packaged into a vector multiplet(Aµ,Φ + iΦ′, ψ1α, ψ2α), and two massless hypermultiplets (φ, φ˜, χα, χ˜α). Once wehave repackaged the content of a theory, we can add certain mass terms to the actionthat break part of the originalN = 4 supersymmetry. We refer to these as massivedeformations. There are actually two unique massive deformations ofN = 4 SYMwhich preserve N = 1 and N = 2 supersymmetry. These are called the N = 1∗and N = 2∗ theories, respectively.3.2 The N = 1∗ Theory and the Holomorphic MatrixModelIn the introduction, I remarked that the N = 1∗ theory has a holomorphic subsec-tor which consists of physical observables computable as derivatives of a particularfunction Weff called the effective superpotential. The effective superpotential in agiven classical vacuum must be a holomorphic function of the underlying param-eters of the theory. Thus, the most pressing matter in such theories becomes com-puting Weff. If one can do so, then a whole class of physical observables are alsoeasily obtainable. In [5], Dijkgraaf and Vafa conjecture that one can compute Wefffor a givenN = 1∗ vacuum by considering a particular holomorphic matrix modelexpanded around the vacuum under consideration. It is this matrix model whichwill be studied. I begin by defining N = 1∗ and studying the classical vacuumstructure of the theory.19The Classical Vacuum StructureDefinition The N = 1∗ theory is realized as a massive deformation of N = 4SYM by giving the same massm to each of the three chiral multiplets Φi. This addsquadratic mass terms to the classical superpotential:Wclassical = Tr(Φ1[Φ2,Φ3] +mΦ21 +mΦ22 +mΦ23). (3.4)This mass deformation preservesN = 1 supersymmetry, which explains the name.The original N = 4 SYM can be recovered by letting m = 0.The Classical Vacuum StructureConsider N = 4 SYM on R4 with gauge group SU(N), and complexified gaugecoupling τ0. Passing to the N = 1∗ massive deformation, we have a classicalsuperpotential of the form (3.4) wherem is the mass given to the chiral superfields.We can make a convenient change of variables Φ+ := Φ1 + iΦ2, Φ− := Φ1− iΦ2,and Φ := Φ3. In these variables, the classical superpotential takes the formWclassical = Tr(iΦ[Φ+,Φ−] +mΦ+Φ− +mΦ2). (3.5)The quantum field theory with action (3.5) has a rich classical vacuum structure.In [7], it is found that the classical vacua correspond to reducible or irreduciblerepresentations of SU(2). In the representation theory of SU(2), one considers theN ×N Hermitian matrix J3, and the raising/lowering matrices J±. Classically, Φis identified with iJ3, while Φ± are related to J±. One can show that classically,the fields satisfy the equations,[Φ+,Φ−] = 2imΦ[Φ ,Φ±] = ± imΦ±, (3.6)which are simply anti-Hermitian versions of the familiar SU(2) commutation rela-tions. In general, we allow for reducible representations. For all p|N (p dividingN ) we can considerN/p direct summands of a p-dimensional irreducible represen-tation. In [7], the authors find that each divisor p ofN corresponds to a theory withreduced gauge group SU(N/p) and N/p physically distinct vacua for this fixed p.This means the total number of classical vacua is,∑p|NN/p =∑p|Np. (3.7)20The Matrix ModelThe Confining and Higgs VacuaThe confining vacuum is defined by taking p = 1, corresponding to N indistin-guishable vacua and the full SU(N) gauge group unbroken. Recalling the repre-sentations consisted ofN/p copies of p-dimensional irreducible representations, inthe confining vacuum, all fields can be taken to vanish,Φ = Φ± = 0. (3.8)Again, this holds only at a classical level; the fields don’t vanish quantum mechan-ically. The Higgs vacuum is defined by taking p = N , which gives only a singlevacuum with gauge group completely broken. In this case, the representation con-sists simply of an irreducibleN -dimensional representation. The confining vacuumis particularly simple, and all other massive vacua will work in fundamentally thesame way [7]. In this thesis, it will suffice to focus on the confining vacuum.The Matrix ModelThe Dijkgraaf-Vafa conjecture [5] is that to compute the effective superpotential inany vacuum, one should take the ’t Hooft limit of a holomorphic three-matrix modelexpanded about the relevant vacuum. This is particularly simple in the case of theconfining vacuum since the classical fields vanish identically. Therefore, we mustconsider the following holomorphic matrix model, which was originally studied inthe thesis of Hoppe [11],ZN (m) =∫DΦ+DΦ−DΦ e−TrWclassical . (3.9)In the above partition function, Φ and Φ± now representN ×N matrices indepen-dent of the classical solutions of the fields. This is because we’re looking at matrixfluctuations about the classical solutions. We can scale all three matrices bym, anddefine the matrix model coupling gs := 1/m3. This produces a partition functionwith gs playing an analogous role to ~ in quantum mechanicsZN (gs) =∫DΦ+DΦ−DΦ e− 1gs Tr(−Φ[Φ+,Φ−]+Φ+Φ−+Φ2). (3.10)The above three-matrix model is well-known in the literature, and was solved ex-actly by Kazakov, Kostov, and Nekrasov in [13]. They showed that one can integrate21The Solution of the Matrix Model in the ’t Hooft Limitover Φ±, leaving simply a one-matrix model,ZN (gs) =∫DΦ e− 1gsTrΦ2det(AdjΦ + i). (3.11)Here, AdjΦ = [Φ, · ] is the adjoint action, and the commutator is the familiar com-mutator on matrices. It is this holomorphic matrix model which I want to study inthe ’t Hooft limit.The Solution of the Matrix Model in the ’t Hooft LimitIn [13] Kazakov, Kostov, and Nekrasov solve (3.11) by passing to the eigenvaluerepresentation, where the partition function reduces to,ZN (gs) =∫ ∏IdλI∏I<J(λI − λJ)2(λI − λJ + i)(λI − λJ − i)e− 1gs∑λ2I . (3.12)The equation of motion (the saddle-point equation) is computed to be,2λI = gs∑I 6=J(2λI − λJ −1λI − λJ + i −1λI − λJ − i). (3.13)Taking the ’t Hooft limit, we let N → ∞ and gs → 0, holding fixed the ’t HooftcouplingS = gsN. (3.14)In general, S is complex-valued. Notice that the above equation of motion has afamiliar interpretation. In the classical limit gs = 0, the righthand side of (3.13)vanishes, indicating that all N eigenvalues are stacked at the critical point, whichhappens to be the origin in this case. Taking the ’t Hooft limit, one expects theeigenvalues to spread out into a continuous cut [−µ, µ]. Since the matrix model isfundamentally holomorphic, the cut may be supported in the complex plane, butlet us assume for now it is contained in the real axis. As usual in matrix modeltechnology, the resolvent is defined byω(x) =∫ µ−µdyρ(y)x− y , (3.15)22The Solution of the Matrix Model in the ’t Hooft Limitwhich is a holomorphic function on the complex x-plane, with a single branch cutcorresponding to the matrix model eigenvalues. The discontinuity of ω(x) acrossthis cut gives the eigenvalue densityρ(x) = − 12pii(ω(x+ i)− ω(x− i)). (3.16)Consider a “probe eigenvalue” located at some point x in the complex plane. Thisprobe feels a force from the eigenvalues within the branch cut. The force is givenby,3f(x) = 2x− S[2ω(x)− ω(x+ i)− ω(x− i)], (3.17)and clearly vanishes when x lies in the branch cut, as can be seen from the equationof motion (3.13). The solution to the model relies on the definition of the functionG(x) := x2 + iS[ω(x+ i2)− ω(x− i2)], (3.18)which is called the generalized resolvent of N = 1∗. It is straightforward to verifythat G(x) is related to the force through,f(x) = −i[G(x+ i2)−G(x− i2)]. (3.19)Since the resolventω(x) has a single discontinuity along the branch cut, the functionG(x) is holomorphic with the exception of two branch cuts. These two branch cutsare the translates of the original branch cut by±i/2. Keep in mind that there is onlythe single, original branch cut in which the eigenvalues live; these mirror branchcuts are merely an artifact arising fromG(x). For x lying within the (real) cut, sincef(x) = 0, we see that for small ,G(x+ i2 ± i) = G(x− i2 ∓ i). (3.20)This relation has the interpretation of a “gluing condition” in the following sense:the top of the upper cut is glued to the bottom of the lower cut, and visa versa. Thiscondition implies that the generalized resolvent G(x) is naturally identified as anelliptic function on an elliptic curve. One can also show thatG(−x) = G(x) which3The above f is not to be confused with (2.17).233.3. The N = 2∗ Theory and its Critical Pointsconstrains the geometry of the eigenvalue plane: the two branch cuts of G(x) mustbe such that the entire eigenvalue plane is symmetric under inversion x→ −x.We’ve now seen that the equation of motion of N = 1∗ on R4 in the ’t Hooftlimit has determined the generalized resolvent G(x) as an elliptic function on anelliptic curve. Later, the physical ’t Hooft coupling S of the matrix model will berelated to the modular parameter τ of the elliptic curve. The remaining solutionof the model requires understanding how the elliptic curve is identified with theeigenvalue plane. I will provide such a construction in the following chapter.3.3 The N = 2∗ Theory and its Critical PointsRecall in N = 2 language, the field content of N = 4 SYM consists of a vectormultiplet (Aµ,Φ + iΦ′, ψ1α, ψ2α) and two massless hypermultiplets (φ, φ˜, χα, χ˜α).The eigenvalues of the VEV of Φ,〈Φ〉 = diag(a1, . . . , aN ), (3.21)are moduli on the Coulomb branch of N = 2. We take the gauge group to beG = SU(N) such that all fields transform in the adjoint representation. In addition,take the spacetime to be the four-sphere S4 of radius R.Definition The N = 2∗ theory is realized as a massive deformation of N = 4SYM by giving equal masses M ∈ R to the two hypermultiplets. This is the uniquemassive deformation which preserves N = 2 supersymmetry. The original N = 4SYM can be recovered by letting M = 0.In the introduction I mentioned Pestun’s seminal method of localization whichallows certain observables in N = 2 supersymmetric gauge theories to be solvedexactly. Here, I will be specifically interested in N = 2∗ compactified on the four-sphere S4, in which case the partition function of the theory localizes to a finite-dimensional matrix model. It is precisely this matrix model which will be studiedin specific limits [18, 19]. I will be most interested in the decompactification limitwhich corresponds to sending the radius R of the sphere to infinity. In a loosesense, this unfolds S4 into R4. Recall it was on R4 that we studied N = 1∗, so thedecompactification limit ofN = 2∗ is where the two deformations may be related.In addition, we take the large-N ’t Hooft limit, where N corresponds to the size ofthe matrices in the matrix model. In passing to the ’t Hooft limit, both the weak24The Matrix Modeland strong coupling regions of the matrix model will be considered. Note that thecoupling appearing in the matrix model partition function below (3.23) actually isthe gauge theory coupling,λ = g2YMN. (3.22)This is to be contrasted with the previous model where the matrix model couplinggs has no a priori connection to the gauge theory. In the ’t Hooft and decompacti-fication limits, there is a phase transition separating the weak and strong couplingregions. In addition, as the ’t Hooft coupling λ increases ever more, we encounter aninfinite sequence of phase transitions which accumulate at infinite coupling [18, 19].There are no such phase transitions in N = 1∗ theory.The Matrix ModelAs discussed above, we have compactified N = 2∗ theory to S4 in order to makeuse of the localization results. The localized partition function then takes the formof the following matrix model [18],ZN (λ,M) =∫dN−1a∏I<J(aI − aJ)2H2(aI − aJ)H(aI − aJ −M)H(aI − aJ +M)e− 8pi2Nλ∑a2I .(3.23)I need to explain the components of this partition function. The ai are the eigen-values of the VEV of the adjoint scalar field Φ in the vector multiplet. These arecoordinates on the Coulomb branch moduli space of the N = 2 theory. The ’tHooft coupling is denoted by λ = g2YMN , and the function H(x) is defined as,H(x) :=∞∏n=1(1 +x2n2)e−x2n . (3.24)In [18], a factor of the Nekrasov instanton partition function Zinst is included, but Iset it to 1 as it will not play a role in the present context. In the ’t Hooft limit, thepath integral above is dominated by a saddle-point, which can be shown to satisfythe equation of motion∫ µ−µdyρ(y)(1x− y −K(x− y) +12K(x− y+M) + 12K(x− y−M))=8pi2λx,(3.25)25The Decompactification Limit at Weak Couplingwhere the eigenvalue densityρ(x) =1NN∑I=1δ(x− aI), (3.26)becomes continuous in the ’t Hooft limit, supported on the cut [−µ, µ] and normal-ized to unity. The function K(x) is essentially the logarithmic derivative of H(x),K(x) := −H′(x)H(x)= 2x∞∑n=1(1n− nn2 + x2). (3.27)The Decompactification Limit at Weak CouplingThe decompactification limit corresponds to letting the radius R of the sphere goto infinity. In this case, the function K(x) can be approximated by its asymptoticsat infinity [18],K(x) = x log x2 +O(x). (3.28)Using this approximation and differentiating the equations of motion (3.25) oncewith respect to x we get∫ µ−µdyρ(y) log(M2x2− 1)2=16pi2λ, (3.29)which can be differentiated once again, resulting in∫ µ−µdyρ(y)(2x− y −1x− y +M −1x− y −M)= 0. (3.30)This is referred to as the equation of motion in the decompactification limit. In thecase of weak coupling, where µM , the last two terms in equation (3.30) cancel,leaving the integral equation,∫ µ−µdyρ(y)1x− y = 0. (3.31)This integral equation is well-known to have as its solution, the inverse square-root,ρ(x) =1pi√µ2 − x2 . (3.32)26The Strong Coupling Region and Phase TransitionsHowever, notice that this doesn’t provide a relationship between the ’t Hooft cou-pling λ and the cut length µ. This relationship is provided by (3.29). Once we havethe expression for the eigenvalue density ρ(x) in (3.32), we can plug it into (3.29),carry out the integration, and we find thatµ = 2Me−4pi2λ . (3.33)This is how the cut-length, and ’t Hooft coupling are related in the weak couplingregion.The solution described above relied on an approximation in the µ  M limit.However, at least in the weak coupling region, the model is actually amenable toan exact solution using elliptic curves in a nearly identical fashion to N = 1∗.To pursue such a solution, an alternative interpretation of the equation of motion(3.30) is needed. In the previous section we encountered the resolvent ω(x) and theeigenvalue density ρ(x). Using these same defintions, I now define the generalizedresolvent of N = 2∗G˜(x) := ω(x+ M2)− ω(x− M2 ), (3.34)which is a holomorphic function on the complex x-plane except now with two mir-ror branch cuts: the single branch cut of ω(x) has been translated by ±M/2. Onecan show by direct computation that the equation of motion (3.30) is equivalent toG˜(x+ M2 ± i)= G˜(x− M2 ∓ i). (3.35)Just as in the previous section, this has the interpretation of a glueing condition: itprescribes that the top of one cut is to be identified with the bottom of the other,and visa versa. Glueing the two cuts together, and adding a point at infinity, theeigenvalue plane is compactified into an elliptic curve. We see that the equationof motion constrains the generalized resolvent G˜ to be defined on an elliptic curve,exactly as in N = 1∗.The Strong Coupling Region and Phase TransitionsWhile the exact solution works for λ small enough, it appears to break down whenthe ’t Hooft coupling reaches a critical point of λ(1)c ≈ 35.4252. This critical pointarises due to the two mirror branch cuts colliding at the origin. In addition, thereis an infinite sequence of phase transitions occurring at critical points λ(2)c ≈ 83,27The Strong Coupling Region and Phase Transitionsλ(3)c ≈ 150, and so on, which appear to be inaccessible to the exact solution, buthave nevertheless been observed numerically [18, 19]. Russo and Zarembo inter-pret these critical points as the appearance of a new massless hypermultiplet in thespectrum. They also give a concise expression for the cut length at the n-th criticalpoint,µ(λ(n)c ) =nM2. (3.36)Recalling the full cut is [−µ, µ], quite simply the critical points of N = 2∗ ariseeach time the cut length coincides with an integer multiple of M .28Chapter 4Embedding the Elliptic Curveinto the Eigenvalue PlaneIn the last chapter, I reviewed two different physical theories whose equations ofmotion imply a deep connection to an elliptic curve. This is another example ofan algebraic variety emerging from a matrix model. It turns out that the solutionto both models will require explicitly constructing a map x embedding the ellipticcurve into the eigenvalue plane, compactified at infinity. There will be virtuallyno physics in this portion of the solution. It is a purely mathematical problem,independent of either theN = 1∗ orN = 2∗ theories. The distinction between thetwo physical theories will be manifested later in computations through the choiceof one of the two generalized resolvents.This chapter is motivated by a construction of Hollowood and Prem-Kumar[10]. My original work consists of attempting to apply this method to an enlargedregion of parameter values inH. I find a large open setH ⊂ Hwhich I refer to as themoduli space, such that for all τ ∈ H andM ∈ C I can determine the configurationof the two mirror branch cuts in the eigenvalue plane as well as the unique non-trivialcycle on the elliptic curve mapping under x into the cut. The compliment H \H isa region where my construction degenerates and is somewhat shrouded in mystery;I can only speculate on its possible physical significance. The region H \ H has aboundary component I call the line of degeneration such that approaching this linefrom withinH, the two mirror cuts are converging to an overlapping configuration(on the real axis, for M ∈ R). On the line of degeneration there are discrete pointswhere the cut length is approaching integer multiples of M . While completelyarbitrary from a mathematical perspective, I will show in the next chapter that thesepoints actually play a distinguished role in theN = 1∗ theory, and perhapsN = 2∗as well.Finally, my geometrical construction in this chapter will allow for the compu-tation of eigenvalue densities later. I reviewed above the idea that the eigenvalue294.1. The Constructiondensity in a matrix model is encoded as the discontinuity of the resolvent across thecut. Given the cycle on the elliptic curve mapping into the cut (which I describein this chapter), the discontinuity can be computed from the generalized resolventrestricted to the cycle. This gives a simple prescription which allows for the com-putation of eigenvalue densities in either the N = 1∗ or N = 2∗ theories.4.1 The ConstructionLet ω1 and ω2 be two complex numbers such that their ratio is not real. Given twosuch parameters, we can form the following lattice within the complex plane,Λ = 2ω1Z⊕ 2ω2Z. (4.1)Definition The complex manifoldC/Λ is called a complex torus with half-periodsω1 and ω2. The modular parameter of the torus is defined to beτ =ω2ω1. (4.2)which is taken to live in the upper-half complex plane H without loss of generality.An elliptic curve4 consists of an underlying complex torus whose points constitutean abelian group [12, 14]. Therefore, an elliptic curve is a complex torus alongwith the choice of a distinguished point to serve as the group identity.Let z be the coordinate on a complex torus, let x be the coordinate on the eigen-value plane, and denote the two branch cuts by C+ andC−. We want to construct amap x(z) which embeds the torus into the eigenvalue plane. Note that we are usingthe same symbol x to denote both the map and the coordinate on the plane. Whatconditions must this map x(z) satisfy? From the geometry of the eigenvalue plane,we see that it must be a quasi-periodic function5 on C/Λ. That is to say,x(z + 2ω1) = x(z), x(z + 2ω2) = x(z) +M, (4.3)4The hyperelliptic curves defined in the first chapter are generalizations of elliptic curves. Anelliptic curve can be realized as a projective algebraic curve embedded in the projective plane P2.Passing to an affine chart, the elliptic curve takes the form y2 = F (x) where F is a cubic polynomialwith complex coefficients.5A quasi-periodic function on C/Λ is a function f : C→ C such that f(z + 2ω1) = f(z) +C1and f(z + 2ω2) = f(z) + C2, where C1 and C2 are complex numbers.304.1. The Constructionwhere ±M/2 are precisely the midpoints of the two branch cuts in the x plane.These conditions uniquely determine x(z) in terms of the Weierstrass ζ-function,x(z) = iMω1pi(ζ(z)− ζ(ω1)ω1z). (4.4)This can be easily shown to obey the conditions in (4.3) by noting that the Weier-strass ζ-function is quasi-periodic along both periods,ζ(z + 2ω1,2) = ζ(z) + 2ζ(ω1,2). (4.5)Without loss of generality, we can gauge fix one of the half-periods to be real. Forconvenience, we make the following choice,ω1 =pi2, ω2 =piτ2. (4.6)With this choice, the embedding becomesx(z) =iM2[ζ(z)− 13E2(τ)z]. (4.7)We observe that the points on the torus z = ±ω2 map into the midpoints of thecuts,6x(±ω2) = ±M2. (4.8)The Weierstrass ζ-function has a simple pole at z = 0 on the torus. It follows thatx(z) itself has a simple pole at z = 0,x(z)∣∣z→0 ≈iM21z+ . . . , (4.9)and we conclude that z = 0 on the torus maps to the point at infinity on the eigen-value plane.Recall that an elliptic curve is simply a complex torus with a distinguishedpoint. The above construction has produced such a distinguished point. As τ varies,generic points inC/Λ take various values under x. However, zero is always mappedto the point at infinity and as such, we have a distinguished point in a natural way.It follows that this is a correspondence to elliptic curves, not merely complex tori.6This statement can be easily verified noting that ζ is an odd function satisfying ω2ζ(ω1) −ω1ζ(ω2) = ipi/2 as well as the relationship with the Eisenstein series 12ω1ζ(ω1) = pi2E2(τ).See Appendix A of [6].314.1. The ConstructionSince it is a torus, an elliptic curve has two linearly independent homologycycles, usually called A- and B-cycles. Under x, the A-cycles will map into theeigenvalue plane such that their image encircles one of the branch cuts, while theimage under x of theB-cycles will be a path connecting the two branch cuts. Theremust exist two special A-cycles, which map into the two branch cuts, respectively.More precisely, the cycles double cover the branch cuts; one should imagine theimages as “infinitely tightly” encircling them. Since the cuts are glued together,these two cycles are really the same, though they will appear translated by 2ω2 onthe fundamental domain of the elliptic curve. Let us call these two specialA-cyclesC+ and C−, not to be confused with the branch cuts C+ and C−. We must haveω2 ∈ C+ and−ω2 ∈ C−, which is to say that C+ maps into the cut with midpoint M2while C− maps into the cut with midpoint −M2 . These special A-cycles are relatedto the branch cuts through the embedding asx(C+) = C+, x(C−) = C−. (4.10)How do we find C+ and C−? The following method is provided in [10]. Choosea fixed value of the modular parameter τ ∈ H. This fixes an elliptic curve withhalf-periods ω1 and ω2, as described above. Since the two cycles C+ and C− mapdirectly into the cuts, and cover them twice, at the endpoints of the branch cuts, thederivative of x(z) must vanish. Thus, the condition that x′(z) = 0 determines thepoints in the fundamental domain of the elliptic curve, which map into the branchpoints. From (4.7), using ζ ′(z) = −℘(z), we find the derivative of the embeddingto be,x′(z) = − iM2[℘(z) + 13E2(τ)]. (4.11)Setting this to zero, and noting that E2(τ) is simply a complex number for fixed τ ,we get a transcendental equation℘(z) = −13E2(τ). (4.12)The Weierstrass ℘-function ℘(z) takes every value in C exactly twice as we rangeover the fundamental domain of the elliptic curve. Therefore, the above equationhas exactly two solutions, call them z1 and z2. It is precisely these two points whichmap to the endpoints of one of the branch cuts. In fact, translating these two pointsby 2ω2 will give two more points satisfying x′(z) = 0, and these will map to theendpoints of the other branch cut. The four points described here, have images in324.2. A Summary of the Method●●■■-2 -1 1 2-1.0-0.50.51.0Figure 4.1: The configuration of the eigenvalue plane for τ = 14 +14 i and M = 1. Noticethat the midpoints of the cuts are ±M2 . As well, note the symmetry of the eigenvalue plane.the eigenvalue plane such that their values all add up to zero. In other words, thebranch points satisfy the symmetry x→ −x of the eigenvalue plane (Figure 4.1).What I have discussed thus far has been entirely independent of the choice ofbranch cut. In principle, there’s some array of possible choices of the (real) cutwhich maintains the symmetry of the eigenvalue plane under inversion, when thecut is translated by ±M2 . For my purposes, it will suffice to take the branch cuts tobe straight line segments.Having decided on the shape of the branch cuts, we will be able to find thespecial cycles which map into the cuts. The cycle C+ must contain the two pointsz1 and z2 mapping to the branch cuts, as well as ω2 and ω2 + 2ω1. Translating C+down by 2ω2 gives us C−. Clearly, C− must contain the translates of z1 and z2 aswell as −ω2 and −ω2 + 2ω1. These two cycles define the fundamental domain ofthe elliptic curve (Figure 4.2), meaning we choose the fundamental domain to bethe region between C+ and C−.4.2 A Summary of the Method1. Choose τ ∈ H and M ∈ C. If τ has too small an imaginary part, themodel may have degenerated. This “region of degeneration” will be de-334.2. A Summary of the MethodC+C−Figure 4.2: The fundamental domain of the elliptic curve defined as the region betweenthe two cycles C+ and C−, shown here for τ = 14 + 14 i. In this figure I use coordinatesω1x + ω2y with 0 ≤ x ≤ 2 and −2 ≤ y ≤ 2. The upper contour is precisely C+ whichmaps into C+ and the lower contour is C− mapping into C−. Notice that the fundamentaldomain contains the points ω2, −ω2, ω2 + 2ω1, and −ω2 + 2ω1.344.3. Schematic Description of the Moduli Space of the Theoryscribed shortly.2. By solving (4.12), this τ determines four points z1, z2, z1 − 2ω2, z2 − 2ω2which map under x(z) into the branch points on the eigenvalue plane. Con-nect them pairwise with straight line segments in the unique way such thatthe midpoints of the two lines are ±M2 . Note that the resulting eigenvalueplane is invariant under x→ −x (Figure 4.1).3. Find theA-cycle C+ on the elliptic curve passing through the pointsω2, z1, z2,ω2 + 2ω1 on the fundamental domain, which maps into the cut whose mid-point is M2 . There will exist another A-cycle C− translated down preciselyby 2ω2 mapping into the other cut.4. If anything goes wrong so far, one has either chosen τ with too small animaginary part, or the points z1 and z2 have not been correctly identified.5. Once the cycles have been found correctly, the fundamental domain of thetorus is defined to be the region between the two A-cycles (Figure 4.2). Thisfundamental domain contains a few special points: the origin 0 of the ellipticcurve maps into the point at infinity of eigenvalue plane, and the points ±ω2map into the midpoints ±M2 of the cuts.6. A key object in my prescription is the image x(C+) = C+ of theA-cycle C+.Obviously, one can plot the real or imaginary parts of this image as a functionof a parameter on C+. Either will suffice to compute which two points inC+ map to any given point in C+. This will allow for the computation ofeigenvalue densities in a later chapter.4.3 Schematic Description of the Moduli Space of theTheoryAbove, I provide a method such that for τ in some allowed region of the upper-halfplane, one can construct the two branch cuts on the eigenvalue plane, as well asthe cycles on the elliptic curve mapping into the cuts. I now want to schematicallydescribe this region of allowed τ ∈ H, which will be referred to as the modulispace. The relevant equations above are explicitly invariant under τ → τ + 1, sowe may restrict to the strip {0 ≤ Re(τ) < 1} ⊆ H. Moreover, there is symmetry354.3. Schematic Description of the Moduli Space of the Theory0.0 0.1 0.2 0.3 0.4 0.50.20.40.60.8Complex τ-Planen=1n=2n=3n=4n=5(Region of Degeneration)(SlicewithoutCriticalPoints)N=2*ABCDEPowered by TCPDF (www.tcpdf.org)Figure 4.3: A schematic diagram depicting (half of) the moduli space H. At the pointlabelled n = 1, the two mirror branch cuts collide at the origin and each have length |M |.At the points n = 2, 3, 4, 5, . . . the mirror branch cuts are approaching overlap, each withlength n|M |, respectively. The points A − E are labeled for future reference to indicatelocations in the moduli space where eigenvalue densities are computed.observed with respect to reflection across the line Re(τ) = 12 , so it suffices toconsider only 0 ≤ Re(τ) ≤ 12 . For all 0 < Re(τ) ≤ 12 , there exists a smallenough Im(τ) such that the branch cuts collide on top of one another on the realaxis (see for instance, Figure 5.2). This defines a one-dimensional path in H whichI call the line of degeneration.7 For yet smaller Im(τ), the model has completelydegenerated. This is referred to as the region of degeneration. The portion of theinfinite strip {0 ≤ Re(τ) < 1} ⊆ H above the line of degeneration is the modulispace of the theory, which I denote by H (Figure 4.3). Note that in the figure, Ionly plot half of the moduli space since H behaves symmetrically with respect toreflection about the line Re(τ) = 12 .Let me briefly summarize some of the schematic features ofH which will playa role in the following chapters, as well as distinguished slices:7I was not able to find an explicit form of this line of degeneration.364.3. Schematic Description of the Moduli Space of the Theory• It is natural to call H the matrix model moduli space: every branch cut con-figuration is realized once, and only once, inH.• Varying M ∈ C does not change the relative position of the two mirror cuts;it merely changes the midpoints and rotates both cuts by Arg(M). With thisin mind, for simplicity I will usually choose M ∈ R.• For M ∈ R, I refer to Re(τ) = 0 as the anti-Hermitian slice, as both branchcuts lie parallel to the imaginary axis in the eigenvalue plane. As such, theycan freely enlarge without colliding.• Again for M ∈ R, the Hermitian slice is Re(τ) = 12 as the branch cuts aresupported symmetrically in the real axis. However, we can see that the cutswill collide at the critical point, labeled n = 1.• Both the Hermitian and anti-Hermitian slices intersect at infinity in the upper-half plane. At this point, the cut length vanishes. Emerging from the point atinfinity along a particular slice through H corresponds to the branch pointsspreading out at a particular angle in the eigenvalue plane. This holds only ina neighborhood of infinity; I will show in the following chapter that there is anon-trivial relationship between the branch cut configuration and the positionwithinH.• There is a region withinH where the two mirror cuts can be made arbitrarilylong and approaching an overlapping configuration. For an arbitrarily smallRe(τ), there will be a correspondingly small Im(τ) giving rise to such a con-figuration. One question pursued in the following chapter is, assuming thereis a Hermitian matrix model whose weak coupling region naturally embedsalong the Hermitian slice, is the strong coupling region embedded along theline of degeneration? Does approaching this line from within H shed lighton the strong coupling behavior of the theory?• Along the line of degeneration, there exists an infinite collection of pointsn = 1, 2, . . . where the two cuts are approaching degeneration in the realaxis each with length n|M |. The first five of these are labeled n = 1, . . . , 5in Figure 4.3. Though seemingly arbitrary at this point, these points actuallyplay a special role in N = 1∗, and possibly N = 2∗ as well.Having completed the construction, I want to re-emphasize that this chapter hasbeen independent of physics. Moreover, the “branch cuts” appearing here need not374.3. Schematic Description of the Moduli Space of the Theoryeven be the cuts of a matrix model in the ’t Hooft limit; this was a purely mathe-matical pursuit. I now want to realize my construction as the underlying structuralbackbone of a physical holomorphic matrix model. One could restrict this model toeither the Hermitian or anti-Hermitian slices, but doing so would veil the bulk ofHwhere I have noted potentially interesting phenomena. My goal is to now reconnectwith the N = 1∗ and N = 2∗ theories and study the physical significance of thesephenomena.38Chapter 5The Quasi-Modular CouplingS(τ )The construction in the previous chapter was independent of any physical theory.In a sense, one should think of it as the geometrical backbone of any matrix modelwhose equation of motion identifies the eigenvalue plane with an elliptic curve.The physical content of a particular theory is then encoded into the generalizedresolvent. It should be expected that all physical quantities (eigenvalue densities,physical couplings, etc.) will be computed using both the geometrical constructionof the last chapter, as well as the generalized resolvents. Eigenvalue densities willoccupy the following chapter but presently, I want to describe a computation of thematrix model ’t Hooft coupling in the N = 1∗ theory, which appeared originallyin [5, 6]. In this context, the physical parameters of the theory appear naturallyin terms of the modular parameter τ of the elliptic curve, perhaps for all τ ∈ H,or perhaps constrained to a particular slice. The only physical parameters in theN = 1∗ matrix model are the massM as well as the ’t Hooft coupling S. The massis more or less a bystander, but the ’t Hooft coupling emerges as a holomorphicfunction S(τ) on the upper-half plane, and a quasi-modular form with respect toSL(2,Z).The Fourier expansion ofS(τ) is really an expansion at infinity in the upper-halfplane, which I will show to be the weak coupling region of the theory. This pointat infinity is the unique zero of S(τ) in the open setH and is also the unique pointwhere the matrix model cut length vanishes. I will argue here that in a neighborhoodof infinity, S(τ) determines the exact branch cut configuration of the matrix model.8This breaks down at higher coupling, but I hope to show that S(τ) actually encodesnon-perturbative data about both the N = 1∗ and N = 2∗ models. A collectionof extrema of S(τ) fall on the line of degeneration. Approaching these points, the8More specifically, S(τ) determines the functionQ(τ) I define in (5.21) which in turn, determinesthe weak coupling configuration of the matrix model.395.1. The ’t Hooft Coupling of N = 1∗matrix model cut length converges to integral multiples ofM . This appears to holdat arbitrarily high coupling. One of the extrema of S(τ) lies on the Hermitian sliceRe(τ) = 12 and remarkably, this is exactly the first critical point of N = 2∗. It isonly by considering the full holomorphic nature of N = 1∗ that one is able to seethe extra data it encodes, which as far as I am aware, is not explicit in the literature.5.1 The ’t Hooft Coupling of N = 1∗In the original papers on the N = 1∗ matrix model [5–7], the authors take M = i.With this choice, equation (4.7) becomesx(z) = −12[ζ(z)− 13E2(τ)z]. (5.1)This agrees precisely with equation (2.14) in [6], noting that ζ(ω1)ω1 = pi212E2(τ)and recalling the choice ω1 = pi2 . Also in [6] Dorey et. al. provide an expressionfor the generalized resolvent. One can easily show that their equation (2.15), usingthe choice of ω1, is given byG(x(z)) =M24[℘(z)− 23E2(τ)]. (5.2)The matrix model ’t Hooft coupling S is related to the generalized resolvent througha contour integral on the eigenvalue plane, surrounding the upper branch cut,S = gsN =12pii∮C+G(x)dx =−12pi∮C+G(x(z))dxdzdz, (5.3)where the final equality follows by changing variables and integrating over the cor-responding A-cycle on the torus mapping into the upper cut under the embedding.It’s relatively straightforward to carry out the integration on the torus. We begin bynoting that, since∮C+ dx = 0,0 =∮C+dx =∮C+dxdzdz =12∮C+[℘(z) +13E2(τ)]dz. (5.4)In the final equality, I have used that ζ ′(z) = −℘(z). Equation (5.4) immediatelyimplies the relationship, ∮C+℘(z)dz = −pi3E2(τ), (5.5)405.1. The ’t Hooft Coupling of N = 1∗where I used∮C+ dz = 2ω1. Similarly, one can show that∮C+℘2(z)dz =pi9E4(τ). (5.6)The reader should be warned that in the above formulae I have used specificallythe choice of ω1 = pi2 ; these formulae may be seen in the literature with explicitdependence on ω1, or may differ by a minus sign thanks to a particular choice.Using the above results, the ’t Hooft coupling S can finally be computed in termsof the modular parameter τ on the elliptic curve:S = − 12pi∮C+G(x(z))dxdzdz= − 12pi∮C+18℘(z)[℘(z) +13E2(τ)]dz= − 116pi∮C+℘2(z)dz − E2(τ)48pi∮C+℘(z)dz=1144(E2(τ)2 − E4(τ)).(5.7)This agrees precisely with [5, 6].S(τ) =1144(E2(τ)2 − E4(τ))(5.8)SinceS is the matrix model ’t Hooft coupling, one should think ofS(τ) as a “changeof variables” from the modular parameter τ to the physical coupling S. I would liketo record a few observations about S(τ):• S(τ) is a quantity belonging specifically to N = 1∗ theory, as is clear in thecomputation where the generalized resolvent of N = 1∗ was chosen.• This function is holomorphic on the upper-half plane H. Using the symme-tries of the Eisenstein series, it’s clear that S(τ + 1) = S(τ). Due to thissymmetry, S descends to a holomorphic function on an infinite strip of widthone. In Figure 5.1 I plot the argument of S(τ) over this infinite strip. Theleft half of Figure 5.1 should be compared to the schematic diagram in Figure4.3.415.1. The ’t Hooft Coupling of N = 1∗Figure 5.1: A plot of the argument of S(τ).• Mathematically, S(τ) is a quasi-modular cusp form of weight four. It is calleda cusp form because it vanishes at the point at infinity in the upper-half plane.• In Figure 5.1 we can see a sequence of “saddle-points” τ (1)c , τ (2)c , . . . withincreasingly small real and imaginary parts which appear to accumulate nearthe origin. One can check that these are the extrema of the ’t Hooft couplingS(τ),dSdτ(τ (n)c ) = 0 (n ≥ 1). (5.9)Applying the methods from the previous chapter, we see that these extrema ofS fall on the line of degeneration of my model and are therefore, technicallyinaccessible. However, we may approach them from withinH and doing so,leads to a non-obvious result: for all M ∈ C, approaching τ (n)c from withinH, we find that the resulting branch cut configuration consists of the twomirror cuts converging to an overlapping configuration, each of length n|M |(Figure 5.2).• I have already noted that S vanishes at infinity in the upper-half plane. This isthe only zero ofS inH and corresponds to the matrix model having vanishingcut length. Below the line of degeneration S has an infinite collection of ze-ros which appear in Figure 5.1 as “blackened” dots. From a purely analytical425.1. The ’t Hooft Coupling of N = 1∗● ●■ ■-3 -2 -1 1 2 3-0.10-0.050.050.10 �=�● ●■ ■-3 -2 -1 1 2 3-0.10-0.050.050.10 �=�● ●■ ■-3 -2 -1 1 2 3-0.10-0.050.050.10 �=�● ●■ ■-3 -2 -1 1 2 3-0.10-0.050.050.10 �=�● ●■ ■-3 -2 -1 1 2 3-0.10-0.050.050.10 �=�Figure 5.2: The configuration of the eigenvalue plane at the first five critical points of S(τ)with M = 1. It’s important to note that the cuts don’t actually overlap here, rather they’reboth converging to the real axis, as the model nears degeneration.435.2. A Relationship between the Gauge Theory and the Matrix Modelperspective, all zeros and extrema of S(τ) are on the same footing, whetherthey are above or below the line of degeneration, which is an artifact of thematrix model. We therefore see that S(τ) seems to exhibit a self-similarityor fractal-like structure9. Recall that the N = 1∗ matrix model arose byconsidering a fluctuation about the confining vacuum. It is tempting to spec-ulate that the structure below the line of degeneration might correspond tothe other massive vacua of N = 1∗. Loosely speaking, a result of [7] is thatsolutions in the various N = 1∗ vacua are related by SL(2,Z) transforma-tions, so perhaps the self-similarity of S we are observing is related to thisfact.The above observations can be summarized in the following relation between ana-lytical data encoded into the ’t Hooft coupling S(τ) and the geometrical configura-tion of the eigenvalue plane:The unique zero of S(τ ) in H corresponds to the unique point in H wherethe matrix model has vanishing cut length. Moreover, the points labeled n =1,2, . . . in Figure 4.3 where the mirror cuts of the matrix model are approach-ing degeneration with cut length n|M | correspond to the extrema τ (n)c ofS(τ ) for allM ∈ C.5.2 A Relationship between the Gauge Theory and theMatrix ModelIn addition to S, the other important quantity arising from the matrix model com-putation is [5, 6],∂F0∂S= i∫BG(x)dx, (5.10)where F0 is the genus-zero part of the free energy expansion, and where B is thecontour on the eigenvalue plane which runs directly from C− to C+. By a com-putation similar to the one in the previous section, one can compute ∂F0/∂S andshow that it is related to S through,∂F0∂S= 2piiτS − 112E2(τ). (5.11)9A possibly related remark was made in [18], albeit in the context ofN = 2∗ theory.445.2. A Relationship between the Gauge Theory and the Matrix ModelThe effective superpotential in the confining vacuum is then given as,Weff = N∂F0∂S− 2piiτ0S, (5.12)where τ0 is the bare gauge theory coupling. One must then minimize the super-potential with respect to τ , which leads to a remarkable relationship between thegauge theory coupling, and the modular parameter of the elliptic curve,τ =(τ0 + k)N, (5.13)where k = 0, . . . , N − 1 labels the N distinct confining vacua. Recalling the formof the bare gauge theory coupling (3.3), we seeτ =kN+θ2piN+4piig2YMN. (5.14)If we take the ’t Hooft limit (in the gauge theory!) then the term involving θ dropsout, but since k can scale with N , the term k/N remains. Since k is an integerbetween 0 and N − 1, if N →∞, then,0 ≤ kN< 1. (5.15)Moreover, since λ = g2YMN is the gauge theory ’t Hooft coupling, we arrive at thefollowing expression for the modular parameter τ in terms of gauge theory quanti-ties,τ =kN+4piiλ, (5.16)where in this expression, k/N is to be interpreted as a real number greater thanor equal to zero, and strictly less than one. This is consistent with the choice 0 ≤Re(τ) < 1 in Figure 5.1. Equation (5.16) seems to give a physical interpretation ofthe coordinate τ on the moduli space H. It expresses τ in terms of the parametersof a gauge theory in the ’t Hooft limit. The real part of τ corresponds to a choiceof scaling of k with N , while the imaginary part of τ is related to the gauge theory’t Hooft coupling λ. S(τ) is then a holomorphic function on this space, and what Inow want to show is that at weak coupling (small λ), S(τ) exactly determines theconfiguration of the matrix model.455.3. Weak Coupling Expansion of S(τ)5.3 Weak Coupling Expansion of S(τ)Recall that the periodicity of the Eisenstein series implies that S(τ) is also invariantunder τ → τ + 1. Of course, it follows that S can be Fourier expanded in τ . It’sconventional to define the complex parameter q = e2piiτ . Using (5.16), we see thatq takes the form,q = e−8pi2λ e2piik/N . (5.17)Thus, it is clear that small |q| is the region of small ’t Hooft coupling λ. This is theregion of large imaginary part of τ . The Fourier expansion takes the formS(τ) =∞∑n=0cnqn. (5.18)To lowest order in qS(τ) = −2q +O(q2), (5.19)and substituting in the form of q above (5.18), we seeS(τ) ≈ 2e− 8pi2λ e2pii(k/N+1/2). (5.20)This holomorphic function onH determines precisely the configuration of the ma-trix model at weak coupling. More specifically, the configuration of the matrixmodel is given simply by providing the angle of the cut, and the length of the cut.The function, which to my knowledge has not appeared in the literature,Q(τ) = −√8M2S(τ) (5.21)has a Fourier expansion which I hope to show recovers the matrix model branch cutconfiguration at weak coupling. The lowest order term in the expansion is,Q(τ) ≈ −(4|M |e− 4pi2λ)eipi(k/N+1/2)+iArg(M). (5.22)Restricting to the Hermitian slice, we can choose kN =12 and M ∈ R. In such acase, the above expansion simplifies toQ ≈ 4Me−4pi2/λ, (k/N = 1/2, M ∈ R). (5.23)465.4. Surprising Connections to N = 2∗ TheoryWe’ve seen essentially the same equation before (3.33) in the context of weak cou-pling cut lengths in theN = 2∗ matrix model. Recalling that equation (3.33) refersto the half -cut length, the lowest order term in the Fourier expansion of Q(τ) re-stricted to the Hermitian slice appears to encode the full cut length 2µ of N = 2∗at weak coupling. Similarly, choosing the anti-Hermitian slice defined by kN = 0along with M ∈ R, the resulting expansion has the same modulus as (5.23) butwith an additional factor of −i. It is therefore natural to speculate that perhaps atweak coupling, the modulus of Q(τ) encodes the matrix model cut length and itsargument encodes the cut angle. I now want to provide graphical evidence that thisis indeed the case.Within H, there are two simple types of slices we can look at. We can fix λ,and run horizontally across the range 0 ≤ Re(τ) < 1/2 (it suffices to look at halfof H by symmetry). In Figures 5.3 and 5.4 we analyze slices of this form for twodifferent couplings. Conversely, we can fix Re(τ) and consider a range of ’t Hooftcouplings. Such slices are analyzed in Figures 5.5 and 5.6.5.4 Surprising Connections to N = 2∗ TheoryThe computation I present at the beginning of this chapter results in a quantity S(τ),and hence Q(τ), which is intrinsically an object inN = 1∗ theory. Nevertheless, Iargue here that these quantities actually encode non-trivial information aboutN =2∗ theory along with its sequence of critical points. I have already stated explicitlythe following conclusion which I repeat here for completeness:For allM ∈ R, the weak coupling expansion ofQ(τ ) restricted to the Hermi-tian slice throughH exactly recovers the well-known weak coupling cut lengthinN = 2∗ theory.This is an example of non-trivial N = 2∗ data encoded into N = 1∗ quantities. Inaddition, there was actually a hint earlier in the chapter of a connection toN = 2∗.Recall that the extrema τ (n)c of S(τ) encode the locations inHwhere the two mirrorcuts are approaching degeneration, each with length n|M | for all M ∈ C. In mybrief review of the N = 2∗ matrix model, I state the result of [18] for the half -cutlengths at the critical points, which I recall hereµ(λ(n)c ) =nM2. (5.24)475.4. Surprising Connections to N = 2∗ Theory����� ��� ������|�(τ)|�μ0.0 0.1 0.2 0.3 0.4 0.50.07469700.07469750.07469800.0746985Re(τ)|Q(τ)|0.0 0.1 0.2 0.3 0.4 0.5-1.5-1.0-0.50.0Re(τ)ArgQ(τ)Figure 5.3: (λ = 2pi,M = 10) Along this horizontal slice, the upper plot shows thematrix model cut length (blue), the modulus of Q (red), and 2µ (green). It appears that|Q| encodes very closely the variation of the matrix model cut length across the slice. Ifwe were to make λ even smaller, the blue and red curves would converge to 2µ. This isthe statement that at incredibly small coupling, the horizontal slices across H correspondto slices of constant cut length. The second plot actually shows both the matrix model cutangle and the argument ofQ, but the match is so precise that the two are indistinguishable.It is reasonable to conclude from these two plots that at extremely small λ, the horizontalslices through the moduli space correspond to constant cut length and linearly increasingcut angle.485.4. Surprising Connections to N = 2∗ Theory����� ��� ������|�(τ)|�μ0.0 0.1 0.2 0.3 0.4 0.591011121314Re(τ)|Q(τ)|����� ��� ���������(τ)0.0 0.1 0.2 0.3 0.4 0.5-1.5-1.0-0.50.0Re(τ)ArgQ(τ)Figure 5.4: (λ = 31.416,M = 10) Also along a horizontal slice, but in this case atlarger coupling λ = 31.416. Notice now that |Q(τ)| deviates more appreciably from theactual cut length. In addition, ArgQ(τ) no longer perfectly fits the cut angle. Thoughthe two agree at the endpoints of the slice, the actual cut angle varies more linearly thanArgQ(τ) predicts. This is the first clue that although S(τ) is deeply connected to the matrixmodel, it may lose contact with the exact configuration at strong coupling.495.4. Surprising Connections to N = 2∗ Theory����� ��� ������|�(τ)|�μ(λ)5 10 15 20 25 30024681012Gauge Theory Coupling (λ)|Q(τ)|����� ��� ���������(τ)5 10 15 20 25 30-1.10-1.05-1.00-0.95Gauge Theory Coupling (λ)ArgQ(τ)Figure 5.5: (Re(τ ) = 0.15,M = 10) One can imagine instead fixing Re(τ) and con-sidering a range of λ. As we can see, at weak coupling 2µ(λ) = 4Me−4pi2/λ, |Q(τ)|,and the actual cut length agree fantastically. As the coupling increases, the three begindeviating. Moreover, the cut angle agrees at weak coupling with ArgQ(τ) and both areapproximately constant, but around λ = 15, ArgQ(τ) begins growing much faster than thecut angle. Thus, at weak coupling on a vertical slice, the cut length obeys the exponentialrelationship 2µ(λ) = 4Me−4pi2/λ while the cut angle remains fixed. Conversely, recallthat on a horizontal slice at weak coupling the cut length was constant in the strict λ→ 0limit, while the cut angle varied linearly.505.4. Surprising Connections to N = 2∗ Theory����� ��� ������|�(τ)|�μ(λ)5 10 15 20 25 300246810Gauge Theory Coupling (λ)|Q(τ)|5 10 15 20 25 30-1.0-0.50.00.51.0Gauge Theory Coupling (λ)ArgQ(τ)Figure 5.6: (Re(τ ) = 0.5,M = 10) As in the previous figure, we fix Re(τ) and considera range of λ. This corresponds to the Hermitian slice above the critical point. Notice thatthe upper plot is very similar to that in Figure 5.5. However, along this slice both the exactcut angle and ArgQ(τ) are identically zero until the mirror cuts collide at the origin andthe model degenerates.515.4. Surprising Connections to N = 2∗ TheoryIn this context, M ∈ R and λ(n)c is the gauge theory ’t Hooft coupling at the n-thcritical point. Multiplying by 2 to account for the full cut length, we see that indeedthe critical cut lengths inN = 2∗ theory appear to arise in my model at the extremaof S(τ).There are however a number of issues. First, in [18] where Russo and Zarembofirst apply the modular solution to the N = 2∗ matrix model, they note that uponencountering the first critical point (n = 1) the modular solution degenerates al-together. Abandoning the modular solution, Russo and Zarembo numerically pushbeyond the frontier of the n = 1 critical point, allowing the matrix model cut to ex-pand freely on the real axis, without any reference to the mirror cuts of the modularsolution. Doing so, they numerically find an infinite sequence of quantum criticalpoints with critical ’t Hooft couplings λ(n)c accumulating at infinite coupling.One of my original hopes was that the holomorphic nature of my model mightbe able to detect these quantum critical points analytically. Perhaps the strong cou-pling region only appeared inaccessible to one who was constraining themselvesto a Hermitian slice. Approaching the extrema τ (n)c of S(τ) from within H, in-deed the branch cut is converging to exactly the critical configuration of Russo andZarembo. Thanks to the above construction of Dijkgraaf and Vafa, we can coor-dinatize the moduli space using gauge theory parameters kN and λ (5.16), whichrelates the imaginary part of τ to λ. I want to compare λ(n)c with λ˜(n)c , where λ(n)care the N = 2∗ critical couplings computed by Russo and Zarembo [10, 18, 19],and λ˜(n)c are my “predicted” couplings defined byIm(τ (n)c ) =4piλ˜(n)c, (5.25)where τ (n)c are the extrema of S(τ). My parameter values are reported in Table 5.1below.The agreement is exact for n = 1! One can check that λ˜(1)c = λ(1)c which meansthat the n = 1 extrema of S(τ) encodes the n = 1 critical coupling of N = 2∗theory. Unfortunately, the couplings do not match for n > 1.A relevant result of [10] is that the N = 2∗ theory is ill-defined off the slicekN =12 and that the quantum critical points indeed lie on this slice, with increas-ingly small imaginary part. Such points lie in the region of degeneration of mymodel, so it appears that maybe the modular solution to N = 2∗ genuinely cannotdetect the critical points for n > 1. However, I offer a highly speculative poten-tial resolution (which I am unable to check): what if the quantum critical points525.4. Surprising Connections to N = 2∗ Theoryτ(n)c λ˜(n)cn = 1 0.5 + 0.35473 i 35.4252 . . .n = 2 0.345 + 0.16075 i 78.1734 . . .n = 3 0.2605 + 0.0928 i 135.413 . . .n = 4 0.209 + 0.0602 i 208.744 . . .n = 5 0.1735 + 0.0429 i 292.922 . . .Table 5.1: The first five critical points of S(τ) and the corresponding gauge theory cou-plings. My value of λ˜(1)c agrees perfectly with [10, 18, 19], however the higher ’t Hooftcouplings disagree with the numerical predictions in [18, 19].on the Hermitian slice can be taken to the extrema of S(τ) by a modular transfor-mation? Perhaps there is an anomalous term in the relationship between τ and λarising when performing a modular transformation, which would account for theabove discrepancy in the couplings. The extrema of S(τ) are reminiscent enoughof the quantum critical points to deserve an explanation. If one were to abandonthe modular solution of N = 2∗ altogether, then the single branch cut centered atthe origin could freely elongate past the first critical point along the real axis, se-quentially hitting cut lengths which are integer multiples ofM . In my model, uponcollision with the first critical point, the rest of the kN =12 slice is off-limits. In-stead, one could travel along (rather, arbitrarily closely to) the line of degenerationwithin the bulk of H. Here, the cut lengths approaching integral multiples of Mwould be detected by the sequence of extrema of S(τ) along this line.All speculations aside, I can at least conclude the following from the abovediscussion:Under the change of variables Im(τ ) = 4pi/λ, the n = 1 extrema of S(τ )exactly encodes the first critical point ofN = 2∗ theory.I want to emphasize that all of the connections to N = 2∗ I have presented in thissection, both exact and speculative, would be invisible if one were to restrict theN = 1∗ theory to the anti-Hermitian slice kN = 0. It is the true holomorphic natureof the model developed in the last two chapters which reveal that N = 1∗ encodesexact weak coupling cut lengths in N = 2∗, as well as at least, the critical ’t Hooftcoupling λ(1)c .53Chapter 6Eigenvalue DensitiesAs a purely geometrical problem, we’ve seen that given any τ ∈ H and M ∈C we can determine the configuration of the branch cuts in the eigenvalue planeas well as the non-trivial cycle on the elliptic curve mapping into the cut. Thisconstruction contained no physics. I now want to compute eigenvalue densities inthe N = 1∗ and N = 2∗ matrix models. Eigenvalue densities clearly depend onthe physical theory one is working in, and therefore naturally should depend onthe choice of generalized resolvent. We’ve encountered the generalized resolventG(x(z)) of N = 1∗ before (5.2) as computed in [6]. Recall,G(x(z)) =M24[℘(z)− 23E2(τ)]. (6.1)The generalized resolvent of the N = 2∗ theory was provided in [10],G˜(x(z)) =4M1℘(z) + 13E2(τ). (6.2)In this chapter I give a prescription for recovering eigenvalue densities, which isnot explicit in the literature as far as I am aware. Given a choice of generalizedresolvent, the eigenvalue densities are encoded into the restriction of this ellipticfunction to the cycle on the elliptic curve mapping into the cut. At weak coupling,my method recovers the inverse square-root density ofN = 2∗ as well as Wigner’ssemi-circular density inN = 1∗. There are numerical and asymptotic indications inthe literature that theN = 1∗matrix model in the limit of large cut length supportedin the real axis should admit a parabolic density. Of course, this region lies on theline of degeneration in my model. Making use of the holomorphic nature of theN =1∗model, I attempt to find evidence of the parabolic density in the region ofHwherethe two mirror cuts are elongated and approaching an overlapping configuration onthe real axis. Indeed, one of the original motivations of this project was to see if theelliptic curve and generalized resolvent encoded the parabolic density, and I presentpartial evidence in favor of this.546.1. The General Method0.5 1.0 1.5 2.0s-0.50.5Re[x(s)]-M/20.5 1.0 1.5 2.0s-0.4-0.20.20.4Im[x(s)]-M/2Figure 6.1: For τ = 14 +14 i, the map from C+ into the real and imaginary parts of the cutC+.6.1 The General MethodAs we have seen, for each τ ∈ H there exists a map x(z) which determines theconfiguration of the eigenvalue plane. In addition, there must exist a distinguishedA-cycle denoted C+ which maps into the cut C+ on the eigenvalue plane with mid-point M2 . In other words, the restriction of the map to C+ takes values in the cutitself. Subtracting M2 we map to the real cut whose midpoint is the origin (recallthe two cuts are the “mirrors” of the single real cut). We choose to parameterize C+with a real parameter 0 ≤ s ≤ 2, such that the points in the real cut are x(s)− M2 ,for all s.In Figure 6.1 above, for s = 0 we begin at the origin (i.e. the midpoint of thecut), then proceed to maximally negative real part, maximally positive imaginarypart, turn around, traverse the cut in the other direction, and finally return to theorigin. The s values at which the maxima and minima occur parameterize the dis-556.1. The General Methodtinguished points in C+ mapping to the branch points. Indeed, both the real andimaginary plots have these extrema occurring at the same parameter values, as theymust. Finally, notice that the range of the real part is slightly larger than the rangeof the imaginary part. Looking back at Figure 4.1 shows this is because the cut isextended more along the real axis than the imaginary axis. Recall the closed formexpression for the eigenvalue density in terms of the discontinuity of the ordinaryresolvent across the branch cut,ρ(x) = − 12pii(ω(x+ i)− ω(x− i)). (6.3)Though the generalized resolvents may appear in a different form than the ordinaryresolvents, when it comes to discontinuities across cuts, they encode the same data.As such, the plan is to exploit the relationship of the generalized resolvent to theordinary resolvent, and then in turn, the relationship of the ordinary resolvent tothe eigenvalue density. This will yield an exact method for extracting eigenvaluedensities.Since the cut is a straight line segment, we can parameterize its points just aswell by their real parts, imaginary parts, or by a length parameter along the cut. Inthis context, it is most natural to choose a length parameter. Let η be the real partof a point in the cut. We define a length parameter ξ by η = ξ cos θ, where θ is theangle of the cut with respect to the real axis. Clearly, the parameter ξ spans the fulllength of the cut, instead of merely the spanning the real or imaginary parts. Thecycle C+ double covers the cut C+. This makes sense if we recall that the imageof A-cycles under x encircle the branch cut. The image x(C+) does so “infinitelytightly”, which we think of as a double cover. Therefore:for all ξ in the branch cut, there exists two points sξ1 and sξ2 in C+ mappingby x to the same point in the cut with parameter ξ. Hence, the discontinuityof the resolvent across the cut can be reformulated as the difference in the twovalues of the generalized resolvent evaluated at these two points on the ellipticcurve. This provides the value of the eigenvalue density at this single point ofthe cut.Sweeping through each point in the cut, I can apply this idea to compute the exacteigenvalue density as a function of ξ along the length of the cut for all τ ∈ H. Exact566.1. The General Methodexpressions for the eigenvalue densities areρ(ξ) =12piiM3S(G(x(sξ1))−G(x(sξ2))), N = 1∗− 12pii(G˜(x(sξ1))− G˜(x(sξ2))), N = 2∗ (6.4)This is rather meager as far as self-contained formulae go, given that for each ξone must do a computation to find sξ1 and sξ2. But it nonetheless shows explicitlythat the eigenvalue densities are encoded into the elliptic curve and the generalizedresolvent, the exact form of which, is known.Note that sξ1 = sξ2 precisely when we are at an endpoint of the cut. By con-struction, we’ve seen the transcendental constraint (4.12) which must hold at anendpoint,℘(sξ1) = ℘(sξ2) = −13E2(τ). (6.5)From (6.1), we see that G(x(s1)) = G(x(s2)) and thus, the difference will vanish.This is the statement that the eigenvalue densities in the N = 1∗ theory vanishat the endpoints. On the other hand, the denominator of (6.2) implies that bothG˜(x(s1)) and G˜(x(s2)) diverge. So we expect eigenvalue densities in the N = 2∗theory to diverge at the endpoints. This is well-known from [18], and in fact is whatmotivated the construction in [10].Reality and Normalization of Eigenvalue Densities?A final remark is in order before presenting results. As noted in [15], eigenvaluedensities in holomorphic matrix models are generally complex. Indeed, this istrue of my construction. For general τ ∈ H, the resulting eigenvalue density willbe complex valued except along distinguished “slices” through the moduli space.For example, the eigenvalue density is real along the entire anti-Hermitian sliceRe(τ) = 0. In addition, along the Hermitian slice Re(τ) = 1/2 (above the criticalpoint) the density is also real valued.Given that the eigenvalue densities are complex, what we mean by normaliza-tion must be adjusted. I have checked in a large number of cases that it is preciselythe absolute value of the eigenvalue densities |ρ(ξ)| which is normalized over thefull length of the cut. This is true of both N = 1∗ and N = 2∗. Therefore, whenthe imaginary part vanishes, the eigenvalue densities will satisfy both reality and576.2. N = 2∗ at Weak Coupling: Shenker’s Inverse Square Rootnormalization conditions. In my recovery of the semi-circle and the inverse square-root, the eigenvalue densities will be real and normalized perfectly. However, inthe bulk of the moduli space H, where I have observed evidence of the parabolicdensity, I must take the absolute value of the complex eigenvalue density.6.2 N = 2∗ at Weak Coupling: Shenker’s Inverse SquareRootWe’ve seen that we expect the eigenvalue densities in the N = 2∗ to diverge at thecut endpoints. In particular, we saw in (3.32) that we expect to recover Shenker’snormalized inverse square-root at weak coupling [8]. Choosing τ = 12 +34 i, aswell as the generalized resolvent G˜(x(z)) in (6.2), I plot the eigenvalue density inFigure 6.2. The normalized inverse square-root I compare to isy(ξ) =1pi√µ2 − ξ2 , (6.6)where µ is the matrix model cut length. Along this slice of the moduli space, theeigenvalue density is inherently real. By symmetry, the eigenvalue density mustbe symmetric under inversion about the midpoint of the cut. Thus, it suffices toplot the density over only the half-cut length [0, µ]. As we can see in Figure 6.2,the eigenvalue density matches the inverse square-root remarkably well consideringλ ≈ 16.76 is not terribly small. The match becomes more exact as λ decreases.6.3 N = 1∗ Eigenvalue DensitiesRecall the form of the N = 1∗ matrix model,ZN (gs) =∫DΦ1DΦ2DΦ3e−1gsTr(Φ1[Φ2,Φ3]+Φ21+Φ22+Φ23). (6.7)This is equation (3.10) written in terms of Φi instead of Φ, Φ±. By completing thesquare in the exponent we can integrate over Φ1,ZN (gs) = C∫DΦ2DΦ3e−NSTr(Φ22+Φ23− 14 [Φ2,Φ3]2), (6.8)586.3. N = 1∗ Eigenvalue Densities0.1 0.2 0.3 0.4 0.5ξ0.51.01.5|ρ(ξ)|Figure 6.2: The exactN = 2∗ inverse square-root eigenvalue density for τ = 12 + 34 i, whichcorresponds to λ = 16pi3 ≈ 16.76. I plot the eigenvalue density computed using my method(orange) alongside the function (black-dotted) in (6.6). For the reader’s convenience, thisoccurs at point A in Figure 4.3. I also note that for this choice of τ , the eigenvalue densityis real and normalized.596.3. N = 1∗ Eigenvalue Densitieswhere C is a complex number, and we’ve used gs = S/N . Since the ’t Hooftcoupling is a finite parameter, we can scale Φ2 and Φ3 by√S, resulting in,ZN (gs) = C∫DΦ2DΦ3e−NTr(Φ22+Φ23−S4 [Φ2,Φ3]2). (6.9)This two-matrix model is identical to the one given in equation (2.1) in [9], providedone identifies the above coupling S/4 with the square of theirs. In [9], Filev andO’Connor note that for strictly vanishing coupling S = 0, we get a two-matrixmodel which is totally decoupled; it’s merely a product of two Gaussian matrixmodels. So integrating over one of them, we have simply a Gaussian one-matrixmodel which should produce a Wigner semi-circular eigenvalue distribution [20].Weak Coupling: The Wigner Semi-CircleIndeed, I show that even at small, but non-vanishing coupling S, my method recov-ers the Wigner semi-circle. Recalling that we use ξ as a parameter on the cut, thenormalized semi-circle to which I will compare my plot is,y(ξ) =1µpi√1− ξ2µ2, (6.10)where µ is the matrix model cut length. Let τ = 12 +34 i, which corresponds to agauge theory ’t Hooft coupling λ = 16pi3 . Recall that on this portion of the modulispace, the cut lives entirely in the real axis. Notice from Figure 6.3 that the fit isexceptional, despite λ ≈ 16.76 not being a terribly small gauge theory ’t Hooftcoupling. The accuracy becomes even better as the coupling becomes yet smaller.Strong Coupling: Evidence for the Parabolic DensityReturning to (6.9), Filev and O’Connor study the same two-matrix model at strongcoupling, after integrating over one degree of freedom to yield a one-matrix model.They show that at large coupling, the leading order approximation to the eigen-value density is parabolic. This parabolic density has appeared quite often in therecent literature [1, 2, 16, 21]. Particularly in [21] it is remarked that the N = 1∗Dijkgraaf-Vafa matrix model in the limit of large cut length supported in the realaxis should reproduce the parabolic density. Along the Hermitian slice throughH, recall that the modular solution to the theory degenerates when the two mirror606.3. N = 1∗ Eigenvalue Densities0.1 0.2 0.3 0.4 0.5ξ0.20.40.60.81.01.2|ρ(ξ)|Figure 6.3: The exact N = 1∗ semi-circular eigenvalue density for τ = 12 + 34 i, whichcorresponds to λ = 16pi3 ≈ 16.76. I plot the exact eigenvalue density (orange) alongsidethe semi-circle (black-dotted) given in (6.10). This occurs at point A in Figure 4.3. Thiseigenvalue density is real-valued and normalized.616.3. N = 1∗ Eigenvalue Densitiescuts collide at the origin. This seems to indicate that my construction in Chapter 4cannot access the region capable of detecting the parabolic density.However, making full use of the holomorphic nature of the N = 1∗ matrixmodel, we may move into the bulk ofH and allow the two mirror cuts to elon-gate without colliding. The mirror cuts can be made arbitrarily long, andthen by approaching the line of degeneration, they can be made to convergeto an overlapping configuration on the real axis. Recalling that the two mirrorcuts are simply translates of the actual cut, this maneuver seems to producea branch cut of arbitrarily long length, nearly supported in the real axis witharbitrarily small imaginary part.In practice, the functions involved become fairly chaotic at the extreme param-eter values and are quite difficult to handle computationally. At such large ’t Hooftcoupling my procedure for recovering eigenvalue densities requires more involvednumerical methods. It is still a completely valid procedure, but its simple appli-cation breaks down. I will reserve myself to approaching the real axis such thatthe imaginary parts of the cuts are as small as possible. I will still be able to findconvincing evidence of the parabola.As we’ve noted, eigenvalue densities in holomorphic matrix models are complex-valued, in general. So when the cuts have non-zero imaginary part, we expect theeigenvalue densities to be complex. As such, we must take the absolute value ofthe density. Upon approaching the real axis, the imaginary part of the densitiesshould approach zero, and so the imaginary contribution to the absolute value playsan increasingly negligible role.It is clear from Figure 6.4 that as the ’t Hooft coupling increases, the eigenvaluedensity more closely resembles a parabola. Near the cut endpoints, the fit breaksdown substantially. The reason is that an eigenvalue density must have a verticaltangent line at its endpoints. This is certainly true of both the semi-circle, and theinverse square-root. So the eigenvalue densities very closely resemble a parabolafor most of the cut, but near the endpoints, they diverge downward to enforce thevertical tangency constraint. Indeed, the region of disagreement near the endpointsrepresents a vanishingly small fraction of the total cut length at large coupling. Thisdisagreement of the eigenvalue density with the parabolic density was also noted in[1].626.3. N = 1∗ Eigenvalue Densities0.2 0.4 0.6 0.8 1.0ξ0.20.40.60.81.0|ρ(ξ)|Figure 6.4: In order of increasing coupling, the red, green, blue, and magenta plots cor-respond to points B, C, D, and E, respectively in Figure 4.3. An exact parabola is shownas a black dotted line. Notice that at the highest coupling, excellent fit to the parabola isobserved over most of the support. As described below, the fit is expected to be imperfectnear the endpoints of the cut. In order to provide a single, illustrative figure, I have scaledmy eigenvalue densities to be supported on the half-length [0, 1] with height 1. Undoing thescaling, my method produces normalized eigenvalue densities.63Chapter 7ConclusionsI have reviewed and studied two main forms of geometry emerging from matrixmodels: an emergent algebraic curve and eigenvalue densities. The N = 1∗ andN = 2∗ theories both give rise to an elliptic curve with generalized resolvents. Ishow explicitly in the previous chapter that the two forms of emergent geometry areintimately intertwined. Namely, the restriction of the generalized resolvent to thespecial cycle C+ (Figure 4.2) on the elliptic curve, allows for an exact constructionof eigenvalue densities. This unification between the two emergent structures is notparticularly surprising, but is important nonetheless.Given τ ∈ H, I attempt to reconstruct the configuration of the mirror cuts onthe eigenvalue plane. This is essentially independent of any physical theory, andwas modeled on [6, 10]. I find there is a region of degeneration in H where thereconstruction of the eigenvalue plane breaks down. The moduli space H (Figure4.3) is the compliment of the region of degeneration in H and the line of degener-ation is the boundary of the region of degeneration, acting as a “line at infinity” ofH. I find that Re(τ) = 1/2 is the Hermitian slice with a critical point when the twomirror branch cuts collide, consistent with [10].The fundamental questions I consider in this thesis are the following.Given a Hermitian matrix model in the ’t Hooft limit whose weak coupling regionembeds naturally along the Hermitian slice up to the first critical point, can thegeometrical solution using the elliptic curve detect the strong coupling region?Does (the neighborhood of) the line of degeneration encode any strong couplingdata?I want to emphasize that exploring the neighborhood of the line of degenerationrelies on the holomorphic nature of the model; instead of terminating at the criticalpoint, we can avoid it and move into the bulk of the moduli space.In the Hermitian matrix model (6.9) studied in [1, 2, 16, 21] the characteristiceigenvalue densities of a Wigner semi-circle and a parabola are discovered at weak64Chapter 7. Conclusionsand strong coupling, respectively. Indeed, I show that along the Hermitian slice,N = 1∗ at weak coupling reproduces the Wigner semi-circular density (Figure 6.3).At strong coupling, I find encouraging evidence (Figure 6.4) that a parabolic densityemerges upon approaching the line of degeneration. This appears to be an exampleof a Hermitian model whose weak coupling region embeds along the Hermitianslice above the critical point, and whose strong coupling region is uncovered in aneighborhood of the line of degeneration. If so, I have shown that an emergentelliptic curve encodes the eigenvalue densities in the model (6.9) which in turn,indicate a particular non-commutative background.I show in Chapter 5 that N = 1∗ restricted to the Hermitian slice throughH encodes the same cut lengths and critical coupling λ(1)c as N = 2∗ before thefirst phase transition. Therefore, theN = 2∗ theory is another example of a theorywhich at weak coupling embeds on the Hermitian slice. On the line of degenerationI find an infinite sequence of points n = 1, 2, . . . (Figure 4.3) reminiscent of theN = 2∗ critical points. These points, and only these points, are the extrema of theN = 1∗ coupling S(τ) in the closure of H. In addition, approaching these pointsfrom withinH, the cut is approaching the real axis exactly with length n|M | (Figure5.2 depicts the mirror cut configurations approaching these points). These coincidewith the critical cut lengths in N = 2∗. The motivation is that by embeddinginto a holomorphic model, instead of terminating at the first critical point, one mayallow the mirror cuts to move slightly off the real axis, enlarge arbitrarily, and thenapproach the real axis once again.However, only for n = 1 does the critical ’t Hooft coupling (Table 5.1) matchthat of N = 2∗. For n > 1, despite also being extrema of S(τ) and having theexpected cut length and configuration, any possible connections toN = 2∗ remaina mystery.65Bibliography[1] David E. Berenstein, Masanori Hanada, and Sean A. Hartnoll. Multi-matrix models and emergent geometry. Journal of High Energy Physics,2009(02):010–010, February 2009. arXiv: 0805.4658.[2] Rodrigo Delgadillo-Blando, Denjoe O’Connor, and Badis Ydri. Geometry intransition: A model of emergent geometry. Physical Review Letters, 100(20),May 2008. arXiv: 0712.3011.[3] Robbert Dijkgraaf and Cumrun Vafa. Matrix Models, Topological Strings, andSupersymmetric Gauge Theories. Nuclear Physics B, 644(1-2):3–20, Novem-ber 2002. arXiv: hep-th/0206255.[4] Robbert Dijkgraaf and Cumrun Vafa. On geometry and matrix models. Nu-clear Physics B, 644(1):21–39, November 2002.[5] Robbert Dijkgraaf and Cumrun Vafa. A Perturbative Window into Non-Perturbative Physics. arXiv:hep-th/0208048, August 2002. arXiv: hep-th/0208048.[6] Nick Dorey, Timothy J. Hollowood, S. Prem Kumar, and AnnamariaSinkovics. Exact Superpotentials from Matrix Models. Journal of High En-ergy Physics, 2002(11):039–039, November 2002. arXiv: hep-th/0209089.[7] Nick Dorey, Timothy J. Hollowood, S. Prem Kumar, and AnnamariaSinkovics. Massive Vacua of N=1* Theory and S-duality from Matrix Mod-els. Journal of High Energy Physics, 2002(11):040–040, November 2002.arXiv: hep-th/0209099.[8] Michael R. Douglas and Stephen H. Shenker. Dynamics of $SU(N)$ Su-persymmetric Gauge Theory. Nuclear Physics B, 447(2-3):271–296, August1995. arXiv: hep-th/9503163.66Bibliography[9] Veselin G. Filev and Denjoe O’Connor. Multi-matrix models at general cou-pling. Journal of Physics A: Mathematical and Theoretical, 46(47):475403,November 2013. arXiv: 1304.7723.[10] Timothy J. Hollowood and S. Prem Kumar. Partition function of N=2* SYMon a large four-sphere. arXiv:1509.00716 [hep-th], September 2015. arXiv:1509.00716.[11] Jens Hoppe. Quantum theory of a massless relativistic surface and a two-dimensional bound state problem. Thesis, Massachusetts Institute of Tech-nology, 1982.[12] Dale HusemÃűller. Elliptic Curves. Springer, New York, softcover reprint ofthe original 2nd ed. 2004 edition edition, November 2010.[13] Vladimir A. Kazakov, Ivan K. Kostov, and Nikita Nekrasov. D-particles, Ma-trix Integrals and KP hierachy. Nuclear Physics B, 557(3):413–442, Septem-ber 1999. arXiv: hep-th/9810035.[14] Neal I. Koblitz. Introduction to Elliptic Curves and Modular Forms. Springer,New York, 2nd ed. 1993 edition edition, April 1993.[15] C. I. Lazaroiu. Holomorphic matrix models. Journal of High Energy Physics,2003(05):044–044, May 2003. arXiv: hep-th/0303008.[16] Denjoe O’Connor, Brian P. Dolan, and Martin Vachovski. Critical Behaviourof the Fuzzy Sphere. Journal of High Energy Physics, 2013(12), December2013. arXiv: 1308.6512.[17] Vasily Pestun. Localization of gauge theory on a four-sphere and supersym-metric Wilson loops. Communications in Mathematical Physics, 313(1):71–129, July 2012. arXiv: 0712.2824.[18] J. G. Russo and K. Zarembo. Massive N=2 Gauge Theories at Large N. Jour-nal of High Energy Physics, 2013(11), November 2013. arXiv: 1309.1004.[19] Jorge G. Russo and Konstantin Zarembo. Evidence for Large-N Phase Transi-tions in N=2* Theory. Journal of High Energy Physics, 2013(4), April 2013.arXiv: 1302.6968.67Bibliography[20] E. P. Wigner. Characteristic Vectors of Bordered Matrices with Infinite Di-mensions II. In Arthur S. Wightman, editor, The Collected Works of EugenePaul Wigner, number A / 1 in The Collected Works of Eugene Paul Wigner,pages 541–545. Springer Berlin Heidelberg, 1993. DOI: 10.1007/978-3-662-02781-3 36.[21] Badis Ydri. Remarks on the eigenvalues distributions of D\leq 4 Yang-Millsmatrix models. International Journal of Modern Physics A, 30(01):1450197,January 2015. arXiv: 1410.4884.68

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