Multiple phases and meromorphic deformations of unitary matrix models
MMULTIPLE PHASES AND MEROMORPHICDEFORMATIONS OF UNITARY MATRIX MODELS
LEONARDO SANTILLI ∗ AND MIGUEL TIERZ † , ‡ Abstract.
We study a unitary matrix model with Gross–Witten–Wadia weight function anddeterminant insertions. After some exact evaluations, we characterize the intricate phase diagram.There are five possible phases: an ungapped phase, two different one-cut gapped phases and twoother two-cut gapped phases. The transition from the ungapped phase to any gapped phaseis third order, but the transition between any one-cut and any two-cut phase is second order.The physics of tunneling from a metastable vacuum to a stable one and of different releases ofinstantons is discussed. Wilson loops, β -functions and aspects of chiral symmetry breaking areinvestigated as well. Furthermore, we study in detail the meromorphic deformation of a generalclass of unitary matrix models, in which the integration contour is not anchored to the unit circle.The ensuing phase diagram is characterized by symplectic singularities and captured by a Hassediagram. Contents
1. Introduction 22. The model 23. Phase structure 64. Wilson loops and instantons 145. Meromorphic deformation of unitary matrix models 206. Outlook 28Appendix A. Technical details of the solution 28Appendix B. Filling fraction fluctuations 31References 32 a r X i v : . [ h e p - t h ] F e b Introduction
The study of the spectral and critical properties of models of random matrices has become awidely popular and interdisciplinary subject in this century, in great part due to the vast scopeof fields where such models appear naturally and play a prominent role [1–3]. One among thepossible many ways to highlight this relevance and versatility of random matrix theory is to simplypoint out the fact that the same model oftentimes appears in remarkably different contexts and,in addition, in a significant manner.A paradigmatic example of this phenomenon could very well be the so-called Gross–Witten–Wadia (GWW) model [4–6]. Originally proposed in the study of gauge theory, it is ubiquitousand pivotal in many other areas, such as combinatorics, representation theory and spectral theory[1–3].With this fact in mind, in this work we will study unitary matrix models, starting, precisely,with the specific case of a generalized form of the Gross–Witten–Wadia model. A possible inter-pretation of the model is as a one-plaquette model of two-dimensional lattice QCD with fermionicor bosonic quarks, which equivalently corresponds to a massive deformation of a model introducedby Minahan [7, 8].We will show that, while quite simple, this model retains several features of a sensible quantumfield theory in the continuum. In turn, its simplicity allows us to exploit standard techniquesfrom random matrix theory to characterize the theory at large N and suggests more general anddeeper problems to consider. Some of them, we will already tackle here, by discussing at lengththe case of meromorphic deformations of unitary matrix model, as we explain below.Before that, a rich phase diagram will be obtained and analyzed in detail. Phase transitionssuch as the ones we obtain in our analysis are relevant to the study of deconfinement transitionsin QCD models in four dimensions [9, 10] and in black holes physics [11–13]. More recently, thistype of phase transitions has been argued to describe the critical behaviour of models exhibitingpartial deconfinement [14–18].We introduce the model in what follows, in Section 2, which includes a discussion on interpre-tations of the model, notation, relationship with other systems and an introductory discussion ofits mathematical properties, including exact evaluations without scaling limits.Then, the main results of the paper are presented and organized as follows: there are threemain contributions, as far as new results are concerned. In Section 3, we fully characterize therich phase structure of the unitary matrix model. In Section 4, we study Wilson loops in thesame setting of Section 3 and discuss at length the physical interpretation of the phase transitions,including the role of instanton contributions.Finally, in Section 5 we study, in a general framework and going beyond the specific modelstudied in the previous sections, the case where the integration contour is deformed in C ∗ awayfrom the unit circle. In spite of the vast body of results on random matrix ensembles, holomorphicmatrix models [19] are arguably understudied.The aim of Section 5 is to adapt the results on holomorphic matrix models to unitary matrixmodels. We do so for a very general set of unitary matrix models and, only as an illustrativeexample, we discuss the particular case of the holomorphic GWW matrix model. Non-traditionaltools in this area, such as symplectic singularities and Hasse diagrams, are introduced here tofully understand the meromorphic models.We conclude with possible avenues for further research in Section 6.2. The model
In this section we present the model and study some of its exact features at finite N .2.1. The model and its interpretations.
Consider a one-plaquette model of two-dimensionallattice gauge theory [20, 21] with gauge group U ( N ) and K pairs of real fields, that can be either bosonic or fermionic. Each pair is called a flavour. We encode the choice of matter fields in thebinary variable (cid:15) = (cid:40) +1 fermions , − . We must impose (anti-)periodic boundary conditions, so the discrete space-time is effectivelyreduced to a point with a loop attached to it, along which the gauge connection travels. Let m f > f th flavour, and introduce the notation µ ( (cid:15) ) f = 1 + √ (cid:15)m f . The partition function of this theory has the matrix model representation [7] Z (cid:15),KU ( N ) ( λ ) = (cid:90) U ( N ) d U K (cid:89) f =1 (cid:20) det (cid:16) µ ( (cid:15) ) f − U (cid:17) det (cid:16) µ ( (cid:15) ) f − U (cid:17) † (cid:21) (cid:15) e Nλ ( Tr U +Tr U † )where λ ≡ N g YM is the ’t Hooft coupling for the bare gauge coupling g YM , U ∈ U ( N ) is theplaquette gauge variable and d U is the normalized Haar measure on U ( N ). Particularizing tothe degenerate case with all equal masses, µ f = µ ∀ f = 1 , . . . , K , the partition function becomes: Z (cid:15),KU ( N ) ( λ ) = (cid:90) U ( N ) d U (cid:104) det ( µ − U ) det ( µ − U ) † (cid:105) (cid:15)K e Nλ ( Tr U +Tr U † )= (cid:73) T N (cid:89) ≤ j Integrable systems interpretation. In the two different limiting cases K = 0 (pure GWW)and λ − = 0 with µ = 1, the matrix model is known to be the integral representation of a τ -function of a Painlev´e system, in particular Painlev´e III (cid:48) and V, respectively [34, 29, 35, 36, 1].In fact, the more general form of the τ -function of PV is very close in form to (2.1), differingthough in having (2.1) with e Nλ z , instead of (2.1) with the full GWW weight. If one inspectsthe integral representation of the τ -function of PVI [29, 35, 1], it is seemingly unrelated to (2.1)and that may partially explain why our model here is essentially unstudied, whereas its twolimiting cases appear in many works, often analyzed simultaneously or in a comparative fashion[34, 36, 29, 35, 1]. Such an expression also appears in the study of spacing distributions [1, Eq. (8.119)]. It is also worth mentioning another possible interpretation of (2.1), from the point of view ofintegrable systems [37]. The study of the so-called Schur flow [38, 39], analogous to Toda flowsbut on the unit circle, precisely entails the generalization of a given weight function of the matrixmodel by multiplication by the GWW weight function.This induces a flow that has many implications. For example, the recurrence coefficients ofthe polynomials, orthogonal with regards to the weight function of the matrix model, satisfy thenon-linear Ablowitz–Ladik equation, with the parameter g = Nλ interpreted as time.Because of this and since our analysis is in a planar limit and centered around the matrix model,the results obtained are not obviously transferable into this integrable systems and spectral theorylanguage [40]. We will further discuss about this at the end, in the outlook Section 6. Gauge theory interpretation. Rather, we consider (2.1) instead as a toy model for latticetwo-dimensional QCD, although we will comment on some of the many other interpretations ofthe model. For example, (2.1) can be regarded as an effective description of two-dimensionalQCD on a small spatial circle [28]. In fact, compactifying the spatial direction generates a massgap for all but the zero-modes. Taking the small circumference limit, we are left with an effectivetheory with integration only over the gauge and matter zero-modes.It was observed in [22] that the massless theory with fermions, that is (cid:15) = +1 and µ = 1, showsa Fisher–Hartwig (FH) singularity. The theory with bosons, on the contrary, yields a singularmatrix model in the massless case. Here we recognize the singularities encountered in [22] as theremnants of the IR singularities due to massless fields, and resolve them via mass deformation. On the parameter µ . Notice that there is a slight difference in the definition of µ between thefermionic and the bosonic theory in (2.1). In the first case µ > µ ∈ C with | µ | > 1. It is easy to see that the phase of µ ∈ C can be reabsorbed in a rotation ofthe integration contour T , and we henceforth restrict our attention to a real µ > µ really means | µ | .Besides, treating µ as a real variable with this caveat in mind, the integrand in (2.1) is analyticin µ > 1, and therefore the results for any other µ ∈ C with | µ | > µ moving alongthe real line, because we would cross the FH singularity. Nevertheless, it is possible to take apath from µ > µ (cid:48) < − Remark . The independence of the partition function on arg µ is the U (1) freedom to choosethe origin of T , and is the incarnation of the residual diagonal U (1) ⊂ U ( N ) gauge symmetry.Our choice arg µ ∈ π Z fixes this residual gauge freedom. Notation. We introduce the notation Y = 1 λ , τ = (cid:15) KN for, respectively, the inverse of the ’t Hooft coupling and a real Veneziano parameter, whose signcarries information on the type of fields we consider.The parameter space of the theory is M = (cid:8) ( µ, τ, Y ) ∈ (1 , ∞ ) × R (cid:9) . Remark . For the role of the mass as a regulator (for the FH singularities on the mathematicalside, for the IR singularities on the field theory side), we do not expect the continuation from M to the sheet { µ = 1 } × R , studied in [22, 23], to be analytic. According to the discussion in the previous paragraph, these well-known facts in random matrix theory can bereinterpreted as stemming from Coleman’s no-go theorem [41] for massless bosons in QCD . Exact finite N evaluations. We now present various analytical results for the matrixmodel (2.1).It is possible to evaluate the partition function of any unitary matrix model at finite N via theHeine–Szeg˝o identity, that for (2.1) gives(2.2) Z (cid:15),KU ( N ) ( λ, µ ) = N ! det ≤ j,k ≤ N [ Z jk ] , with Z jk = K (cid:88) p =0 K ! p !( K − p )! ( − µ ) p (1 + µ ) K − p d p d x p I k − j (2 x ) | x = Nλ where I k ( x ) is the modified Bessel function. We have simplified the expression assuming (cid:15) = +1,although a similar determinant expression exists for (cid:15) = − Z (cid:15),KU ( N ) exactly for fixed N and K . This is done in Table 1 in Appendix A.1.The partition function for generic masses, encoded in the parameters ( µ , . . . , µ K ), can be alsorelated to the expectation value of Wilson loops in arbitrary representations in the pure GWWmodel, thanks to the Cauchy identity (see [42] for a similar procedure). For the fermionic theorywe write Z +1 ,KU ( N ) = K (cid:89) f =1 µ f (cid:73) T N (cid:89) ≤ j This section is dedicated to the large N analysis of the matrix model (2.1) and the determina-tion of its phase diagram.Write the partition function (2.1) as(3.1) Z U ( N ) = (cid:73) T N e − N S eff ( z ,...,z N ) N (cid:89) j =1 d z j π i z j where the effective action S eff is the sum of a potential and the Coulomb interaction betweeneigenvalues: S eff ( z , . . . , z N ) = 1 N N (cid:88) j =1 V eff ( z j ) + 1 N N (cid:88) j =1 (cid:88) k (cid:54) = j V int ( z j , z k ) V eff (cid:16) e i θ (cid:17) = − λ cos θ − τ log (cid:0) µ − µ cos θ (cid:1) V int (cid:16) e i θ , e i ϕ (cid:17) = − log 2 sin (cid:18) θ − ϕ (cid:19) . writing z ∈ T as z = e i θ , − π < θ ≤ π .The potential V eff (cid:0) e i θ (cid:1) admits isolated minima at each point in M . Besides, there existsa surface { λ = λ ∗ ( µ, λ ) } ⊂ M at which it passes from a single-well to a double-well profile.Explicitly, these two regimes are separated by λ ∗ ( µ, τ ) = ( µ − τ µ , and the potential develops stationary points at θ = ± θ ∗ withtan θ ∗ = ± (cid:112) µ (2 − λ τ ) + 2 λµ τ + 2 λµτ − µ − − λµτ + µ + 1 . We stress that λ ∗ is not a critical value of the model (2.1). The potential is plotted in Figure 1 ( τ > 0) and in Figure 2 ( τ < Figure 1. V eff ( e i θ ) at µ = 3 and τ = 2. Left: λ = . Right: λ = . Figure 2. V eff ( e i θ ) at µ = 3 and τ = − 2. Left: λ = − . Right: λ = − .Now that we have set the ground, we are ready to discuss the large N limit of the model (2.1).3.1. Large N . We now take the large N ’t Hooft and Veneziano limit of the partition function(3.1). This means that we consider the planar limit with both λ and τ fixed. The leadingcontributions to the integral at large N come from the saddle points of the effective action:(3.2) ∂S eff ∂θ j = 0 j = 1 , . . . , N. Introducing the eigenvalue density ρ ( θ ) = 2 πN N (cid:88) j =1 δ (cid:16) e i θ − e i θ j (cid:17) , with normalization chosen so that(3.3) (cid:90) π − π d θ π ρ ( θ ) = 1 , we can collect the system (3.2) of N coupled equations in a single singular integral equation atlarge N . The saddle point equation then reads(3.4) P (cid:90) π − π d ϕ π ρ ( ϕ ) cot (cid:18) θ − ϕ (cid:19) = 2 Y sin θ − µτ sin θ µ − µ cos θ . The solution to (3.4) must satisfy the non-negativity constraint(3.5) ρ ( θ ) ≥ , − π < θ ≤ π, that follows from the compactness of the integration domain. Ungapped solution: Phase 0. We begin assuming ρ ( θ ) is supported on the whole circle, − π < θ ≤ π . We exploit µ > ρ ( θ ) = 1 + 2 Y cos θ − τ µ cos θ − 11 + µ − µ cos θ . The derivation is standard [4, 45], thus we omit it. We call Phase 0 the region of M for whichthe solution (3.6) is valid. Critical loci. In those regions of M for which the solution (3.6) violates the constraint (3.5), weshould drop the assumption supp ρ = ( − π, π ] and look for a new solution, whose support has oneor more gaps on the unit circle. The arcs on which ρ is supported are called cuts.We find a phase transition with a gap opening at θ = ± π at the critical surface(3.7) Y cr , a = 12 + τµ + 1 . Another phase transition, with a gap opening at θ = 0, takes place at the critical surface(3.8) Y cr , b = − 12 + τµ − . Besides, there exists a multi-critical point at the value τ = τ cr , + ( µ ) at which ρ ( ± π ) = 0 = ρ (0),determined as the unique point at which Y cr , a and Y cr , b meet:(3.9) τ cr , + ( µ ) = µ − . Examples of the limiting cases of ρ ( θ ) are shown in Figure 3. Figure 3. ρ ( θ ) at the transition point. Left: µ = 3, τ = − and λ = λ cr , a .Centre: µ = 3, τ = 1 and λ = λ cr , b . Right: µ = 3, τ = 4 and λ = .Besides the two critical surfaces just described, looking at ρ ( θ ) for negative Y and τ we alsofind values at which it attains zero value at two distinct, symmetric points in the interior of( − π, π ), as in Figure 4. We expect a new phase transition into a two-cut solution. One-cut solution: Phase Ia. We now solve Equation (3.4) dropping the assumption that ρ ( θ )is supported on the whole T , and replace it by the assumption that the support is an arc Γ ⊂ T .The derivation is standard and we relegate it to Appendix A.2.Introduce the trace of the resolvent in the large N limit,(3.10) ω ( z ) = (cid:90) Γ d w π i w (cid:37) ( w ) z + wz − w , z ∈ C \ Γ . We adopt the standard notation ω ± ( e i θ ) ≡ lim (cid:15) → + ω ( z = (1 ± (cid:15) ) e i θ ) . Figure 4. ρ ( θ ) at the transition point. µ = 2, τ = − λ = − . Thesupport of ρ will break in two disjoint cuts beyond this critical value.Then ω + ( e i θ ) − ω − ( e i θ ) = 2 (cid:37) ( e i θ ) , e i θ ∈ Γ . We find (see Appendix A.2 for the details) ω Ia ( z ) = − i W ( z )+ (cid:113) ( e i θ − z ) ( e − i θ − z ) (cid:34) Y (cid:18) z (cid:19) − τ (cid:112) µ − µ cos θ (cid:18) µz − µ − z − µ − (cid:19)(cid:35) . The first term is regular and, taking the discontinuity at z = e i θ ∈ Γ, we arrive at(3.11) ρ Ia ( θ ) = 2 cos θ · (cid:112) θ − θ · (cid:34) Y − τ µ ( µ − (cid:112) µ − µ cos θ (1 + µ − µ cos θ ) (cid:35) The angle θ is fixed by normalization:(3.12) Y (1 − y ) + τ (cid:32) µ − (cid:112) µ − µy − (cid:33) = 1 . where y := cos θ . Equation (3.12) admits a unique real solution, thus the problem is completelydetermined. One-cut solution: Phase Ib. The solution above has been derived assuming that Γ is an arcalong T joining e − i θ to e i θ running counter-clockwise, thus the gap has opened around θ = π .For the gap opening at θ = 0, the procedure is identical, but now Γ is an arc from ˜ θ > π − ˜ θ . The procedure of Appendix A.2 leads us to ρ Ib ( θ ) = 2 (cid:12)(cid:12)(cid:12)(cid:12) sin θ (cid:12)(cid:12)(cid:12)(cid:12) (cid:113) θ − θ − Y + τ µ ( µ + 1) (cid:113) µ − µ cos ˜ θ ( µ + 1 − µ cos θ ) which is non-negative definite. There is, however, a more direct route to get the correct answer.Looking back at the matrix model (2.1) we can chose a different parametrization 0 ≤ θ < π ,and the solution with the gap opening at θ = 0 is recovered from the solution (3.11) in Phase Iaupon replacement Y (cid:55)→ − Y , µ (cid:55)→ − µ and eventually θ + π (cid:55)→ θ .In conclusion, we have two different phases with a one-cut solution, as expected: one for Y > Y cr , a ( τ, µ ), that we have called Phase Ia, and one for Y < Y cr , b ( τ, µ ), that we have calledPhase Ib. Two-cut solution: Phase II. We have seen that at τ = τ cr , + = ( µ − / Y = Y cr , a and Y = Y cr , b meet. Thus, we expect a new phase characterized by a two-cut solutionin the region (cid:26) ( µ, τ, Y ) : µ > , τ > µ − , Y cr , a < Y < Y cr , b (cid:27) ⊂ M . with gaps around θ = 0 and θ = ± π , and eigenvalue density supported onsupp ρ II = Γ ∼ = Γ u (cid:116) Γ d := (cid:110) e i ϕ ∈ T : ˜ θ ≤ θ ≤ θ (cid:111) (cid:116) (cid:110) e i ϕ ∈ T : − θ ≤ θ ≤ − ˜ θ (cid:111) . That is, Γ is the union of two disjoint arcs, Γ u and Γ d , as in Figure 5. Figure 5. The two-cut support Γ in Phase II.To determine ρ II ( θ ) it is simpler to adopt a different strategy, detailed in Section 3.4 below. Two-cut solution: Phase III. The fact that the potential V eff ( e i θ ) develops a double well fornegative Y and τ in a given range hints at the existence of a two-cut solution in that regionof M , with the eigenvalues sitting around the two minima. This observation is corroboratedlooking at the shape of ρ Ia ( θ ) and ρ Ib ( θ ) in the negative quadrant, where they become negativein Y cr , b < Y < Y cr , a for τ below a certain threshold.We find a transition from Phase 0 to a two-cut phase in Y cr , c+ < Y < Y cr , a and Y cr , b < Y < Y cr , c − where the critical surfaces Y = Y cr , c ± are given by Y cr , c ± ( τ, µ ) = µ ( µ + 1) (cid:104) µ ( τ − − τ − ± (cid:112) − τ [2 τ ( µ − 1) + ( µ − (cid:105) . The two curves Y cr , ± form an ellipse in each ( τ, Y )-leaf of M at fixed µ , with the physical criticalcurve being the first branch of the ellipse encountered when decreasing Y from 0.In this phase, that we call Phase III, the eigenvalues distribute along a contour Γ which consistsof two cuts, with gaps opening around ± θ ∗ , see Figure 6.The eigenvalue density is ρ III ( θ ) = 2 (cid:112) [cos ( θ ∗ − δθ ) − cos θ ] [cos ( θ ∗ + δθ ) − cos θ ] × (cid:34) − Y + τ µ ( µ + 1)( µ − (cid:112) ( µ + 1 − µ cos ( θ ∗ − δθ )) ( µ + 1 − µ cos ( θ ∗ + δθ )) [ µ + 1 − µ cos θ ] (cid:35) . (3.13)Note that the argument of the outer square root is non-negative definite. The value of θ ∗ isknown explicitly, as obtained from Phase 0, and the dependence of δθ on the parameters is fixed Figure 6. The two-cut support Γ in Phase III.by normalization. Equivalently, we can fix cos ( θ ∗ + δθ ) and cos ( θ ∗ − δθ ) comparing the large z behaviour of ω ( z ) computed in this phase with its definition.For multi-cut solutions, the dependence on the number of eigenvalues filling each cut should betaken into account when computing physical observables [46]. We analyze the role of the fillingfractions in Appendix B: the upshot is that our conclusions are unaltered, both in phase II andIII, although for different reasons.3.2. Phase diagram. Putting all the information together, the following phase diagram emerges.0) When both Y and τ are small, Phase 0 holds, with the eigenvalues spread on the wholecircle.Ia) When Y > Y cr , a the system is in a new phase, Phase Ia, with a one-cut solution gappedaround θ = ± π .Ib) Likewise when Y < Y cr , b the system is in Phase Ib, with a one-cut solution gapped around θ = 0.II) At τ > µ − the two critical surfaces cross each other. In the region Y cr , a < Y < Y cr , b thesystem is in Phase II, a two-cut solution with density of eigenvalues gapped both around θ = 0 and θ = π .III) The system develops a new two-cut phase, Phase III, in the region Y cr , b < Y < Y cr , a and also bounded by an arc of ellipse determined by Y cr , c ± . The density of eigenvalues isgapped around θ = ± θ ∗ , with θ ∗ → π as Y → Y cr , a and θ ∗ → Y → Y cr , b .See Figure 7 for a slice of M at fixed µ .Taking the massless limit µ → + , the critical surface Y cr , b is rotated onto the vertical axis.Using the analytic dependence on µ , we can also reach µ → − − by first going to the negativereal axis walking through C outside of the unit disk and then taking the limit | µ | → + . In thatcase, it is Y cr , a that is rotated onto the vertical axis.3.3. Free energy and massless theory. Before delving in the analysis of Wilson loop vevs inthe next section, we comment on the free energy of the model, defined as F = 1 N log Z . Ia Ib IIIII - - τ - - Y μ = Figure 7. Phase diagram of the model in the ( τ, Y ) plane, at µ = 3. The bluestraight lines are Y = Y cr , a and Y = Y cr , b , the black curve is Y = Y cr , c ± , thered dot is the multi-critical point at τ = µ − . The gray shaded region in theungapped phase, Phase 0. The other light shaded regions are the two-cut phases,Phase II and III.The free energy in Phase 0 is easily obtained, and corresponds to the analytic continuation ofSzeg˝o’s strong limit theorem in the bulk of the ’t Hooft parameter space [47]. It takes the value(3.14) F = Y − Y τµ − τ log (cid:18) − µ (cid:19) . It is clearly separated into three contributions: pure gauge ( Y ), matter only ( ∝ τ ) and theinteraction. At strong coupling λ → ∞ ( Y → 0) we are left with a matter contribution whichcounts gauge singlets: indeed, the integral over the gauge group projects onto gauge invariantstates. Massless theory. As we have stressed, a core assumption of our analysis is | µ | > 1, and themassless limit | µ | → + can only be taken at the end. Due to the non-analyticity for µ ∈ T , theresulting model will differ from a model with massless matter [22, 23].A main consequence of this non-analyticity is the spontaneous chiral symmetry breaking, thatwe will discuss in Section 4.4. On the other hand, it is well known that the large N limit and themassless limit do not commute.In the na¨ıve | µ | → | µ | = 1 from the beginning, and arg µ = ˜ θ , 0 < ˜ θ ≤ π , thepartition function acquires a FH singularity and the large N limit cannot be understood bystandard methods. We use known results on Toeplitz determinants to derive the free energy inPhase 0 in the massless theory [48]: F (cid:16) µ = e i˜ θ (cid:17) = Y − Y τ cos ˜ θ − τ log (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) θ (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) + τ log N. That is, the contribution from matter fields has an additional factor of log N and dominatesat large N . Remarkably, this matches the logarithmic divergence of the na¨ıve massless limit of(3.14). The result is in fact much more general [48] and directly extends to the case of various Veneziano parameters τ , . . . , τ n associated to different µ , . . . , µ n that approach the unit circlefrom outside at different angles ˜ θ , . . . , ˜ θ n .3.4. Stereographic projection. To better understand Phase II and the transition from a one-cut to a two-cut phase, we map the model onto the real line and study the resulting Hermitianmatrix model at large N . It can be interpreted as a massive deformation of the model in [49].We conformally map the unit circle on the real line through the stereographic projection, seeFigure 8. The drawback of the stereographic map is that it introduces a puncture on the circleat θ = ± π : this has no effect at finite N , but the Hermitian matrix model will fail to reproducePhase 0 of the unitary matrix model because of this change in topology [45]. Phase 0 and itsassociated transitions are well understood from the unitary matrix model side, and we use theconformally mapped model as yet another way to gain further insight into the one-cut to two-cuttransition. Figure 8. The stereographic projection. The red cross is the puncture on thecircle, the ticker line is the cut Γ.Our choice of coordinates is consistent with Phase Ia on the circle, but Phase Ib is easilyretrieved rotating T by e i π , so that the puncture is placed at θ = 0. The Hermitian matrix modelis(3.15) Z p .U ( N ) = ( µ − K (cid:90) R N (cid:89) ≤ j The saddle point equation for the Hermitian matrix model (3.15) isP (cid:90) d y π ρ p . ( y ) x − y = x (cid:20) Y ( x + 1) + τ + 11 + x − η τ η x (cid:21) . The solution is found by standard large N techniques [50]. Using a one-cut ansatz for the density ρ p . ( x ) supported on [ − A, A ] ⊂ R we find(3.16) ρ p . I ( x ) = 2 (cid:112) A − x (cid:34) τ + 1 √ A (1 + x ) − τ η (cid:112) η A (1 + η x ) − Y A (cid:0) x − (cid:1) − A ) (1 + x ) (cid:35) . The value of A is fixed by normalization: (cid:90) A − A d x π ρ p . I ( x ) = 1 = ⇒ τ + 1 √ A − τ (cid:112) η A + 2 Y A (1 + A ) = 0 . As a cross-check, turning off the mass deformation, µ → 1, sends η → ∞ and we recover theeigenvalue density found in [49]. Besides, sending A → ∞ and expanding at leading order in A the normalization becomes the consistency condition Y = 12 + τµ + 1 , correctly reproducing the critical surface Y cr , a in the limit in which A is back-projected to e i π . Westress that, requiring that ρ p . ( x )d x descends from a measure on T , the non-negativity constraint ρ ( x ) ≥ ρ p . I (0), we find that the critical point is fixed by the condition2 Y (cid:18) A A (cid:19) + τ + 1 − τ η (cid:115) A η A = 0 . Phase II. From the result above as well as from the analysis of the unitary matrix model, wefind a phase transition to a two-cut solution, with a gap opening at x = 0. The new phase is theconformal image of Phase II of the unitary matrix model.We look for a new eigenvalue density, supported on [ − A, − B ] ∪ [ B, A ]. The result is ρ p . II ( x ) = 2 (cid:112) ( A − x )( x − B ) | x | (cid:34) − Y x ( A + B + 2) + 3( A + B ) + 2 A B + 4[(1 + A )(1 + B )] (1 + x ) − τ + 1 (cid:112) (1 + A )(1 + B )(1 + x ) + τ η (cid:112) (1 + η A )(1 + η B )(1 + η x ) (cid:35) . (3.17)The parameters A and B are fixed by normalization, − Y ( A − B ) [(1 + A )(1 + B )] − ( τ +1) (cid:32) A + B + 2 (cid:112) (1 + A )(1 + B ) − (cid:33) + τ (cid:32) η A + η B + 2 (cid:112) (1 + η A )(1 + η B ) − (cid:33) = 1 , and by an additional self-consistency condition on ω ( z ),2 Y A + B + 2(1 + A )(1 + B ) + τ + 1 − τ η (cid:115) (1 + A )(1 + B )(1 + η A )(1 + η B ) = 0 , which reproduces the criticality condition for B → Wilson loops and instantons We continue the investigation of the features of the phase transitions and establish their orderby evaluating the vacuum expectation value (vev) of the Wilson loop in the fundamental repre-sentation. Moreover, we further discuss the different physics of the various transitions by lookingat the different contributions by instantons. Wilson loops. Wilson loops are order operators in gauge theories that, for simple connectedgauge group, describe the holonomy of the gauge connection around a closed path. For our one-plaquette model, we consider the Wilson loop in the fundamental representation wrapping theplaquette, and compute its vev. It is given by (cid:104)W(cid:105) = (cid:28) N Tr U + 12 N Tr U † (cid:29) = (cid:42) N N (cid:88) j =1 cos θ j (cid:43) with the average taken in the unitary ensemble (2.1). We use the eigenvalue density at large N found in each phase to evaluate the Wilson loop. Wilson loops: Generalities. From the matrix model (2.1) we immediately get the relation (cid:104)W(cid:105) = 12 N Z ∂∂ ( N Y ) Z = 12 ∂ F ∂Y . Therefore, all the information about the order of the transition can be extracted from the Wilsonloop vev. This is precisely what we expect from an order parameter, and follows from the Wilsonloop belonging to the class of order operators of QCD .Being ρ ( θ ) continuous on the whole M , the Wilson loop vevs are continuous as well, implyingthat every phase transition we find must be at least second order. Wilson loops: Evaluation. We focus now on the Wilson loop vev at large N . In the ungappedphase we find(4.1) (cid:104)W(cid:105) = (cid:90) π − π d θ π ρ ( θ ) e i θ = Y − τµ . This reproduces the GWW result as τ → 0, but also as µ → ∞ , as expected when the matterbecomes non-dynamical. For a Wilson loop winding k > (cid:104)W k (cid:105) = − τµ k . In Phase Ia the Wilson loop vev is (cid:104)W(cid:105) Ia = (cid:90) π − π d θ π ρ Ia ( θ ) e i θ = 2 π (cid:90) y d y y (cid:114) y − y − y (cid:34) Y − τ µ ( µ − (cid:112) µ − µy (1 + µ − µy ) (cid:35) = Y (1 − y )(3 + y )4 − τ µ (cid:34) µ + 1 + 1 + µ ( µ − y − µ (cid:112) µ − µy (cid:35) (4.2)where we have used the change of variables y = cos θ , with y = cos θ . The value of y as afunction of the gauge theory parameters is known from (3.12).The Wilson loop vev in Phase Ib is obtained likewise,(4.3) (cid:104)W(cid:105) Ib = Y (1 + ˜ y )(3 − ˜ y )4 − τ µ (cid:34) µ + 1 + µ ( µ + 1)˜ y − µ − (cid:112) µ − µ ˜ y (cid:35) , where ˜ y = cos ˜ θ .To study the derivative of (cid:104)W(cid:105) Ia and establish the order of the phase transition, it suffices tonotice that dd Y (cid:104)W(cid:105) Ia = 1 + (cid:34) Y − y − 2) + τ · ( µ − µ ( y + 1)(1 + µ − µy ) / (cid:35) ∂y ∂Y . This implies lim y →− dd Y (cid:104)W(cid:105) Ia = 1 which matches the derivative of (cid:104)W(cid:105) . The computations are identical for the transition betweenPhase 0 and Phase Ib. Taking a further derivative, d d Y (cid:104)W(cid:105) vanishes identically in Phase 0, butdoes not vanish at the critical loci when computed in Phases Ia and Ib.We conclude that the Wilson loop vev is an order parameter of class C at the critical surfaces Y = Y cr , a ( τ, µ ) and Y = Y cr , b ( τ, µ ), thus the system shows a pair of third order phase transitions.In particular, both the GWW transition [4, 5] and the transition in [27] are special points on thecritical locus of the present model.Crossing from a one-cut to a two-cut phase, the first derivative of the Wilson loop is notprotected. Indeed, in Phase II the derivative of the Wilson loop vev has the schematic form(4.4)dd Y (cid:104)W(cid:105) II = (cid:90) ˜ y y y ∂∂Y f ( y, y , ˜ y )d y + ∂y ∂Y (cid:90) ˜ y y y ∂∂y f ( y, y , ˜ y )d y + ∂ ˜ y ∂Y (cid:90) ˜ y y y ∂∂ ˜ y f ( y, y , ˜ y )d y, with the first term coming from the derivative of the explicit dependence on Y , and the other twofrom the dependence on Y through y and ˜ y . The integrand evaluated at the endpoint vanishes,hence those contributions do not appear.In (4.4), f ( y, y , ˜ y ) is known explicitly from Section 3.1, f ( y, y , ˜ y ) = 2 π (cid:115) ( y − y )(˜ y − y )(1 + y )(1 − y ) (cid:34) − Y + τ µ ( µ + 1)( µ − (cid:112) (1 + µ − µy )(1 + µ − µ ˜ y )(1 + µ − µy ) (cid:35) , but the difficulty comes from the only implicit knowledge of the dependence of y, ˜ y on Y .Passing from Phase II to Phase Ia, the first term in (4.4) matches continuously with thecorresponding expression in Phase Ia, as ˜ y → 1. The integral in the second summand in (4.4)also agrees with the corresponding contribution in Phase Ia at ˜ y → 1. Both facts follow fromlim ˜ y → y | II = y | Ia . Moreover, the symmetries of the integrand allow to combine the third term in (4.4) with thesecond term, in a simpler expression. Moreover, the symmetric form of the equations fixing y , ˜ y can be used to show that ∂ ˜ y ∂Y = ∂y ∂Y (cid:12)(cid:12)(cid:12)(cid:12) ˜ y ↔ y . By this we mean that the expressions on the two sides agree upon exchanging all ˜ y with y .Due to the complicated dependence on the parameters, the derivatives of the boundaries y , ˜ y are not continuous at the transition point. The differentiability of (cid:104)W(cid:105) above followed by thevanishing of the term multiplying such derivatives. This does not happen for the transition froma two-cut to a one-cut phase. Therefore, the sum of the second and third terms in (4.4) gives anobstruction to the differentiability of (cid:104)W(cid:105) , so we expect a second order transition. In a sense, theobstruction arises from taking a limit that breaks explicitly the y ↔ ˜ y symmetry of Phase II.The proof is very similar for the transition from Phase II to Phase Ib or from Phase III toeither Phase Ia or Ib.The argument fails at the critical surface Y cr , c and at the multi-critical point at which Y cr , a = Y cr , b . Indeed, when passing directly from Phase 0 to a two-cut phase, the simplifications that arisefrom closing both gaps simultaneously imply that the Wilson loop vev is C . This is consistentwith the observation of the previous paragraph, as these transitions preserve the Z -symmetry ofthe two-cut phase.4.2. Phase structure and remarks. Summing up the results extracted from the analysis ofWilson loop vev, we find that • the transition from Phase 0 to any other phase is third order, but • the transition from a one-cut to a two-cut phase is second order. In the rest of this subsection we gather comments on various aspects of the phase structure weuncovered, insisting on the role of the second order phase transitions. Remark . As obtained in the previous subsection, the second order discontinuities are finitejumps, not divergences. The correlation lengths remain finite at each transition. These finitediscontinuities vanish in the limit | µ | → Metastability. While the third order transitions we find are a continuation of the GWW tran-sition in M , it is worth to further comment on the second order transitions we obtain. Thephase transition to a two-cut solution happens slightly beyond the values of τ where the potentialdevelops a double-well structure. The proposal in [51] states that a second order transition canbe associated with tunneling from a metastable vacuum to a stable one. Our analysis confirmsthat picture in the one-plaquette model we consider. In Section 4.5 we study instanton effects,expanding this discussion leading to a further refined distinction between second and third orderphase transitions in this model, from the instantonic point of view. Critical behaviour. It is worthwhile to notice that the phase diagram in Figure 7 resemblesthat in [52], where a unitary matrix model with potential Y cos( θ ) + Y cos(2 θ ) was analyzed.The critical behaviour close to a transition to a two-cut phase in our model differs fromthat found in similar models in the literature, for matrix models with potentials of the form (cid:80) Kn =1 Y n cos( nθ ). This is so because the potential in (2.1) includes both a polynomial and a log-arithmic part, requiring different scaling approaching the critical regime from the two-cut phase.This distinction, however, fades away approaching the multicritical point. Double-scaling limit. The statements above can be refined exploiting the double-scaling limit.In particular, we can zoom in the critical regime, tuning Y towards a transition to the ungappedphase. In the double-scaling limit, the dynamics is governed by Painlev´e II equation. The prooffollows from [53, 27] with minimal variations. An alternative proof can be given using orthogonalpolynomials [54]. We have checked explicitly that, in the double-scaling limit, the problem reducesto the analogous one for the pure GWW model.For the transition from a one-cut to a two-cut phase, however, there is no double scaling thatgives Painlev´e II. Other gauge groups. Throughout the work, we focus on gauge theories with gauge group U ( N )or SU ( N ). Nevertheless, by direct computation or by universality arguments, it can be shownthat the phase diagram of Figure 7 and the associated phase transitions carry over to theorieswith gauge group G , that is to say, matrix models like (2.1) with the integration over U ( N )replaced by integration over G , for any G ∈ (cid:8) SO (2 N ) , SO (2 N + 1) , Sp ( N ) , O − (2 N ) , O − (2 N + 1) (cid:9) . Here, O − ( n ) ⊂ O ( n ) consists of n × n orthogonal matrices with determinant − 1, and Sp ( N ) isthe compact symplectic group.4.3. Continuum limit and β -function. The β -function of the theory, as a function of the ’tHooft coupling λ , can be computed using the chain rule through [4](4.5) β ( λ ) = 2 λ (cid:104)W(cid:105) log (cid:104)W(cid:105) ∂∂Y (cid:104)W(cid:105) . This quantity can be used to test whether our model reproduces the expected features of QCD in the continuum limit. The fixed points of the RG flow, that capture the continuum physics, aregiven by β ( λ ) = 0, which, from (4.5), can only happen at (cid:104)W(cid:105) = 0 or at (cid:104)W(cid:105) = 1.Direct computations in Phases 0, Ia, Ib, show that only the solution to (cid:104)W(cid:105) = 0 is consistent,while the solution to (cid:104)W(cid:105) = 1 always falls out of the phase in which it has been computed,and thus should be discarded. It is a nice consistency check that the solution to be discarded is precisely the one that would violate Elitzur’s theorem [55], and the one to be retained is inagreement with the confining nature of QCD [4].The continuum limit of a lattice theory consists in sending the lattice spacing to zero whileapproaching a critical curve [56]. In particular, this requires | µ | → Y cr , a or Y cr , b , we find that theunique consistent solution is Y = 0 (i.e. λ = ∞ ). This is physically meaningful for a toy modelof QCD : the theory flows to a strongly interacting theory in the deep infrared.Taking the continuum limit from Phase Ia close to the transition to Phase II, we find a trivialsolution with λ = 0 = τ , describing a theory of free gauge bosons without matter. The continuumlimit approaching the critical surface between Phase Ib and Phase II, instead, yields a non-trivialfixed point at λ = 1 Y ≈ . . Remark . The existence of a continuum theory is not established by our analysis, becausecorrelation lengths remain finite. While this has no effect in our model, which consists of a singleplaquette, it may (and most likely shall) wash away the fixed point in the continuum limit of amore realistic lattice model.4.4. Chiral symmetry breaking. Let us focus now on the model with fermionic matter. Thefermion two-point function is by definition (cid:104) ¯ ψ f ψ f (cid:105) = − N Z ∂∂µ f Z = − ∂∂µ f N log Z . Due to our degenerate choice of masses, we can only compute the average over flavours of suchquantity: (cid:42) K K (cid:88) f =1 ¯ ψ f ψ f (cid:43) = 1 τ ∂ F ∂µ . In Phase 0 we find 1 τ ∂ F ∂µ = 2 µ (cid:18) Y + τ µµ − (cid:19) . This quantity diverges as µ → 1, therefore we expect the chiral symmetry to be spontaneouslybroken in the continuum, consistently with the analysis of the β -function in Phase 0.In Phases Ia and Ib, we can move along Y = 0 and study the behaviour of (cid:104) K (cid:80) f ¯ ψ f ψ f (cid:105) onthat subspace of M . The result is read off directly from [45]:1 τ ∂ F Ia ∂µ (cid:12)(cid:12)(cid:12)(cid:12) Y =0 = − µ + 1 + 4 τ τ µ ( µ − τ ∂ F Ib ∂µ (cid:12)(cid:12)(cid:12)(cid:12) Y =0 = − − µ + 4 τ τ µ ( µ + 1) , again non-vanishing and continuous at the transition point, and goes to 1 in the µ → + limit.This latter result, in turn, hints at a transition to a free theory: the free energy of a theory of K free flavours of mass m goes as F free ∝ Km , whence (cid:104) K (cid:80) f ¯ ψ f ψ f (cid:105)| free = 1. Note that thiscomputation has been done at infinite gauge ’t Hooft coupling, which has the physical meaningof governing the theory in the deep infrared.To sum up, we have observed that the phase transition from Phase Ib to Phase 0 is accompaniedwith spontaneous chiral symmetry breaking. Instantons. We discuss non-perturbative effects in the unitary matrix model, coming fromunstable saddle point configurations [57].An instanton configuration is characterized by a collection of integers { N , N , . . . } with (cid:80) k N k = N . For example, the d -instanton configuration is associated with the symmetry break-ing pattern U ( N ) → U ( N ) × U ( N ) × · · · × U ( N d ) , with the eigenvalues z j ∈ T of U ∈ U ( N ) grouped in d different sets, sitting at d different extremaof the potential. For the one-instanton configuration, Z U ( N ) ( ν ) = N (cid:88) (cid:96) =0 e − ν(cid:96) Z (cid:96) with Z (cid:96) the partition function of a U ( N − (cid:96) ) × U ( (cid:96) ) model, and we have turned on a chemicalpotential ν > (cid:96) -sector leads to non-perturbative corrections to the free energy of the matrix model, ofthe form e − N(cid:96)S inst ( (cid:126)λ ) f (cid:96) ( (cid:126)λ )where (cid:126)λ generically denotes the couplings of the theory, and the functions { f (cid:96) } (cid:96) admit themselvesa N expansion. Instanton effects and third order transitions. Let us consider our model (2.1) at large N andfocus on the one-cut phase, in which the interpretation of instanton effects is more transparent.We discuss them in Phase Ia, being the corresponding analysis in Phase Ib completely analogous.Most of the details are just an extension of the thorough analysis in [57].The contribution of an instanton excitation, obtained moving one eigenvalue from the minimumof V eff to the maximum at θ = ± π is found to be πS inst = 2 Y (cid:20)(cid:113) − x − x cosh − (cid:18) x (cid:19)(cid:21) + τ (cid:34) tanh − (cid:32) ( µ − (cid:115) − x ( µ − + 4 µx (cid:33) + µ − (cid:112) ( µ − + 4 µx log (cid:32) x (cid:112) − x (cid:33)(cid:35) where cosh − and tanh − are the inverse of the hyperbolic functions, and x = sin θ . One of theresults in [57] (already conjectured in [58]) is that the GWW transition is triggered by instantons.We see that the result carries over to the present model, aslim x → S inst = 0and the instanton excitations cease to be suppressed at the critical point when the gap closes.Analogous conclusions are found if we go to Phase III, in which the effective potential hasdeveloped a double well, and consider the instanton configuration with a few eigenvalues takento the local maximum at θ ∗ . Approximating close to the transition to Phase 0, we find πS inst = ( δθ ) sin θ ∗ (cid:20) − Y + τ µ ( µ + 1)( µ − 1) (1 + µ − µ cos θ ∗ ) (cid:21) + O (cid:16) ( δθ ) (cid:17) . At the critical surface, δθ → Figure 9. Instantons in Phase III. A single eigenvalue (blue dot) is moved ontop of the maximum of V eff , while all the others (gray sea) fill the minima. Instanton effects and second order transitions. We now turn our attention to the analysisof instanton effects in the two-cut phase, starting from Phase III. We consider a single eigenvalueplaced on the maximum of the potential, as sketched in Figure 9.We find that the instanton action is the sum of two pieces, S inst ,L = (cid:90) y ∗ y L d yπ (cid:115) ( y − y L )( y R − y )1 − y (cid:34) − Y + τ µ ( µ − (cid:112) (1 + µ − µy L )(1 + µ − µy R )[1 + µ − µy ] (cid:35) ,S inst ,R = (cid:90) y R y ∗ d yπ (cid:115) ( y − y L )( y R − y )1 − y (cid:34) − Y + τ µ ( µ − (cid:112) (1 + µ − µy L )(1 + µ − µy R )[1 + µ − µy ] (cid:35) , where y ∗ = cos θ ∗ and y L,R = cos( θ ∗ ± δθ ). The two are associated with the eigenvalue escapingfrom the left and right cut, respectively. There exists a third relevant quantity, namely thetunneling from one cut to the other, S tunnel = (cid:90) y R y L d yπ (cid:115) ( y − y L )( y R − y )1 − y (cid:34) − Y + τ µ ( µ − (cid:112) (1 + µ − µy L )(1 + µ − µy R )[1 + µ − µy ] (cid:35) . All the three effects are non-perturbatively suppressed by a factor e − NS inst , with S inst the cor-responding action. The three contributions are still suppressed at the critical loci, although thetunneling term will coalesce with one of the other two.The situation is slightly different in Phase II, where the two wells have equal depth, see Figure10. In this case, S tunnel is simply twice S inst , and both go to zero as a gap closes. The phasetransition takes place when the tunneling between the two cuts ceases to be suppressed in onedirection (e.g. passing through θ = 0 in Figure 10) but remains non-perturbative in the otherdirection (e.g. passing through θ = π in Figure 10).The picture we infer is that the third order phase transitions are associated with releasingnon-perturbative instabilities, while the second order transitions correspond to release only thoseinstabilities in one direction. This is also in agreement with the proposal in [51, 18] relatingsecond order phase transitions in GWW-type models to partial deconfinement.5. Meromorphic deformation of unitary matrix models We now depart from the model (2.1) with the aim of setting the stage for the study of mero-morphic deformations of unitary matrix models, in which the integration contour is deformed in C ∗ and not bound to be the unit circle. This consists of an adaptation of the theory of holo-morphic matrix models [19] to unitary matrix models and, as we shall show, is instrumental in Figure 10. Instantons in Phase II. A single eigenvalue (blue dot) is moved ontop of the local maximum of V eff at θ = 0, while all the others (gray sea) fill theminima.understanding their phase diagram from new angles. This section can be read independently ofthe rest of the paper.A unitary matrix model is characterized by a weight function e − Nλ V ( z ) , with V ( z ) admittingthe expansion(5.1) V ( z ) = (cid:88) n ≥ (cid:18) t n n z n + t − n n z − n (cid:19) . The function e − Nλ V ( z ) is singular at z ∈ { , ∞} ⊂ P and possibly has other zeros and polesin C ∗ ∼ = P \ { , ∞} . The Vandermonde determinant appearing in a unitary matrix model isconveniently rewritten in meromorphic form (cid:89) ≤ j Let N ∈ N , λ ∈ C ∗ and V ( z ) as in (5.1). A meromorphic matrix model Z is theintegral(5.2) Z = (cid:73) C N (cid:89) ≤ j A critical locus C is an irreducible component of the locus in M at which two rootsof P ( z ) coalesce.The critical loci C ⊂ M necessarily have positive complex codimension, and the hyperel-liptic fibration is singular along them. Singularities in higher codimension, placed at the (self-)intersection of critical loci, correspond to multicritical points of the matrix model.The theory of Abelian differentials provides a suitable framework to analyze the genus g hy-perelliptic curve (5.13) [60]. At this stage, the analysis of the spectral curve works exactly as inthe holomorphic deformation of Hermitian matrix models, thus we omit the details and refer to[60, 61]. Remark . We are now in the position to elaborate more on Remark 5.1, from a point of viewadvocated in [62]. Let (cid:8) A (cid:96) , B (cid:96) (cid:9) be a basis of one-cycles in the hyperelliptic complex curve. The A -cycles are chosen to go around the cuts Γ (cid:96) . Therefore (cid:73) A (cid:96) y ( z )d z = (cid:73) A (cid:96) ω ( z )d z = N (cid:96) N =: ξ (cid:96) . The first equality follows from the definition (5.7) noting that y ( z ) and ω ( z ) only differ by aregular term. Introducing chemical potentials for the filling fractions ξ (cid:96) and extremizing theaction with respect to these quantities gives their saddle point value as a function on M . Moreprecisely, one gets a set of equations analytic in the ratios s (cid:96) := ξ (cid:96) λ [62]. At this point, it ispossible to invert the relations and express the moments { ρ k } in terms of the complex variables s (cid:96) , keeping the latter as free parameters.Note that only g out of the g + 1 of both quantities are free.The study of the phases of the model (5.2) leads to a stratification of the parameter space M .We postpone the analysis to Section 5.4, discussing explicit models first. Genus 0. The unique way to obtain a genus 0 spectral curve is from the holomorphic deformationof the CUE. In that case, the model has no couplings and, as opposed to g ≥ 1, the additionalcondition derived from the definition of ω ( z ) is automatically fulfilled, leaving ρ − as unique,unconstrained parameter. Then, (5.13) describes a P fibered over C . If we try to get a lesstrivial model by considering the insertion of (det U ) τN , the consistency condition, which in g ≥ { ρ k } , imposes τ = 0.5.2. Holomorphic GWW. We now put the machinery at work and revisit the phase structureof the holomorphic GWW model. The phase diagram of this model has been obtained in [63]for λ ∈ R , while the behaviour at complex coupling has been partially analyzed in [64], althoughwithout fully exploiting the holomorphic deformation.The GWW model has t − = t = 1, and t n (cid:54) = ± = 0, whence d + = 0, d − = 2, g = 1 and onlythe moments ρ − , ρ − appear in (5.12). Fixing ρ − as a function of λ and ρ − , the spectral curveof the holomorphic GWW model is [63](5.14) ˆ y = z + 2 λz + (cid:2) ( ρ − + 1) λ − (cid:3) z + 2 λz + 1 . It is an elliptic curve. Following the strategy outlined above, we think of (5.14) as an ellipticfibration over C ∗ × C , with coordinates on the base λ and ρ − , and identify the phase transitionswith singularities of the fibration. The discriminant of (5.14) is(5.15) ∆ = λ (cid:0) λ ρ − − (cid:1) ( λ ( ρ − + 1) − 4) ( λ ( ρ − + 1) + 4) , from which the critical loci are { ∆ = 0 } = C ∪ C (cid:48) ∪ C , C := (cid:26) ρ − = − λ , λ ∈ C ∗ (cid:27) , C (cid:48) := (cid:26) ρ − = − − λ , λ ∈ C ∗ (cid:27) , C := (cid:26) ρ − = 4 λ , λ ∈ C ∗ (cid:27) . In Kodaira’s classification [65], C and C (cid:48) are singularities of type I and C is of type I . TheGWW critical points ( λ, ρ − ) = ( ± , 1) are singled out as the codimension-two singularities atwhich C intersects one of the other two critical curves. Besides, we recognize the elliptic curve(5.14) as the Seiberg–Witten curve of N = 2 supersymmetric four-dimensional SU (2) gaugetheory with two flavours [66].The original GWW transition is thus, from the perspective of the holomorphic deformation, oneof the possible ways to approach the codimension-two singularity from a generic direction. Thesingularity at the multicritical points ( λ, ρ − ) = ( ± , 1) is of Kodaira type III. The correspondingsymmetry is A , which is precisely the symmetry of Painlev´e II, that is known to control theGWW phase transition [54, 53]. If, instead, we approach the multicritical point not from ageneric direction but moving along a critical locus, the singularity type is enhanced to I ∗ .It is possible to allow t − (cid:54) = t . This corresponds to introduce a θ -term in the GWW latticeaction, θ π = t − t − . The procedure goes through with only minor modifications, the uniquedifference being that the singularities C , C (cid:48) are placed at ρ − = − t − ± λ √ t − , so in particularthe Z -symmetry is preserved.5.3. Meromorphic deformations. The formulation can be extended to include weight func-tions with zeros and poles in C ∗ . For concreteness, we consider the illustrative example of ouroriginal model (2.1) at λ − = 0. In this case W (cid:48) ( z ) = − τ (cid:20) z − µ + 1 z − µ − (cid:21) + 1 + τz ,f ( z ) = τ (cid:20) ˜ ρ − ( µ ) z − µ + ˜ ρ − ( µ − ) z − µ − (cid:21) − (1 + τ ) z ρ − . In the second line, we have defined ˜ ρ − ( µ ) := (cid:90) Γ ρ ( w ) w − µ d w, with, in particular, ˜ ρ − (0) = ρ − . Comparing the definition of y ( z ) with the spectral curve atlarge | z | , we find a pair of consistency conditions, fixing ˜ ρ − as a function of the other parameters,˜ ρ − ( µ ) = − ρ − ( τ + 1) + µτ ( τ + 6) + µ µ − τ . Note that the two conditions fix ˜ ρ − ( µ ± ) independently, and the solutions are consistentlymapped into each other under µ ↔ µ − . We get(5.16) y = P ( z )4 z ( z − µ ) ( z − µ − ) , with P ( z ) a polynomial of degree 4. The spectral curve thus describes again en elliptic fibrationover the moduli space C ∗ × (cid:8) µ ∈ P : 1 < | µ | < ∞ (cid:9) × C , parametrized by ( τ, µ, ρ − ). The dis-criminant takes the form ∆ = µ ( µ + 1) ( µ − ( τ + 1) (cid:101) ∆, the last term being a cumbersomepolynomial of degree 6 in ρ − , degree 8 in τ and degree 10 in µ . The critical points of the unde-formed model become higher-codimensional singularities, at which two roots of P ( z ) collide. Thecollection of all critical loci in this model is C ∪ C ∪ C (cid:48) (cid:91) j =1 C j , with the subscript indicating the order of vanishing of ∆ along the component C . Taking whatwe have called the continuum limit in Section 4.3, that is, sending τ → τ cr ( µ ) and then µ → ± ρ − = − τµ set to its undeformed value, yields a non-minimal singularity ∆ ∝ ( µ ± .5.4. Stratification of the moduli space. The critical loci and their intersections endow theparameter space M with additional structure.The stratification of an algebraic variety V is a collection of open sets { V I } , with V a pointand V max = V , with a partial order given by the inclusion of the closures of { V I } . The parameterspace M of the model (5.2) is the union of M reg , C reg I , C reg IJ , . . . where the superscript means the regular part, and C IJ = C I ∩ C J , and so on. The inclusionrelations C reg IJ = C I ∩ C J ⊂ C I are obvious.The partial order can be represented with the aid of a Hasse diagram: M reg C reg I C reg J · · · C reg K C reg IJ C reg IK · · · ... ... ...In general, this does not define a full-fledged stratification of M because multiple final pointsmay exist. Nonetheless, whenever the potential (5.1) has a Z -symmetry, the Hasse diagraminherits it. This Z -symmetry acts as an automorphism of the Hasse diagram, which is mappedinto itself under reflection along the vertical axis. By construction, the Hasse diagram resultingfrom folding the initial diagram via this Z -symmetry determines a stratification of M / Z .We draw the Hasse diagram of the holomorphic GWW model of Section 5.2:(5.17) M reg C reg1 C reg2 C reg1 (cid:48) λ = 2 λ = − The diagram of the meromorphic model of Section 5.3 is schematically M reg C reg2 C reg4 C reg2 (cid:48) C reg1 C reg1 C reg1 C reg1 C reg1 C reg1 µ = 1 µ = − Z reflection symmetry. Folding thediagram along that line yields the stratification of M / Z . Remark . The results of Section 5.2 with t − (cid:54) = t show that, even for models in which a Z reflection symmetry is not manifest from the potential but emerges at large N , the Z -foldingyields a stratified moduli space. Symplectic singularities. Recall that H (Σ g , C ), the first homology group of a hyperellipticcurve Σ g of genus g , is a symplectic space. The A - and B -cycles that we have implicitly usedin the study of the spectral curve (5.8) can be chosen to be Darboux coordinates in H (Σ g , C ).Moving along M reg corresponds to vary the symplectic structure without changing the topologyof Σ g . At the critical loci C I , however, either • a B -cycle collapses, or • an A -cycle collapses.Both situations correspond to a singularity of the symplectic form. Therefore, the analysis ofthe phase structure of the meromorphic matrix models can be rephrased in terms of symplecticsingularities in the sense of Kaledin [67].The appearance of symplectic singularities is not entirely unexpected. The consideration ofholomorphic matrix models in their large N limit lead to the Seiberg–Witten curves [66] of certain N = 2 gauge theories [59]. The so-called Coulomb branch of these theories is a symplecticsingularity and is stratified [68]. In fact, the use of Hasse diagrams in the present work wasinspired by [69, 70].As a final remark, we emphasize that the structure uncovered in this section is not specific ofthe meromorphic matrix models. The parameter spaces of unitary or Hermitian matrix modelsinherit it, as they can be realized as slices inside the parameter space of our meromorphic models.As an example, the phase diagram of the GWW model is(5.18) •• • M reg | GWW = (cid:8) λ ∈ R \ { } , λ (cid:54) = 4 (cid:9) C | GWW = { λ = 2 } C (cid:48) | GWW = { λ = − } It is found by fixing ρ − = 1, taking the slice λ ∈ R \ { } and identifying the intersection of suchsubspace with the strata in (5.17). Accidentally, this is precisely the Hasse diagram of the reduction to three dimensions of the SU (2) theory withtwo flavours, captured by the holomorphic GWW of Section 5.2, cf. [70, Eq.(4.2)]. It should be stressed, however,that the strata in (5.18) are real, not hyperK¨ahler. Outlook We conclude by commenting, in a qualitatively and non-exhaustive fashion, on avenues thatwe have considered at some point but not pursued here.It would be interesting to know if the results obtained have a meaning from the point of viewof integrable systems such as the Schur flow. While both the matrix models considered and thestudy of such flows have in common an associated system of orthogonal polynomials, the waythis association actually works is quite different. For example, the recurrence coefficients of thepolynomials, central in the integrable systems description, are not directly relevant in the typeof matrix model analysis presented.On the other hand, for what concerns matrix models on the real line, the spectral propertiesof Jacobi matrices can be more directly related to matrix models, since it is known that, underrather general conditions, a suitably normalized counting measure of the zeroes of the orthogonalpolynomials converges weakly, in the large N limit, to the density of states of the matrix model[71]. With this in mind, a question would be whether our results have any implication in thestudy of the spectral properties of CMV matrices [40], for example. The large N planar limitstaken make this possibility not obvious.Another reason to further study any eventual implications of the planar limit and the ensuingphase structure, from the point of view of integrable systems, would be the connection obtained,presented in Section 5.4, between the phase diagram of meromorphic matrix models and thesymplectic foliation of singular varieties [67]. Again, the ’t Hooft scaling of the couplings involvedat large N obscures the relation between the symplectic structures we naturally find and thosein the integrability literature [72].Also, from a mathematical point of view, it would be interesting if proofs of the order ofthe phase transition, in particular for the second order phase transitions, can be obtained inalternative or more rigorous ways.Regarding more physical considerations, when discussing the model with fermionic matter andits interpretation in terms of chiral symmetry breaking, it is worth mentioning that recently [73],the chiral symmetry breaking phase transition in four-dimensional QCD has been studied fromthe point of view of thermodynamic geometry [74, 75]. The argument is based on the observationthat the grand canonical partition function (cid:98) Z ( v ) = ∞ (cid:88) N =0 e − Nv Z U ( N ) , v > g thermo on a two-dimensional parameter space with coordinates ( F , v ) [74],where F is the free energy and v the grand canonical chemical potential. Then, a second orderphase transition is triggered by the instability at det g thermo = 0.Any eventual use of this observation or other ideas from information geometry to furtherunderstanding phases in matrix models would be of interest. Acknowledgements. We thank Jorge Russo for a careful reading and valuable commentaries.The work of LS is supported by the Funda¸c˜ao para a Ciˆencia e a Tecnologia (FCT) through thedoctoral grant SFRH/BD/129405/2017. The work is also supported by FCT Project PTDC/MAT-PUR/30234/2017. Appendix A. Technical details of the solution A.1. Exact expressions at finite N via Toeplitz determinants. Table 1 collects the ex-plicit expressions for Z (cid:15) =+1 ,KU ( N ) for the first few values of N and K , computed using the Toeplitzdeterminant formulation detailed in Section 2.2. N K Z +1 ,KU ( N ) µ +1 ) I (2 Y ) − µI (2 Y ) µ +6 µ +1 ) I (2 Y ) − µY ( Y ( µ +1 ) + µ ) I (2 Y ) Y [ Y ( Y ( µ +15 µ +15 µ +1 ) +4 µ ) I (2 Y ) − µ ( Y ( µ +10 µ +3 ) +3 Y ( µ + µ ) +2 µ ) I (2 Y ) ]2 1 ( µ + µ +1 ) I (4 Y ) − µI (4 Y ) (( µ +1 ) I (4 Y )+ µI (4 Y ) ) − ( µ − ) I (4 Y ) − µ I (4 Y ) +4 µ ( µ +1 ) I (4 Y ) I (4 Y ) Y )4 [ Y ( Y ( µ +8 µ +30 µ +8 µ +1 ) − Y ( µ + µ ) − µ ) I (4 Y ) − Y µ ( Y ( µ +1 ) + µ ) I (4 Y ) (( Y ( µ +6 µ +1 ) − µ ) I (4 Y )+2 Y µ ( Y ( µ +1 ) + µ ) I (4 Y ) ) − (cid:16) Y ( µ − ) − Y ( µ + µ ) +8 Y ( µ +4 µ + µ ) + µ (cid:17) I (4 Y ) − Y µ ( Y ( µ +1 ) + µ ) I (4 Y ) +8 Y µ ( Y ( µ +7 µ +7 µ +1 ) +4 Y ( µ +6 µ + µ ) +4 Y ( µ + µ ) + µ ) I (4 Y ) I (4 Y ) ]2 3 Y )6 [ Y ( − Y ( µ +62 µ +27 ) µ − Y ( µ − µ − µ +5 ) µ +16 Y ( µ +21 µ +195 µ +334 µ +195 µ +21 µ +1 ) − Y ( µ + µ ) − µ ) I (4 Y ) − Y µI (4 Y ) (cid:16) Y µ ( Y ( µ +10 µ +3 ) +6 Y ( µ + µ ) +3 µ ) I (4 Y )+ ( Y ( µ +55 µ +198 µ +198 µ +55 µ +3 ) +16 Y µ ( µ +60 µ +130 µ +60 µ +3 ) − Y µ ( µ +3 µ +3 µ +2 ) − Y µ ( µ +22 µ +9 ) − Y ( µ + µ ) − µ ) I (4 Y ) ) − (cid:16) Y ( µ +26 µ +9 ) µ +96 Y ( µ − ) ( µ +4 µ +1 ) µ +64 Y ( µ − ) − Y ( µ + µ ) +48 Y ( µ +11 µ +11 µ + µ ) +36 Y ( µ + µ ) +9 µ ) I (4 Y ) − Y µ ( Y ( µ +10 µ +3 ) +6 Y ( µ + µ ) +3 µ ) I (4 Y ) +8 Y µ ( Y ( µ +55 µ +198 µ +198 µ +55 µ +3 ) +16 Y µ ( µ +60 µ +130 µ +60 µ +3 ) +96 Y ( µ +8 µ +8 µ + µ ) +24 Y µ ( µ +10 µ +3 ) +36 Y ( µ + µ ) +9 µ ) I (4 Y ) I (4 Y )] (cid:16) − ( µ ( − I (6 Y ))+ ( µ +1 ) I (6 Y ) − µI (6 Y )+ µI (6 Y ) − I (6 Y ) )(( µ +1 ) I (6 Y ) − µ ( I (6 Y )+ I (6 Y )) ) + (( µ +1 ) I (6 Y ) − µI (6 Y ) ) (cid:16) (( µ +1 ) I (6 Y ) − µI (6 Y ) ) − (( µ +1 ) I (6 Y ) − µ ( I (6 Y )+ I (6 Y )) ) (cid:17) + (( µ +1 ) I (6 Y ) − µ ( I (6 Y )+ I (6 Y )) ) (cid:16) (( µ +1 ) I (6 Y ) − µ ( I (6 Y )+ I (6 Y )) ) − (( µ +1 ) I (6 Y ) − µI (6 Y ) )(( µ +1 ) I (6 Y ) − µ ( I (6 Y )+ I (6 Y )) ))) Table 1. Analytic evaluation of Z +1 ,KU ( N ) in terms of modified Bessel functions forvarious N and K .A.2. Large N limit: Gapped solutions. In this appendix we sketch the computation of ω ( z ),defined in (3.10), which allows to extract the eigenvalue density in the phases with one or moregaps. The procedure is standard and we follow closely [76, 47], glossing over many details. Wework in Phase Ia, since all other phases are analyzed in similar fashion.Introduce the function (cid:37) ( z ) of complex variable z ∈ C such that (cid:37) ( e i θ ) = ρ ( θ ) for e i θ ∈ Γ. Thesaddle point equation (3.4) is rewritten as(A.1) P (cid:90) Γ d w πw (cid:37) ( w ) z + wz − w = W (cid:48) ( z )where W (cid:48) ( z ) = − i (cid:20) Y (cid:18) z − z (cid:19) − τ (cid:18) µz − µ + µ − z − µ − (cid:19)(cid:21) . Equation (A.1) is valid for z ∈ Γ, and is complemented by the normalization condition(A.2) (cid:90) Γ d w π i w (cid:37) ( w ) = 1 . Recall that we have started with a Z -symmetric system, invariant under z (cid:55)→ z − for z ∈ T . Wewill thus find an eigenvalue density with symmetric support, and in particular ∂ Γ = (cid:8) e − i θ , e i θ (cid:9) in a one-cut phase. Then, depending on whether the gap opens at θ = π or θ = 0, Γ will be thearc on the unit circle connecting − θ to θ or θ to − θ , respectively, with orientation alwaystaken counter-clockwise.Recall from the definition (3.10) that ω + ( e i θ ) − ω − ( e i θ ) = 2 (cid:37) ( e i θ ) , e i θ ∈ Γ . In turn, from the definition of Cauchy principal value and (A.1) we immediately get(A.3) ω + ( e i θ ) + ω − ( e i θ ) = − W (cid:48) ( e i θ ) . The normalization (A.2) and the definition (3.10) imply that ω ( z ) → | z | → ∞ . We havethen reduced the problem of finding the eigenvalue density to the problem of determining thediscontinuity of ω ( z ) along Γ, from the knowledge of its regular part and the boundary condition ω ( z → ∞ ) = 1. It is standard procedure to reduce the problem (A.3) to a discontinuity equationfor a new, auxiliary function Ω( z ) related to ω ( z ) via(A.4) ω ( z ) = (cid:113) ( e i θ − z ) ( e − i θ − z )Ω( z ) . We take the square root with positive value, but any potential ambiguity in the intermediatesteps and definitions from now on, would drop out from the final answer.Writing (cid:20)(cid:113) ( e i θ − z ) ( e − i θ − z ) (cid:21) ± = (cid:104) √ z · (cid:112) θ − θ (cid:105) ± = ∓ e i θ/ (cid:112) θ − θ for z = e i θ ∈ Γ, we obtain from (A.3) the discontinuity equation for Ω( z ):(A.5) Ω + ( e i θ ) − Ω − ( e i θ ) = 2 e − i θ/ i W (cid:48) ( e i θ ) √ θ − θ . For a multi-cut phase, withΓ ∼ = { θ , − ≤ θ ≤ θ , + } ∪ { θ , − ≤ θ ≤ θ , + } ∪ · · · ∪ { θ k, − ≤ θ ≤ θ k, + } the procedure is the same, but with Ω( z ) defined via ω ( z ) = (cid:118)(cid:117)(cid:117)(cid:116) k (cid:89) j =0 (cid:0) e i θ j, + − z (cid:1) (cid:0) e i θ j, − − z (cid:1) Ω( z ) . Let us now introduce a closed contour C Γ which is a Jordan curve enclosing Γ but not z , andoriented counter-clockwise. See Figure 11 for the contour C Γ in Phase Ia.From the definitions (3.10) and (A.4) it follows that Ω( z ) falls off (at least) as 1 /z at infinity.Then, for z lying in the exterior of C Γ , Cauchy’s theorem together with (A.5) impliesΩ( z ) = (cid:73) C Γ d w π i i W (cid:48) ( w )( z − w ) (cid:112) ( e i θ − w ) ( e − i θ − w )On the other hand, because W (cid:48) ( w ) is meromorphic we can deform the contour C Γ into an infinitelylarge circle, picking the poles of the integrand. We findΩ( z ) = − i W (cid:48) ( z ) (cid:112) ( e i θ − z ) ( e − i θ − z ) − (cid:73) C ∞ d w π i i W (cid:48) ( w )( z − w ) (cid:112) ( e i θ − w ) ( e − i θ − w )+ (cid:88) z p ∈{ ,µ,µ − } Res w = z p i W ( w )( z − w ) (cid:112) ( e i θ − w ) ( e − i θ − w )(A.6) Figure 11. Contour C Γ encircling the cut Γ.where the first term is the residue at w = z , the second term is the remaining contour integralalong a circle at infinity, which in our case simply contributes Y , and the last term includes theresidues at the poles z p of W (cid:48) ( w ).In Phase Ia, explicit computation of each term leads to ω Ia ( z ) = − i W (cid:48) ( z )+ (cid:113) ( e i θ − z ) ( e − i θ − z ) (cid:34) Y (cid:18) z (cid:19) − τ (cid:112) µ − µ cos θ (cid:18) µz − µ − z − µ − (cid:19)(cid:35) . The solution in the other phases is found likewise. Appendix B. Filling fraction fluctuations This appendix contains the analysis of the effect of taking into account fluctuations of thefilling fractions around the equilibrium configuration.For a generic matrix model in a two-cut phase, the dependence of the filling fractions on theparameters of the theory should be taken into account when computing physical observables [46]. Below we briefly review how this effect comes about, and study it for the model at hand. Westart with Phase III, and look at Phase II projected onto the real line, as in Section 3.4.B.1. Phase III. For the two-cut solution in Phase III, let N L be the number of eigenvalues inthe left arc around θ = π , 0 ≤ N L ≤ N , and N R = N − N L the number of eigenvalues in the rightarc around θ = 0. Let also ξ = N L N and 1 − ξ = N R N denote the corresponding filling fractions.The values of y L = cos ( θ ∗ + δθ ) and y R = cos ( θ ∗ − δθ ) can be fixed, as functions of ξ and ofthe other parameters, through the equations2 π (cid:90) y L − d y (cid:115) ( y R − y )( y L − y )1 − y (cid:34) − Y + τ µ ( µ − (cid:112) (1 + µ − µy L )(1 + µ − µy R )(1 + µ − µy ) (cid:35) = ξ, π (cid:90) y R d y (cid:115) ( y − y R )( y − y L )1 − y (cid:34) − Y + τ µ ( µ − (cid:112) (1 + µ − µy L )(1 + µ − µy R )(1 + µ − µy ) (cid:35) = 1 − ξ, that come from the definition of ξ after changing variables y = cos θ . Then, the value of ξ is fixedby the equilibrium condition(B.1) d S eff d ξ (cid:12)(cid:12)(cid:12)(cid:12) ξ = ξ sp = 0 . The original work [46] dealt with Hermitian matrix models, but the argument extends to the present setting. For example, approximating close to the critical surface diving Phase III from Phase Ib, we find ∂ξ sp ∂y R (cid:12)(cid:12)(cid:12)(cid:12) y R =1 = √ − y L π (cid:32) − τµ − τ µ ( µ + 1)( µ − (cid:112) µ − µy L (cid:33) where we have also substituted Y = Y cr , b . It has been shown in [46] that the quantum fluctuationsaround the saddle point ξ sp contribute to the free energy a term of the form − N log ϑ ( N ξ sp ),where ϑ ( z ) is the Jacobi theta function. See Appendix B.2 below for more details and a veryshort review of the derivation. This is a sub-leading contribution to the free energy but, due tothe dependence on N ξ sp , each derivative generates a factor of N . Therefore, the ξ sp -dependentpart becomes of the same order as the leading order term when differentiating the Wilson loopvev, and must be taken into account. The relevant part of the derivative is (cid:88) y ∈{ y L ,y R } (cid:20) dd z log ϑ ( z ) | z = Nξ sp (cid:21) (cid:18) ∂y∂Y ∂ξ sp ∂y (cid:19) , which yields a non-trivial contribution to the derivative of the Wilson loop vev in Phase III.However, when approaching the critical loci, ξ → Y → Y cr , a or ξ → Y → Y cr , b , and thederivative of the theta function evaluated at an integer vanishes.This shows that the effect of the filling fractions does not play any role in determining theorder of the phase transition, despite being non-trivial in the bulk of Phase III.B.2. Phase II. We now discuss the same effect in Phase II. It is more convenient and akin tothe work [46] to do this in the alternative, Hermitian matrix model presentation of Section 3.4.The argument can be succinctly summarized as follows.Consider a two-cut solution with support supp ρ p . II = Γ L ∪ Γ R , and denote by ξ = N L N and1 − ξ = N R N the corresponding filling fractions, as above. The saddle point value ξ sp of ξ is fixedby (B.1). Then, in the large N approximation, the partition function takes the form [46] Z = N (cid:88) N L =0 e − N F pert − N ( ξ − ξ sp ) ∂ ξ S eff | ξ = ξ sp + O (( ξ − ξ sp ) ) where F pert is the perturbative free energy to all orders in the N expansion. This yields [46, 77] − N log Z = F + 1 N F nlo − N log ϑ ( N ξ sp ) + 12 N log (cid:16) π ∂ ξ S eff (cid:12)(cid:12) ξ = ξ sp (cid:17) + O ( N − ) , where F is the leading order or planar free energy, F nlo the next-to-leading order correction,and (after an implicit resummation) we have recognized the Jacobi theta function ϑ ( N ξ sp ). 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