Thermodynamics of theories with sixteen supercharges in non-trivial vacua
Gianluca Grignani, Luca Griguolo, Nicola Mori, Domenico Seminara
aa r X i v : . [ h e p - t h ] J u l Thermodynamics of theories with sixteen superchargesin non-trivial vacua
Gianluca Grignani a , Luca Griguolo b , Nicola Mori c and Domenico Seminara ca Dipartimento di Fisica and Sezione I.N.F.N., Universit`a di Perugia,Via A. Pascoli I-06123, Perugia, Italia [email protected] b Dipartimento di Fisica, Universit`a di Parma, INFN Gruppo Collegato di Parma,Parco Area delle Scienze 7/A, 43100 Parma, Italy [email protected] c Dipartimento di Fisica, Polo Scientifico Universit`a di Firenze,INFN Sezione di Firenze Via G. Sansone 1, 50019 Sesto Fiorentino, Italy [email protected], [email protected]
Abstract
We study the thermodynamics of maximally supersymmetric U ( N ) Yang-Mills theoryon R × S at large N . The model arises as a consistent truncation of N = 4 superYang-Mills on R × S and as the continuum limit of the plane-wave matrix model ex-panded around the N spherical membrane vacuum. The theory has an infinite numberof classical BPS vacua, labeled by a set of monopole numbers, described by dual super-gravity solutions. We first derive the Lagrangian and its supersymmetry transformationsas a deformation of the usual dimensional reduction of N = 1 gauge theory in ten di-mensions. Then we compute the partition function in the zero ’t Hooft coupling limit indifferent monopole backgrounds and with chemical potentials for the R -charges. In thetrivial vacuum we observe a first-order Hagedorn transition separating a phase in whichthe Polyakov loop has vanishing expectation value from a regime in which this order pa-rameter is non-zero, in analogy with the four-dimensional case. The picture changes inthe monopole vacua due to the structure of the fermionic effective action. Depending onthe regularization procedure used in the path integral, we obtain two completely differ-ent behaviors, triggered by the absence or the appearance of a Chern-Simons term. Inthe first case we still observe a first-order phase transition, with Hagedorn temperaturedepending on the monopole charges. In the latter the large N behavior is obtained bysolving a unitary multi-matrix model with a peculiar logarithmic potential, the systemdoes not present a phase transition and it always appears in a “deconfined” phase. ontents R × S from D = 10
63 BPS vacua and their gravitational duals 114 Free SYM partition functions in monopole vacua 14 A . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 184.4 Fermions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 194.5 Partition functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21 A.1 Supersymmetry variations . . . . . . . . . . . . . . . . . . . . . . . . . . . 44
B Computing the one loop partition function 45
B.1 Computing determinants: the master-formula . . . . . . . . . . . . . . . . 45B.2 The scalar determinant . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48B.3 The vector/scalar determinant . . . . . . . . . . . . . . . . . . . . . . . . . 49B.4 The spinor determinant . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52
C U(1) truncation of N = 4 super Yang Mills 55 C.1 Scalars . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55C.2 Vectors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56C.3 Fermions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57
D Solving the matrix model 59 Introduction
In the context of the AdS/CFT correspondence [1, 2, 3, 4] an interconnected familyof theories with sixteen supercharges has been recently studied [5]. They all have amass gap and a discrete spectrum of excitations. These theories can be obtained fromconsistent truncations of N = 4 super Yang-Mills on R × S and have many BPS vacua.Remarkably, smooth gravity solutions corresponding to all these vacua can be describedrather explicitly. At large ’t Hooft coupling some properties of the dual string theory havealso been examined according to the pioneering proposal of [6].From the gauge theoretical point of view it seems particulary appealing to investigatethe properties of one specific theory belonging to this class, namely the maximally su-persymmetric U ( N ) Yang-Mills theory on R × S . This theory already appeared in [7]where it arises from the fuzzy sphere vacuum (membrane vacuum) of the plane-wave ma-trix model by taking a large N limit that removes the fuzzyness. The model can also beconstructed from the familiar N = 4 SYM theory by truncating the free-field spectrumon R × S to states that are invariant under U (1) L ⊂ SU (2) L , where SU (2) L is one of the SU (2) factors in the SO (4) rotation group of the three-sphere. Geometrically this corre-sponds to a dimensional reduction of the four-dimensional supersymmetric theory alongthe U (1) fiber of S seen as an Hopf fibration over S . The resulting model lives in one di-mension less and maintains supersymmetry through a rather interesting mechanism. Theparticular dimensional reduction breaks the natural SO (7) R -charge symmetry to SO (6),singling out one of the seven scalars of the maximally supersymmetric Yang-Mills theory,which then behaves differently from the others. It combines with the gauge fields to forma peculiar Chern-Simons-like term that is crucial to preserve the sixteen supercharges,balancing the appearance of mass terms for fermions and scalars. The BPS vacua aregenerated by the same term that allows to combine the field strength and the scalar intoa perfect square whose zero-energy configurations are determined by N integers n , ..., n N associated to monopole numbers on the sphere.The model represents an interesting example of a supersymmetric non-conformal gaugetheory, with smooth gravitational dual and non-trivial vacuum structure, defined on acompact space. The last feature is particulary appealing in the study of the thermalproperties of the theory. Recently the thermodynamics of large N theories on compactspaces has attracted much attention. On compact spaces the Gauss’s law restricts physicalstates to gauge singlets. Consequently, even at weak ’t Hooft coupling the theories arein a confining phase at low temperature and undergo a deconfinement transition at acritical temperature. For example, the partition function of N = 4 super Yang-Millstheory on R × S was computed at large N and small coupling in [8, 9, 10]. It wasshown that the free energy is of order O (1) at low temperature and of order O ( N )above a critical temperature. At strictly zero ’t Hooft coupling the transition is a first-order Hagedorn-like transition. At small coupling a first or a second order transition isexpected, depending on the particular matter content of the theory. The computation inthe N = 4 maximally supersymmetric case has never been performed but in [11] it wasargued that the maximally supersymmetric plane-wave deformation of Matrix theory and2 = 4 SYM should show similar behavior, including thermodynamics. The plane wavematrix model is a theory with sixteen supercharges and it was argued in [7] to be dualto a little string theory compactified on S . For a small sphere, this theory is weaklycoupled and one may study the little string theory thermodynamics rather explicitly [12].The phase transition for this model was shown to remain first order in [13] indicating thatthis might also be the case for N = 4 SYM. This was shown by computing the relevantparts of the effective potential for the Polyakov loop operator to three loop order [13].With the same procedure it was shown in [14] that also for pure Yang-Mills the phasetransition remains first-order up to three loops. The phase transition at weak couplingis basically driven by a Hagedorn-like behavior of the spectrum in the confining phase,suggesting a possible relationship with the dual description of large N gauge theoriesin terms of strings. For N = 4 the relevant string theory lives on an asymptotic AdSspace and, at large ’t Hooft coupling, the deconfinement phase transition corresponds to aHawking-Page transition [15, 16]. The thermal AdS space dominates at low temperatureand the AdS-Schwarzschild black hole is the relevant saddle-point in the high-temperatureregime. The original proposal presented in [8, 9] to connect the phase transitions at smallcoupling on compact spaces with the gravitational/stringy physics stimulated a largenumber of investigations. Lower-dimensional theories on tori were examined in [17, 18],while the inclusion of chemical potentials for the R -charges was discussed in [19, 20] and,more recently, pure Yang-Mills theory on S [21] was found to have a second order phasetransition at small ’t Hooft coupling.In this paper we study the thermodynamics of N = 8 super Yang-Mills theory on R × S . We first derive the Lagrangian and its supersymmetry transformations as a de-formation of the usual dimensional reduction of N = 1 gauge theory in ten dimensions.Actually our procedure will generate a larger class of three-dimensional theory: accordingto the particular choice of the generalized Killing spinor equation we obtain also theorieson AdS with peculiar Chern-Simons couplings. Then we compute the N = 8 parti-tion function in the zero ’t Hooft coupling limit, for different monopole vacua. In thetrivial vacuum we observe a first-order Hagedorn transition separating a phase in whichthe Polyakov loop has vanishing expectation value from a regime in which this order pa-rameter is non-zero, in complete analogy with the four-dimensional case. The Hagedorntemperature is also obtained in the presence of chemical potentials for the R -charges.Discussions on the dual gravitational picture [5] and the possibility of matching the gaugetheory Hagedorn transition with a stringy Hagedorn transition, by exploiting for examplea decoupling limit as in [20, 22, 23, 24] postponed to a forthcoming investigation.The situation is very different in the non-trivial monopole vacua. The original U ( N )gauge group is broken to a direct product U ( N ) × U ( N ) × ..U ( N k ) and the constituentfields transform, in general, under bifundamental representations of U ( N I ) × U ( N J ). Be-cause of the Gauss’s law on a compact manifold, however, the only allowed excitationsare SU ( N I ) × SU ( N J ) singlets. Different selection rules are instead possible for the U (1)charges in three dimensions, depending on the definition of the fermionic Fock vacuumin the presence of background monopoles [25]. The appearance of fermionic zero-modesmakes possible, in general, to assign a non-trivial charge to the Fock vacuum, as clearly3xplained in [26]. In the path-integral formalism this corresponds to precise choices inregularizing fermionic functional determinants which might produce Chern-Simons termsin the effective action. In our case the different possibilities are clearly manifested in thematrix model describing the partition function. We recall that, in the trivial vacuum, thethermal partition function is reduced to an integral over a single U ( N ) matrix [8, 9] Z ( β ) = Z [ dU ] exp h − S eff ( U ) i (1.1)where U = e iβα ( α is the zero mode of the gauge field A on S × S and β = 1 /T theinverse of the temperature). In the non-trivial monopole vacuum Z ( β ) is given insteadby a multi-matrix model over a set of unitary matrices U I ( N I ), i = 1 , , ..k , reflecting thebreaking of the U ( N ) gauge group. More importantly the effective action S eff ( U I ), at zero’t Hooft coupling, can be modified by the presence of logarithmic terms N Q I Tr log( U I )that implement selection rules on the U (1) charges. The large N analysis is highly affectedby these new interactions: they contribute at order N and can drive the relevant saddle-point always at a non-zero value of the Polyakov loop. Unitary matrix model of the kindwe encountered in our analysis have been previously considered in the eighties [27, 28], butwith an important difference: in those studies the coefficient weighting the logarithmicterm Tr log( U ) in the action was taken independent on N . Conversely the large N saddle-points were not modified by its presence, being determined by the rest of the action. Inour case, instead, we have to cope with a linear dependence on N and we cannot simplyborrow those results. We have therefore performed an entirely new large N analysis ofthese kind of models, starting from an exact differential equation of the Painlev´e typethat describes the finite N partition function [29].The paper is organized as follows. In section 2 we construct the supersymmetricYang-Mills theory on R × S using a different strategy with respect to [5] and [7] (seealso [30] for a careful derivation of the Hopf reduction and [31] for an extension to moregeneral fiber bundles). We start from N = 1 super Yang-Mills theory in ten dimensionsand consider its dimensional reduction on R × S . We find the relevant Killing spinorsthat generate the rigid supersymmetry, generalizing to our case the approach developedin [32]. We further determine the deformations of the original ten dimensional Lagrangianand of the supersymmetry transformations ensuring the global invariance of the action.Interestingly, using the same strategy it is possible to construct two other maximallysupersymmetric gauge theories on three-dimensional curved spacetimes, living both onAdS and differing from the theory introduced in [7] in the structure of the Chern-Simonsterms. In section 3 we briefly examine the BPS vacua of the model, we comment on theirgravitational description and the related instanton solutions.We then turn to study the thermodynamics at zero ’t Hooft coupling. Following theanalysis in [8, 9], we obtain the partition function of the theory in a generic vacuum, interms of matrix integrals. In section 4 we present the results of the relevant functionaldeterminants in the background of a gauge flat-connection and of a monopole poten-tial, recovering the appropriate single-particle partition functions for scalars, spinors andvectors. Careful ζ -function evaluations are deferred to the appendices. We discuss the4merging, on the monopole background, of new logarithmic terms in the effective action,directly related, in this formalism, to the appearance of fermionic zero-modes. We explaintheir dependence on the regularization procedure and remark their interplay with a typ-ical three-dimensional phenomenon, the induction of Chern-Simons terms. We interprettheir effect as a part of the projection into singlets of the gauge group, as required by theGauss’s law. Section 5 is devoted to discuss the large N thermodynamics in the trivialvacuum. We determine the critical temperature at which the first-order phase transitiontakes place and we generalize the result to the case of non vanishing chemical potentialsfor the R -charges. Finally, in sections 6 and 7, we study the large N theory on the non-trivial monopole backgrounds: we consider a large class of vacua, characterized by theset of integers n , .., n k and large N degeneracies N , .., N k . According to the discussionof section 4, we study two different choices for the logarithmic terms, within our regu-larization procedure. First, in section 6, we discuss the “uncharged” case, that amountsto make a particular choice of branch cuts, in the ζ -function regularization procedure[33, 34], that cancels the Chern-Simons like contributions. In turn we get a non-vanishingCasimir energy, depending explicitly on the monopole background. The resulting uni-tary multi-matrix model is an obvious generalization of the trivial case. We find again afirst-order phase transition, with an Hagedorn temperature explicitly depending on themonopole numbers. We discuss also some particular class of vacua, characterized by largemonopole charges, whose Hagedorn temperature approaches the one of the theory on S / Z k in trivial vacuum. In section 7 we discuss the opposite situation of a “maximally”charged fermionic vacuum: we have a non-trivial modification of the unitary multi-matrixmodel due to appearance of the new logarithmic terms and vanishing Casimir energy.For the sake of clarity we will restrict our discussion to a particular simple background( n, n, .., n, − n, − n.., − n ). We show the existence of a non-trivial saddle-point for the ef-fective action for a wide range of temperatures starting from zero, within the assumptionthat we can disregard higher windings contributions in this regime. This implies that thetheory is always in a “deconfined” phase. We have to face the problem of computing thefree energy and the phase structure of the matrix model Z ( β, p ) = Z DU exp (cid:0) βN (Tr( U ) + Tr( U † )) (cid:1) det( U ) Np , (1.2)that is a non-trivial deformation of the familiar Gross-Witten model [35]. Its large N behavior is carefully studied in section 7.1 , obtaining the exact free energy in terms ofthe solution of a fourth-order algebraic equation: we prove that there is no phase transitionas long as p = 0, in contrast with the usual p = 0 case, that appears as a singular pointin the parameter space. In section 7.2 we use the results of our analysis to derive a set ofsaddle-point equations for the partition function which describes the “deconfined” phase.The disappearance of the confining regime is consistent with the known results on finitetemperature 2+1 dimensional gauge theories where, once a topological mass (a Chern-Simons term) is turned on, there cannot be a phase transition [36, 37, 38]. In section 8we briefly draw our conclusions and discuss future directions. Several appendices aredevoted to technical aspects and to an alternative derivation of the partition functions.5n appendix A we report some details on supersymmetry transformations. In appendixB we give the details of the computation of functional determinants. In appendix Cwe recover the results for the single-particle partition functions from those of the parent N = 4 theory by explicitly constructing the projector into the U (1) invariant modes. Wealso check the consistency of our results with those of [39], where the theory on R × S / Z k has been studied. Appendix D is instead focused on some technical aspects, related tothe solution of the large N matrix integrals. R × S from D = 10 There are many ways to construct the Lagrangian of the gauge theory with sixteen super-charges on R × S and its supersymmetry transformations. For instance, in [7] this theorywas obtained from the plane-wave matrix model action expanded around the k -membranevacuum in the large N limit. Subsequently, in [5] it was derived as a U (1) truncationof the spectrum of the N = 4 gauge theory on R × S . Since here we shall be mainlyconcerned with the field theoretical features of this N = 8 model, we shall follow a moreconventional (and maybe pedagogical) approach: the Lagrangian and its supersymmetrytransformations will be derived as a deformation of the standard toroidal compactificationof N = 1 gauge theory in ten dimensions.We first consider the theory on the flat Minkowski space in three dimensions, M (1 , .The N = 8 theory in this case is the straightforward dimensional reduction of the N = 1theory in D = 10. The most convenient and compact way to present its Lagrangian isto maintain the ten-dimensional notation and to write (see appendix A for a summary ofour conventions ) L (0) = − F MN F MN + iψ Γ M D M ψ. (2.1)All the fields in (2.1) only depend on the space-time coordinates ( x , x , x ). In particular,from the three-dimensional point of view, the gauge field A M contains the reduced gaugefield A µ and seven scalars ( φ m ) = ( φ , φ , · · · , φ ) ≡ ( φ , φ m ). The flat ten dimensionalspace-time metric is diagonal and it has the factorized structure T × M (1 , .Our goal is now to promote the supersymmetric theory in the flat 2+1-dimensionalspace-time to a supersymmetric theory on the curved space R × S . It is useful tokeep a ten-dimensional notation where the above space-time is viewed as a submanifoldembedded in T × R × S with the metric ds = − dt + R ( dθ + sin θdϕ ) + X i =1 dη i . (2.2)Here the coordinates θ and ϕ span the sphere S of radius R , while the internal angularcoordinates η i parameterize the torus T . The action (2.1) in the background (2.2) is In general we shall omit the trace over the gauge generator in our equations, unless it is source ofconfusion. ǫδ (0) A M = − iψ Γ M ǫ,δ (0) ψ = F MN Γ MN ǫ . (2.3)Constant spinors however do not exist, in general, on a curved space. For a space-time ofthe type (2.2), the notion of a constant spinor should be replaced with that of a Killingspinor [32]. Its specific definition may depend on the detail of the geometry, but, for us,it will be a spinor satisfying an equation of the type ∇ µ ǫ = K νµ Γ ν Γ ǫ , (2.4)where the Greek indices run only over the three-dimensional space-time since the trans-verse coordinates η i are flat and we can always choose ǫ to be a constant along thesedirections. In (2.4) we have also inserted an additional dependence on the Γ matricesthrough a monomial factor Γ
123 2 . This has double role: (a) it makes (2.4) compatiblewith the ten-dimensional chirality conditions; (b) it generates, as we shall see, the rele-vant massive deformations for our fields. Finally the tensor K νµ expresses an additionalfreedom in constructing the Killing spinors. In a curved space, there is in fact no a priori reason to treat all the coordinates symmetrically. In the R × S curved space-time ge-ometry there is a natural splitting between space and time and thus it is quite natural toweight them differently by choosing K νµ = α (cid:2)(cid:0) δ νµ + k µ k ν (cid:1) − B k µ k ν (cid:3) , (2.5)where k µ is the time-like Killing vector of (2.2) and α, B are two arbitrary parameters.The parameter α is fixed by imposing the necessary integrability condition (the first) [40],which arises from the commutator [ ∇ µ , ∇ ν ] ǫ . This can be either expressed in terms of thespace-time curvature scalar R = 2 /R or, through (2.4), in terms of K νµ and consequentlyof α . We thus get for α α = 12 R . (2.6)The parameter B , instead, remains free and it will be determined in the following.The variation of the action (2.1) with respect to the supersymmetry transformations(2.3) written in terms of a non-constant supersymmetry parameter ǫ does not vanish.Terms depending on the covariant derivatives of ǫ (2.4) are in fact generated (see appendix The direction (1 ,
2) span the tangent space to the sphere S , while the index 3 is along the first ofthe compactified dimensions. δ (0) L (0) =2 R e { iψF MN Γ µ Γ MN ∇ µ ǫ } =2 R e { i B αψ [Γ ij F ij − i F i + 2Γ jm D j φ m − m D φ m − ig Γ mn [ φ m , φ n ]]Γ ǫ + iαψ [ − ij F ij + 4Γ D φ m − ig Γ mn [ φ m , φ n ]]Γ ǫ } . (2.7)where in the second equality we have used (2.4) and (2.5). This undesired variation canbe compensated by adding the following deformations to the original Lagrangian L (1) = iM αψ Γ ψ + N αφ F , L (2) = V α φ m + W α φ , (2.8)and by adding new terms to the supersymmetry transformations of the fermions δ (1) ψ = P α Γ m Γ φ m ǫ + Gα Γ Γ φ ǫ, (2.9)where M, N, V, W, P, G are arbitrary parameters to be fixed by imposing the invarianceof the complete action. The size of the deformations is tuned by the natural mass scale α = 1 / (2 R ) provided by the radius of the sphere.Some comments on the form of (2.8) and (2.9) are in order. The addition of massterms for the scalars ( L (2) ) is a common and well-known property for supersymmetrictheories in a background admitting Killing spinors. Some of the mass terms can alsobe justified with the requirement that the conformal invariance originally present in flatspace is preserved. In four dimensions, for N = 4 super Yang-Mills, this is the onlyrequired modification of the Lagrangian because of an accidental cancellation. Since weare in three dimensions, we are also forced to introduce a non-standard mass term for thefermions (the first term in L (1) ). The natural supersymmetric companion for a fermionicmass in D = 3 is then a Chern-Simons-like term (the second term in L (1) ). Its unusualform, φ F , mixes the scalar φ with the gauge-fields and is inherited from the particularchoice of the monomial Γ in (2.4). Then the modifications (2.9) in the supersymmetrytransformations are the only possible ones with the right dimensions and compatible withthe symmetries of the theory.The most convenient and simple way to analyze the effect of the additional terms inthe Lagrangian (2.8) and in the supersymmetry transformations (2.9) is to single out, inthe variation of the Lagrangian, different powers of the deformation parameter α . Westart with the linear order in α , the zeroth order being automatically absent since ourtheory is supersymmetric in flat space-time. At this order we have three contributions:the original variation (2.7), the variation of the new Lagrangian L (1) with respect to theold transformations (2.3) δ (0) L (1) =2 M α R e { iψ ( F ij Γ ij − F i Γ i − D φ Γ +2 D i φ Γ i +2 D φ m Γ m − D i φ m Γ im ++ 2 i [ φ , φ m ]Γ m − i [ φ m , φ n ]Γ m n )Γ ǫ } + iN α ( F ij ψ Γ ij + 2 D i φ ψ Γ i )Γ ǫ (2.10)8nd finally the variation of L (0) with respect to (2.9) δ (1) L (0) = 2 R e { iαψ ( P Γ µm D µ φ m − igP Γ mn [ φ m , φ n ]+ G Γ µ D µ φ − igG Γ m [ φ m , φ ])Γ ǫ } . (2.11)See appendix A for all the different index conventions. It is quite straightforward to derive(2.10) and (2.11) since at this order in α we can consider ǫ as a constant spinor, namely ∇ µ ǫ = 0. Imposing that δ (0) L (0) + δ (0) L (1) + δ (1) L (0) = O ( α ) gives a linear system of eightequations in the five unknowns M, N , P , G and B . The details are given in appendixA.1. Quite surprisingly, this system is still solvable and it fixes the value of the aboveconstants as M = − , N = 4 , P = − , G = − , B = 12 . (2.12)The next and final step is to consider the order α in our supersymmetry variation. Thesituation is much simpler now since we need to evaluate only few terms. We have in factto consider the effects of the corrected transformation (2.9) on L (1) δ (1) L (1) = iM αδ (1) ( ψ Γ ψ ) = 2 R e { iα ψ (Γ m φ m − φ ) ǫ } (2.13)and we have to take care of the terms coming from δ (1) L (0) originated from the covariantderivative of the Killing spinor ǫ . We obtain δ (1) L (0) = − R e { iα ψ [3Γ m φ m + 6Γ φ ] ǫ } . (2.14)These two contributions are easily compensated by the variation of L (2) , δ (0) L (2) = − iα ( V φ m ψ Γ m ψ + ( V + W ) φ ψ Γ ψ ) , (2.15)By setting V = − W = − O ( α ) term, because there is neither an α -dependent term in the variation of bosons(which might produce a O ( α ) term in the variation of L (2) ) nor α term in the variationof fermions.We have thus reached our original goal: to promote the N = 8 theory in flat space inthree dimensions to an N = 8 theory in the curved background R × S . Its Lagrangianin a ten-dimensional language is thus given by L = − F MN F MN + iψ Γ M D M ψ − i µ ψ Γ ψ + 2 µφ F − µ φ m − µ φ , (2.16)and it is invariant under the supersymmetry transformations δA M = − iψ Γ M ǫ,δψ = F MN Γ MN ǫ − µ Γ m Γ φ m ǫ − µ Γ Γ φ ǫ, (2.17)where µ is the mass-scale µ = 1 /R . Notice that the mass for the scalars φ m (with m =4 , , . . . ,
9) in (2.16) is that required by conformal invariance on R × S : m conf. = R =9 R = µ . The mass of the scalar φ is, instead, different because φ mixes with the gaugefields. This mixing also breaks the original SO (7) R -symmetry present in flat spaceto the smaller group SO (6) R ( ≃ SU (4) R ): the bosonic symmetries R × SO (3) × SO (6) R combine with the supersymmetries into the supergroup SU (2 | K νµ and by considering a monomial factor Γ mixing one of the transverse compactdirections with the two spatial directions of the actual space-time of the theory.The Lagrangian (2.16) written in terms of the three-dimensional fields becomes L = − F µν F µν + 2 iλ i γ µ D µ λ i − D µ φ ij D µ φ ij − D µ φ D µ φ − igλ i [ φ , λ i ]+ − g √ (cid:16) λ iT [ φ ij , ελ j ] − λ i [ φ ij , ελ Tj ] (cid:17) + 18 g [ φ ij , φ kl ][ φ ij , φ kl ]+ 12 g [ φ , φ ij ][ φ , φ ij ]+ − µ λ i γ λ i − µ φ ij φ ij − µ φ + 2 µφ F . (2.18)This is the N = 8 SYM Lagrangian on R × S that will be used in computing the ther-modynamic partition function of the model. We have cast the contribution of the scalarfields ( φ , . . . , φ ) in an SU (4) R manifestly covariant form, by rewriting their Lagrangianin terms of the representation of SU (4) R , φ ij . The spinor fields λ i are four Dirac spinorsin D = 3 originating from the dimensional reduction of ψ .Since we will be mainly interested in the finite temperature features of the model, theEuclidean version of (2.18) will be more relevant. It is given by L = 12 F µν F µν − iλ i γ µ D µ λ i + 12 D µ φ ij D µ φ ij + D µ φ D µ φ ++ g √ (cid:16) λ iT [ φ ij , ελ j ] − λ i [ φ ij , ελ Tj ] (cid:17) + 2 igλ i [ φ , λ i ]+ − g [ φ ij , φ kl ][ φ ij , φ kl ] − g [ φ , φ ij ][ φ , φ ij ]++ iµ λ i γ λ i + µ φ ij φ ij + µ φ − µφ F . (2.19)We conclude by noting that, in the above analysis, we have made a particular choicein considering the form of the Killing spinor equation. A careful reader might wonderif there are other possibilities. Unfortunately, different choices in (2.4) generally lead toinconsistencies: the Killing equation is not integrable or no consistent supersymmetricdeformation exists. For example, the second type of inconsistency would occur if wehad simply chosen K νµ = δ νµ . It is however intriguing to note that the choice K νµ =10 νµ becomes consistent if we alter the background geometry from R × S to AdS andsubstitute Γ with Γ or Γ . In the former case, we would have found a maximallysupersymmetric version of the topologically massive theory, with bosonic symmetry group SO (1 , × SO (7). In the latter we would have instead reached a massive deformationof the maximally supersymmetric Yang-Mills with the peculiar interaction Tr( φ [ φ , φ ])and symmetry group SO (1 , × SO (3) × SO (4). This case was already considered in [32].It would be nice to understand better their relations with higher dimensional theories andto explore the possible existence of gravitational duals. In this section we shall briefly review the structure of the BPS vacua of the N = 8 theoryon R × S [5] that will be the main ingredients of the thermodynamical investigation ofsection 6 and 7. More specifically, we shall be interested in those vacua that maintainboth the R -invariance and the geometrical symmetries.In order to have an SU (4) R invariant vacuum, we have to choose φ ij = 0. More-over, to preserve the invariance under time translations and the SO (3) rotations of thebackground geometry, we require that all the fields are time-independent and that thechromo-electric field E i = F i vanishes, respectively. The BPS condition can be derivedfrom the requirement that on the supersymmetric invariant vacuum the supersymmetryvariations should vanish. Fermions must be set to zero to saturate the BPS bound andconsequently the supersymmetry variations of bosons automatically vanish on the vac-uum. The supersymmetry variation of fermions, instead, must be set to zero and withthe above assumptions it reads0 = δψ = [2( F θϕ − µ sin θφ )Γ θϕ + 2 D µ φ Γ µ ] ǫ , (3.1)( θ and ϕ are coordinates on S ) which translates into two simple equations F θϕ − µ sin θφ = 0 , D µ φ = 0 . (3.2)The reader familiar with YM will immediately recognize in these equations, those ofYang-Mills theory on the sphere S , for which a complete classification of the solutionsexists [41, 42]. The general solution for a U ( N ) theory is given by a stack of N independent U (1) Dirac monopoles of arbitrary charges. In detail, we have φ = µ f F θϕ = f θ A = f − cos θ )sin θ (sin θdϕ ) ≡ f A , (3.3)where f is a diagonal matrix with integer entries, for which we shall use the short-handnotation f = ( n , N ; n , N ; . . . ; n k , N k ) . (3.4)11ach n I represents the Chern-class of the corresponding Dirac monopole and it assumesvalues in Z , while N I is the number of times that this charge appears on the diagonal. Thevacuum (3.4) then breaks the original U ( N ) gauge symmetry to a direct product U ( N ) × U ( N ) × . . . U ( N k ). However, since all fields in (2.18) are in the adjoint representation,this breaking will affect the dynamics only through the relative charge ( n I − n J ) betweendifferent sectors, while the global charge Q = P kI =1 N I n I will play no role.The gravitational backgrounds dual to the vacua of these theories were derived in [5]and further discussed in [61] (where also the relations between vacua of theories with SU (2 |
4) symmetry group are studied): they have an SO (3) and an SO (6) symmetry andthereby the geometry contains S and S factors, the remaining coordinates being time,a non-compact variable η , −∞ ≤ η ≤ ∞ , and a radial coordinate ρ . These backgroundsare non-singular because the dual theories have a mass gap. The relevant supergravityequations can be reduced to a three-dimensional electrostatic problem where ρ is theradius of a charged disk. The ten dimensional metric and the other supergravity fieldsare completely specified in terms of the solution V of the related Laplace equation . Theregularity condition requires that the location where the S shrinks are disks at constant η i (in the ρ, η space) while S shrinks along the segment of the ρ = 0 line between twonearby disks. The geometry therefore contains three-cycles connecting the shrinking S and six-cycles connecting the shrinking S , supporting respectively non-trivial H and ∗ F fluxes. There is a precise relation between these quantized fluxes and the data of theelectrostatic problem, namely the electric charges Q i of the disks are related to the RRfluxes while the distance (in the η direction) between two disks bounding a three cycleis proportional to the NS flux. To be more specific, this electrostatic description of anon-trivial vacuum generically contains k disks, whose positions are parameterized by k integers n I through the relations η I = πn I . (3.5)These integers are identified with the monopole charges n I in (3.4). Moreover each diskcarries a charge Q I given by Q I = π N I , (3.6)where N I are the same integer numbers counting the degeneracy of each monopole chargein the gauge theory. At the level of supergravity data, the above picture realizes k groupsof D N I elements, wrapping different two-spheres. This is the geometricmanifestation of the breaking of the gauge symmetry to a direct product U ( N ) × U ( N ) ×· · · × U ( N k ). The charges n I instead combine into NS5-fluxes given by n I − n J . Againthe total charge seems to play no role.In our field theoretical analysis we have neglected the time component of the gaugefield A , which disappears from (3.2) when considering the solutions (3.3). Its dynamicsis implicitly governed by the requirement that E i = 0, which, for a time-independentbackground, becomes D i A = 0. It is a trivial exercise to show that the most general This problem has been recently tackled in [43] and [44], searching for a dual description of LittleString theory on S A = 0 when the topology of the time directionis R . In the finite temperature case where time is compactified to a circle S , the mostgeneral solution is, instead, given by A = a , where a is a constant diagonal matrix,namely a flat-connection living on S . This will play a fundamental role in studying thethermodynamical properties of the theory.It is instructive to look at the BPS vacua also at the level of the Euclidean Lagrangian:this will elucidate the emerging of an interesting class of instanton solutions thoroughlystudied in [45]. If we focus on the bosonic sector of our model and we set φ ij = 0 topreserve the SU (4) R symmetry, we can write √ g L = √ g F αβ F αβ + √ gD α φ D α φ + √ gµ φ − µφ F θϕ . (3.7)This Lagrangian can be easily arranged in a BPS-form , i.e. as a sum of squares and totaldivergences. In fact, after some algebraic manipulation, the Euclidean Lagrangian can becast in the following form √ g L = ± µ sin θD t ( φ ) ∓ D α ( φ F βρ ǫ αβρ ) + sin θ (cid:18) F tθ ± θ D ϕ φ (cid:19) ++ 1sin θ ( F tϕ ∓ sin θD θ φ ) + µ sin θ (cid:18) F θϕ − µ sin θ ( µφ ∓ D t φ ) (cid:19) . (3.8)Consequently, the minimum of the action is reached when the fields satisfy the followingBPS-equations( a ) : F tθ ± D ϕ φ sin θ = 0 ( b ) : F tϕ ∓ sin θD θ φ = 0 ( c ) : F θϕ − µ sin θ ( µφ ∓ D t φ ) = 0 , (3.9)or in a compact and covariant notation √ gǫ ρνλ F νλ = ∓ D ρ φ + 2 µk ρ φ , (3.10)where k ρ is the Euclidean version of the time-like Killing vector of the metric on R × S .The vacuum equations (3.2) are just a particular case of (3.9) or equivalently (3.10). Theyemerge when we add the requirement of time-independence and vanishing of the chromo-electric field E i . From (3.8) it is manifest that all our vacua (3.3) possess a vanishingaction and they are all equivalent from an energetic point of view.It is natural to ask now what is the meaning of the Euclidean time-dependent solutionsof (3.9). The action on these solutions reduces to S class = ∓ µ Z S dθdϕ sin θ Z ∞−∞ dt∂ t Tr( φ ) , (3.11)which is finite, and thus relevant for a semiclassical analysis of the theory, if and onlyif φ ( t = −∞ ) = f −∞ µR and φ ( t = ∞ ) = f ∞ µR . In other words, these solutions are13nteresting if and only if they interpolate between two vacua: one at t = −∞ and theother at t = + ∞ . Their finite action is then given by S class = ∓ µ Z S sin θdθdϕ Z ∞−∞ dt∂ t Tr( φ ) = ∓ πg Y M R (Tr( f ∞ ) − Tr( f −∞ )) , (3.12)where we have reintroduced the relevant coupling constant factors. We recognize the char-acteristics of instantons in these (Euclidean) time-dependent solutions. At the quantumlevel, they will possibly induce a tunneling process between the different vacua. At zerotemperature Lin [45] discussed the effect of these instantons from the gauge theoreticalside, at weak coupling, and from the gravity side, that should describe the strong-couplinglimit of the theory (see also [46]), finding precise agreement in both regimes. Moreoverhe argued, in analogy with the plane-wave matrix model, that because of the presenceof fermionic zero-modes around these instanton solutions, the path-integral for the tun-neling amplitude is zero. The vacuum energies would not be corrected and the vacuaare exactly protected at the quantum mechanical level: in particular they should remaindegenerate. This kind of instantons has also been recently considered in [47].In the rest of the paper, in any case, we shall neglect the effect of these solutionssince we shall work at zero-coupling and in this limit the probability of tunneling isexponentially suppressed anyway. In this section we shall derive the finite temperature partition function in the BPS vacua(3.3), taking the limit g Y M R →
0. We follow a path-integral approach where the compu-tation is reduced to the evaluation of one-loop functional determinants in the monopolebackgrounds. Since at finite temperature the Euclidean time is a circle S of length β = 1 /T , we can also allow for a flat-connection a wrapping this S . The mode a willplay a very special role because it is the only zero-mode in the decomposition into Kaluza-Klein modes on S × S . Consequently, as stressed in [9], the fluctuations described by a are always strongly coupled, including in the limit g Y M R → U ( N ) gauge symmetry andthe final result for the partition function is given by a matrix integral over the unitarymatrix U = exp (cid:2) iβa (cid:3) Z ( β ) = Z [ dU ] exp ( ∞ X n =1 n (cid:2) z B ( x n ) + ( − n +1 z F ( x n ) (cid:3) Tr( U n )Tr( U − n ) ) . (4.1)The functions z B,F ( x ) are respectively the bosonic and fermionic single-particle partitionfunctions (here x = e − β ), counting the one-particle states of the theory without the The instantons are 1/2 BPS solutions and therefore we expect 8 fermionic zero-modes associated tothe broken supersymmetries
Adj in our case) and without any gauge invariant constraint z B,F ( x ) = X i e − βE ( B,F ) i . (4.2)The explicit form of the thermal partition function is obtained by integrating over thematrix U [8, 9] Z ( β ) = ∞ X n =0 x n E B ∞ X n =0 x n E B .. ∞ X m =0 x m E F ∞ X m =0 x m E F ... × { sym n ( Adj ) ⊗ sym n ( Adj ) ⊗ · · ·⊗ antisym m ( Adj ) ⊗ antisym m ( Adj ) ⊗ · · · } : (4.3)the partition function is expressed as a sum over the occupation numbers of all modes,with a Boltzmann factor corresponding to the total energy, and a numerical factor thatcounts the number of singlets in the corresponding product of representations. Particlestatistics requires to symmetrize (antysimmetrize) the representations corresponding toidentical bosonic (fermionic) modes.The same result can also be obtained starting from Z ( β ) = Tr (cid:2) e − βH (cid:3) ≡ Tr (cid:2) x H (cid:3) , (4.4)where H is the Hamiltonian of the theory. To calculate (4.4) at zero coupling we need acomplete basis of states of the free theory or, thanks to the state-operator correspondence,of gauge-invariant operators and we should count them weighted by x to the power oftheir energy. A complete basis for arbitrary gauge-invariant operators follows naturallyafter we specify a complete basis of single-trace operators. At the end, one can write (4.4)in terms of single-particle partition functions z RB,F ( x ) [9] as Z ( β ) = Z [ dU ] exp (X R ∞ X n =1 n (cid:2) z RB ( x n ) + ( − n +1 z RF ( x n ) (cid:3) χ R ( U n ) ) , (4.5)where the sum is taken over the representations R of the U ( N ) gauge group and χ R ( U )is the character for the representation R . The result (4.1) is reproduced when all fields arein the adjoint representation: the variable U has to be identified as the holonomy matrixalong the thermal circle, i.e. the Polyakov loop. The path-integral approach providestherefore a physical interpretation for the unitary matrix U , otherwise missing in theHamiltonian formalism. On the other hand the Hamiltonian construction explains howthe group integration forces the projection into color singlets and how it emerges thestructure of the full Hilbert space. We consider the possibility to have fields in an arbitrary representation. z RB,F . However, the structure of the gauge group is more complicated on monopolebackgrounds, consisting into a direct product of U ( N I ) factors: consequently our con-stituents fields transform also under bifundamental representations, producing additionalcomplications for the explicit expression of the matrix model. We also remark that bifun-damental fields can transform non-trivially under U (1) rotations and implementing theGauss’s law hides some subtleties in three dimensions, when background monopole fluxesare present [25]: this potential additional freedom could affect non-trivially the spectrumof physical operators in our theory. For the theory we are investigating, however, thefree-field spectrum is simply obtained by truncating the four-dimensional parent theory,suggesting that the N = 8 counting is conveniently performed through the relevant U (1)projection on the N = 4 single-particle partition functions. This is what we do in ap-pendix C, where we construct the projector that eliminates all the fields which are notinvariant under the U (1) and we derive, even in the non-trivial vacuum, the single-particlepartition functions for bosons and fermions. While this is certainly the quickest way toobtain these quantities, we prefer to adopt here a path integral approach which in turnprovides also the contributions of fermions and bosons to the Casimir energy and allowsfor a careful treatment of the fermion zero modes. In the path-integral computation allthe subtleties will be treated in the well-defined framework of the ζ -function regularizationprocedure and in this section we present only the final results, referring for the technicaldetails to appendix B. Let us first describe the contribution of the six SU (4) R scalars φ ij to the partition functionin the background (3.3) and in presence of the flat-connection a : it amounts to the eval-uation of the determinant of the scalar kinetic operator. We have to solve the associatedeigenvalue problem, i.e. − ˆ (cid:3) φ ij + µ φ ij + [ ˆ φ , [ ˆ φ , φ ij ]] = λφ ij , (4.6)where the hatted quantities are computed in the relevant background. In the followingwe shall drop the subscript ij and we shall consider just one field denoted by φ . Thetotal result at the level of free energy is then obtained by multiplying by six the single-component contributions. Since φ is a matrix-valued field, we shall expand it in theWeyl-basis, whose elements are the generators H i of the Cartan subalgebra and the ladderoperators E α φ = N − X i =1 φ i H i + X α ∈ roots φ α E α . (4.7)16e shall also expand the background fields in this basis and define the following twoaccessory quantities a α = h α | a i and q α = h α | f i . (4.8)Here a α denotes the projection of the flat-connection a along the root α and q α is theeffective monopole charge measured along the same root. Once the time-dependenceis factored out, the original eigenvalue problem splits into two subfamilies: N ( N − N − a α , q α →
0. The relevant eigenvalue equation can be solved algebraicallyif we introduce the angular momentum operator in the presence of a U (1) monopole ofcharge q α , as explained in appendix B, and the resulting spectrum does not depend onthe sign of q α . By using ζ -function regularization, the scalar contribution to the effectiveaction can be easily computed asΓ Sc. = X α ∈ roots | q α | (cid:0) | q α | − (cid:1) βµ + ∞ X n =1 z scal.q α ( x n ) n e inβa α ! +( N − ∞ X n =1 z scal. ( x n ) n , (4.9)where the scalar single-particle partition function is given by z scal.q α ( x ) = x | q α | +1 / (cid:18) x (1 − x ) + 2 | q α | − x (cid:19) . (4.10) Evaluating the contribution of the system ( A µ , φ ) is more subtle and involved: the fieldsare coupled through the Chern-Simons term and the Lagrangian for A µ requires a gauge-fixing procedure, with the consequent addition of a ghost sector. A convenient choice forsuch a gauge-fixing appears to be L g.f. = ( ˆ D ν A ν − i [ ˆ φ , φ ]) , (4.11)where ˆ φ = µ f and the hatted derivative is defined in (B.19). With this choice someof the mixing-terms in the Euclidean quadratic Lagrangian cancel and we obtain therelevant eigenvalue-problem for computing the vector-scalar contribution to the partitionfunction: it is defined by the system of coupled equations, written explicitly in (B.26).Since both the geometrical and the gauge background are static, the time-componentof the vector field A decouples completely from the eigenvalue system and satisfies themassless version of the scalar equation previously studied. For the moment we shall forgetabout A since its contribution will be cancelled by the ghost determinant. We are left witha purely two-dimensional system where all the indices run only over space: the spectrumis again conveniently determined by factoring out the time-dependence and projecting theeigenvalue equations on the Weyl basis. We remark that the equations involve also the17aplacian on vectors in the background of a monopole of charge q α , besides the Laplacianon scalars. The full computation of the spectrum is reported in appendix B: we obtainedthree families of eigenvalues, denoted by λ + , λ − and λ . The contribution of λ will becancelled by the ghost determinant and we just consider, at the moment, the first twofamilies λ ± , which instead yield the actual vector determinant in the roots sectorΓ Vr = X α ∈ roots − (cid:0) q α + 5 q α (cid:1) βµ − ∞ X n =1 z vec.q α ( x n ) n e inβa α ! , (4.12)where z vec.q α ( x ) = x q α (cid:20) x (1 − x ) − q α x − x (cid:21) . (4.13)We remark that the results (4.12) and (4.13) were shown to hold under the initial as-sumption q α ≥
1. The extra-cases to be considered are q α = ,
0. By recomputing thespectrum for q α = 1 / q α = 0 and we getΓ Vr ( q α = 0) = − ∞ X n =1 z vec. ( x n ) n e inβa α with z vec ( x ) = 4 x (1 − x ) , (4.14)a factor − N − q α by simply replacing q α with | q α | . A Let us discuss now the contributions to the partition function of the eigenvalues λ , ofthe field A and of the determinant of ghost operator − ˆ (cid:3) · +[ ˆ φ , [ ˆ φ , · ]] : (4.15)they do not cancel completely but, importantly, they give a measure of integration for theflat-connection. It is possible to show that when q α = 0 we have a complete cancellationof the different contributions: crucially for q α = 0 this does not happen and a modificationof the measure for the flat-connection is induced Y α ∈ rootswith qα =0 ie − i βaα sin (cid:18) βa α (cid:19) = Y α ∈ positive rootswith qα =0 (cid:18) βa α (cid:19) . (4.16)The meaning of this measure is quite transparent: the monopole background breaks theoriginal U ( N ) invariance to the subgroup Q kI =1 U ( N I ), (4.16) being the product of theHaar measure of each U ( N I ) component, as can be easily checked by recalling the explicitform of the roots and the definition of q α . As a matter of fact, in non-trivial monopolebackgrounds, when we shall write the integral over the flat-connections we will be naturallyled to consider a unitary multi-matrix model instead of an ordinary one.18 .4 Fermions
The contribution of the fermions to the total partition function needs a careful analysis.At first sight, apart from having antiperiodic boundary conditions along the time circle,the computation of the fermion determinants seems to follow closely the bosonic cases.We have again N ( N −
1) independent eigenvalues coming from each direction along theladder generators and N − D ( α ) (see app. B.4) on the two-sphere, in the effective monopole backgrounds provided by q α . The spectrum of D ( α ) , asexpected in two dimensions, consists in a set non-vanishing eigenvalues, symmetric withrespect the zero, and in a finite kernel, as predicted by the Atiyah-Singer theorem. Thesezero-modes are chiral and can be classified by using the eigenvalues of the operator ( σ · ˆ r ),playing the role of γ : we shall denote ν ± the number of zero modes with eigenvalue ± ν + = | q α | − q α and ν − = | q α | + q α ,namely for positive q α we have only zero modes with negative chirality and viceversa. Asshown in appendix B.4, the contribution of the first set of eigenvalues to the effectiveaction can be easily evaluatedΓ S = X α ∈ roots − βµ (cid:0) | q α | + 3 | q α | + | q α | (cid:1) − ∞ X n =1 ( − n n z spin.q α ( x n ) e iβna α ! , (4.17)with z spin.q α ( x ) = 2 x | q α | +1 (cid:18) − x ) + | q α | − x (cid:19) ( x + x − ) . (4.18)Next we consider the contribution of the zero-modes of the effective Dirac operators:in a monopole background, this subsector originates the spectral asymmetry [48] of thethree dimensional fermionic operator and therefore the potential appearance of a parityviolating part in the effective action. In particular, we could expect the generation of theChern-Simons anomalous term (we refer to [33, 34] for a complete discussion of this issue).Concretely, in our case, the explicit computation of the zero-mode contribution amountsto evaluate a family of one-dimensional massive fermion determinants, in a flat-connectionbackground (see appendix B.4). It is well-known that the ζ -function regularization schemecarries an intrinsic regularization ambiguity in this case, depending on the choice of somebranch-cuts in the s -plane, affecting the local terms in the effective action [33, 34]. Forus all the different possibilities boil down to two alternatives: we can regularize thecontributions associated to the zero-modes of negative and positive chirality by choosingopposite cuts in defining the complex power of the eigenvalues (one on the real positiveaxis and the other on the real negative axis) or by choosing the same cut. We find quite This ambiguity is not something peculiar of the ζ -function regularization, but it appears in differentforms also in other regularizations: in the usual Pauli-Villars approach, for example, this ambiguitytranslates into a dependence of the local terms in the effective action on the sign of the mass of theregulator. all the four fermions present in the theory: we surelypreserve the R -symmetry and the global non-abelian symmetry in this way. Within thischoice, the following results hold from our one-dimensional fermion determinants: takingopposite cuts we getΓ S ,A = X α ∈ roots (1 − r ) βµ (cid:18) q α + | q α | (cid:19) − X α ∈ roots ∞ X n =1 ( − n n | q α | x n | q α | e iβna α x n . (4.19)Here r = ± S ,B = X α ∈ roots h βµ (cid:18) | q α | + | q α | (cid:19) + irβa α q α i − X α ∈ roots ∞ X n =1 ( − n n | q α | x n | q α | e iβna α x n . (4.20)Again r = ± ir X α βq α a α = irβ ( N Tr( a f ) − Tr( a )Tr( f )) . (4.21)We immediately recognize the SU ( N ) part of the usual Chern-Simons term, calculatedin our particular background. The related regularization choice is therefore consistentwith the intrinsic parity anomaly of three dimensional gauge theories. We stress that theabove contribution arises just in the monopole vacua and it is related to non-perturbativeproperties of the fermion determinants. We also observe that the two results differ in thecharge-dependent contribution linear in β , and we will see this to modify crucially theCasimir energy.Summing now, in both cases, the kernel contribution to Γ S we getΓ SA = X α ∈ roots − βµ (cid:0) | q α | + 12 r | q α | + (3 r + 1) | q α | (cid:1) − ∞ X n =1 ( − n n z spin.q α ( x n ) e iβna α ! , (4.22)with the first choice andΓ SB = X α ∈ roots − βµ (cid:18) | q α | + | q α | (cid:19) + irq α a α − ∞ X n =1 ( − n n z spin.q α ( x n ) e iβna α ! , (4.23)in the latter. Happily the single-particle partition function is the same for both theregularization choices z spin.q α ( x ) = x | q α | (cid:18) x (1 − x ) + 2 | q α |√ x − x (cid:19) ( x + x − ) . (4.24)The contribution of the Cartan components is of course obtained from the above resultsby simply setting q α = 0. 20 .5 Partition functions The next step is to collect the different contributions, coming from the functional deter-minants, and write down the total result as a compact integral over unitary matrices.According to the previous discussion, we must distinguish two cases, depending on theform of the spinor determinant (4.22) or (4.23). We shall first consider the choice (4.22).The complete effective action, obtained by including roots and Cartan contributions withthe appropriate multiplicities, can be expressed as S eff. = − βV + X α ∈ roots ∞ X n =1 n (6 z scal.q α ( x n ) + z vec.q α ( x n ) + ( − n +1 z spin.q α ( x n )) e inβa α ++ ( N − ∞ X n =1 n (6 z scal. ( x n ) + z vec. ( x n ) + ( − n +1 z spin. ( x n )) ≡≡ − βV + X α ∈ roots ∞ X n =1 n z tot.q α ( x n ) e inβa α + ( N − ∞ X n =1 n z tot. ( x n ) , (4.25)where we have introduced the total single-particle partition functions and the Casimirenergy V of the configuration V = r X α ∈ roots (4 | q α | + | q α | ) . (4.26)The matrix structure hidden in (4.25) appears manifest when writing the original Polyakovloop U = exp( iβa ), associated to the diagonal flat-connection a , through k sub-matrices U I acting on the invariant subspaces implicitly defined by the monopole background (3.4).The N I × N I unitary matrices U I have the form U I = diag( e iβa I , . . . , e iβa INI ), where wehave parameterized the original flat connection a as follows: a = diag( a , . . . , a N | {z } N , a , . . . , a N | {z } N , . . . . . . , a I , . . . , a IN I | {z } N I , · · · ) . (4.27)Let us consider now the subset A IJ of the positive roots of SU ( N ) whose first and secondnon vanishing entries belong respectively to the I th and J th invariant subspace of f . Theeffective charges q α = h α | f i = n I − n J and, consequently, the z tot.q α take always the same valuefor this class of roots. The sum over roots on this subset reduces to X α ∈A IJ e inβa α = N I X i =1 N J X j =1 e inβ ( a Ii − a Jj ) = Tr( U nI )Tr( U † nJ ); (4.28)the analogous subsector ¯ A IJ given by the negative roots yields Tr( U † nI )Tr( U nJ ). We remarkthat the pre-factor z tot.q α is however the same for both cases since it depends just on the The roots of SU ( N ) are all the N ( N −
1) permutations of the N − vector (1 , − , , · · · ,
0) and theycan be separated in positive and negative according to the sign of the first non zero entry. B I whose first and secondnon vanishing entries live in the same I th invariant subspace of f have instead effectivemonopole charge zero. Then the contribution of this subsector is simply given by ∞ X n =1 n z tot. ( x n ) X α ∈B I e inβa α = ∞ X n =1 n z tot. ( x n ) N I X i = j =1 e inβ ( a Ii − a Ij ) == ∞ X n =1 z tot. ( x n ) n (Tr( U † nI )Tr( U nI ) − N I ) . (4.29)Because of the results (4.28) and (4.29), it is convenient to change our notation andto define the k × k matrix-valued single-particle partition function z tot.IJ : the diagonalelements are z tot.II = z tot. , the off-diagonal ones are instead identified with the function z tot.q α ,associated to the charge n I − n J . The matrix z tot.IJ is symmetric since everything dependsjust on the modulus of the charge. The complete effective acton takes the elegant form S eff. = − βV + X IJ ∞ X n =1 n z tot.IJ ( x n )Tr( U nI )Tr( U † nJ ) − ∞ X n =1 n z tot.II ( x n ) . (4.30)The last term drops if we consider U ( N ) instead of SU ( N ). Remarkably the structureof the matrix action is perfectly consistent with the measure found in (4.16), which isexactly the Haar measure for this multi-matrix model.The above analysis is practically unaltered when considering the fermionic contribution(4.23) in the effective action, except on a couple of points. It changes the value of theCasimir energy V , which now vanishes identically, and we have a new important additionto (4.30), that can expressed in terms of the determinants of the unitary matrices U I irβ X α ∈ roots q α a α = log k Y I =1 det( U I ) r ( Nn I − Q ) ) ! = r k X I =1 ( N n I − Q ) log(det( U I )) , (4.31)where Q = P kI =1 N I n I . As a first remark, we notice that new contributions depends stillon the differences n I − n J , consistently with the decoupling of the total U (1) charge of themonopole configuration. Then we observe that the two different values r = ±
1, relatedto our regularization choice, produce the same result when integrating over the unitarygroup: the difference can be reabsorbed just changing integration variable U I ( U I ) − ,which leaves the measure and (4.30) unaltered. From now on, we shall set r = 1.In the trivial vacuum we obtain a partition function that is a straightforward gener-alization of the unitary matrix model discussed in [9] Z = Z dU exp ∞ X n =1 n z tot. ( x n )Tr( U n )Tr( U † n ) ! (4.32)where the function z tot. ( x n ) encodes the dynamical content of the three-dimensional su-persymmetric theory. Notice that the Casimir energy is identically zero, since it vanishesfor each contribution both bosonic and fermionic.22he situation changes in non-trivial monopole vacua: we get respectively Z A = Z k Y I =1 [ dU I ] exp − βV + X IJ ∞ X n =1 n z tot.IJ ( x n )Tr( U nI )Tr( U † nJ ) ! (4.33)and Z B = Z k Y I =1 [ dU I ] exp X IJ ∞ X n =1 n z tot.IJ ( x n )Tr( U nI )Tr( U † nJ ) ! k Y I =1 det( U I ) ( Nn I − Q ) , (4.34)depending on our regularization choice. First of all we see that the partition functionis related to a unitary multi-matrix model: the gauge group is broken in factors andstates in the bifundamental representation are present, with energies clearly encoded intothe off-diagonal entries of the single-particle partition function z tot.IJ . Let us discuss ongeneral grounds the effects of the different choices for the fermion determinants. A firstmild diversity arises in the Casimir energies: from (4.22) we have a non-vanishing V ,with arbitrary sign, while (4.23) leads to a vanishing result. We recall that the Casimirenergy is supposed to correspond to the mass of the dual geometry [39]: in the firstcase it seems that different backgrounds supports different, monopole dependent, masses,suggesting a possible lifting of the vacua degeneracy at quantum level. The second choiceis instead consistent with the believed degeneracy: unfortunately no computation fromthe gravitational side seems to be available up to now and we do not have further insightson the meaning of the different results.The presence of the new terms (4.31) in the matrix model (4.34) can be, instead, betterunderstood at the level of partition functions. First of all we notice that the matrixintegral implementing the Gauss’s law is actually over unitary matrices U I : the U (1)phases contained into the the U I ’s play a non-trivial role in the monopole background.This has to be contrasted with the trivial vacuum: there the effective action is invariantunder U (1) transformations and we can simply forget the integration over the center. Inthe non-trivial vacuum the resulting effective action (4.30) is not invariant under phaserotations, as an effect of the off-diagonal terms in the single-particle partition function,and the U (1) integrations precisely correspond to selection rules in the bifundamentalsector. It is not difficult to realize that within the first regularization the matrix integralsselect states having vanishing U (1) charge, with respect to all U ( N I ) group factors. Tounderstand the effect of the new terms in (4.34) instead, we simply observe that thedeterminants depend just on the U (1) phases and modify non-trivially the selection rulesof the bifundamental sectors, according to the charges of the monopole background. Weshall say in this case that our regularization procedure correspond to the choice of acharged vacuum, as discussed in [26], while we will refer to the first possibility as tothe uncharged vacuum. Since at the quantum field theory level both choices seems tobe allowed, we think it is instructive to investigate the thermodynamics in both cases,deferring a deeper understanding of the different possibilities to future studies, in thecontext of supersymmetry and gravitational duals.23e end this section introducing the simple modification to the effective action due tochemical potentials for the SU (4) R -charge. In the path integral approach their effectamounts to simply adding an imaginary SU (4) flat connection A R = i (Ω Q R + Ω Q R +Ω Q R ) in the Euclidean time direction. Here Q R i are the Cartan generators of SU (4) and R denotes the relevant representation: for the spinors and for the scalars. One findsthe new partition functions4 z spin.q α z spin.IJ = x | q α | x (1 − x ) X p =1 (cid:16) x y − e Ω p + x − y e Ω p (cid:17) ++2 | q α | x − x X p =1 (cid:16) y − e Ω p + x y e Ω p (cid:17)! , z scal.q α z scal.IJ = x | q α | +1 / (cid:18) x + 1(1 − x ) + 2 | q α | − x (cid:19) X p =1 (cid:0) y Ω p + y − Ω p (cid:1) , (4.35)with y = e − β and˜Ω = 12 (Ω + Ω + Ω ) ˜Ω = 12 (Ω − Ω − Ω )˜Ω = 12 ( − Ω + Ω − Ω ) ˜Ω = 12 ( − Ω − Ω + Ω ) . (4.36) We have seen in the previous section that the thermodynamics in the trivial vacuumis governed, in the zero-coupling approximation, by the one-component unitary matrixmodel Z = Z dU exp ∞ X n =1 n z tot. ( x n )Tr( U n )Tr( U † n ) ! (5.1)where the function z tot. ( x n ) encodes the dynamical content of the three-dimensional su-persymmetric theory. Notice that the Casimir energy is identically zero, since it vanishesfor each contribution both bosonic and fermionic.When N is large we can trade the integration in (5.1) over the unitary group for anintegration over the normalized distribution function ρ ( θ ) of the continuous eigenvalues e iθ of U , with − π < θ ≤ π . More precisely we can write the integral over the unitarymatrices in terms of the Fourier-modes ( ρ n , ¯ ρ n ) defined as ρ ( θ ) = 12 π + ∞ X n =1 ( ρ n e inθ + ¯ ρ n e − inθ ) . (5.2)Following [8, 9], we can then reduce the integral to the standard form Z = Z Dρ n D ¯ ρ n exp − N ∞ X n =1 ρ n ¯ ρ n V ( x n ) ! with V ( x n ) = 1 n (1 − z tot. ( x n )) . (5.3)24n the large N limit, (5.3) is dominated by the absolute minimum of the quadratic action S = P ρ n ¯ ρ n V ( x n ) which is reached for ρ n = 0 for every n if V ( x n ) is positive definite. Forsmall temperatures, namely small x , the function V ( x n ) is positive for any n and closeto 1 /n since V ( x n ) ∼ n for x ≪
1. (Recall that z tot. ( x n ) vanishes as x approaches zero.)Therefore the partition function is 1 at the leading order and it is simply given by thesmall fluctuation around the minimum at the subleading order: Z ∝ ∞ Y n =1 − z tot. ( x n )) . (5.4)When we increase the temperature, x approaches 1 and the above description is reliableup to the smallest value x c where V ( x n ) becomes negative. Since z tot. ( x ) is a monotonicfunction ranging from 0 to infinity, this value always exists and it is reached for n = 1,namely V ( x c ) = 1 − z tot ( x c ) = 0 . (5.5)This algebraic condition, whose explicit form is V ( x c ) = 1 − z tot. = 1 − (4 z spin. + 6 z scal. + z vec. ) == (cid:0) √ x c + 1 (cid:1) (cid:0) x c − x c / + 4 √ x c − √ x c + 1 (cid:1) (1 − x c ) = 0 , (5.6)can be exactly solved, since it can be reduced to an equation of fourth degree. It possessesjust one solution in the interval [0 ,
1] given by x c = (cid:18) √ − q
11 + 8 √ (cid:19) ≃ (0 . . (5.7)It is interesting to compare this value with the critical temperature computed in [39] for N = 4 on S / Z k . This theory should in fact reproduce our model when k goes to infinity.However, the three-dimensional theory obtained in this limit lives on a S sphere whoseradius is half of the radius of the original S : this means that x c = lim k →∞ x c ( k ). Tofacilitate the comparison with the four dimensional literature and in particular with theresults of [20, 39] in what follows we shall replace the basic variable x with x . In [39] the x c ( k ) for k = 10 is 0 . ρ n = ¯ ρ n = 0 and one has to look for other saddle-points [8, 9]. Following [9], onecan easily show that above x c the dynamics is governed by a distribution different fromzero only in the interval [ − θ , θ ] and given, in first approximation , by ρ ( θ ) = cos (cid:0) θ (cid:1) π sin (cid:0) θ (cid:1) s sin (cid:18) θ (cid:19) − sin (cid:18) θ (cid:19) with cos (cid:18) θ (cid:19) = s − z tot. ( x ) . (5.8)This behavior at x c produces a first-order transition with the same qualitative character-istics of the four-dimensional model. We are assuming that the relevant features are completely captured by the first mode n = 1. .1 Chemical potentials A natural and intriguing generalization is to add chemical potentials for the R -charges,while maintaining the trivial vacuum as a gauge background.The critical equation has still the form (5.6) but 4 z spin. and 6 z scal. are substituted by4 z spin. x (1 − x ) X p =1 (cid:16) x y − e Ω p + x − y e Ω p (cid:17) z scal. x / x + 1(1 − x ) X p =1 (cid:0) y Ω p + y − Ω p (cid:1) , (5.9)which is (4.35) for q α = 0. The effect of small chemical potentials can be easily computedby treating them as a perturbation and expanding around (Ω , Ω , Ω ) = (0 , , T H (Ω) = T H (0) − . X i =1 Ω i − . Y i =1 Ω i − . X i =1 Ω i − . X i 0) and(Ω , Ω , Ω ) = (Ω , Ω , Ω). In all three cases, the behavior around Ω = 1, in a trivial vac-uum background, is similar to that of the N = 4 theory in four dimensions discussedin [19, 20]. We find, in fact:(Ω , Ω , Ω ) = (Ω , , 0) : T H = − − Ω) (cid:20) − log ( − log[1 − Ω])log(1 − Ω) + . . . (cid:21) , (Ω , Ω , Ω ) = (Ω , Ω , 0) : T H = 1 − Ωlog 2 (cid:20) − e − log 22(1 − Ω) + O ( e − log 2(1 − Ω) ) (cid:21) , (Ω , Ω , Ω ) = (Ω , Ω , Ω) : T H = 1 − Ωlog 4 (cid:20) − e − log 4(1 − Ω) + O ( e − log 4(1 − Ω) ) (cid:21) , (5.11)which have the same qualitative behavior of the analogous equations found in [20] forthe N = 4 theory. This similarity suggests the possibility to consider decoupling limitsanalogous to those performed in [20] for the N = 4 theory. This might help to singleout some subsectors of the present model with simple properties at the (full) quantumlevel [22, 23, 24]. However, this analysis is left for future research.26 .2 0.4 0.6 0.8 1 W c Figure 1: The continuous, dashed and dot-dashed lines correspond to (Ω , Ω , Ω ) = (Ω , , , Ω , Ω ) = (Ω , Ω , 0) and (Ω , Ω , Ω ) = (Ω , Ω , Ω) respectively. All the curves reach Ω = 1 when x c approaches zero. In the high temperature regime the eigenvalue distribution becomes almost like a delta-function [9]. Therefore ρ n = 1 and the free energy can be evaluated by looking at theexpression of the functional determinants in the background of vanishing flat-connections.When the chemical potentials are strictly zero the leading contribution to the free energy F = − T log Z is (see (B.9), (B.14) and (B.15)) F = − π ζ (3) V ( S ) N T + O ( T ) . (5.12)We see that the limiting free energy density here coincides precisely with that of the N = 8 super Yang-Mills theory in flat three-dimensional space. Taking the dimension-less parameter T R to infinity is equivalent to taking the limit of large volume at fixedtemperature, loosing in this way any memory of the original deformed supersymmetry.We can also notice that no dependence appears, at the leading order, on the particularmonopole vacuum on which the expansion has been performed and the result (5.12) isactually general.It is interesting to consider the corrections to this result when chemical potentialsare taken into account. The first non-trivial contribution is easily evaluated by using theexpansions of Li ( z ) presented in (B.15): we simply notice that chemical potentials appearas imaginary parts of the flat-connections and are contained in the variable z introduced inthe appendix B.1. Summing carefully the contributions coming from bosons and fermions,we obtain the free energy F = − V ( S ) N T " π ζ (3) + X i =1 y i π (cid:18) − log y i (cid:19) + O ( T ) , (5.13)27here we introduced the relevant combination y i = Ω i /T . This result is perfectly con-sistent with the computation performed in [49], for a system of N free D2 branes in thepresence of chemical potentials. We shall first consider the multi-matrix model, (4.33), which originates from the unchargedvacuum. We recall that in this case the partition function is defined by the matrix integral Z A = Z k Y I =1 [ dU I ] exp − βV + X IJ ∞ X n =1 n z tot.IJ ( x n )Tr( U nI )Tr( U † nJ ) ! , (6.1)where V = r X α ∈ roots (4 | q α | + | q α | ) (6.2)is the Casimir energy. The value of the Casimir energy is puzzling not only for the r dependence, making its sign ambiguous, but also because it depends on the charge of thevacuum q α so that it is different for different vacua. At the supergravity level we expectinstead these vacua to be degenerate. This last feature is reproduced within our secondregularization choice, giving a vanishing Casimir energy and consequently degeneratevacua: the price we pay is the introduction of the logarithmic interactions (4.34) that willbe studied in the next section.The large N -limit of the matrix-model (6.1) is investigated by generalizing to a multi-dimensional case the technique presented in the previous section: we introduce the densityfunctions ρ I ( θ I ) associated to the matrices U I and in terms of the Fourier-modes ρ In ρ I ( θ I ) = 12 π + ∞ X n =1 ( ρ In e inθ I + ¯ ρ In e − inθ I ) , (6.3)the matrix integral (6.3) reduces as well to an infinite set of independent gaussian integrals Z A = Z k Y I =1 Dρ In D ¯ ρ In exp − βV − N X IJ ∞ X n =1 ρ In ¯ ρ Jn n ( δ IJ − z tot.IJ ( x n )) s I s J | {z } V IJ ( x n ) , (6.4)where we have introduced the filling fractions s I = N I /N . In the large N limit (6.4) isdominated by the absolute minimum of the quadratic action S = X IJ ∞ X n =1 ρ In ¯ ρ Jn V IJ ( x n ) , (6.5)which is given by ρ In = 0 for every I and n if the quadratic form V IJ ( x n ) is positive definite.For small temperatures, namely small x , the eigenvalues of the matrix V IJ ( x n ) are all28ositive and close to 1 /n since V IJ ( x n ) ∼ n δ IJ for x ≪ z tot.IJ ( x n ) vanishesas x approaches zero). Therefore the partition function is simply given by the Casimircontribution at the leading order and by the small fluctuation around the minimum atthe subleading order Z A ∝ e − βV ∞ Y n =1 V IJ ( x n )) . (6.6)When we increase the temperature, x approaches 1 and the above description is reliableuntil the quadratic form V IJ ( x n ) develops the first negative eigenvalue. This occurs at thesmallest x c for which one of the eigenvalues of V IJ ( x n ) vanishes, or equivalently for whichdet ( V IJ ( x nc )) = 0 . (6.7)The smallest x c = e − /T c , namely the smallest critical temperature, is obviously obtainedfor n = 1 which provides the strongest condition. Moreover this critical value alwaysexists since z tot.IJ ( x ) is a monotonic function ranging from 0 to infinity when x ∈ [0 , f with just twosectors of equal length. It is given by f = ( n , . . . , n , n , . . . , n ) . (6.8)The z IJ and thus the critical temperature depend only on the absolute effective charge,namely q = | n − n | / 2. This property reflects the fact that the global U (1) sector ofcharge ( n + n ) / N limit, since thereare no degree of freedom which couples to it. We also observe that the critical equationis independent of the filling fractions s I and it is obtained by requiring the vanishing ofthe determinantdet (cid:18) − z tot. ( x ) − z tot. ( x ) − z tot. − z tot. ( x ) (cid:19) = (1 − z tot. ) − ( z tot. ) = 0 , (6.9)where we have used that the matrix V IJ is symmetric ( z tot. = z tot. ) and that z tot. = z tot. = z tot. is the partition function in the trivial vacuum. This equation naturally splits intotwo simpler equations ( a ) : λ − ( x ) = 1 − z tot. ( x ) − z tot. ( x ) = 0 (6.10)( b ) : λ + ( x ) = 1 − z tot. ( x ) + z tot. ( x ) = 0 . (6.11)The critical temperature is determined by the lowest zero of these two equations. Since λ + − λ − = 2 z ≥ λ + (0) = λ − (0) = 1, λ − ( x ) reaches its zero at a smaller tempera-ture: in determining x c we can then neglect λ + ( x ).From the structure of the critical equation, λ − ( x ) = 0, we can deduce two generalproperties of the critical temperature. First, the positivity of z tot also ensures that λ − ( x ) ≤ λ ( x ) = (1 − z tot ). This means that the critical temperature in a non-trivial monopole29ackground will always be smaller than the corresponding one in the trivial vacuum.Second, the function z tot. decreases with the monopole charge q (in the interval x ∈ [0 , q approaches infinity the value of the critical temperature becomes that of the trivialvacuum. Below we present a table for the critical temperature, where the behaviorsdescribed above are manifest q x c T c / . . . . / . . . . / . . . . / . . Table 1: x c and T c in the two sectors situation as a function of the relative monopole charge q . When the number k of sectors grows, the dependence of the critical temperature T c onthe relative monopole charges becomes quite intricate. However, some general behaviorscan be anticipated. Consider, for example, a generic background of the form f = ( n , . . . , n , n , . . . , n , . . . . . . , n k , . . . , n k ) , (6.12)where the induced relative monopole charges q IJ = | n I − n J | n I and n J are very different from each other. Then the Hagedorn tem-perature is dominated by the smallest charge and the off-diagonal terms associated to theother charges can be considered as small perturbations. The determinant is approximatelygiven by det( V IJ ) ≈ (1 − z ) k − ((1 − z tot ) − ( z totq min ) ) . (6.14)Exploiting what we have learned for the k = 2 system, the lowest transition temperatureis an approximate solution of the equation 1 − z tot − z totq min = 0.Another interesting family of configurations is built by considering long sequences ofsectors with equal length and monopole charge increasing by a fixed value q , namely f = ( n , . . . , n , n + q , . . . , n + q , n + 2 q , . . . , n + 2 q , . . . . . . , n + k q , . . . , n + k q ) . (6.15)When the number of sectors k goes to infinity, the Hagedorn temperature in these vacuaapproaches that of N = 4 on the Lens space S / Z q in the sector described by a vanishingflat-connection. For example for q = 1, a simple numerical analysis shows that T c goes tothat of pure N = 4, T D =4 c = − / log(7 − √ ≃ . x c T c . . . . . . . . . . . . . . Table 2: x c and T c in the k sectors situation at Ω = 0. The vacua are labelled by f k = diag( k − , ..., k − , ..., Analytically, this result can be argued by noting that the matrix V IJ , of which we have tocompute the determinant, is of Toeplitz type, namely a matrix in which each descendingdiagonal from left to right is constant. Consequently its entries do not depend on I and J separately, but only on the difference I − J . For this kind of matrices, when thedimension is large, the determinant is approximated by that of their circulant version [50].This means that the smallest zero of the determinant can be found as a solution of1 − ∞ X k = −∞ z tot. ( k q , x ) = 0 , (6.16)which is the smallest eigenvalue of the corresponding circulant matrix. In (6.16) z tot. ( k q , x )is the single-particle partition function in the sector of charge k q . It is now possible toshow that this infinite sum produces the single-particle partition function of the N = 4SYM theory in the trivial vacuum of S / Z q (see [39] for comparison). In other words (6.16)coincides with the critical equation for the N = 4 SYM theory in the trivial vacuum of S / Z q .Finally we consider the addition of chemical potentials to a monopole configurations.Their introduction does not alter significantly the picture and a numerical analysis isgiven in fig. 2. To understand what happens when we cross the critical temperature, we shall now focusour attention on the two-sectors configuration (6.8). In this case, if we introduce thecombination ρ ± = 12 ( ρ ± ρ ) , (6.17)the action takes a diagonal form S = 2 ∞ X n =1 (cid:18) n λ − ( x n ) ¯ ρ + n ρ + n + 1 n λ + ( x n ) ¯ ρ − n ρ − n (cid:19) . (6.18)Above the critical temperature, λ − ( x ) is negative and the dominant saddle-point is nolonger realized by a flat distribution ρ n = ρ n = 0 ( ρ + n = ρ − n = 0). In fact, as31 .2 0.4 0.6 0.8 1 W c Figure 2: Transition lines for three sectors vacuum: f = diag( i, ..., , ..., − i ). Narrow lines correspondsto i = 1, thick lines to i = 10. The convention for continuous, dashed and dot-dashed are those of fig. 1.The qualitative behavior is the same for every number of sectors. the temperature is increased, the attractive term in the pairwise potential continues toincrease in strength, so the eigenvalues become increasingly bunched together, occupying,at the end, only a finite interval I = [ − θ , θ ] on the circle (we arbitrarily choose themiddle of this interval to be at θ = 0 for convenience). However, since λ + ( x ) is stillpositive, we can safely assume that the new dominant saddle point satisfies ρ − n = 12 ( ρ n − ρ n ) = 0 , i . e . ρ = ρ = ρ + . (6.19)In other words, the problem reduces to an effective one matrix model governed by theaction S =2 Z dθdθ ′ ρ + ( θ ) ρ + ( θ ′ ) ∞ X n =1 (cid:20) ( λ − ( x n ) − n cos( n ( θ − θ ′ )) (cid:21) ++ 2 Z dθdθ ′ ρ + ( θ ) ρ + ( θ ′ ) log (cid:12)(cid:12)(cid:12)(cid:12) sin θ − θ ′ (cid:12)(cid:12)(cid:12)(cid:12) , (6.20)where the distribution function has support in the interval [ − θ , θ ]. In complete analogywith what we found in trivial vacuum case (6.21), we have ρ ( θ ) = cos (cid:0) θ (cid:1) π sin (cid:0) θ (cid:1) s sin (cid:18) θ (cid:19) − sin (cid:18) θ (cid:19) with cos (cid:18) θ (cid:19) = s λ − ( x ) λ − ( x ) − . (6.21)Near the critical temperature, for T > T H we have the following expansion for the partitionfunction FN = T H λ − ( x ) + O (( λ − ) ) = T H T − T H ) ∂λ − ∂T (cid:12)(cid:12)(cid:12)(cid:12) T = T H + O (( T − T H ) ) , (6.22)32hich gives the characteristic first-order transition, already found in the four-dimensionalmodel. We discuss now the thermodynamical behavior arising when the second regularizationscheme, considered for the fermions in section 4, is adopted. As previously derived, anon-trivial logarithmic deformation of the multi-matrix model (6.1) has to be considered Z B = Z k Y I =1 [ dU I ] exp X IJ ∞ X n =1 n z tot.IJ ( x n )Tr( U nI )Tr( U † nJ ) ! k Y I =1 det( U I ) ( Nn I − Q ) . (7.1)To illustrate the effect of the new interactions on the large N dynamics, we shall make avery drastic assumption and we shall focus our attention just on the first winding, n = 1.With this choice the original matrix integral reduces to Z k Y I =1 DU I exp X IJ z tot.IJ ( x )Tr( U I )Tr( U † J ) ! k Y I =1 det( U I ) Nn I − Q . (7.2)It is useful, as a first step, to introduce a set of k complex Lagrange multipliers λ I andthe partition function can be written as Q kI =1 N I (det( z IJ )) k Z k Y J =1 dλ J d ¯ λ J exp( − X IJ N I N J ¯ λ I z − IJ ( x ) λ J ) × k Y I =1 Z DU I exp (cid:16) ¯ λ I N I Tr( U I ) + λ I N I Tr( U † I ) (cid:17) det( U I ) Nn I − Q . (7.3)Next we use the polar decomposition λ I = γ I e iα I for each Lagrange multipliers. Thephases e iα I are then decoupled from the matrix integration by means of the change ofvariables U I U I e iα I . This procedure yields the following integral Q kI =1 N I (det( z IJ )) k Z k Y J =1 dγ I dα I exp − X IJ N I N J γ I z − IJ ( x ) e i ( α J − α I ) γ J + i k X I =1 N I ( N n I − Q ) α I ! × k Y I =1 Z DU I exp (cid:16) γ I N I (Tr( U I ) + Tr( U † I )) (cid:17) det( U I ) Nn I − Q . (7.4)In (7.4) the group integrations over the unitary matrices U I are completely decoupled.Each matrix integration corresponds to a Gross-Witten model [35] with a coupling γ I and an additional logarithmic potential proportional to log(det( U I )). We remark thatthese kinds of deformations for unitary matrix models were widely considered in the33arly eighties (see e.g. [27, 28]). The determinant operator was expected to act as anorder parameter for the large N phase transitions characterizing this class of models[51]: unfortunately, we cannot simply borrow the old results. In (7.4) in fact we havea new and decisive ingredient with respect to the original investigations: the power ofthe determinant is not a fixed number, but it grows linearly with N . This last featuredramatically alters the usual large N dynamics since the integral (7.4) is not dominatedanymore by the same family of saddle-points of the familiar Gross-Witten model, as wewill see in the following. The phase structure of (7.4) can be naturally studied along the lines proposed in [52].We will first perform the integration over the unitary matrices and then the integrationover the Lagrange multipliers. We will then start by studying the large N properties ofthe reduced model Z ( γ, p ) = Z DU exp (cid:0) γN (Tr( U ) + Tr( U † )) (cid:1) det( U ) Np , (7.5)where N p is an integer, whose sign is irrelevant because we can transform N p into − N p by performing the change of variable U U † . For this reason, from now on, we shall take p to be positive. The first important effect of the new logarithmic interaction concernsthe small γ behavior of (7.5): differently from the Gross-Witten model ( p = 0), where Z ( γ, p ) is finite as γ approaches zero, here Z ( γ, p ) vanishes as γ N p . This leading behavioris determined by expanding the exponential around γ = 0 and performing the integralterm by term. The first non-vanishing contribution is fixed by the selection rule imposedby the U (1) factor present in U ( N ) and it is given by Z ( γ, p ) ≈ ( γN ) N p ( N p )! Z DU Tr( U † ) N p det( U ) Np = ( γN ) N pN − Y i =0 i !( i + N p )! = (2 γ ) N p e N C , (7.6)where the constant C in the large N limit is given by C = − (cid:0) (log(4) − p + ( p + 1) log( p + 1) − p log( p ) (cid:1) . (7.7)In other words, the free energy F ( γ, p ) = log Z ( γ, p ) = N F ( γ, p ) + .. of the presentunitary matrix model starts, at leading N order, with a logarithmic singularity similarto the one of the usual Penner model [53, 54] F ( γ, p ) = p log(2 γ ) + C + O ( γ ) . (7.8)This new behavior suggests that the usual strong-coupling expansion of (7.5) might beradically different from that of the Gross-Witten model, which is simply given by e N γ .34o explore this idea, one could perform a full strong-coupling expansion and to resumthe resulting series in the large N limit; however the presence of the determinant factormuch complicates this approach. Here, we shall choose a simpler path and consider adifferent expansion, peculiar of the present model, namely p very large. In this limit wecan perform a semiclassical analysis on the integral (7.5): the relevant classical potentialis, in this case, pV ( θ i ) = pN γp N X i =1 cos θ i + i N X i =1 θ i ! . (7.9)The equations for the critical point are easily derived and solved (we will denote fromnow on 4 γ = t ) − √ tp sin θ i + i = 0 ⇒ θ i = i sinh − (cid:18) p √ t (cid:19) . (7.10)The semiclassical approximation is then obtained by expanding the classical action aroundthe critical point up to the quadratic order N (cid:18)p p + t − p sinh − (cid:18) p √ t (cid:19)(cid:19) − N (cid:16)p p + t (cid:17) N X i =1 ˆ θ i + O (cid:16) ˆ θ i (cid:17) , (7.11)with ˆ θ i ≡ θ i − i sinh − (cid:18) p √ t (cid:19) . (7.12)We remark that this is a good approximation as long as p p + t ≫ 1: in this limit thegaussian integration covers the whole real line and the Haar measure over the unitarymatrices becomes the usual measure over the hermitian matrices. We can easily performthe integration over the angles θ i and up to a constant independent of p we get F ( t, p ) = (cid:18)p p + t − p sinh − (cid:18) p √ t (cid:19) − / p p + t ) (cid:19) == p p + t − p log p √ t + r p t + 1 ! − / p p + t ) ! . (7.13)For p large and t finite or small, we finally arrive to the following expansion F ( t, p ) = p (cid:18) log( t )2 − log ( p ) − log(2) + 1 (cid:19) − 12 log( p ) + t p − t (cid:18) p (cid:19) −− t (cid:18) p (cid:19) + 18 t (cid:18) p (cid:19) + O (cid:18) p (cid:19) ! . (7.14)This result is quite remarkable: we see that the above expansion reproduces exactly thelarge p limit of (7.8) and contains a infinite series of corrections in powers of t . Since357.14) holds also for small t , we must conclude that the strong-coupling expansion ofour deformed Gross-Witten model leads to a non-trivial function of t and p , eventuallyencoding an intriguing modification of the p = 0 result.It is quite easy to repeat the same analysis taking t large, exploring in this way thedeformation of the weak-coupling phase of the familiar unitary model. In this limit weshould obtain, at leading order in t , the very same result for the free energy as in theGross-Witten case: again we could expect a non-trivial deformation due to the presenceof p . Actually, performing the same steps as before, we get again (7.13) , which expandedfor large t gives F ( t, p ) = − 34 + √ t − 14 log( t ) − p r t − p t + 124 p (cid:18) t (cid:19) / + 18 p (cid:18) t (cid:19) −− p (cid:18) t (cid:19) / − p (cid:18) t (cid:19) + O (cid:18) t (cid:19) / ! . (7.15)We recognize in the first three terms the exact large N result of the Gross-Witten weak-coupling phase: it does not come as a surprise, being the semiclassical approximation exactin this phase. As expected, we also observe an infinite series of corrections, depending on p , that modify non-trivially the usual spherical free energy of the weak-coupling phase.We do not expect, of course, that the above expansions yield the exact large N freeenergy: these results are semiclasssical, in the sense that we missed the contribution ofthe Vandermonde determinants associated to the measure over unitary matrices, that isessential in recovering the correct spherical free energy. Nevertheless they should capturethe leading order behavior at large p or t of the complete large N answer, and also acertain series of subleading terms (as we will explicitly check in the following).These computations suggest an intriguing possibility: we observe non-trivial deforma-tions of both strong and weak-coupling expansion of the Gross-Witten model, involvingcomplicated functions of p and t . It is quite natural to conjecture, at this point, that aunique non-trivial analytic function F ( t, p ) exists, reproducing for p = 0 both behaviorsand being the large N free energy of the model. This is also suggested by the fact thatthe same free energy (7.13) describes smoothly either the large p or the large t region(see footnote 9). If this is the case, the presence of the logarithmic interaction wouldsmooth out the third-order phase transition of the Gross-Witten model, the parameter p providing an analytic interpolation between the strong and the weak-coupling phase.In order to prove this idea, we have to solve exactly the large N dynamics: we shallexploit the beautiful relation between our model and the Painlev´e III system illustratedin [29]. In that paper the authors have shown that it is possible to construct an auxiliaryfunction, σ ( t ) = − t ddt log (cid:16) ( tN ) N p / e − N t/ Z ( t, p ) (cid:17) , (7.16) The semiclassical computation really holds for p p + t >> p or t large. Therefore we have to obtain the same free energy (7.13) in the large t case as well. N , the following non-linear differential equation − p N + (cid:0) p − (cid:1) σ ′ ( t ) N + σ ′ ( t ) (cid:0) σ ′ ( t ) − N (cid:1) ( σ ( t ) − tσ ′ ( t )) + t σ ′′ ( t ) = 0 . (7.17)In the large N limit, the spherical ansatz for the partition function Z ( t, p ) = e N F ( t,p ) dictates the following scaling for the auxiliary σ ( t ) σ ( t ) = N ρ ( t ) . (7.18)Thus, at the leading order in N , we obtain a nice first-order differential equation for thereduced function ρ ( t ) − tρ ′ ( t ) + (cid:0) p + t + 4 ρ ( t ) − (cid:1) ρ ′ ( t ) − ρ ( t ) ρ ′ ( t ) − p 16 = 0 . (7.19)The analysis for small and large t given in (7.8) and (7.15) provides two possible boundaryconditions for the above equation:(s): ρ ( t ) | t =0 = − ( p + p );(w): ρ ( t ) | t →∞ = t − √ t .Since (7.19) is a first-order differential equation, these boundary values will correspond,in general, to two different solutions: the former, which satisfies ( s ), is denoted with ρ s ( t ) and it is supposed to describe the strong-coupling regime ; the latter, ρ w ( t ), obeys( w ) and it is expected to hold in the weak-coupling regime. The two corresponding freeenergies F s,w ( t, p ) are then constructed by integrating the simple relation d F s,w ( t, p ) dt = (cid:18) − p t − ρ s,w ( t ) t (cid:19) (7.20)which follows from (7.16) once we have used the spherical ansatz Z ( t, p ) = e N F s,w ( t,p ) .The above simple picture works very well at p = 0, where our model reduces tothe usual Gross-Witten model. In this case the differential equation becomes extremelytractable, factorizing into two simple first-order equations: the solution F s ( t, 0) and F w ( t, 0) can be obtained explicitly and they exactly coincides with the well-known freeenergies of the model at strong and weak coupling. The condition F s ( t, 0) = F w ( t, t c = 1). When p = 0, thesituation reserves some surprises as we shall illustrate below.As thoroughly described in appendix D, the general case can be solved exactly, in spiteof the apparent difficult non-linearity of the differential equation. In particular there are In the matrix model language, γ is conventionally identified with the inverse of the fundamentalcoupling constant. Thus small values of t = 4 γ are in the strong-coupling region. ρ s ( t ) and ρ w ( t ) foundin the Gross-Witten case. Integrating (7.20) we get a candidate F s ( t, p ) given by F s ( t, p ) = − (cid:0) (log(4) − p + ( p + 1) log( p + 1) − p log( p ) (cid:1) + t p ) − p t ) , (7.21)while F w ( t, p ) has the form F w ( t, p ) = f w + (cid:18) p ρ ′ w − p 64 ( ρ ′ w ) + 12 (cid:18) log ( ρ ′ w ) p − p tanh − (cid:18) p + 4 (cid:18) p − p (cid:19) ρ ′ w (cid:19) ++ log (1 − ρ ′ w ) + 21 − ρ ′ w (cid:19)(cid:19) , (7.22)where the constant f w is given by f w = − 34 + 14 p (( − p − p − 1) + 2 log( p + 1)) . (7.23)Here ρ ′ w ( t ) is the solution of the fourth order algebraic equation (D.25), which respectsthe large t behavior implied by the boundary condition ( w ). One can easily check that F w ( t, p ) smoothly reduces, as p goes to zero, to the free energy of the Gross-Witten modelin the weak-coupling phase, and accurately reproduces the semiclassical expansion (7.15),up to higher order terms in p n /t n + m/ , coming from the exact large N solution encodedinto the differential equation. It is also evident from (7.21) that F s ( t, p ) reproduces, inthe limit of vanishing p , the Gross-Witten strong-coupling result.On the other hand, we already know that F s ( t, p ), as given by (7.21), cannot providethe right solution describing the small t regime! The large p expansion of (7.21) is quiteboring and does not reproduce the non-trivial series (7.14), obtained from the semiclassicalapproximation. On the other hand it is possible to show that F w ( t, p ), for p = 0, alsosatisfies the right boundary condition to describe the strong-coupling region (see appendixD) and, more importantly, correctly reproduces (7.14) in the large p limit (up to higherorder corrections in t n /p n + m , coming from the exact large N solution of the model).We arrive therefore to the conclusion that the critical behavior of the standard unitarymatrix model is completely modified by the addition of our logarithmic interaction. Aslong as p = 0 the system is always in a “weak-coupling” phase, described by the freeenergy F w ( t, p ): this solution has the correct boundary condition both at small andat large t and smoothly interpolates between them. We also identify F s ( t, p ) with an unphysical solution of the differential equation (7.19) and therefore we neglect it. Thesituation drastically changes for p = 0: it is possible to show that, starting from F w ( t, p ),the limiting behavior changes discontinuously at t = 1. On the other hand, taking p = 0at level of the differential equation (7.19), the strong-coupling phase is instead encodedinto the solution ρ s ( t ). 38 .2 Phase-structure in non-trivial vacua In this subsection, we shall explore the consequences of the previous results on the phasestructure of the theory. After having performed the integration over the unitary matricesin the deformed Gross-Witten models, we are left with the integration over the Lagrangemultipliers Z k Y J =1 dγ I dα I exp N k X I =1 ( is I ( n I − q ) α I + s I F ( γ I , p )) − N X IJ s I s J γ I z − IJ ( x ) e i ( α J − α I ) γ J ! (7.24)with q = Q/N . Since N is large, we can perform this integral in the saddle-point ap-proximation as well. The saddle-points which dominate this integration are determinedby 2 k X I =1 s I γ I z − IJ γ J sin( α J − α I ) + i ( n J − q ) = 0 − k X I =1 s I γ I z − IJ cos( α J − α I ) + s J F ′ ( γ J , p ) = 0 . (7.25)To be concrete, we shall consider only the case k = 2: here the relevant combinations ofthe parameters are given by n − q = s ( n − n ) ≡ s n and n − q = − s ( n − n ) ≡ − s n ( n = n − n > γ z − γ sin( α − α ) + in = 0 and 2 γ z − γ sin( α − α ) − in = 0 . (7.26)These two equations are obviously equivalent and they are solved bysin( α − α ) = − i n γ z − γ ⇒ cos( α − α ) = ± s n γ z − γ ) . (7.27)Substituting this result into (7.25), the second equation provides two relations, whichdetermines γ , γ − s γ z − ∓ s r ( γ z − γ ) + n s γ F ′ ( γ , p ) = 0 , ∓ s r ( γ z − γ ) + n − s γ z − + s γ F ′ ( γ , p ) = 0 . (7.28)In the following, we shall further simplify our example and we shall choose two sectors ofequal length, namely we shall set s = s = 1 / 2. Then, by taking the difference of the twoequations and using the fact that F ( t, p ) is a monotonic function, one can immediatelyshow that γ = γ . We remain with just one equation, which determines t = 4 γ ∓ r 14 ( z − t ) + n − t z − + 2 t F ′ ( t , p ) = 0 , (7.29)39hich is conveniently rewritten in terms of ρ w ( t ) as follows f ± ( t ) ≡ ± r 14 ( z − t ) + n + t − z − ) + n = 2 (cid:16) ρ w ( t ) + n n + 1) (cid:17) . (7.30)When t runs from zero to infinity, the r.h.s of (7.30) spans the same region. Thus anecessary condition for having a non-trivial solution is that the l.h.s. of (7.30) is notnegative definite. Let us discuss the first equation: f + ( t ) has the following properties f + (0) = 2 n, f ′ + (0) = 12 det z − z det zf ′ + ( t ) = 0 ⇒ t = − n (det z − z ) ( z ) det z det(1 − z ) . (7.31)Moreover we have that for large tf + ( t ) → t z + z − z + z + n + O (1 /t ) . (7.32)We immediately conclude that for temperatures near zero ( x ≪ f + ( t ) is always de-creasing: for T < T H , where T H is the Hagedorn temperature defined by the equation z + z = 1 (7.33)as in (6.10), we have that det(1 − z ) ≥ 0, implying that f ′ + ( t ) never vanishes for t > t = 0. At the Hagedorn temperature T H the function f + ( t ) is still decreasingbut becomes positive definite, asymptotically approaching the value n . Above T H we seethat f + ( t ) develops a minimum at finite t and then becomes monotonically increasing.The minimum disappears at the temperature T defined bydet z = z (7.34)and the function becomes monotonically increasing for any T > T .In spite of these changes of behavior, one can check that there is always one solutionto the saddle-point equation as shown, in different regimes, in figure 3. Moreover theposition of this saddle-point changes smoothly as function of the temperature (fig. 4).Let us examine the second saddle-point equation, the one involving f − ( t ). We canrepeat the same analysis: the main conclusion is that for temperature T < T we see f − ( t )being monotonically decreasing and therefore, because f − (0) = 0, there is no solution for t = 0 to the saddle-point equation. We notice that t = 0 is not acceptable because of(7.26). For T > T it is not easy to see analytically if f − ( t ) provides new solutions tothe saddle-point equation: we have done a numerical study, showing that a new solutionappears for x ≥ . f + is theonly relevant saddle-point in the large N limit.40 t f + Figure 3: Plot of f + ( t ) for different values of T and n = 1, q = 1 / 2. Going bottom-up, the solid linesillustrate the behavior for T < T H , T = T H (lower thick line), T H < T < T , T = T (upper thick line), T > T . The dashed line is the r.h.s. of (7.30) as a function of t . x t Figure 4: Saddle point as a smooth function of the temperature x for n = 1, q = 1 / 2. The graphcovers all the different regimes, the Hagedorn temperature being x H = 0 . T correspondingto x = 0 . .2 0.4 0.6 0.8 1 x Ρ ' x F (cid:144) N Figure 5: On the left side the saddle-points (for n = 1, q = 1 / 2) in terms of ρ ′ ( x ) associated to f + (continuous line) and f − (dashed line) are shown. At x = 0 . f − solution intersects with the t = 0 unphysical solution (dotted line). On the right the free energies for both cases. We conclude therefore that within our approximation, that consisted in taking justthe first winding in the matrix model action ( n = 1), we have always a non-trivial saddle-point giving a free energy F B = log Z B of order N . Moreover this saddle-point variescontinuously with the temperature: in particular at the Hagedorn temperature T H , repre-senting the point of the first-order phase transition in our first regularization scheme, thefree energy remains smooth and no discontinuous behavior appears in this second scheme. In this paper we have studied the maximal supersymmetric gauge theory on R × S ,with particular attention to its thermodynamical properties in the limit of zero ’t Hooftcoupling. In the case of the trivial vacuum, we found a behavior similar to the parentfour-dimensional theory, with a first-order Hagedorn transition separating a “confining”phase from a “deconfined” one, with non-trivial expectation value for the Polyakov loop.We have repeated the analysis for monopole vacua and we have apparently differentbehaviors, depending on the regularization procedure: this actually reflects the particularchoice of the fermionic three-dimensional vacuum, that is related to generation of Chern-Simons terms when monopole are present on the sphere. We have presented two oppositechoices, both allowed at quantum field theory level, generating different unitary multi-matrix models describing the thermal partition function. The critical behaviors we found,under suitable assumptions on the relevant contributions at small temperature, are verydifferent: in particular we have observed that no Hagedorn transition seems to be presentwithin our second regularization choice. Further studies are surely necessary to elucidatethe situation: first of all we expect that supersymmetry should play a role in order todistinguish between the different regularization choices and consistency with the SUSYalgebra could probably select a preferred “vacuum charge”. On the other hand the relation42ith the gravitational duals should also be investigated to provide a physical interpretationof the Casimir energies and of the Chern-Simons contributions. Apart from solving thepuzzles arisen in this paper, there are a lot of potential interesting developments involvingthe study of the N = 8 three-dimensional supersymmetric theory considered here. Itwould be important of course to determine the nature of the phase transition beyondzero ’t Hooft coupling and to discuss the issue of exact decoupling limit using chemicalpotentials, in the spirit of [20, 23, 24]. We also plan to consider the phase diagram in thepresence of background scalars as in [55, 56]. More generally one could try to explore ifsome remnant of four-dimensional integrability persists in three dimensions and to makesome quantitative connection, in the strong-coupling limit, between the gauge theory andits gravity dual. It would also be very interesting to study BPS Wilson loops on S : in fourdimensions there have been exact results for particular classes of loops, the computationsreducing to matrix integrals [57, 58] . It is natural to ask if a similar phenomenon takesplace in three dimensions too. Acknowledgments We thank Valentina Forini for a fruitful collaboration at the early stages of this work.We also thank Leonardo Brizi, Gianni Cicuta, Filippo Colomo, Troels Harmark, CarloMeneghelli, Marta Orselli, Ettore Vicari for useful discussions. We wish to thank theGalileo Galilei Institute for Theoretical Physics for hospitality during the last stages ofthis work. A Conventions and supersymmetry variations Before discussing in more details the supersymmetry variations considered in section 2,we shall briefly summarize our conventions and identities on Γ-matrices. Metric and gauge conventions: The metric is taken diagonal and with Minkowskiansignature: η MN = {− , + , ..., + } . The capital letters M, N, . . . will span the ten dimen-sional spacetime indices (0 , , . . . , µ, ν . . . will denote the threedimensional spacetime indices (0 , , i, j, k are associated to the directions(1 , 2) along the sphere S , while the directions (3 , . . . , 9) transverse to S are indicatedwith m, n, . . . . Finally a special index notation is also reserved to the set of directions(4 , . . . , 9) for which we shall use the overlined letters ¯ m, ¯ n, . . . .The gauge fields A = A a t a are taken to be hermitian and the generator t a are nor-malized so that Tr( t a t b ) = δ ab . The covariant derivatives are then defined as follows D µ = ∇ µ − ig [ A µ , · ], where ∇ µ is the geometrical covariant derivative. In general weshall omit the trace over the gauge generators in our expressions, unless it is source ofconfusion. 43 ome useful Γ -identities: For convenience, here we have collected some Γ-identities,which are useful in checking the supersymmetry invariance of the Lagrangian of our model:Γ i Γ jk Γ i = − jk , Γ Γ jk Γ = − Γ jk , Γ i Γ j Γ i = 0 , Γ Γ j Γ = Γ j , Γ i Γ jm Γ i = 0 , Γ Γ jm Γ = − Γ jm , Γ i Γ m Γ i = 2Γ m , Γ Γ m Γ = Γ m , Γ i Γ mn Γ i = 2Γ mn , Γ Γ mn Γ = − Γ mn . (A.1)Summation over repeated index is understood. Here Γ M denotes the ten dimensionalmatrices, while the symbol γ µ is used for the three dimensional Dirac matrices. Thesymbol Γ M M ...M N defines the completely antisymmetrized product of the matrices Γ M ,Γ M ,. . . ,Γ M N . Three-dimensional fields: The scalar field φ ij is antisymmetric in i , j , which are SU (4) R indices and it satisfies reality condition: φ ij ≡ ( φ ij ) † = 12 ǫ ijkl φ kl . (A.2)It is defined in terms of the old fields φ m by the relations: φ = φ + φ √ , φ = − φ + φ √ , φ = φ + φ √ ,φ = i φ − φ √ , φ = i φ + φ √ , φ = i − φ + φ √ . (A.3)The spinor fields λ i (again, i is an SU (4) R index) denote the Dirac spinors in D = 3 orig-inating from the dimensional reduction of ψ M , while A µ describes the three-dimensionalgauge field. A.1 Supersymmetry variations In this appendix, for completeness, we shall write the conditions for the vanishing of thevariation at the order α and at the order α . At the linear order the complete variation44an be summarized by the following table: Term Condition R e { αg [ φ m , φ n ] ψ Γ m n Γ ǫ } B + 2 + P + M = 02 R e { αg [ φ , φ m ] ψ Γ m Γ ǫ } B + 4 + 2 P + G − M = 02 R e { αiD φ ψ Γ Γ ǫ } − B + P + G − M = 02 R e { αiD φ m ψ Γ m Γ ǫ } − B + P + 2 M = 02 R e { αiD i φ ψ Γ i Γ ǫ } B + P + G + 2 M + N = 02 R e { αiD i φ m ψ Γ im Γ ǫ } B + P − M = 02 R e { αiF i ψ Γ i Γ ǫ } − B − M = 02 R e { αiF ij ψ Γ ij Γ ǫ } B − M + N = 0 (A.4)There are eight different kind of terms, listed in the first column, and they must vanishseparately: this leads to the conditions in the second column.At the quadratic order in α we have simply Term Condition R e { iα φ m ψ Γ m ψ } − V + (2 − βα ) P + M P = 02 R e { iα φ ψ Γ ψ } − V + W ) + (2 − βα )( P + G ) − M ( P + G ) = 0 (A.5) B Computing the one loop partition function Here we give all the details of the calculation of the partition function in a monopolebackground. For the free model the one-loop contribution of each field is a functionaldeterminant, giving the single-particle partition function. B.1 Computing determinants: the master-formula We illustrate our regularization scheme: readers who are not interested in these detailscan take (B.8) and (B.17) as main results, and skip to next subsection.All the determinants appearing in the evaluation of the free partition function contains,as a key ingredient, the evaluation of the following infinite productΣ( η, ρ, β, w ) := ∞ Y j =0 ∞ Y n = −∞ (cid:20) ( j + η ) + 4 π β ( n + w ) (cid:21) j + ρ . (B.1)45his quantity is divergent and it must be regularized. Here, we shall adopt the standard ζ − function regularization and we shall defineΣ( η, ρ, β, w ) := e − ζ ′ (0) , (B.2)where ζ ( s ) = ∞ X j =0 ∞ X n = −∞ j + ρ h ( j + η ) + π β ( n + w ) i s . (B.3)Notice that (B.3) defines the function ζ ( s ) only for | s | > 1. In order to compute ζ ′ (0), wehave to consider its analytical continuation to a neighborhood of the origin in the s -plane.This is achieved through a standard technique: firstly, we shall use the Mellin-Barnesrepresentation and subsequently we shall perform a Poisson-resummation in nζ ( s ) = 1Γ( s ) ∞ X j =0 (2 j + ρ ) ∞ X n = −∞ Z ∞ dt t s − e − t ( j + η ) − t π β ( n + w ) == β √ π Γ( s ) ∞ X j =0 (2 j + ρ ) ∞ X n = −∞ Z ∞ dt t s − e − t ( j + η ) e − β n t − πiwn == β Γ( s − )2 √ π Γ( s ) ∞ X j =0 (2 j + ρ )( j + η ) s − ++ 2 − s β s + √ π Γ( s ) ∞ X j =0 ∞ X n =1 (2 j + ρ ) n − s ( j + η ) s − K − s ( n ( j + η ) β ) cos(2 nπw ) == β Γ( s − )2 √ π Γ( s ) (2 ζ (2 s − , η ) − (2 η − ρ ) ζ (2 s − , η ))+ 2 − s β s + √ π Γ( s ) ∞ X j =0 ∞ X n =1 (2 j + ρ ) n − s ( j + η ) s − K − s ( n ( j + η ) β ) cos(2 nπw ) . (B.4)The only contribution to ζ ′ (0) in (B.4) arises when the derivative acts on 1 / Γ( s ) sincethis quantity vanishes as s approaches 0. We obtain ζ ′ (0) = − β (2 ζ ( − , η ) + ( ρ − η ) ζ ( − , η )) + ∞ X j =0 ∞ X n =1 e − nβ ( j + η ) (2 j + ρ ) cos(2 nπw ) n . (B.5)From the final expression (B.5) we can deduce two equivalent representations of this result,which are both useful for our goals. Firstly we can perform the sum over j , which yields ζ ′ (0) = β (cid:18) B ( η ) + 12 ( ρ − η ) B ( η ) (cid:19) + ∞ X n =1 x nη ( ρ − x n ( ρ − n ( x n − cos(2 nπw ) , (B.6)with x := e − β and B k ( η ) being the Bernoulli polynomial. Next, we shall define the“single-particle” partition function z single ( x ) := x η ( ρ − x ( ρ − − x ) , (B.7)46nd finally writelog (Σ( η, ρ, β, w )) = − β (cid:18) B ( η ) + 12 ( ρ − η ) B ( η ) (cid:19) − ∞ X n =1 z single ( x n ) n cos(2 nπw ) . (B.8)This representation will be the most natural when discussing the matrix model and theposition of the Hagedorn transition.Alternatively, in (B.5) we can first sum over nζ ′ (0) = β ( 23 B ( η ) + 12 ( ρ − η ) B ( η )) − ∞ X j =0 (2 j + ρ ) (cid:0) log (cid:0) − ¯ zx j (cid:1) + log (cid:0) − zx j (cid:1)(cid:1) (B.9)where z := e − βη +2 iπw . If we define η ( z, x ) := ∞ Y j =0 (cid:0) − zx j (cid:1) and M ( z, x ) := ∞ Y j =0 (cid:0) − zx j (cid:1) j , (B.10)we can recast the above result in a very compact formΣ( η, ρ, β, w ) = e − β ( B ( η )+ ( ρ − η ) B ( η )) | η ( z, q ) | ρ |M ( z, q ) | . (B.11)This second representation will be the most suitable when discussing the high temperaturebehavior. In this limit the leading contribution is encoded in the function F ρ ( z, x ) F ρ ( z, x ) = ∞ X j =0 (2 j + ρ ) log (cid:0) − zx j (cid:1) . (B.12)The x → F ρ ( z, x ) as F ρ ( z, x ) = − ∞ X m =1 z m m (cid:20) ( ρ − 2) 11 − x m + 2(1 − x m ) (cid:21) , (B.13)and expanding in β , at fixed z , we get F ρ ( z, x ) = − β ) Li ( z ) − ρβ Li ( z ) + ( ρ − 22 + 56 ) log(1 − z ) + O ( β ) . (B.14)To recover (5.13), where the contribution of chemical potentials to the high-temperaturelimit has been presented, we need further expand Li ( z ) for z → 1: we are interested inthe case when w = 0 and w = 1 / 2, appearing respectively in the bosonic and fermioniccase, and with zero flat-connection ( z = e − y , y → ( e − y ) = ζ (3) − π y + (cid:18) − 14 log y (cid:19) y + O ( y ) , Li ( − e − y ) = − ζ (3) + π y − 14 log(4) y + O ( y ) . (B.15)47 ermionic zero modes: In order to compute the contribution of the fermion zeromodes, we need to compute the product F = ∞ Y n = −∞ (cid:20) πβ ( n + w ) (cid:21) ρ . If we adopt the zetafunction regularization as before, we are led to compute the following accessory sum G ( s ) = β s (2 π ) s ∞ X n = −∞ ρ ( n + w ) s = β s (2 π ) s ρ ( ζ ( s, w ) + e iπs ζ ( s, − w )) . (B.16)Then log( F ) = − G ′ (0) = − ρ ∞ X n =1 e − πinw n . (B.17) B.2 The scalar determinant Let us discuss the solution of the eigenvalue problem (4.6). Since our background isstatic, we can factor out the time-dependence in the eigenfunction by posing φ ( t, θ, φ ) ∼ φ n ( θ, φ ) e − πinβ t . Then the eigenvalue problem in the Weyl basis (4.7) takes the form X α ∈ roots " π β (cid:18) n + βa α π (cid:19) φ αn − ˆ △ φ αn + µ φ αn + µ q α φ αn E α ++ N − X i =1 (cid:18) π n β φ in + µ φ in − △ φ in (cid:19) H i = λ r X i =1 φ in H i + λ X α ∈ roots φ αn E α , (B.18)where △ denotes the geometrical Laplacian for a scalar on the sphere. The symbolˆ △ instead represents the geometrical Laplacian in the background of a U (1) magneticmonopole of charge q α . This Laplacian is constructed with the covariant derivativeˆ D µ = ∇ µ − iq α A µ , (B.19)where ∇ µ is the geometrical covariant derivative. In (B.18) the components along thedifferent directions in the Lie algebra do not interfere and we can consider them as inde-pendent. This allows us to split the original eigenvalue problem into two subfamilies, wehave: (a) N ( N − 1) independent eigenvalues coming from each direction along the laddergenerator 4 π β (cid:18) n + βa α π (cid:19) φ αn − ˆ △ φ αn + µ φ αn + µ q α φ αn = λ αn φ αn , (B.20)and (b) N − π n β φ in + µ φ in − △ φ in = λ in φ in . (B.21)To begin with, we shall focus our attention on the family (a), since the family (b) canbe obtained from (a) as a limiting case for a α , q α → 0. The solution of the eigenvalue48quation (B.20) can be translated into an algebraic problem if we introduce the angularmomentum operator in the presence of a U (1) monopole of charge q α . Its form [59] is L ( α ) i = ǫ ijk x j ( − i∂ k − q α A k ) − q α x i | x | ≡ ǫ ijk x j P k − q α x i | x | . (B.22)Here x i are the Cartesian coordinates of a flat R where our sphere S is embedded. Interms of this auxiliary operator, the kinetic operator in (B.20) takes the form µ ( L ( α ) ) φ αn + " π β (cid:18) n + βa α π (cid:19) + µ φ αn = λ αn φ αn . (B.23)Thus our task is reduced to finding the eigenvalues and the eigenfunctions of this dressed angular momentum operator ( L ( α ) ) . Its spectrum was determined thirty years ago byWu and Yang [59] and it is formally equal to that of the usual angular momentum: theeigenvalues are j α ( j α + 1) and their degeneracy is 2 j α + 1. What changes is the rangespanned by the index j α , which now is | q α | , | q α | + 1 , | q α | + 2 , · · · . Putting everythingtogether the spectrum of the kinetic operator (B.20) turns out to be λ αn = µ (cid:18) j α + 12 (cid:19) + 4 π β (cid:18) n + βa α π (cid:19) with j α = | q α | , | q α | + 1 , | q α | + 2 · · · , (B.24)and each eigenvalue has degeneracy 2 j α + 1. Notice that the spectrum does not dependon the sign of q α . The contribution of the family (a) to the effective action is given bythe infinite productΓ Sc. ( a ) = log Y α ∈ roots ∞ Y j α = | q α | ∞ Y n = −∞ " µ (cid:18) j α + 12 (cid:19) + 4 π β (cid:18) n + βa α π (cid:19) j α +1 , (B.25)which is easily computed by using the results of appendix B.1 (with ρ = 1 + 2 | q α | , η = 1 / | q α | , w = βa α π ). Setting x = e − βµ , we obtain (4.9) and (4.10) The contributionof the family (b) is then obtained from the above results by setting q α = a α = 0. B.3 The vector/scalar determinant The eigenvalue problem for the coupled system ( φ , A ) can be simplified by choosing thegauge-fixing (4.11). This choice allows us to cancel some of the mixed terms ( φ A ) in theEuclidean quadratic Lagrangian and to obtain L (2)( A µ ,φ ) = − A ν ˆ D µ ˆ D µ A ν + R µν A µ A ν − i ˆ F νµ [ A ν , A µ ] − [ A ρ , ˆ φ ][ A ρ , ˆ φ ]++ ˆ D ρ φ ˆ D ρ φ + µ φ − [ ˆ φ , φ ] − µ √ g φ ǫ ρνλ k ρ ˆ D ν A λ . (B.26) The eigenfunctions are also known and they are given by the so-called monopole harmonics Y qjm ( θ, ϕ ).They are a straightforward generalization of the usual spherical harmonics, but we shall not need theirexplicit form here. We refer the reader to [59] for more details. − ˆ (cid:3) φ + µ φ + [ ˆ φ , [ ˆ φ , φ ]] − µ √ gǫ ρνλ k ρ ˆ D ν A λ = λφ , (B.27) − ˆ (cid:3) A ν + R µν A µ + i [ ˆ F νµ , A µ ] + [ ˆ φ , [ ˆ φ , A ν ]] + µ √ gǫ ρλν k ρ ˆ D λ φ = λA ν . (B.28)Since both the geometrical and the gauge background are static, the time-component ofthe vector field ω = k ρ A ρ = A decouples completely from the above system. It satisfiesthe massless version of the scalar equation studied in B.2, namely the eigenvalue problemassociated to this component is − ˆ (cid:3) ω + [ ˆ φ , [ ˆ φ , ω ]] = λ ω. (B.29)We shall forget about ω since its contribution is cancelled by the ghost determinant. Weare left with the system given by (B.27) and (B.28) where the indices run only over space.We expand the coupled system (B.27) and (B.28) in the Weyl basis and we factor outthe time-dependence of the eigenfunctions: A µ ( t, θ, φ ) ∼ A nµ ( θ, φ ) e − πinβ t and φ ( t, θ, φ ) ∼ φ n ( θ, φ ) e − πinβ t . Along the directions associated to the ladder operators E α we find − ˆ △ A iαn + m n A iαn + iµ q α √ gǫ ij A jαn + µ q α A iαn + µ √ gǫ ji ˆ D j φ αn = λ αn A iαn , − ˆ △ φ αn + m n φ αn + µ q α φ αn − µ √ gǫ ij ˆ D i A jαn = λ αn φ αn , (B.30)where m n = (2 πn/β + a α ) + µ . In the first equation, the symbol ˆ △ denotes the Laplacianon vectors in the background of a monopole of charge q α , while in the second representsthe Laplacian on scalars. Along the Cartan directions we shall again get the system (B.30)but for q α = 0.To find explicitly the spectrum of system (B.30), it is convenient to decompose ourvector A iαn in its selfdual part A + iαn and anti-selfdual part A − iαn . Consequently we shallintroduce the differential operators O ( α ) ± mapping (anti-)selfdual vectors into scalars andtheir adjoints, mapping scalars into (anti-)selfdual vectors. They are defined by O ( α ) ± V ± ≡ O i ( α ) ± V ± i = 1 √ g ǫ ij ˆ D i V ( ± ) j , O ( α ) †± φ ≡ O i ( α ) †± φ = ∓ i (cid:18) g ij ± i √ g ǫ ij (cid:19) ˆ D j φ, (B.31)where ˆ D as in (B.30) stands for the covariant derivative in the background of a monopoleof charge q α . In terms of these operators, the system (B.30) takes the form O ( α ) † + O ( α )+ A + αn + q α µ A + αn + ℓ n A + αn − µ O ( α ) † + φ αn = λ αn A + αn ,O ( α ) †− O ( α ) − A − αn + q α µ A − αn + ℓ n A − αn − µ O ( α ) †− φ αn = λ αn A − αn , (B.32)12 ( O ( α ) − O ( α ) †− + O ( α )+ O ( α ) † + + q α µ + ℓ n + µ ) φ αn − µ O ( α )+ A + αn − µ O ( α ) − A − αn = λ αn φ αn . Here we have dropped the index i because it is immaterial and we have set m n = ℓ n + µ .At first sight the eigenvalue problem might appear cumbersome, but in this representation50t is quite simple to provide a basis where our problem reduces to diagonalizing an infiniteset of three by three matrices. In fact, let us take q α ≥ and consider the following basis for scalars, selfdual and anti-selfdual vectors on the sphere e + αjm = O ( α ) † + Y q α jm for j ≥ q α + 1 , e − αjm = O ( α ) †− Y q α jm and e αjm = Y q α jm for j ≥ q α . (B.33)Here Y q α jm are the monopole harmonics, namely the eigenfunctions of the angular mo-mentum (B.22). For the anti-selfdual vector we have to add also 2( q α − 1) + 1 elementscoming from the zero modes of O ( α ) †− O ( α ) − . We shall denote them as e − α ( q α − m . For a de-tailed proof that (B.33) with the addition of the zero modes is a basis, we refer the readerto [60], where the following two useful identities are also shown to hold: O ( α )( ± ) O ( α ) † ( ± ) e αjm = µ L ( α ) ) − q α ∓ q α ) Y q α jm = µ j ( j + 1) − q α ∓ q α )) e αjm , (B.34)and O ( α ) † ( ± ) O ( α )( ± ) e ± α ± jm = O ( α ) † ( ± ) O ( α )( ± ) O ( α ) † ( ± ) e α ± jm = µ j ( j + 1) − q α ∓ q α )) e ± αjm . (B.35)Because of (B.34) and (B.35) and the definitions (B.33), e ± αjm and e jm for fixed j ≥ q α + 1and fixed m generate an invariant three-dimensional linear subspace for the eigenvalueproblem (B.32). The original problem can be then separately solved in each subspace,where it reduces to diagonalizing the following three by three matrix m n − µ + j ( j + 1) µ − q α µ − µ m n − µ + j ( j + 1) µ + q α µ − µ − µ (cid:0) − q α − q α + j ( j + 1) (cid:1) − µ (cid:0) − q α + q α + j ( j + 1) (cid:1) m n + j ( j + 1) µ . (B.36)The three distinct eigenvalues of this matrix are given by λ + = ℓ n + j µ , λ − = ℓ n + ( j + 1) µ , λ = ℓ n + j ( j + 1) µ , with j ≥ q α + 1 . (B.37)For j = q α self-dual vectors do not exist and the invariant subspace is generated onlyby e − αq α m and e αq α m . Instead of (B.36), we have the two by two matrix that is obtainedfrom (B.36) by dropping the first row and the first column. Its diagonalization producesthe following two eigenvalues λ − = ℓ n + ( q α + 1) and λ = ℓ n + q α ( q α + 1). Finally, wehave to consider j = q α − 1. In this case, we are left with a one-dimensional invariantsubspace generated by e − α ( q α − m . The eigenvalue is simply λ − = ℓ n + q α . Summarizing wehave λ − = ℓ n + ( j + 1) µ for j ≥ q α − λ = ℓ n + j ( j + 1) µ for j ≥ q α so that wehave extended the range of existence of the eigenvalues (B.37). The degeneracy is always2 j + 1.In the following we shall neglect the family with eigenvalue λ , since its contributionis cancelled by the ghosts. We shall just consider the first two families λ ± , which instead The case q α ≤ − q α = ± / q α = 0 will be discussed separately. λ + is obtained from the resultsof appendix B.1 by setting w = βa α / (2 π ), η = q α + 1 and ρ = 2 q α + 3Γ V + = X α ∈ roots − βµ (cid:0) q α + 18 q α + 10 q α + 1 (cid:1) − ∞ X n =1 z vect.q α + ( x n ) n e inβa α ! , (B.38)with z vect.q α + ( x ) = x q α +1 (cid:20) (3 − x )(1 − x ) + 2 q α − x (cid:21) . (B.39)The contribution of λ − is instead obtained setting w = βa α / (2 π ), η = q α and ρ = 2 q α − V − = X α ∈ roots − βµ (cid:0) q α − q α + 10 q α − (cid:1) − ∞ X n =1 z vect.q α − ( x n ) n e inβa α ! , (B.40)with z vect.q α − ( x ) = x q α (cid:20) x (1 + x )(1 − x ) − q α − x (cid:21) . (B.41)When adding these two contributions, we obtain (4.12) and (4.13). For what concernsthe non-negative values of the monopole charge, there are still two cases to be considered: q α = 1 / q α = 0. In both cases, the elements of the basis coming from the additionalzero modes of the operator O ( α ) †− O ( α ) − disappear [60]. For q α = 0, in the basis (B.33) theelement e − αjm with j = q α = 0 is absent. The net effect is to reduce the range of theexistence of the eigenvalues λ − = ℓ n + ( j + 1) µ to j ≥ q α for q α = 0 , / λ = ℓ n + j ( j + 1) µ to j ≥ q α = 0. By recomputing the contribution of λ − , for q α = 1 / ρ and η in appendix B.1: η = 3 / ρ = 2. This does not happen, instead,for q α = 0 : by using η = 1 and ρ = 1, we get (4.14). B.4 The spinor determinant In determining the contribution to the total partition function of the spinors λ i , we shallfollow closely the steps of the previous appendix. The fermion kinetic operator expandedaround the background (3.3) has the following eigenvalue problem − iγ µ ˆ D µ λ + i [ ˆ φ , λ ] + i µ γ λ = ρλ, (B.42)where we dropped the SU (4) R index since all the components give the same contribution.Expanding the matrix-valued field λ in the Weyl basis and separating the time-dependencewe get λ = N − X ℓ =1 λ ℓn H ℓ + X α ∈ roots λ αn E α ! e − πiβ ( n + ) t . (B.43)The only real difference with the scalar and vector cases is that fermions have antiperiodicboundary conditions along the time circle. The usual procedure will, in turn, disentangle52he different components along the Lie algebra and it will divide the eigenvalue problem(B.42) into two subfamilies. As in the scalar case, we have: (a) N ( N − 1) independenteigenvalues coming from each direction along the ladder generator ( αn ) λ αn ≡ − γ (cid:20) πβ (cid:18) n + 12 (cid:19) + a α − i µ (cid:21) λ αn − iγ i ˆ D i λ αn + iµq α λ αn = ρ αn λ αn , (B.44)and (b) N − ( ℓn ) λ ℓn ≡ − γ (cid:20) πβ (cid:18) n + 12 (cid:19) − i µ (cid:21) λ ℓn − iγ i ∇ i λ ℓn = ρ ℓn λ ℓn . (B.45)In (B.45) the symbol ∇ denotes the geometrical covariant derivative on spinors while ˆ D i in (B.44) is the covariant derivative in the background of a U (1) magnetic monopole ofcharge q α , i.e. ˆ D i = ∂ i + i abi Σ ab − iq α A i . (B.46)We shall first consider the family ( a ). The problem of diagonalizing the operator (B.44)can be solved algebraically by exploiting the unitary transformation U = e i θσ e i ϕσ . Infact, after performing this transformation, the operator (B.44) becomes directly relatedto the total angular momentum J ( α ) = L ( α ) + σ in the monopole background S ≡ U † ( αn ) U = − (cid:20) πβ (cid:18) n + 12 (cid:19) + a α − i µ (cid:21) ( σ · ˆ r ) + µǫ ijk ˆ r i σ j J ( α ) k + iµq α . (B.47)Here ˆ r stands for the usual radial unit vector in three dimensions while σ i are the Paulimatrices. In (B.47), the operator S is the sum of three contributions. There is a reducedDirac operator D ( α ) ≡ µǫ ijk ˆ r i σ j J ( α ) k = iµ ˆ r · σ + µǫ ijk ˆ r i σ j L ( α ) k , (B.48)which is the standard two-dimensional massless Dirac operator in the presence of amonopole, but written in an unusual basis. Then we have a “chiral” mass term pro-portional to ( σ · ˆ r ), which plays the role of the two-dimensional γ (we have in fact { ( σ · ˆ r ) , D ( α ) } =0). Finally there is a constant shift proportional to the charge q α .Now, we can focus our investigation just on the operator (B.48), since the spectrum of(B.47) follows from that of D ( α ) . For each eigenfunction ψ of D ( α ) with eigenvalue ˆ ρ = 0there exists another eigenfunction ( σ · ˆ r ) ψ with eigenvalue − ˆ ρ . The possible values of ˆ ρ can then be computed by considering the eigenvalues of ( D ( α ) ) . This operator has thefollowing simple form ( D ( α ) ) = µ (cid:20) ( J ( α ) ) + 14 − q α (cid:21) , (B.49)and it is diagonal on the basis ψ jm ± of the total momentum eigenfunctions, which satisfy( σ · ˆ r ) ψ jm ± = ψ jm ∓ . The eigenvalues are ˆ ρ jα = µ (( j + 1 / − q α ). The positivity of theoperator ( D ( α ) ) imposes ( j + 1 / − q α ≥ 0, and in turn j ≥ | q α | − . The degeneracy53f each eigenvalue is 2(2 j + 1). On this basis, the operator D ( α ) is also diagonal and itpossesses the following spectrum D ( α ) ψ jm + = ˆ ρ jα ψ jm + and D ( α ) ψ jm − = − ˆ ρ jα ψ jm − . (B.50)In (B.50) each eigenvalue has degeneracy (2 j + 1). The above analysis does not directlyextend to the kernel of the operator D ( α ) , which is obtained for j = | q α | − . These zero-modes can be classified by using the eigenvalues of the operator ( σ · ˆ r ): we shall denote ν ± the number of zero modes with eigenvalue ± 1. Then a simple application of the indextheorem shows that ν + = | q α | − q α and ν − = | q α | + q α , namely for positive q α we haveonly zero modes with negative chirality and viceversa.We now turn back to the problem of diagonalizing the operator S defined in (B.47).The operator S on the basis provided by the eigenvectors of D ( α ) is not diagonal. However,on the subspace spanned by the eigenfunctions of non-vanishing eigenvalue, it factorizesin an infinite series of two by two matrices. Each matrix acts on the space generated bythe eigenfunctions ψ jm ± and it has the form (cid:18) ρ jα + iµq α − πβ (cid:0) n + (cid:1) − a α + i µ − πβ (cid:0) n + (cid:1) − a α + i µ − ρ jα + iµq α (cid:19) . (B.51)Since we are only interested in the determinant of the operator S , we shall not really needto convert this matrix into a diagonal form, but it is sufficient the evaluate its determinant µ ( j α + 1 / + 4 π β (cid:18) n + 12 + βa α π − i βµ π (cid:19) with j α = | q α | + 12 , | q α | + 32 , . . . , (B.52)and to recall that there are 2 j + 1 determinant with the same value. Then by using themaster formula of appendix B.1 (with ρ = 2 + 2 | q α | , η = 1 + | q α | and w = + βa α π − i βµ π ),the contribution of this part of the spectrum gives (4.17) and (4.18). On the kernel of D ( α ) , the operator S is instead diagonal and it has the following spectrum ρ nα + = 2 πβ (cid:20) − n − − βa α π + i βµ π + i βµq α π (cid:21) , with degeneracy | q α | + q α ,ρ nα − = 2 πβ (cid:20) n + 12 + βa α π − i βµ π + i βµq α π (cid:21) , with degeneracy | q α | − q α . (B.53)Now we have to deal with the regularization ambiguity discussed in section 4.4. In ourcase, all the different choices for the cuts in the s -plane are encoded in the two followingsituations: (I) we regularize the determinants associated to the “ zero-modes ” of negative andpositive chirality by choosing opposite cuts in defining the complex power (one on thereal positive axis and the other on the real negative axis). With the help of appendix(B.1), we then obtain (4.19); (II) we regularize the determinants associated to the “ zero-modes ” of negative andpositive chirality by choosing the same cut in defining the complex power. A similaranalysis yields (4.20).The appearance of a Chern-Simons term for case II and the total fermionic contributionto the effective action for both cases are discussed in section 4.4 as well.54 U(1) truncation of N = 4 super Yang Mills In this appendix we show that the previous results can be easily recovered from N = 4super Yang Mills theory on R × S by a suitable U (1) projection which gives the maximallysupersymmetric theory on R × S .The single-particle partition function in the representation R , z R ( x ), is given by z R ( x ) = X E x E , (C.1)where E is the energy eigenvalue subtracted of the Casimir energy, which can be derivedfor example with the procedure described in the body of the paper. The eigenvalue E canbe computed most directly by noting that the Laplacian on the sphere may be written interms of angular momentum generators which can be diagonalized by means of generalizedspherical harmonics on S . The isometry group of S is SO (4) ≃ SU (2) × SU (2) and wewill need the spherical harmonics for scalars, vectors and fermions, which will be denotedby S j,m, ¯ m (Ω), V j,m, ¯ m (Ω) and F j,m, ¯ m (Ω), respectively. Here m and ¯ m are the eigenvaluesof J and ¯ J for SU (2) and SU (2) and Ω represents the coordinates of S . We followhere the notation of [30]. Having determined the single-particle partition functions on S we may then perform a U (1) projection to derive the single-particle partition functionson S . Such projection amounts in a consistent truncation of N = 4 super Yang Mills asdiscussed in [30], and it can be realized by taking into account that the only modes thatactually contribute to the partition function on S are those for which the eigenvalue of¯ J is equal to half the monopole charge. The projection onto S can thus be performedintroducing into the N = 4 partition functions a U (1) projection operator of the form Z π − π dθ π e iθ ( ¯ J − q ) (C.2)where 2 q is the integer monopole charge of the BPS vacua on S and as a notation weshall assume q ≥ S to S rescales the radius of the sphere by 1 / S of radius R = 1 / C.1 Scalars Scalars on S can be expanded in scalar spherical harmonics S j,m, ¯ m (Ω) where m and ¯ m take the values − j/ , − j/ , . . . , j/ − , j/ 2. The energy of a scalar on S withradius R S = 1, conformally coupled to curvature, is E = j + 1. The partition functionfor a scalar on S then is z scal. ( x ) = ∞ X j =0 j/ X m = − j/ j/ X ¯ m = − j/ x j +1 = ∞ X j =0 ( j + 1) x j +1 = x (1 + x )(1 − x ) (C.3)where the lower index on z denotes the spacetime dimension.55nserting the projector (C.2) we easily get the partition function for a scalar on S .The scalar partition function in the presence of a monopole of charge q becomes z scal. ( x, q ) = ∞ X j =0 j/ X m = − j/ j/ X ¯ m = − j/ Z π − π dθ π e iθ ( ¯ m − q ) x j +1 . (C.4)Performing the sums we end up with an integral z scal. ( x, q ) = Z π dθπ x (1 − x ) cos(2 qθ )(1 + x − x cos θ ) , (C.5)that can be easily done and gives z scal. ( x, q ) = x q +1 (cid:20) (1 + x )(1 − x ) + 2 q − x (cid:21) . (C.6)We can now reintroduce the appropriate dependence on the radius R = 1 /µ . Keepinginto account that the partition function (C.6) is defined on an S with radius R = 1 / R = 1 /µ amounts in simply replacing x → x ≡ e − βµ (C.7)without having to compute a single determinant. C.2 Vectors Vectors on S can be expanded in vector spherical harmonics V ± j,m, ¯ m (Ω) which belong tothe representations ( j , j ) = ( j +12 , j − ) and ( j , j ) = ( j − , j +12 ), respectively. The energyfor both the representations is given by E = j + 1. The partition function on S for the+ vector component is then z vect. ( x ) = ∞ X j =1 ( j +1) / X m = − ( j +1) / j − / X ¯ m = − ( j − / x j +1 = ∞ X j =1 j ( j + 2) x j +1 = x (3 − x )(1 − x ) ; (C.8)for the − vector component we obviously have the same result z vect. + ( x ) = z vect. − ( x ) andthe sum of these two quantities gives the partition function for a vector on S z vect. ( x ) = z vect. ( x ) + z vect. − ( x ) = x (6 − x )(1 − x ) . Inserting now the projector (C.2) into (C.8) and into the analogous one for V − we getfor the + and − vector components respectively z vect. + ( x, q ) = ∞ X j =1 ( j +1) / X m = − ( j +1) / j − / X ¯ m = − ( j − / Z π − π dθ π e iθ ( ¯ m − q ) x j +1 Z π dθπ (3 + x − x cos θ ) cos 2 qθ (1 + x − x cos θ ) = x q (cid:20) x (3 − x )(1 − x ) + 2 q x − x (cid:21) (C.9)and z vect. − ( x, q ) = ∞ X j =1 ( j − / X m = − ( j − / j +1) / X ¯ m = − ( j +1) / Z π − π dθ π e iθ ( ¯ m − q ) x j +1 = Z π dθπ (1 + x − x cos θ + 2 cos 2 θ ) cos 2 qθ (1 + x − x cos θ ) . (C.10)For q = 0 this integral gives z ( vec. ) − ( x, q = 0) = x (1 + x )(1 − x ) , (C.11)and for q = 0 z vec. − ( x, q ) = x q (cid:20) x (1 + x )(1 − x ) − q − x (cid:21) . (C.12)The limit q → − vector partition functions for q = 0give z vec. ( x, q ) = z vec. + ( x, q ) + z vec. − ( x, q ) = x q (cid:20) x (1 − x ) − q (cid:18) x − x (cid:19)(cid:21) (C.13)whereas for q = 0 z vec. − ( x, q = 0) = z vec. + ( x, 0) + z vec. − ( x, 0) = 4 x (1 − x ) . (C.14)Again, with the substitution (C.7) we immediately get back the results (4.13,4.14). C.3 Fermions Fermions on S can be expanded in spinor spherical harmonics F ± j,m, ¯ m (Ω) which belongto the representations ( j , j ) = ( j , j − ) and ( j , j ) = ( j − , j ), respectively. The energyfor both the representations is given by E = j + 1 / 2. Therefore on S we get z spin. ( x ) = ∞ X j =1 j/ X m = − j/ j − / X ¯ m = − ( j − / x ( j +1 / = ∞ X j =0 j ( j + 1) x j +1 / = 2 x / (1 − x ) (C.15)for the + fermion component and the same result for the − fermion component. The sumof these two quantities gives the partition function for a fermion on S z spin ( x ) = z spin. ( x ) + z spin. − ( x ) = 4 x / (1 − x ) . F − one gets thepartition functions for a + or − spinor on S z spin + ( x, q ) = ∞ X j =1 j/ X m = − j/ j − / X ¯ m = − ( j − / Z π − π dθ π e iθ ( ¯ m − q ) x j +1 / = Z π dθπ x / (1 − x cos θ ) cos 2 qθ (1 + x − x cos θ ) = x q (cid:20) x / (1 − x ) + 2 qx / − x (cid:21) , (C.16) z spin. − ( x, q ) = ∞ X j =1 ( j − / X m = − ( j − / j/ X ¯ m = − j/ x ( j +1) Z π − π dθ π e iθ ( ¯ m − q ) x j +1 / = Z π dθπ x / ( − x + cos θ ) cos 2 qθ (1 + x − x cos θ ) = x q (cid:20) x / (1 − x ) + qx / − x (cid:21) . (C.17)Adding (C.16) and (C.17) we get the partition function for a fermion on S in the non-trivial background z spin. ( x, q ) = z spin + ( x, q ) + z spin. − ( x, q ) = x q (cid:20) x ( x / + x − / )(1 − x ) + 2 q (cid:18) x / (1 + x )1 − x (cid:19)(cid:21) . (C.18)With the substitution (C.7) we get back the result (4.18).The complete partition function for our theory can now be constructed using (4.5). Aswe showed before the presence of the monopole background (3.4) breaks the original U ( N )invariance to the subgroup Q kI =1 U ( N I ) so that the positive definite charge 2 q , appearingin the single-particle partition functions, is actually a function of the integers labellingthe sectors into which the monopole field splits. It can be written here as q → q IJ = | n I − n J | . We easily get Z A ( x ) = Z [ k Y I =1 dU I ] exp ( k X I,J =1 ∞ X n =1 n (cid:2) z IJB ( x n ) + ( − n +1 z IJF ( x n ) (cid:3) Tr( U nI )Tr(( U † J ) n ) ) . (C.19)Here k is the number of sectors into which the monopole field splits and reintroducing theappropriate dependence on the radius R = 1 /µ with the substitution (C.7), we recoverfor the bosonic partition function z IJB ( x, q ) = 6 x q IJ +1 / (cid:20) (1 + x )(1 − x ) + 2 q IJ − x (cid:21) + x q IJ (cid:20) x (1 − x ) − q IJ (cid:18) x − x (cid:19)(cid:21) , (C.20)and for the fermionic one z IJF ( x, q ) = 4 x q IJ (cid:20) x ( x / + x − / )(1 − x ) + 2 q IJ (cid:18) x / (1 + √ x )1 − x (cid:19)(cid:21) . (C.21)58e thus reobtain with a very simple and straightforward procedure the result (4.33),up to the constant (temperature-independent) Casimir contribution. Of course the pathintegral approach has the advantages of giving to Tr( U I ) the meaning of matrix holonomyalong the thermal circle and of providing an explicit derivation of the Casimir energies. D Solving the matrix model The solution of the matrix model in the presence of a logarithmic interaction has beenreduced, in section 7.1, to solve the non-linear differential equation (7.19) with a givenset of boundary conditions. Surprisingly, this equation can be explicitly integrated. Toachieve this goal, we first express ρ ( t ) in terms of t and ρ ′ ( t ) by means of (7.19) ρ ( t ) = 64 tρ ′ ( t ) − 16 ( p + t − ρ ′ ( t ) + p ρ ′ ( t ) (4 ρ ′ ( t ) − . (D.22)Subsequently we take the derivative of with respect to t on both sides. The differentialequation (7.19) factorizes into two factors, which can be set separately to zero. In fact,we obtain (cid:0) tρ ′ ( t ) − tρ ′ ( t ) + 16 (cid:0) p + t − (cid:1) ρ ′ ( t ) − p ρ ′ ( t ) + p (cid:1) ρ ′′ ( t ) = 0 , (D.23)which implies ρ ′′ ( t ) = 0 ⇒ ρ ( t ) = At + B (D.24)and 256 tρ ′ ( t ) − tρ ′ ( t ) + 16 (cid:0) p + t − (cid:1) ρ ′ ( t ) − p ρ ′ ( t ) + p = 0 . (D.25)Consider first (D.24). This solution can only satisfy the boundary condition ( s ) associatedto the strong-coupling region (see section 7.1) and thus it seems the natural candidate togenerate F s ( t, p ). This implies that the integration constant B is fixed to be − p ( p + 1).The constant A is instead determined by imposing that (D.24) actually solves (7.19) .We obtain ρ s ( t ) = pt p + 1) − p ( p + 1) . (D.26)The free energy F s ( t, p ) is evaluated by integrating (7.16) with the boundary condition(7.8) F s ( t, p ) = − (cid:0) (log(4) − p + ( p + 1) log( p + 1) − p log( p ) (cid:1) + t p ) − p t ) . (D.27)As discussed in section 7.1, (D.27) is not the right solution at small t because cannotreproduce the series obtained from the large p expansions. We come now to (D.25). Itis an algebraic quartic equation, which determines ρ ′ ( t ) as a function of t and p . We Since we have taken a derivative of 7.19, we could have potentially added spurious solutions t = − (4( p − ρ ′ ( t ) − p ) (4(1 + p ) ρ ′ ( t ) − p )16 ρ ′ ( t ) (4 ρ ′ ( t ) − (D.28)and by drawing its plot. Since we are interested in positive t and in real solutions, wecan focus our attention just on the interval [ p p +1) , p p − ]. The plot is given in fig. 6. We p €€€€€€€€€€€€€€€€€€€€€€ H p + L €€€€ p €€€€€€€€€€€€€€€€€€€€€€ H p - L Figure 6: Plot of the r.h.s of (D.28). It diverges for ρ ′ = 1 / 4. For any positive t we have two solutions. immediately recognize that there are two potential solutions in this region. At small t ,they are both finite and their values at t = 0 are respectively ρ ′ = p p + 1) and ρ ′ = p p − . (D.29)For large t , both solutions approach 1 / ρ ∼ / bt α in (D.28), we immediately find ρ ′ ( t ) = 14 − √ t + O ( t ) and ρ ′ ( t ) = 14 + 14 √ t + O ( t ) . (D.30)The actual functions ρ , ( t ) can be easily recovered by exploiting (D.22), which provides ρ in terms of ρ ′ and t (and p ). It is easy to check that the solution ρ ( t ) can be dropped sinceits behavior at small and large t is in contrast with the boundary conditions. Instead,we can identify ρ ( t ) with the weak-coupling solution ρ w ( t ) and by integrating (7.20) toevaluate F w ( t, p ). The integration constant is fixed by requiring that our free energycoincides with that of the Gross-Witten model for large t . The logarithmic interaction60s in fact sub-leading for t ≫ 1. Nicely the integration over t can be performed withoutan explicit knowledge of ρ w ( t ). In fact (D.28) defines an invertible mapping in the range p p +1) ≤ ρ ′ ≤ (see fig. 6). Thus, by means of (D.28), we can write F w ( t, p ) = f w + Z dt (cid:18) − p t − ρ w ( t ) t (cid:19) == f w + Z dρ ′ w (cid:0) p (4 ρ ′ w − − 64 ( ρ ′ w ) (cid:1) (cid:0) p (4 ρ ′ w − + 16 ( ρ ′ w ) (4 ρ ′ w + 1) (cid:1) 32 (1 − ρ ′ w ) ( ρ ′ w ) (cid:0) p (1 − ρ ′ w ) − 16 ( ρ ′ w ) (cid:1) == f w + 132 (cid:18) p ρ ′ w − p ρ ′ w ) + 16 (cid:18) log ( ρ ′ w ) p − p tanh − (cid:18) p + 4 (cid:18) p − p (cid:19) ρ ′ w (cid:19) ++ log (1 − ρ ′ w ) + 21 − ρ ′ w (cid:19)(cid:19) . (D.31)Here f w is the arbitrary constant of integration. Requiring that we reobtain the usualGross-Witten model for t ≫ f w = − 34 + 14 p (( − p − p − 1) + 2 log( p + 1)) . (D.32)With this choice expansion of the free energy F w ( t, p ) for large t takes the form F w ( t, p ) = √ t + 14 (cid:18) log (cid:18) t (cid:19) − (cid:19) − p r t − p t + 124 p (cid:0) p − (cid:1) (cid:18) t (cid:19) / ++ 18 p (cid:0) p − (cid:1) (cid:18) t (cid:19) − (cid:0) p (cid:0) p − p + 8 (cid:1)(cid:1) (cid:18) t (cid:19) / + O (cid:18) t (cid:19) . (D.33)The leading behavior is independent of p and it coincides with that of the Gross-Wittenmodel. The above expression contains also the result of the semiclassical approximation(7.15), up to higher orders in p n /t n + m/ . We can also compute the small t behavior ofthis solution and it is given by F w ( t, p ) = 12 (cid:0) log( p ) p + (3 − log 4) p − ( p + 1) log( p + 1) (cid:1) ++ p t ) + t p + 4 − pt p + 1) + ( p − pt p + 1) + O (cid:0) t (cid:1) . (D.34)Surprisingly, we see that F w ( t, p ) satisfies also the boundary condition (7.8) for small t and reproduces, in that regime, the result of the large p expansion. 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