FZZT branes and non-singlets of Matrix Quantum Mechanics

We explore the non-singlet sector of matrix quantum mechanics dual to $c=1$ Liouville theory. The non-singlets are obtained by adding $N_{f}\times N$ bi-fundamental fields in the gauged matrix quantum mechanics model as well as a one dimensional Chern-Simons term. The present model is associated with a spin-Calogero model in the presence of external magnetic field. In chiral variables, the low energy excitations-currents satisfy an ${SU(2N_{f})}_{\tilde{k}}$ K\v{a}c-Moody algebra at large $N$. We analyse the canonical partition function, discuss a Gross-Witten-Wadia phase transition at large $N, N_f$ and study different limits of the parameters that allow us to recover the matrix model of Kazakov-Kostov-Kutasov conjectured to describe the two dimensional black hole. The grand canonical partition function is a $\tau$- function obeying discrete soliton equations. We comment on questions related to the dichotomy between integrability and chaos that such models pose and conjecture a possible dynamical picture for the formation of a black hole in terms of condensation of long-strings in the strongly coupled region of the Liouville direction.


Introduction
The duality between the singlet sector of gauged Matrix Quantum Mechanics (MQM) and c = 1 Liouville theory is long known to provide a very powerful complementary description of the physics of this low-dimensional version of string theory. A plethora of observables such as the partition function, scattering amplitudes of tachyons and loop operator correlators have been computed and shown to match in both sides of the correspondence, with the matrix model providing further predictions for the Liouville theory. Arguably the biggest difficulty was in establishing the dictionary between the two formalisms, a programme which has been brought close to completion at least for the singlet sector and the corresponding linear dilaton background 1 . Much less is known though for the non-singlet sectors of MQM. The importance of understanding these sectors as well, comes from pertinent questions related to the presence of non-trivial states in the theory such as black holes. The existence of the two dimensional black hole solution [1,2] and generalisations thereof [3], is long known from the string theory side and has provided a rare understanding of the role that stringy α effects have in the physics of black holes. There exists also a proposal for a matrix model dual to the Euclidean black hole involving the addition of Polyakov-Wilson lines around the thermal circle [29]. The main issue with this description is that it is inherently Euclidean and one does not have a control over the dynamical aspects in the physics of the Lorentzian black hole. In addition there is still some confusion about the thermodynamic interpretation of this model in comparison with results from the effective action of the 2-d string theory [33], and whether it can really describe two dimensional black holes of arbitrary radius, since its derivation was based on the FZZ-duality [31] which holds for the specific radius R = 3/2 near the so-called black hole string correspondence point [32].
In addition to possibly providing new input to these questions, a real time description would in principle serve as an arena for attacking problems related to the black hole formation and evaporation (information paradox) [4], understanding the fast scrambling/chaotic behaviour of black holes, elucidating the nature of the black hole interior [5] and the role of state-dependence, and finally might even shed some light on spacelike singularities. We believe that even though we are quite far from achieving such a goal in the context of c = 1 Liouville theory, nevertheless the first steps have already been laid out in the works connecting the simplest non-trivial matrix model sectors such as the adjoint, with the presence of the so called FZZT branes [6,7] that extend along the Liouville direction φ. Some related quantum mechanical models that involve non trivial representations are of the spin-Calogero type [55,56] and have considerably enriched symmetries, spectrum and dynamics. An interesting fact about these models is that they can be derived by adding extra fundamental/antifundamental fermionic or bosonic degrees of freedom in addition to the N × N matrix M ij of MQM, and can thus also be thought of as models of open-closed string theory [36,40].
In this paper we will analyze further the properties of models of this kind containing a set of U (N ) fundamental/antifundamental N f × N fermionic {ψ αi , χ βj }, or bosonic {V αi , W βj }, matrices interacting indirectly with the N × N matrix M ij of MQM through a non dynamical gauge field A ij . We will find that our specific model can be also recast as a SU (2N f ) spin-Calogero model in the presence of an external magnetic field proportional to the masses of the (anti)-fundamental fields. The magnetic field/masses break the SU (2N f ) symmetry, which in the case of equal number of (anti)-fundamentals becomes SU (N f ) × SU (N f ). In several steps, we will compare the physical properties of considering fermionic versus bosonic fields. The low energy excitations-current operators are then found to satisfy an SU (2N f )k Kǎc-Moody algebra at large N . The levelk arises from adding a Chern-Simons term k TrA in the action and is affected by a normal ordering shift at the quantum-level. By analysing the partition function of the model, we will also provide further evidence that one can describe the two dimensional black hole of [29] and possible generalisations thereof, by resorting to a double scaling limit involving N f and the mass of the fundamentals/antifundamentals in addition to taking the usual double scaling limit of MQM. In such a scaling limit the partition function exhibits a Gross-Witten-Wadia thirdorder phase transition [65,66]. The grand-canonical partition function is found to obey a specific form of the discrete Hirota-Miwa equation [91,92], which in the afforementioned limit reduces to the Toda lattice studied in the work of [29].
The main difficulty again is into developing an appropriate dictionary between string theory and our matrix model that will allow for asking sharp questions for the gravitational dual. In our work we take a preliminary step into filling this gap. An analysis of winding modes on the two sides of the duality and of the canonical/grand-canonical partition function, allows us to identify the parameter of the open string boundary cosmological constant µ B with the mass of the (anti-fundamental) fields via the simple relations σ = 2m and µ B = √ µ cosh(πσ) where µ is the closed string cosmological constant. The Chern-Simons levelk seems to be related to the flux sourced by the bulk FZZT branes. The closed string chemical potential µ and the Chern-Simons levelk are then found to form a natural complex string coupling g −1 str ∼ µ + ik. To further check this preliminary analysis one could compute some correlation functions or scattering amplitudes on the two sides. In the usual MQM there are two main ways to further understand the dual target space physics, namely the collective field formalism [11] which should be thought of as the string field theory of tachyons and the bi-local fermionic field theory formalism [12] that is in principle more powerful but harder to manipulate for explicit computations. The collective field theory formalism for spin Calogero models and adjoint/fundamental Matrix models has been considered already in [13,14,16] but is also quite complicated. In particular one should find new interesting states that solve the non-linear string field equations and then expand in fluctuations of the string fields. We believe that further work on this topic could give an impetus in answering the aforementioned questions from a target space point of view.
On the contrary, it is important to note that in order to answer questions related to thermalisation and a possible chaotic behaviour of such models, one does not need a fully developed dictionary and it is enough to compute appropriate two and four point correlators [19] and analyse their properties along the lines of what has been achieved for the SYK model [20]. Even though the class of models we study are generically integrable, one can imagine a slight deformation that could provide for such a chaotic behaviour, such as a small anisotropy in the magnetic field of the spin-Calogero model. We hope to present a detailed analysis of two and four-point correlation functions in a future publication. In relation to this point, we should also mention that similar models of adjoint interacting with fundamental degrees of freedom have been studied in the past and recent years [21][22][23] and the calculation of correlators in these models indicated that one needs a large number of flavors in order to have a chance to satisfy the refined criteria of chaotic behavior stemming from the four point function, a conclusion which seems in line with the scaling limit we take (where both N , N f → ∞), in order to have a strong backreaction effect that reproduces the matrix model partition function of the two dimensional black hole and some generalisations thereof.

The Matrix Model
In this section we will briefly review the gauged version of matrix quantum mechanics (MQM) that describes a set of N unstable ZZ branes [37,38], before adding to the model N f FZZT branes described by the addition of extra fundamental/antifundamental fermionic or bosonic fields into the action. The gauged MQM model is defined by where the covariant derivative with respect to the gauge group is with V (M ) = − 1 2 M 2 in the relevant case of c = 1 Liouville theory described by the double scaling limit of MQM that focuses in the unstable inverted harmonic oscillator potential near the tip [25,27]. The theory has an SU (N ) gauge symmetry 2) The conserved current associated to this symmetry is J = −i[M,Ṁ ], note that it is traceless and its diagonal U (1) part is trivial as one can also see from the transformation One can set the gauge field to zero, but needs to impose the Gauss law constraint δS/δA = −i[M,Ṁ ] = 0. This has the effect of projecting to the singlet sector where J = 0. In a more general context, the ungauged system decomposes into different irreducible representations of SU (N ) algebra classified by J [61]. More precicely, the allowed representations R should be such that they can be constructed taking products of the adjoint [41]. A constraint on the possible representations comes from the fact that the diagonal part of A does not appear in the action which leads to the selection rule that only zero weight states should be considered [27]. This is again in accord with the fact that J is traceless. For the general representation, after diagonalising For the singlet sector (K = 0) the second Calogero-type term is not there and the system reduces to N-free particles in the potential V. In the Hamiltonian picture, after redefining the wavefunctions asΨ(λ) = ∆(λ)Ψ(λ) one findŝ withP a projector to zero weight 2 states. In particular the singlet sector describes N non interacting fermions in the potential V (λ). This singlet sector has been shown to describe the physics of c = 1 Liouville theory upon taking a special double scaling limit of WKB type [24]. The bosonic string is described by an unstable cubic potential while the 0B superstring by a quartic potential. Taking the double scaling limit is equivalent to filling the fermi sea near the unstable tip of the potential and in this limit the model becomes Gaussian and exactly solvable. For more details we refer the reader to the excellent existing reviews [25,26].
We will now consider an extension of this model. We want to "feed in" non-singlet representations while still keeping the model gauged. To achieve this we can add extra fields into the action. In particular we will be interested into adding two extra fermionic or bosonic matrices, , as well as under a global SU (N f ) × SU (N f )-flavor like symmetry, which we symbolise with the Greek indices. The relevant fermionic action is with D t ψ α = ∂ t ψ α − iAψ α the covariant derivative for the fundamental. Similarly the bosonic action (V, W are complex) is Note that the fundamentals/antifundamentals are coupled only indirectly through the gauge field A to the matrix M . We keep in mind the general possibility of having different masses for each flavor and different number of flavors for the (anti)-fundamentals but we will mostly discuss the equal number of flavors/equal masses case. This affects the global symmetry of the system which varies from SU (N f ) × SU (N f ) when all masses are equal to U (1) N f +N f in case that all the masses are different. Although it is possible to add terms of the form ψ † F (M )ψ, in the double scaling limit where we focus at the tip of the potential, they would lead to a renormalization of the mass term of the N f × N matrix fields. Nevertheless it might be interesting to study such extensions in more detail. We are now also in the position to augment the SU (N ) symmetry to U (N ), by adding the 1d Chern-Simons term S CS = k dtTrA [57] 3 . It is inconsistent to add this term in the usual MQM action, since one cannot fullfill the Gauss-law constraint J + kI = 0 with finite dimensional matrices. In contrast, adding extra fermionic/bosonic degrees of freedom into the action makes it possible [57,59] as we will soon describe. Similar models have been proposed in the past for the description of open-closed string theory [36] (albeit the single flavored non-gauged version) as well as in the context of the matrix model description of FZZT-branes [42] (where again the single flavored model is studied in some detail). The extra Chern-Simons term is not encountered in these studies 4 , but it can certainly be present and we would like to understand its physical implications. This term recently appeared in [62,63] and we will comment on the connection with these works in section 2.1. Finally, let us also note that S f arises in more modern studies of Grassmann matrix models [52][53][54] which consider quartic interactions between the Grassmann matrices. In these later models after performing a Hubbard-Stratonovich transformation, the theory becomes quadratic in the presence of an "effective" gauge field A and thus similar to the fermionic sector of our model 5 . We thus conclude that the present model is a generalization of such models studied previously in the literature and blends several of their features. In the end of this section we will find out that it belongs to the general category of spin-Calogero models which are reviewed in [61] with extra kinetic terms for the fundamentals and in the presence of an external magnetic field.
We will now consider the total action with the fermionic and Chern-Simons term S M QM + S CS +S f and keep an equal number of flavors for simplicity. The bosonic case can be treated in the same way just by replacing anti-commutators with commutators in the appropriate formulae. One can immediately find the Gauss-law constraint to be where in the quantum version one can interpret this constraint as projecting to the singlet representation of the global Hilbert space. Upon quantizing the theory the fermions acquire the usual anticommutation relations [ψ αi , ψ † βj ] + = δ αβ δ ij and [χ αi , χ † βj ] + = δ αβ δ ij . One then also needs to consider the normal-ordered form of the constraint. By taking its trace, it is easy to see that the normal ordering ambiguity can shift/renormalise the value of the Chern-Simons level. Our normal ordering should be defined along the lines of [36] and in the quantum version of the model we explicitly have Taking its trace one finds which means that the parameter k should be quantised. This equation implies that the number of boxes in the Young tableaux should be an integer multiple of N. In addition we also have the stronger form (2.10) We will now show how one can connect this model with the well studied spin Calogero models [61]. Using fermionic bilinears, one can define the following SU (N f ) flavor-current operators (i is like a "lattice site"-not to be summed over) which satisfy the SU (N f ) algebra [T αβ , T γδ ] = T αδ δ βγ − T γβ δ αδ independently and transform in the same way under U (N ). One can also combine them in a single operator . Using a Schwinger-Wigner representation, one then defines also the following spin operators which satisfy an SU (2) algebra for each lattice site and flavor with σ a mn , a = (1, 2, 3) the three Pauli matrices and Ψ † m = (ψ † , χ) 6 . The nomenclature stems from the fact that this SU (2) can also be thought of as a spin SU (2) for which Ψ ↑ = ψ and Ψ ↓ = χ † . Finally, one can define the spin-flavor operators The set of all these operators satisfies an SU (2N f ) algebra, much similarly as in the nonrelativistic quark model [82], in the form 14) with f ABC the totally antisymmetric structure constants and d ABC the totally symmetric ones.
Integrating out the gauge field in the path integral after completing squares [36], or passing again to the Hamiltonian picture and imposing the constraint, one can then write the Calogero-type interaction term in terms of the above operators. In particular using the constraint 2.8 the form of the Calogero interaction term is found to bê Then, using the operators above, one can also write this expression as follows ( 1 in terms of spin, flavor and spin-flavor operators with ferromagnetic long-range couplings. As a comparison the bosonic model carries through with very similar steps, the difference being in that the Calogero interaction term the long-range couplings are now antiferromagnetic. This part of the Hamiltonian has an SU (2N f ) symmetry, which is broken through the mass terms m α Tr ψ † α ψ α + χ † α χ α . Since this form of the Calogero Hamiltonian is a bit non standard, it is quite illuminating to note that one can transform it in a more conventional spin Calogero form 7 using the observation that we can equivalently define it in terms of 2N f -flavors of the fundamental fields Ψ † αi = (ψ † αi , χ αi ) by renormalizing both the vacuum energy and the Chern-simons level of the coupling ask = k + N f so that the constraint reads This needs also to be supplemented with the condition mα = (m α , −m α ) for the mass term mαΨ † αi Ψα i . The transformation in the bosonic case differs in the shiftk = k − N f . After this transformation the relevant terms in the Hamiltonian become which is an SU (2N f ) spin-Calogero model in the presence of a non-abelian "magnetic" field B = Ã BÃTÃ with TÃ,Ã = 1, ...(2N f ) 2 − 1 the SU (2N f ) generators. In our specific case, B = diag{mã} is a diagonal matrix. The mass term-magnetic field partially lifts the degeneracy of the energy states of the spin Calogero model, for more details see [58]. These models although integrable, have a highly intricate spectrum. In terms of allowed representations, eqn. 2.17 indicates that one should consider representations havingk × N boxes. Then one should decompose thek-fold symmetric/anti-symmetric tensor product rep of the N spins into SU (2N f ) irreps, having at most 2N f rows/columns for the bosonic/fermionic case. The full model then decomposes into a direct sum of standard spin Calogero models, where the spins carry different admissible representations according to the previous rules. We also conclude that in terms of the Young tableaux that characterize the allowed representations of U (N ), the bosonic/fermionic cases are simply related by flipping the tableaux along the diagonal. Let us also finally mention that one can also take and combine symmetric reps for boxes with antisymmetric ones for anti-boxes or equivalently taking bosonic fundamentals with fermionic antifundamentals. One then has a supersymmetric Calogero model. In section 3 we will elaborate more on the simplest gauge invariant excitations-currents that one can form and discuss their algebraic properties.

Chiral variables
It turns out that one can simplify the description of MQM and elucidate the role of the constraint for the algebra of the simplest gauge invariant excitations by passing to the so-called chiral variables introduced in [45] and further studied in [46,47,49,50]. We defineX with the commutation relations [(X + ) ij , (X − ) kl ] = −iδ il δ jk . The hamiltonian becomeŝ (2.20) and the action for the usual MQM part in chiral variables can be written as In this variables the action is first order in time derivatives and the normal ordered constraint acquires the form Similar first order actions had been proposed in studies of non-commutative Chern-Simons theory in relation to the quantum-Hall system [59,60] with a revived interest in the recent papers [62,63] that studied the connection with the SU (N f ) k WZW model in the large N-limit. Our model is a slight extension of these recent works, since we consider both fermionic and bosonic fundamental and antifundamental U (N ) matrices 8 , ψ αi , χ αi or V αi , W βj , instead of only complex fundamental bosons and most importantly the operatorŝ X ± are hermitean and not hermitean conjugates which in that case made them act as creation/annihilation operators. This is owed to the fact that we use an inverted harmonic oscillator potential. Had we used the usual harmonic oscillator we would have ended up with the term dtTriZ † D t Z − 1 2 TrZ † Z with Z a complex matrix acting as an annihilation operator to physical states. The similarities of the two models though pinpoint to the fact that our model is also naturally related to the WZW model at large N. To further corroborate this, one should study in detail the partition function and the algebra of the simplest gauge invariant excitations-currents to which we now turn to.

Gauge invariant excitations and Collective fields
Given the gauge invariant vacuum |0 , one can build other U (N ) invariant states acting with the following operators-U (N ) singlets. To keep the discussion simple, we will describe the fermionic case in terms of the 2N f variables Ψα i = Ψ mαi with m = (↑, ↓).
• One first defines the U (1) currents These are known to correspond to tachyon vertex operators (closed strings) in the dual string theory.
• Using X − , one can then define the following SU (2N f ) spin currents (positive graded) with U (N ) indices contracted In our original description one can split these into flavor, spin and flavor-spin currents • One can also define similar currents using X + We immediately see that the n = 0 currents are common and form the operators of the previous section. If we keep the two sets of currents separate, it is easy to check that they obey a form of the SU (2N f ) Kǎc-Moody algebra without the central extension, for example (T n )Ã , (G m ) bB = ifÃBC(G n+m ) bC . Upon identifying the negative grading currents as J −m =J m , one then expects that a central extension term will arise, coming essentially from terms containing commutators of the form [(X − ) n , (X + ) m ] when one substitutes the constraint 2.22. We have not managed to show this in full generality due to technical complexity, but in [63] it was shown that at large N things simplify and one indeed recovers a central extension term from these commutators 9 . Moreover, using the techniques in [63] with some minor adjustments, we have also checked in appendix C that the partition function is indeed related to the SU (2N f )k WZW model at large N.
We currently cannot prove to what type of excitations they correspond to in the dual string theory, but SU (N f ) singlets should correspond to open strings streched between the same FZZT brane, while non-singlets to open strings streched between different FZZT branes. The role of the sandwiched powers of X ± seems to correspond to gravitational (tachyonic) dressing. A way to properly identify the corresponding open/closed string theory excitations that these operators correspond to, is to develop the collective field theory description of this model. This will also indicate the symmetry algebra of the excitations in the double scaling limit, since in the simplest MQM case one can perturb the fermi sea with the operators W n,m = x n + x m − which are known to obey a W 1+∞ algebra [17] [W n,m , W n ,m ] = (nm − mn )W n+n −1,m+m −1 . (3.5) The case of matrix-vector models has been treated in [13], where a very interesting nonlinear algebraic structure was found that involves Kǎc-Moody and Virasoro subalgebras. A similar non-linear algebraic structure for all the non-singlet sectors is discussed in [15] and termedŴ ∞ algebra, whose generators describe the joining and splitting of loop operators. We will now give some more details on the collective field theory of matrix-vector models.
One first defines the following collective fields The first is the usual bosonic collective field capturing the fermi sea eigenvalue density fluctuations while the second is a SU (2N f ) current providing extra flavor like degrees of freedom to this fermi sea. α ± (x, t) are chiral collective fields, analogous to the matrix variables X ± , that are given by α ± (x, t) = ∂ x Π(x, t) ± πφ(x, t). They can be defined unambiguously only if the fermi-sea does not form crests, otherwise they are multivalued functions 10 . If one neglects the flavors the collective field Hamiltonian is Notice that even though the fermions are free, the bosonization of the theory results in an interacting string-field theory, with a cubic interaction term. The extra terms arising from the flavors are is the stress tensor arising from the currents (dÃB is the SU (2N f ) killing form). One needs to supplement the Hamiltonian with extra cubic terms due to the Jacobian of the transformation to collective fields. In a further development [14], these extra cubic terms were found to be related to higher spin operators 11 and the total theory to possess a Yangian symmetry. One notices that this string field theory description is background independent. Furthermore it seems to describe the usual closed string fields interacting with a Sugawara stress tensor, with a cubic vertex for the closed string fields and a vertex where the closed string field is coupled to the stress tensor (or equivalently an interaction between closed and open string fields with the SU (N f ) indices playing the role of Chan-Paton factors), the currents being also non-locally coupled. The classical static solutions that correspond to time independent backgrounds satisfy along with an equation for the J 's. It is easy to see that in the case of no stress energy deformation, the only time independent solution is πφ 0 (x) = µ − 1 2 x 2 , which is known to correspond to the linear dilaton background. The stress tensor can provide deformations of this fermi sea. It will be very interesting to further understand the properties of our model from the collective or bilocal fermionic field theory point of view and try to connect more general solutions/excitations with target space backgrounds/ perturbative fields. One drawback of the collective field theory approach is that typically one can find the target space metric up to a conformal factor, thus it is not clear if one can derive an exact metric for a specific fluid profile. We are thus leaving the development of a more systematic treatment for the future.

Liouville Theory and Long Strings
In this section we will briefly review the connection between FZZT branes, long strings and the adjoint representation of MQM. As we mentioned in the introduction, the FZZT brane is extending along the Liouville direction and can be thought of as a D1 brane with 10 One can overcome such obstacles with the formalism of [12]. The collective field theory is a "hydrodynamic" approximation of that formalism. 11 These are current tri-linear operators W3 of spin-3.
with K the extrinsic curvature and the parameters µ, µ B the bulk-boundary cosmological constants. The length of the boundary (a loop) is l = dse bφ . The following relations between the parameters hold (µ KP Z the KPZ scaling parameter of correlation functions) .
For c matter = 1 ⇒ b = 1 and one finds a renormalization of µ KP Z , µ B such that µ B = √ µ cosh(πσ) becomes the correct relation between physical parameters.
We can now study qualitatively the dynamics of open and closed strings, see fig.1. For more details the reader can consult [42,43,[49][50][51]. The dynamics of closed strings (tachyons) is governed by the bulk Liouville potential µe 2bφ . The tachyons start from the asymptotic region φ = −∞, move towards bigger values of φ until they reach the Liouville wall at φ ∼ − 1 2b log µ, where they get reflected and return back to the asymptotic region. The dynamics of open strings is more complicated since they also feel the boundary potential µ B e bφ . For large σ, one has two regimes. Let us consider a very energetic open string that starts at the asymptotic region. It will first reach the boundary potential wall at φ ∼ −πbσ − 1 2b log µ, much before the bulk wall. It will then loose some of its energy, and get its end-points trapped due to the open string potential. If the string is energetic enough, the bulk of the string will nevertheless continue moving until it looses all the kinetic energy at φ ∼ − 1 2b log µ and returns back. It is then clear that for large σ, we have the formation of a long folded string, with endpoints stuck at the FZZT brane far away from the closed string scattering region. The reflection amplitude of this motion was computed in [49] both in Liouville theory and in the matrix model where it was found that the adjoint representation of MQM contains precicely one long folded string. In terms of the fermi-sea this long string should be thought of as an impurity interacting with the rest of the fermions via a Calogero interaction term. States containing n folded strings should then be described by considering irreducible respresentations with a Young-Tableaux of n-boxes and n-anti-boxes. This brings forward the possibility of describing a large number of FZZT branes/impurities using the model we described in section 2. The statistics of the long-strings are then to be understood from the structure of the Young-Tableaux. Some further details on partitions and Young diagrams can be found in appendix B.

Cylinder partition function
One obstacle one needs to surpass when considering possible matrix quantum mechanics models of FZZT branes, is that there is no well established dictionary between the parameters on the two sides. We will now describe a computation at the Liouville theory side that will indicate such a connection with our matrix model at least for c = 1. Either from c = 1 Liouville theory or the matrix quantum mechanics description [9,26,28] one finds the correlator of two macroscopic loops (cylinder partition function) with Neumann boundary conditions in Euclidean time to be with p the momentum conjugate to Euclidean time and σ is related to the boundary cosmological constant with the formulas 4.2. For winding modes around a compact Euclidean circle with Dirichlet boundary conditions, the analogous computation was performed in [49] (up to normalization) We can compute this integral by extending the integration range to E ∈ (−∞, ∞), writing the cos 2 in terms of e ±2iπσ and picking up the appropriate poles. For the purely winding modes around the compact Euclidean direction we need to pick the poles at E = inR. One then finds The first term has the interpretation of a factorised vortex anti-vortex correlator of winding number ±1, the higher terms correspond to higher windings. It also seems that the extra poles at E = im correspond to other string excitations that exist both in the compact and non-compact case. In the compact case these might be interpreted as giving the contribution of interactions between these vortices. The partition function of the first term in the expansion above, has been shown to be proportional to the matrix model partition function in the adjoint representation [49]. More precicely Z M QM adjoint ∼ Z singlet Z imp with Z imp (β) the first term in the expansion 4.5 describing the partition function of a single impurity. The total sum can also be interpreted as a grand-canonical free energy of impurities corresponding to vortices anti-vortices , essentially the inverted harmonic oscillator partition function. This expression describes vortices -antivortices in terms of bosonic particles with no interactions between them. The extra poles would give rise to interactions among these bosonic excitations. For large σ which is the regime of having long-strings, one can express the fugacity in more natural variables as This expression also matches the genus -0, µ → ∞ limit of the expression given in [77] up to the expected leg-pole factor normalisation of the winding operators between the matrix model and Liouville side of the duality with T nR the Liouville vortex operator describing a string wound n-times around the thermal circle.
By introducing N f flavors of FZZT branes, one just multiplies 4.5 by N 2 f due to the Chan-Paton factors at the open string endpoints and it is then possible to take a double scaling limit such that the flavored FZZT branes backreact on the geometry produced by the ZZ branes keeping only the first winding mode. One could also take a similar scaling limit, which has an advantage of separating the µ, µ B contributions [49] These scaling limits have the additional property that relate the number of FZZT branes to the size of the thermal circle. One might then sensibly try to recover a semiclassical backreacted geometry for large R. This limit will be further discussed in the next chapters, where we will also perform a matching of the mass of the fundamental/antifundamental matrix model fields with the parameter σ of the FZZT branes. We will also discuss the expectations that taking such a limit one can access the physics of two dimensional black holes, and corroborate this with an explicit computation of the partition function on the dual matrix model.

Relation between ZZ and FZZT
We will now discuss a relation between the ZZ and FZZT boundary state, that will later on provide a further check of our proposal for the matching between Liouville theory and Matrix model parameters. In the work of [6][7][8] one finds two sets of natural boundary states, the FZZT brane boundary state (|ν are Ishibashi states) and the ZZ-brane boundary state 12 (m, n are integers) It was noticed in [9] that one can derive the ZZ-brane boundary state, from the FZZT one for a specific choice of imaginary σ For b = 1, this also results in a simple relation of the boundary and bulk cosmological constants µ = µ 2 B .

Canonical ensemble
In this section we will analyze the partition function of the model with S = S M QM + S CS + S f . We chose to integrate out the Grassmann matrices ψ, χ as well as the matrix M to derive an effective action for the gauge field A to be reduced to eigenvalues. Equivalently, one might try to compute the partition function with Hamiltonian methods as in [63]. We will need to work in Euclidean time τ with period β. The gauge field A and the Matrix M are periodic functions of Euclidean time, while the fermions antiperiodic. We will choose to consider a refinement of the partition function where each fermion flavor has its own mass m α . In the end we can always restrict to the same mass case where m α = m, ∀α.
Integrating out the fermions we get The case of complex bosons similarly gives These are functional determinants in the space of anti-periodic/periodic functions on S 1 respectively. One can also integrate out matrix M to find Explicit computations with similar functional determinants can be found in [44,73]. For completeness we also carry out their computation in appendix A. Let us also note that all the resulting functional determinants are invariant under the original gauge symmetry This has the schematic form DAf (A). It can be reduced to an integral over the zero modes of A. A thorough derivation can be found in [83].
It is then convenient to use the angle variables with the 1/N ! corresponding to the discrete Weyl-group S N of permuting the N eigenvalues and the (2π) N coming from the stability group U (1) ⊗N . Upon assembling the various terms from appendix A, one finds the following form of the partition function as an angular integral in the case of fermions (one simply obtains the bosonic expression by replacing the cosines with sines in the denominator) where q = e −βω13 and the normalization now reproduces the correct normalization of the matrix harmonic oscillator [27] if we neglect the fermionic fields ψ, χ. One notices that the integral is symmetric in m α ↔ −m α . We can further massage this expression into One can also arrive to the same expression via a different route, that provides a good check for this expression and gives a more clear basis-independent interpretation of the various terms. In particular one trades the integration over the gauge field to a U (N ) twist of the boundary conditions of the fields and an integral over the U (N ) twist [27,44]. . We now treat all the terms in a similar fashion and find for the full partition function (det is now a matrix determinant) which matches 5.8 upon diagonalising U (in the denominator we have the determinant in the tensor product space). The first interesting fact to notice is that the fermions induce winding perturbations, whose strength is governed by m α and β, in the form of determinant operators. This can be made even more explicit by exponentiating 10) with z w (β) = N f α e −βmα , the single winding mode fugacity. A similar winding perturbation involving all winding modes in the context of Liouville theory was also recently found in [44], having only a β dependence. Fundamental/antifundamental complex bosons can be treated similarly and give some inverse determinant factors of the form These terms can be interpreted as a grand canonical partition function of positive/negative winding Wilson lines with fugacities e −βmα , in a similar spirit to that of section 4. In the first case the statistics is fermionic whereas in the latter bosonic as expected from the nature of the fundamentals. Moreover if one tunes the total number of bosons and fermions to be equal with equal masses, one finds that the vacuum energy term e βM N from the numerator cancels with a similar term in the denominator which is a hint for a supersymmetric point in the moduli space of these kind of models and thus of the supersymmetric version openclosed string theory 14 . From a more formal standpoint, these expressions are also the generating functions of characters of the fundamental/antifundamental representation of U (N ) , TrU n = χ f (U n ), Tr(U † ) n = χ af (U n ), see appendix B.2. In the same spirit one can also exponentiate the term in the denominator of 5.9 using [73] det( This is to be expected since the matrix M ij is just another matter field that transforms in the adjoint representation of U (N ). This formalism allows to treat matter fields with Gaussian action in any representation, by just replacing with the appropriate character. Similar models with fundamental and adjoint characters can be found in QCD studies [67,68] and more recently in [74] and display a quite interesting phase structure at large N . We provide more details in the next subsection.
Finally we now turn to the interesting factor det U k arising due to the topological Chern-Simons term. This term has already appeared in the Leutwyler Smilga matrix model [69] describing the IR physics of finite volume QCD [70]. Similarly to that case, one should interpret the parameter k as labelling different superselection/topological sectors. In addition it is also possible to sum over distinct topological sectors to derive a partition function with θ angle dependence by Z(θ) = k e iθk Z k . To understand the string theory meaning of this θ angle, let us note that in the case of type 0A supersymmetric string theory a similar Chern-Simons term has been discussed in [48]. In that work it was understood that it arises from the presence of a different number of D0-D0 branes/anti-branes. From the point of view of supersymmetric 2-d string theory it has the interpretation of adding k-units of RR electric flux to the system through a bulk term of the form k dτ dφF τ φ . It is natural for us to expect that our term has a similar string theory interpretation, albeit our theory is bosonic or 0B and the flux should be understood to be sourced by the spacetime filling FZZT branes. This is in accord with the constraint whose trace measures the difference between the number of U (N ) fundamental/anti-fundamental fields that describe the FZZT branes/anti-branes. The θ angle corresponds to the flux, which can be changed continuously in the Z(θ) ensemble.

Partition function and symmetric polynomials
One can rewrite the partition function in terms of symmetric polynomials. The relevant computation is carried out in appendix C. The computation follows [62] with a slight extension. The result for bosons reads with a similar result for fermions given in appendix C. The sum is over all partitions/representations λ, K λ,(k N ) (q) are Kostka polynomials and s λ (X) are Schur-polynomials of the variables xã = e −βmã . One should also keep in mind the relations for bosons/fermionsk = k ∓ N f and thatm a = (m a , −m a ). For more details see also the appendix B.

Large N, N f limits
One can take a large N limit of eqn. 5.12. Then one finds a single rectangular partition/representation to contribute. The result is where χ Rk ,C is the character of theÂ 2N f affine Lie algebra at levelk associated to the rep . This is thek-fold symmetrization of the C th antisymmetric rep. The partition function for the fermions is quite similar and contains the character of thek-fold antisymmetrization of the C th symmetric rep.
This result is nevertheless too simple to exhibit interesting phenomena such as phase transitions. One would expect that there should be a competition between different representations as was found in the Douglas-Kazakov phase transition [76]. In that context one finds the Young tableaux to acquire a continuous shape for large reps and a phase transition when the rectangular shape with a corner "melts" to a smooth shape with no sharp edges.
To this end, another very interesting limit that one can examine in our construction is the following (of Veneziano type) N and N f → ∞ with N f /N = 2L fixed (5.14) In such a limit one expects a large number of FZZT branes to condense causing a large backreaction on the closed string background of the ZZ branes. It would be very interesting to study this limit from the point of view of partitions and it is easy to see that one needs indeed to understand the limit of large partitions for expression 5.12, since bothk, N are large. Some recent results might be of help here [64], since they allow a representation for the partition function in which N and N f are treated in a similar fashion. Instead we will follow a more standard procedure that involves the density of eigenvalues on the circle The normalization is π −π ρ(θ) = 1. We also define the moments Following [73,74] one can show that the adjoint characters in this limit behave in a similar fashion to the fundamental ones. In particular one needs to find the equilibrium configuration based on the action 1 n a n ρ n + N f N b n TrU n + TrU −n , a n = q n , b n = (±1) n+1 x n ,  where we kept the case of equal masses and the ± refer to bosons/fermions. The saddle point is given by self-consistently solving where now all expressions have support on a contour C, since the eigenvalues can generically have support on several segments of the unit circle. In this equation, the Vandermonde makes the eigenvalues to spread, while the potential terms clump them. The result depends on this competition. The two simplest cases are to have support on the full circle (A 0 case) or in an arc of the circle (A 1 case), see fig. 2. It could also saturate in some arcs. The most general case is described in [75]. If one keeps only the first winding mode, one finds a GWW phase transition [65,66] between the A 0 and A 1 phase. This third order phase transition is encountered before one reaches the Hagedorn temperature. From the dual string theory point of view according to the analysis of [79,80], it should correspond to a string-black hole transition that is really a crossover for finite values of the parameters. In our model it happens for It is easy to see that the phase transition can exist only in the Veneziano limit and not in the simple large N limit. An interesting novel extension would be to analyse all the winding modes together to understand better the global phase structure of the model.

Matching parameters and limit of the two dimensional Black Hole
In order to compare computations between Liouville theory and the matrix model, one should perform a matching of the parameters on the two sides of the correspondence. A preliminary understanding can be provided through the study of winding modes that also allows to establish a connection with the matrix model of [29], conjectured to describe the physics of the two dimensional black hole.
We can now take again the double scaling limit that picks the first winding modes (assuming m α = m, ∀α), We thus see that the only winding modes surviving in this case, i.e. exp t TrU +tTrU † , are identical to those studied in the matrix model of [29], conjectured to describe the physics of the SL(2, R)/U (1) black hole coset via the black hole -sine Liouville correspondence of [31]. This limit can also be extended to a triple scaling limit of the Veneziano type by sending both N, N f → ∞ as in the previous section. We note that in this limit bosons and fermions behave in the same way, so it is universal for both realizations of open-closed string theory. In section 4, we discussed a very similar limit in the Liouville theory side that again decouples higher windings. The Liouville and MQM limits turn out to produce the same winding modes if we identify the parameters σ = 2m, so that the mass of the fermions is naturally related to the open string chemical potential. This leads to the identification oft in terms of string theory parameters as This analogy can be extended further for all the higher winding modes. A further check of the matching σ = 2m goes through the relation between the ZZ and FZZT branes described in 4.2 for σ = i(n 1 + n 2 ). The matrix model parameter e −βm describing the winding mode fugacity becomes then e −iβ(n 1 +n 2 )/2 which for the (1, 1) brane corresponds precicely to the parameter q = e −iβ of the inverted oscillator. Moreover in the next section we will see that upon realising the partition function as a τ function of the Toda hierarchy, the natural Toda time variables are nott l = N f e −lβm /l, but the rescaled t l =t l /2i sin(πnR), see eqn. 5.24. This will allow for a physical interpretation of the Toda time variables as the partition function of non interacting winding modes/vortices. We thus interestingly find that couplings/parameters of the partition function [29] can be interpreted as the partition function of some more microscopic variables.
The only parameter thus left to be matched on the two sides is the Chern-Simons level k. Although we have discussed its relation to spacetime flux sourced by the FZZT branes, the formalism of integrable hierarchies described in Appendix F indicates that the partition function depends on the complex combination µ + ik that should be roughly thought of as an inverse complex string coupling g −1 st = µ + ik along the lines of [49].
Let us finally note that it is expected that "higher spin" generalizations of the 2-d black hole exist [3], for which the discrete states [25,81], remnants of the higher spin excitations in higher dimensions, are turned on together with the usual 2-d black hole operator. In [78] it was shown that a more general version of the FZZ correspondence relates these discrete states with the simultaneous perturbation of higher winding modes. In our case the fundamental fermions resulted into winding perturbations of arbitrary order and thus keeping all these modes, could mean that the model might be able to capture a higher spin generalisation of the 2-d black hole in a limit where one keeps all of them.

Grand Canonical ensemble
We will now discuss the grand-canonical ensemble of the model. The reason for doing this, is because the connection to c = 1 Liouville theory through the double scaling limit is most easily performed there, whereby one tunes the level of the fermi-sea through the chemical potential µ. One could in principle introduce chemical potentials both for N f and N . Since we do not understand the purpose it would serve for the FZZT branes very well, we will refrain from doing so in this work and therefore introduce only an extra chemical potential conjugate to N to be later identified with the closed string cosmological constant µ 15 . Later on we will also consider the double scaling limit of the previous section as N f → ∞. In our case one is thus lead to compute It is easy to pass over to the grand canonical ensemble using the Cauchy identity as in [27,29,44] to write the integrand as a determinant. The final result for the grand canonical ensemble takes the form of a Fredholm determinant where the integral kernel is defined via its action on test functions f (z) as (z = e iθ ) |n| z w (|n|β) , z w (β) = N f α e −βmα and t 0 = βµ. It is hard to compute the spectrum of such a kernel 16 , nevertheless in appendix E we express the kernel in the energy basis. These expressions might prove useful in deriving the density of states for the partition function.
To proceed further it can be useful to exploit the integrable structure inherent in such models. In particular, it was found [29], that in the case of k = 0 and a finite number oft n this grand canonical partition function can be interpreted as a τ function of Toda integrable hierarchy with the rescaled couplings t n =t n q −n/2 − q n/2 , (5.24) playing the role of Toda "times". In our example we see that we also have the extra conjugate-zero Toda "time" term k log z arising essentially from the presence of the Chern-Simons term 17 . Some more details about integrable hierarchies and free fermions and how we can still realise our partition function as a τ function for arbitrary k are given in appendix F. The result is that 15 A more detailed treatment can be found in the review [25]. 16 Although q-deformed polynomials might be of help here. 17 The reason why k is conjugate to t0 can be very easily deduced from the form of Virasoro constraints, see appendix D and eqn. 5.31.
which is a τ -function with a total charge dictated by the Chern-Simons level. In appendix F it is shown that shifts in the chemical potential are related to imaginary shifts in the charge l of the τ function µ → µ + il, whereas the Chern-Simons level k is directly related to the charge of the τ function k ≡ l. This points to the possibility of defining a complex string coupling as g −1 str = µ + ik. We also observe that the free energy normalization prefactor has some very interesting combinatorial interpretation, since it is also based on the completeness relation) B.3 for Schur's symmetric polynomials (or characters) where the sum is over all partitions λ. In this formula we have i the oscillator energy levels (with ω = i for the inverted oscillator). If we then restrict to the case of equal masses, we precisely match the Liouville computation of the winding modes free energy 4.5 and 4.6, since is precicely the grand free energy of non-interacting vortices 4.5, 4.6 upon identifying 2m = σ. This piece of the free energy scales as ∼ N 2 f , which points to the presence of the N f × N f fundamental/antifundamental degrees of freedom. Another interesting point is that it is the same irrespective if we use fundamental bosons or fermions. We would like to understand in more detail the physical significance of this vortex free energy term in the large N f limit, since it has the correct scaling for a deconfined phase when N f ∼ N → ∞ which means that it could account for a 2-d black hole entropy in a combinatorial microscopic fashion arising from the wound string degrees of freedom.
The τ function or grand canonical partition function obeys the difference equation F.29 discussed in appendix F which is actually a particular discrete analogue of the Toda equation (also known as Hirota-Miwa equation from the work of [91,92]). The dispersionless continuum limit of this equation would then suffice to obtain the genus-0 contribution to the grand-canonical free energy along with the associated spectral curve.
In the double scaling limit defined in the previous chapter, we find that only t +1 = −t −1 , t 0 and k remain. In this case the Hirota-Miwa equation reduces to the Toda differential equation (see [29] and appendix F) which is obeyed ∀k. Both equations 5.28 and F.29 can be supplemented with extra conditions that relate the partition function for different values of k. From the integrability and GL(∞) point of view, it turns out that k is a parameter of the point of Sato's Grassmannian, while the "times" do not affect it. The appropriate k-dependent equations are the so-called Virasoro constraints, from which the most important is the so-called string equation which is the lowest of them [71]. From a practical viewpoint, since the Toda equation has derivatives of second order a single initial condition is not enough to fully determine the solution, one needs to supplement it with the Virasoro constraints for a unique solution to be found. To derive these constraints, one can study our partition function in the abstract form (here βµ =t 0 = t 0 ) . This is an extension of the known constraints in the case of fundamental plus adjoint representation. We derived these constraints that can be found in appendix D. The result is in terms of differential operatorsL n , obeying the centerless Virasoro algebra that annihilate the partition function.
Since we want to find their action in the grand-canonical ensemble, we note that we getL where we also included the "zero-time" term et 0 N = e βµN , since the Virasoro constraints act on this term as well. When using these equations together with the Toda differential equation, one should be careful to expresst in terms of t properly. The simplest constraint isL 0 which readsL In case that k = 0 theL 0 constraint imposes a total level matching or winding conservation condition. When k = 0 there is an imbalance between winding modes. It is also easy to see from this formula that k is conjugate to the zero time t 0 . We hope to analyse further solutions to the combined system of equations in the future. Some further help to reduce such equations can be provided using KPZ-DDK scaling arguments along the lines of [29].

Discussion
In this paper, we explored a model that captures the physics of non singlet sectors of matrix quantum mechanics dual to c = 1 Liouville theory. The field content of the model is a N × N matrix that transforms in the adjoint of SU (N ) which describes the dynamics of N unstable ZZ branes and N f ×N fundamental and anti-fundamental fields (either bosonic or fermionic) that describe the dynamics of open strings streched between the ZZ and FZZT branes. We also considered the presence of a 1d Chern-Simons term that enhances the symmetry of the model from SU (N ) to U (N ).
In section 2 we write down the Hamiltonian of the model and show the connection with a spin -Calogero model in the presence of an external constant magnetic field. Introducing chiral variables and by defining the currents associated to the symmetries of our model one finds that at large N they satisfy a Kǎc-Moody SU (2N f ) k algebra, relating the present model to the WZW model, in accordance with the study of [62].
Then, we write down the canonical partition function using different methods. The most compact expression is in terms of symmetric polynomials. Another way to proceed is to write it in terms of U (N ) characters. This form makes it clear that the fundamental and anti-fundamental fields induce winding perturbations of arbitrary winding number.
The expression of the partition function in terms of symmetric polynomials allows to take the large N limit. One then finds that only a single rectangular representation contributes. More interesting physics is provided by the Veneziano scaling limit N, N f → ∞ and N f /N = const. To study this limit, it is more convenient to introduce the density of eigenvalues on the unit circle in line with [73,74]. At this limit and keeping the first winding mode, one finds a Gross-Wadia-Witten [65,66] type phase transition that is really a crossover for finite values of the parameters which is expected to correspond to the string black hole crossover [79,80] in the Liouville dual. It would be interesting to study the contribution of all winding modes simultaneously in this limit, such an analysis might be possible using the techniques that relate a similar partition function to Nekrasov partition functions [64].
It is important that one can also take an extra scaling limit to isolate the first winding modes in the expression for the partition function which relates our model to the one of [29] proposed to describe the physics of the two dimensional black hole. In the Grand-canonical ensemble, the partition function can be interpreted as a τ function of the Toda integrable hierarchy. Generically it obeys discrete soliton equations, that reduce to the simpler Toda differential equation when one focuses on the first winding mode. These equations need to be supplemented by Virasoro constraints that involve the Chern-Simons parameter k. We found the relevant constraints, but we leave as a future work the task to solve the lowest ones together with the soliton equations.
This study revealed a preliminary connection between matrix model and Liouville theory/ FZZT brane parameters. In a short summary we found that one can define a complex string coupling parameter through the combination µ + ik that takes into account fluxes sourced by the FZZT branes and that the mass m of the extra fundamental fields are related to the boundary Liouville theory parameter σ through σ = 2m. This relation seems to also pass the test that relates ZZ with FZZT branes for complex integer σ (for the (1,1) brane).
One novelty of our work is that the parameters of the partition function/Toda times seem to correspond to partition functions themselves (of wound strings) when expressed in terms of more microscopic physical quantities of the theory. This opens the possibility to understand better the thermodynamical properties of this model in a microscopic fashion and elucidate possible connections with two dimensional black hole physics. To achieve this it might be enough to study the dispersionless limit of the difference/differential equations, that is known to reproduce the genus-0 contribution to the Free energy.
Let us now, present in more detail what future directions one can follow: Thermodynamics: We described an explicit connection between the proposed Matrix Quantum Mechanics description of the two dimensional black hole, our model and of a condensate of FZZT branes. As we briefly discussed in the previous sections, there is still some confusion about the thermodynamic interpretation of the matrix model of [29]. The reason is two-fold. First in that work, it was not possible to fully solve the Toda equations but only the Toda equations with a single initial condition coming from the unperturbed circle partition function. The string equation was solved in the dispersionless limit in [30], and provided an exact relation for the genus zero part of the free energy. Even more important is that the microscopic origin of the Toda time parameter t that controls the strength of the vortex perturbations is obscure and one does not know whether it should be regarded as an independent physical parameter or if it should depend on µ, R or any other parameters when discussing the thermodynamics of the model.
In this work we obtained some understanding on the possible microscopic origin of t (through the presence of N f FZZT branes/ non-singlet representations of MQM) and elucidated its relation to Liouville theory quantities such as N f , µ B , µ KP Z . We should also mention at this point that the FZZ correspondence in which the model [29] is based on, is known to hold for the radius R = 3/2 close to the so called black hole-string correspondence point [32], which poses some trouble into extrapolating this result to arbitrary radii. This is where we think that since our coupling is manifestly dependent on the size R of the compact dimension and the number of flavors N f , the thermodynamics might prove more reasonable to understand and it would be interesting to repeat a thermodynamic analysis and compare with the results of the free-energy computed via semiclassical effective action approaches [33].
Black holes -Integrability vs Chaos: Connected to the previous discussion is the possibility that such models could describe non-trivial target space states such as the two dimensional black hole. An immediate conceptual clash arises between the integrability of Spin-Calogero models and the apparent chaotic and thermal behaviour of black holes exemplified in the recent studies of four-point (OTOC) correlators. Even though we cannot provide a complete answer to this dichotomy we would like to make some preliminary comments.
First of all Spin-Calogero models exhibit quite interesting and highly degenerate spectra. As an example if we focus only on the spin degrees of freedom of our Hamiltonian 2.18, corresponding to the so-called Polychronakos-Frahm spin chain [58], one finds a highly degenerate (but equispaced spectrum) and for large number of "sites" N the level density is approximated by a Gaussian distribution. For this spin-chain it is also known that the unfolded spectrum statistics is neither of the Poissonian nor of the Wigner type indicative of the usual discrepancy between integrable and chaotic systems. For more details see [85] and references within. To this end one can imagine the possibility of highly degenerate spectra acquiring chaotic behaviour with the addition of small perturbations into the system. The highly degenerate levels start to split and form an approximate continuum around a given reference state. A correlator of probe operators computed on a high energy state that had a quite simple evolution law for the degenerate equidistant spectrum might then start to show approximate thermal characteristics 18 . In our model, such a behaviour might be provided by the external magnetic field coupling to the spin operators Ai B A S A i with i the N lattice sites and A the SU (2N f ) generators. This magnetic field is proportional to the masses of the anti/fundamental degrees of freedom and points in relative opposite directions.
One could further argue for some analogy with the well studied case of AdS 2 , which admits no finite energy excitations but a ground state degeneracy [18]. Nevertheless once one studies the case of nearly-AdS 2 through the SYK model [20], where the conformal symmetry is broken spontaneously as well as explicitly by UV effects, one indeed finds an exponentially large number of states contributing to the low energy physics and a chaotic behaviour to emerge.
A preliminary picture of the formation of a Black Hole from a target space long-string point of view involves a large number of long-strings that start from infinity where the string coupling is small and they can be considered to be approximately free. As they move in the bulk they start interacting and slow down, and if they have enough energy it is then possible for them to form a condensate and backreact on the spacetime by curving it near the strongly coupled region in the bulk. If the backreaction is strong, we expect a singularity with the associated horizon to form. A part of the long strings then remains outside the horizon (where the string coupling is small) while the tip condensate in the strongly coupled region has formed the singularity. Extra closed strings can scatter off such a configuration. This is reminiscent of the Susskind -Uglum picture of open strings with both ends stuck on the horizon [86]. The analogous process in the Spin-Calogero model, would involve a separation of two properties of the dynamics: the scattering off the inverted oscillator potential and the spin-chain part of the Hamiltonian. The first part is known not to provide any black hole characteristics by itself and is related to the Liouville wall on which closed strings scatter back to infinity. The magnetic field part due to the masses of the extra (anti)-fundamental fields is related to the open string Liouville wall, through the identification σ = 2m. One should then consider the evolution of a state with a large number of (anti)-fundamental excitations on top of the vacuum equivalent to a large number of long strings. The limit of large N f is similar to the "freezing" limit of Polychronakos since the spin chain part of the Hamiltonian becomes parametrically large. From the point of view of representations, one is interested in the limit of large representations with a big number of (anti)-boxes. Since the magnetic field breaks the big degeneracy of the high energy state approximate chaotic evolution might then arise from the large number of energy states available near the original high energy state. It would be nice to check such a picture using the spin-chain dynamics at least numerically.
State dependence: As a final comment, it would be interesting to consider the issue of state dependence in the context of two dimensional string theory through the dual matrix model formulation. This is because on the one hand one still finds the presence of non trivial states such as the two dimensional black hole and on the other hand one can hope for a much better handling of the duality at a perturbative and non-perturbative level. What the matrix model indicates is that the full Hilbert space decomposes into different sectors from which the singlet one is only able to capture the physics of the linear dilaton background and local excitations thereof. If one wants to describe other backgrounds one inevitably needs to enlarge the Hilbert space with the non-singlet sectors. There does not seem to be any fundamental reason to expect any need to discuss state dependence at this microscopic level. Nevertheless state dependence could arise for practical purposes from the non trivial mapping to target space quantities (or bulk reconstruction), once for example one wishes to describe local operators in the two dimensional black hole background in the interior and exterior regions. The existence of the collective field theory description as a "hydrodynamic" description of the tachyonic scattering indicates that a form of state dependence could arise once one uses more coarse grained variables, since currently it does not seem possible to uniquely define and describe bulk tachyons in any background with this formalism. In addition the tachyonic field operators are related via a complicated non-local transform to the string theory loop operators, a relation which is known in detail only for the linear dilaton background. One expects this construction to be modified for an arbitrary background and many further subtleties to arise. We thus believe that in order to clarify such issues one should address the mapping between matrix model non-singlets and target space physics in more detail.
To evaluate the determinant, one wants to solve for the spectrum of the matrix equation klQ ij,kl f kl (τ ) = λ ij f ij (τ ). It is convenient to expand the periodic functions in the fourier modes of S 1 , i.e. f (τ ) = f 0 + n f n e 2πniτ /β . One then has a discrete set of eigenfunctions and det Q = det matrix n λ n . Using these modes, one can write where α is a matrix and the determinant is with respect of this matrix structure. The normalization can be set N = 2 to conform with the usual harmonic oscillator. Otherwise one can adopt a regularisation procedure and keep the finite piece 19 . If the gauge field is α N ×N = diag(α 1 , α 2 , ..., α N ), then α adj ij,kl = (α i − α j )δ ik δ jl . Thus we finally get Similarly let us define the differential operatorQ acting on functions transforming in the fundamental representation (with a general mass m) Upon diagonalising the matrix α we get α f ij = α i δ ij . Similarly to the previous case we find (for periodic functions) Since we are also interested in the Grassmann case (anti-periodic fermions), we find The case of anti-fundamental can be treated in the same way and the result is obtained simply by sending θ → −θ. We will also set N = 2, with similar arguments as above.

B.1 Partitions
We provide some terminology on partitions. The reader can consult [87] for more details.
• A partition λ is a sequence of non-increasing integers such that The number of non-zero elements (λ) is called the length of the partition. The sum of all the elements |λ| = i≥1 λ i is called the weight of the partition.
• The multiplicity m j (λ) of the positive integer j is how many times the number j appears in the partition λ (such that λ i = j).
• The partitions are labelled graphically using Young diagrams. They are an array of boxes where the i'th row contains λ i boxes. This means that the number of rows is the length of the partition and the number of columns is just λ 1 . The total number of boxes is then equal to the weight |λ|.
• Another way of representing a partition is in the form (2 m 2 , 3 m 3 , ...9 m 9 , ...), which just means that the number j appears m j times.
• The conjugate or transpose of a partition λ or λ T is obtained by either reflecting the Young diagram along the diagonal exchanging rows and columns. As an example one obtains λ 1 = (λ).
As a simple example to have in mind the partition (7, 5, 3 2 , 1 2 ) corresponds to the following Young tableaux

B.2 Characters and Schur polynomials
In this section we will follow the review [84]. Schur polynomials s λ (X) are symmetric polynomials in N variables that form a linear basis for the space of all symmetric polynomials. Seen from a representation theory point of view they are characters for the irreducible representations of the general linear groups.
To be more concrete the representations R of GL(∞) are labeled by Young diagrams, or ordered integer partitions R : λ 1 ≥ λ 2 .... ≥ 0 with |R| the number of boxes in the diagram. We will thus use the representation index R or the partition index λ interchangeably in this case. We will represent the characters either through the use of time or through auxiliary Miwa variables of a matrix X as χ R (t) or χ R (X). The definition of Miwa variables is t k = TrX k /k = i x k i /k with x i the eigenvalues of the matrix X.
We define the Schur polynomials/characters as We have the following useful sum rule over all representations/partitions We also have the generating functions of characters for the (anti)-symmetrised products of arbitrary copies of the representation R [73] These expressions describe the formulae 5.10, 5.11 and 5.12 in terms of character expansions of the fundamental/antifundamental and adjoint representations of U (N ).

C Canonical ensemble and symmetric polynomials
It is possible to compute the partition function using the technology of Hall-Littlewood symmetric polynomials. This method is powerful enough to treat the case of different masses/chemical potentials for each flavor and simplifies the method of [27] in terms of characters, since one needs to perform less summations over partitions. The canonical reference is [87]. This method was also used to compute the partition function of the matrix model in [63].
We start by first defining the q-Hall inner product for two symmetric functions f (Z), g(Z) The orthogonal polynomials with respect to this measure are the Hall-Littlewood polynomials where λ ∈ P denotes the partition and the normalization is with m j (λ) the multiplicity of the positive integer j in the partition λ and m 0 = N − (λ) ≥ 0. One also defines the Q-Hall polynomials as Q λ (X; q) = b λ (q)P λ (X; q) with b λ (q) = P λ , P λ −1 q = j≥1 φ m j (λ) . The orthogonality relation can then be written as We next define the Schur polynomials through eqn. B.2, which are also a limit of the Hall-Littlewood polynomials for q = 0 and orthonormal under the inner product C.1 upon setting q = 0. One also has the relation with K λ,µ (q) the Kostka-Foulkes polynomials. The inverse relation defines the Modified Hall-Littlewood polynomials.
There is also a relation between the Modified Hall and the Q-Hall polynomials that reads Let us note some useful properties of the Kostka polynomials • K λ,µ (q) = 0 unless |λ| = |µ|. All the non-zero coefficients are positive.
We will also use the following Cauchy identities with λ the conjugate partition to λ. These should be thought of as completeness relations with respect to the inner product C.1, the first two for q = 0 and the last for non-zero q (q-Hall-inner product). The summands need to vanish unless (λ) ≤ min{N, N f }.
To exploit these identities, it is convenient to use the representation in terms of 2N ffundamental fields only. One needs to remember the shiftsk = k ∓ N f of the Chern-Simons level. For fundamental bosons one can write down their contribution in terms of symmetric functions as follows where xã = e −βmã behave like fugacities for each flavor. The Chern-Simons term can be written as with (k N ) the partition with N non-zero parts equal tok.
The full partition function in the case of bosons is then For fermions we will need to use the dual cauchy identity to simplify The bosonic partition function can also be written in terms of characters of with a similar expression for the fermionic case.
The ground state energy for the bosonic expression was first computed in [63] and upon using the parametetrization N = 2LN f + C and keeping the leading term in |q| << 1 we find E b 0 (k, L, N f , C) =kL(L − 1)N f +kLC + 1 2 N 2 (C.14) A limit one could take in expression C.13 is L → ∞ with N f , C fixed. This is the standard large N limit. In this limit the Kostka polynomial becomes a branching function that relates theÂ 2N f −1 affine characters with the A 2N f −1 characters, see [88]. The partition function in the large L limit corresponds to where χ Rk ,C is the character of theÂ 2N f affine Lie algebra at levelk associated to the rep . This is thek-fold symmetrization of the C th antisymmetric rep. For C = 0 the character can be also identified with the vacuum character corresponding to the partition function of the W ZW model [63]. The partition function for the fermions is quite similar and contains the character of thek-fold antisymmetrization of the C th symmetric rep.

D Virasoro constraints
In this appendix we briefly discuss the derivation of Virasoro constraints for our model that contains fundamental and adjoint characters. It is based on the thorough derivation of Virasoro constraints for the case of fundamental characters [72]. We start with the general action (one should remember that these t's are actuallyt's to match with the main text) and then perform variations of the matrix U consistent with the Unitary symmetry of the model One then needs to pinpoint the transformation of the various terms in the path integral and write them in terms of differential operators acting on the action S. In particular for the fundamental character term one finds One also finds from the variation of the measure that it is expressible in terms of second derivatives of e S L +me n = n m=1 ∂ 2 ∂t n−m ∂t m + ∂ 2 ∂t m−n t −m , for n ≥ 1 , We now come to the adjoint character terms which are interestingly again quadratic similarly to the terms in the measure Finally there is the Chern-Simons term with level k, det U k for which the appropriate operators read The total Virasoro constraints are simply the sum of the various terms L n = L a n +L f n +L me n + L k n . Taking linear combinations of L ± n one can show that they indeed obey the Virasoro algebra. This means that imposing the lowest constraints one automatically satisfies the higher ones. In particular one needs to satisfy L 0 and L ±1 , L ±2 . For L 0 we get The model we study is a particular reduction of this general case since the q's and t's are related due to the fact that they are expressed in a Miwa-like parametrization q m = q m /m , t m = (±) m+1 N f x m /m. This means that we need to impose correctly the constraints in the one dimensional subspace of possible q's and t's respectively.

E Kernel in the energy basis
It will be useful to study also the integral kernels in the energy basis. This is interesting, since the information about the spectrum of the theory should be in principle extractable from such a representation. A paper where kernels with a similar structure are studied is [89] and a detailed review can be found in the lectures by van Moerbecke in [90]. The context of those studies is "Probability on Partitions and the Plancherel measure".
In our context, one can understand the energy basis from the basis of polynomials on the circle. It is a discrete basis where the energy levels are conjugate variables to the θ's. In order to change basis one needs to use n|θ = e −inθ or n|z = z −n , z = e iθ . Using the resolution of the identityÎ = dz 2πiz |z z|, a generic vortex perturbed Kernel can be written in the energy basis as: with the most general u(z) = − k 2 log z + 1 2 lt l z l (k is the Chern-Simons coupling). One can expand the denominator in powers of q and perform the integrals term by term with the basic integral I l (t) = dz 2πi z −l−1 e u(z) . The simplest case is when u(z) = 0 i.e. no vortex perturbations. Then one finds the result m|K|n = δ mn q m+ 1 2 , the harmonic oscillator propagator in the energy basis. Turning on just the Chern-Simons coupling k, one finds the kernel is no longer diagonal in the energy basis and acts more like a lowering/raising operator depending on the sign of k (one should remember that it is incosistent to just add the C-S term).
To cover the deformations caused by the presence of the fermions, in a case where e u(z) = (1 + ξz) b (1 + ξz −1 ) b , |ξ| < 1 and l ≥ 0 we can find [89] with (a) x = Γ(x + a)/Γ(x) the rising factorial and we are interested in particular for the symmetric case with (b = b = N f /2 , ξ = e −βm ). Fundamental bosons can be treated in a similar fashion, just inverting the signs of ξ, b, b . In the case of the 2d black hole u(z) = 1 2 t(z + z −1 ), one can use the generating function for Bessel functions to find with J the Bessel-J function. Upon setting ξ = t/N f and taking the limit N f → ∞, we can obtain the Bessel-kernel from the Hypergeometric one. The double scaling limit here is just the limit that connects Hypergeometric with Bessel functions.
To connect with the work of [89], one instead defines the rescaled variables ζ = q 1/2 e iθ , w = q −1/2 e iθ , by which It is easy then to see that the kernel m|K|n is written as with m = n . (E.7) The second line of E.7, is derived by rescaling z → tζ, w → tw and taking derivatives with respect to t. The above integrals are around the annular region with the width dictated by β, or upon considering the inverse oscillator where q = e iβ , one should consider a small regulator such that |q| < 1. Unfortunately in contrast with the cases considered in [89] where T k = −S k , our kernel does not take the form of a Christoffel-Darboux kernel. Nevertheless, it might be useful to study further the kernel in this basis to extract information about the spectrum of the theory.

F Integrable Hierarchies
This section is a brief discussion on the Toda integrable hierarchy and how one identifies the grand canonical partition function as a Toda τ function. We will follow the conventions in [29,95]. We extended the discussion of those references in two ways. First we included the possibility of turning on the "conjugate zero-time" k, and second we cover also the case of having singularities in the measure for z = 0, ∞, coming from terms of the form (1 + az) N f (1 + a/z) N f , which lead to discrete equations for the τ function as we will see.
Our focus is on enabling the reader to perform the calculations step by step and not into giving a rigorous definition of the construction. Some more complete introductions to τ functions and free fermions are the classic review [94] and the modern [96].

F.1 Fermionic Algebra
To set up our conventions we define ψ n , n ∈ Z + 1 2 to be free fermionic operators satisfying which in particular means that ψ * creates particles and ψ creates holes and the fermi sea has particle-states filled up to charge l. With these definitions one finds the fermionic correlator Bilinear expressions of the fermions generate an infinite dimensional Lie algebra GL(∞) and a generic element of the group will take the form G = exp m,n∈Z+ 1 2 b mn ψ m ψ * n . (F.4) We will be interested in the specific case of b mn = q iµ+n δ m,n (case with zero Chern-Simons level) and b (k) mn = q iµ+n δ m,n−k for the case of C.S. level k. One also defines the current operators (or Hamiltonians) that generate the Toda time flows as J n = where t's are Toda "time" parameters. In our case, the analogous formula for a τ function where the GL(∞)-element carries the chargeQ operator is Expanding the GL(∞) operator F.12 in a series and commuting the fermions past the 20 Notice that in string theory notation one labels this operator withp, since the charge then would be the momentum of the string. where we related the correct Toda times t m =t m q −m/2 −q m/2 in terms of the apparentt as in the main text. The chemical potential is related to a shift to the charge l, that is real for the usual oscillator but imaginary for the case of the inverted oscillator (q = e iβ ). Notice also that this formula proves that the Chern-Simons level k can be identified with the charge l of the τ function and therefore allows to define the complex string coupling g −1 str = µ + ik.

F.2 Hirota equations
The τ function satisfies an infinite set of difference-differential equations known as Hirota equations. We now give the main formulas for their derivation. where the contour C 0 encircles all the singularities of z l−l exp k<0 (t k − t k )z k and similarly the contour C ∞ those coming from z l−l exp k>0 (t k − t k )z k . In particular when a finite number of t's are turned on, one should enclose the essential singularities either at ∞ or 0. Such cases lead to differential equations. In the case that the essential singularity splits into poles, we need to take the residues around these poles and this will result in discrete equations. As a general rule, the differential/difference equations that the τ function obeys are uniquely determined from the explicit form of the measure, or in other words from the evolution of the fermionic operators ψ(z).
One can also write Hirota's equations in a more explicit form (essential singularity case) by introducing the Schur polynomials s j = ∂ x f (t n + x)g(t n − x)| x=0 (F.25) Using the notationD ± = (D ± , D ±2 /2, D ±3 /3, ....) , y ± = (y ± , y ±2 , y ±3 , ....) , y n = 1 2 (t n − t n ) , (F. 26) one can rewrite Hirota's equations as a hierarchy of partial differential equations ∞ j=0 s j+i (−2y + )s j (D + ) exp The Hirota equations can also be thought of as equations for the correlators of operators generating the Toda flows. In this form they make practical sense if a finite number of Toda times are turned on as we discussed.
(F. 28) This corresponds to having only the two first Toda-times turned on.
(F. 29) with u(l, m, n) = τ (l,m,n) /τ (l+1,m,n) . This is the simplest difference equation in this category of deformations and relates the partition function for different values of N f and complex string coupling g −1 str and a specific version of the general Hirota's DAGTE [91]. One can again take a continuous limit that sends eqn. F.29 to eqn. F.28, see [93].