Many eigenvalue matrix models possess a peculiar basis of observables which have explicitly calculable averages. This explicit calculability is a stronger feature than ordinary integrability, just like the cases of quadratic and Coulomb potentials are distinguished among other central potentials, and we call it superintegrability. Aa a peculiarity of matrix models, the relevant basis is formed by the Schur polynomials (characters) and their generalizations, and superintegrability looks like a property $<character>\,\sim character$. This is already known to happen in the most important cases of Hermitian, unitary, and complex matrix models. Here we add two more examples of principal importance, where the model depends on external fields: a special version of the complex model and the cubic Kontsevich model. In the former case, straightforward is a generalization to the complex tensor model. In the latter case, the relevant characters are the celebrated $Q$ Schur functions appearing in the description of spin Hurwitz numbers and other related contexts.
We analyze the well-known equivalence between the quadratic Kontsevich-Penner and Hermitian matrix models from the point of view of superintegrability relations, i.e. explicit formulas for character averages. This is not that trivial on the Kontsevich side and seems important for further studies of various deformations of Kontsevich models. In particular, the Brezin-Hikami extension of the above equivalence becomes straightforward.
We reconsider the simple matrix model description of Hurwitz numbers proposed by S. Natanzon and A. Orlov, which uses the superintegrability property of the complex matrix model, and discuss a way of its possible supersymmetric extension to approach spin Hurwitz numbers.
This paper is devoted to the phenomenon of superintegrability. This phenomenon is manifested in the existence of a formula for character averages, expressed through the same characters at special points and of its various generalization. In this paper we develop a method of proving such formulas from first principle from Virasoro constraints and $W$-representation. We apply it to prove the formula for the Jack functions averages - appropriate analogue of characters for the $β$-deformed Hermitian Gaussian matrix model. We also sketch the construction of $W$-operators from Calogero-Ruijsenaars H…
We find that superintegrability (character expansion) property persists in the exotic sector of the monomial non-Gaussian matrix model, with potential $\Tr X^r$, in pure phase, where the naive partition function $\langle 1 \rangle$ vanishes. The role of the (anomaly-corrected) partition function is played by $\left\langleχ_ρ\right\rangle$ -- the Schur average of the suitably chosen \textit{square} partiton $ρ$; such partitions are well-known to correspond to singular vectors of the Virasoro algebra. Further, non-zero are only Schur averages $\left\langle χ_μ\right\rangle$ for such $μ$ that hav…
For ordinary matrix models, the eigenvalue probability density decays rapidly as one goes to infinity, in other words, has "short tails". This ensures that all the multiple trace correlators (multipoint moments) are convergent and well-defined. Still, many critical phenomena are associated with an enhanced probability of seemingly rare effects, and one expects that they are better described by the "long tail" models. In absence of the exponential fall-off, the integrals for high moments diverge, and this could imply a loss of (super)integrability properties pertinent to matrix and eigenvalue m…
Once famous and a little mysterious, AGT relations between Nekrasov functions and conformal blocks are now understood as the Hubbard-Stratanovich duality in the Dijkgraaf-Vafa (DV) phase of a peculiar Dotsenko-Fateev multi-logarithmic matrix model. However, it largely remains a collection of somewhat technical tricks, lacking a clear and generalizable conceptual interpretation. Our new claim is that the Nekrasov functions emerge in matrix models as a straightforward implication of superintegrability, factorization of peculiar matrix model averages. Recently, we demonstrated that, in the Gaussi…
Relation between the Virasoro constraints and KP integrability (determinant formulas) for matrix models is a lasting mystery. We elaborate on the claim that the situation is improved when integrability is enhanced to super-integrability, i.e. to explicit formulas for Gaussian averages of characters. In this case, the Virasoro constraints are equivalent to simple recursive formulas, which have appropriate combinations of characters as their solutions. Moreover, one can easily separate dependence on the size of matrix, and deduce superintegrability from the Virasoro constraints. We describe one …
We deform N-dimensional (Euclidean, spherical and hyperbolic) oscillator and Coulomb systems, replacing their angular degrees of freedom by those of a generalized rational Calogero model. Using the action-angle description, it is established that maximal superintegrability is retained. For the rational Calogero model with Coulomb potential, we present all constants of motion via matrix model reduction. In particular, we construct the analog of the Runge-Lenz vector.
The wonderful formulas by I.Dumitriu and A.Edelman rewrite $β$-ensemble, with eigenvalue integrals containing Vandermonde factors in the power $2β$, through integrals over tridiagonal matrices, where $β$-dependent are the powers of individual matrix elements, not their differences. These potentially useful formulas are usually deduced from rather complicated and non-transparent combinatorics and are not as widely known as they deserve. We explain that the superintegrability property, i.e. a simple expression of the Gaussian averages of arbitrary Jack polynomials through the same Jack polynomia…
We construct the ($β$-deformed) partition function hierarchies with $W$-representations. Based on the $W$-representations, we analyze the superintegrability property and derive their character expansions with respect to the Schur functions and Jack polynomials, respectively. Some well known superintegrable matrix models such as the Gaussian hermitian one-matrix model (in the external field), $N\times N$ complex matrix model, $β$-deformed Gaussian hermitian and rectangular complex matrix models are contained in the constructed hierarchies.
We review some relations occurring between the combinatorial intersection theory on the moduli spaces of stable curves and the asymptotic behavior of the 't Hooft-Kontsevich matrix integrals. In particular, we give an alternative proof of the Witten-Di Francesco-Itzykson-Zuber theorem --which expresses derivatives of the partition function of intersection numbers as matrix integrals-- using techniques based on diagrammatic calculus and combinatorial relations among intersection numbers. These techniques extend to a more general interaction potential.
We continue investigating the superintegrability property of matrix models, i.e. factorization of the matrix model averages of characters. This paper focuses on the Gaussian Hermitian example, where the role of characters is played by the Schur functions. We find a new intriguing corollary of superintegrability: factorization of an infinite set of correlators bilinear in the Schur functions. More exactly, these are correlators of products of the Schur functions and polynomials $K_Δ$ that form a complete basis in the space of invariant matrix polynomials. Factorization of these correlators with…
The Brezin-Gross-Witten (BGW) model is one of the basic examples in the class of non-eigenvalue unitary matrix models. The generalized BGW tau-function $τ_N$ was constructed from a one parametric deformation of the original BGW model using the generalized Kontsevich model representation. It is a tau-function of the KdV hierarchy for any value of $N\in{\mathbb C}$, where the case $N=0$ reduces to the original BGW tau-function. In this paper, we present a representation of $τ_N$ in terms of the $W_{1+\infty}$ operators that preserves the KP integrability. This naturally establishes a connection …
We investigate the physical interpretation of the Riemann zeta function as a FZZT brane partition function associated with a matrix/gravity correspondence. The Hilbert-Polya operator in this interpretation is the master matrix of the large N matrix model. Using a related function $Ξ(z)$ we develop an analogy between this function and the Airy function Ai(z) of the Gaussian matrix model. The analogy gives an intuitive physical reason why the zeros lie on a critical line. Using a Fourier transform of the $Ξ(z)$ function we identify a Kontsevich integrand. Generalizing this integrand to $n \times…
In this paper we introduce a new family of the KP tau-functions. This family can be described by a deformation of the generalized Kontsevich matrix model. We prove that the simplest representative of this family describes a generating function of the cubic Hodge integrals satisfying the Calabi-Yau condition, and claim that the whole family describes its generalization for the higher spin cases. To investigate this family we construct a new description of the Sato Grassmannian in terms of a canonical pair of the Kac-Schwarz operators.
A study of the intersection theory on the moduli space of Riemann surfaces with boundary was recently initiated in a work of R. Pandharipande, J. P. Solomon and the third author, where they introduced open intersection numbers in genus 0. Their construction was later generalized to all genera by J. P. Solomon and the third author. In this paper we consider a refinement of the open intersection numbers by distinguishing contributions from surfaces with different numbers of boundary components, and we calculate all these numbers. We then construct a matrix model for the generating series of the …
Recently we explained that the classical $Q$ Schur functions stand behind various well-known properties of the cubic Kontsevich model, and the next step is to ask what happens in this approach to the generalized Kontsevich model (GKM) with monomial potential $X^{n+1}$. We propose to use the Hall-Littlewood polynomials at the parameter equal to the $n$-th root of unity as a generalization of the $Q$ Schur functions from $n=2$ to arbitrary $n>2$. They are associated with $n$-strict Young diagrams and are independent of time-variables $p_{kn}$ with numbers divisible by $n$. These are exactly the …
The Ward identities in Kontsevich-like 1-matrix models are used to prove at the level of discrete matrix models the suggestion of Gava and Narain, which relates the degree of potential in asymmetric 2-matrix model to the form of $\cal W$-constraints imposed on its partition function.
Loop equations of matrix models express the invariance of the models under field redefinitions. We use loop equations to prove that it is possible to define continuum times for the generic hermitian {1-matrix} model such that all correlation functions in the double scaling limit agree with the corresponding correlation functions of the Kontsevich model expressed in terms of kdV times. In addition the double scaling limit of the partition function of the hermitian matrix model agree with the $τ$-function of the kdV hierarchy corresponding to the Kontsevich model (and not the square of the $τ$-f…
The technique of $Q$-polinomials are used to derive the $w$- constraints in the two-matrix and Kontsevich-like model at finite $N$. These constraints are closed and form Lie algebra. They are associated with the matrices, $λ^n{\partial}_λ^m$ with $n,m\geq 0$. In the case of two-matrix model they can be reduced to the $W$-constraints of \cite{8}. For the case of Kontsevich-like model and two-matrix model with the finite polinomial potential, the number of constraints are limited by the power of the finite matrix potential i.e. the spin of $w$-s coincide with that power. This statement is the na…
In arXiv:hep-th/0310113 we started a program of creating a reference-book on matrix-model tau-functions, the new generation of special functions, which are going to play an important role in string theory calculations. The main focus of that paper was on the one-matrix Hermitian model tau-functions. The present paper is devoted to a direct counterpart for the Kontsevich and Generalized Kontsevich Model (GKM) tau-functions. We mostly focus on calculating resolvents (=loop operator averages) in the Kontsevich model, with a special emphasis on its simplest (Gaussian) phase, where exists a surpris…
Earlier we explained that partition functions of various matrix models can be constructed from that of the cubic Kontsevich model, which, therefore, becomes a basic elementary building block in "M-theory" of matrix models. However, the less topical complex matrix model appeared to be an exception: its decomposition involved not only the Kontsevich tau-function but also another constituent, which we now identify as the Brezin-Gross-Witten (BGW) partition function. The BGW tau-function can be represented either as a generating function of all unitary-matrix integrals or as a Kontsevich-Penner mo…
We represent the partition function of the Generalized Kontsevich Model (GKM) in the form of a Toda lattice $τ$-function and discuss various implications of non-vanishing "negative"- and "zero"-time variables: the appear to modify the original GKM action by negative-power and logarithmic contributions respectively. It is shown that so deformed $τ$-function satisfies the same string equation as the original one. In the case of quadratic potential GKM turns out to describe {\it forced} Toda chain hierarchy and, thus, corresponds to a {\it discrete} matrix model, with the role of the matrix size …
We study the algebraic geometrical background of the Penner--Kontsevich matrix model with the potential $Nα\tr {\bigl(- \fr 12 ŁXŁX +\log (1-X)+X\bigr)}$. We show that this model describes intersection indices of linear bundles on the discretized moduli space right in the same fashion as the Kontsevich model is related to intersection indices (cohomological classes) on the Riemann surfaces of arbitrary genera. The special role of the logarithmic potential originated from the Penner matrix model is demonstrated. The boundary effects which was unessential in the case of the Kontsevich model are …
We advocate a new approach to the study of unitary matrix models in external fields which emphasizes their relationship to Generalized Kontsevich Models (GKM) with non-polynomial potentials. For example, we show that the partition function of the Brezin-Gross-Witten Model (BGWM), which is defined as an integral over unitary $N\times N$ matrices, $\int [dU] e^{\rm{Tr}(J^\dagger U + JU^\dagger)}$, can also be considered as a GKM with potential ${\cal V}(X) = \frac{1}{X}$. Moreover, it can be interpreted as the generating functional for correlators in the Penner model. The strong and weak couplin…
We argue that topological matrix models (matrix models of the Kontsevich type) are examples of exact open/closed duality. The duality works at finite N and for generic `t Hooft couplings. We consider in detail the paradigm of the Kontsevich model for two-dimensional topological gravity. We demonstrate that the Kontsevich model arises by topological localization of cubic open string field theory on N stable branes. Our analysis is based on standard worldsheet methods in the context of non-critical bosonic string theory. The stable branes have Neumann (FZZT) boundary conditions in the Liouville …
In the first part of the talk, I review the applications of loop equations to the matrix models and to 2-dimensional quantum gravity which is defined as their continuum limit. The results concerning multi-loop correlators for low genera and the Virasoro invariance are discussed. The second part is devoted to the Kontsevich matrix model which is equivalent to 2-dimensional topological gravity. I review the Schwinger--Dyson equations for the Kontsevich model as well as their explicit solution in genus zero. The relation between the Kontsevich model and the continuum limit of the hermitean one-ma…
We give an explicit demonstration of the equivalence between the Normal Matrix Model (NMM) of c=1 string theory at selfdual radius and the Kontsevich-Penner (KP) model for the same string theory. We relate macroscopic loop expectation values in the NMM to condensates of the closed string tachyon, and discuss the implications for open-closed duality. As in c<1, the Kontsevich-Miwa transform between the parameters of the two theories appears to encode open-closed string duality, though our results also exhibit some interesting differences with the c<1 case. We also briefly comment on two differe…
We derive a Kontsevich-type matrix model for the c=1 string directly from the W-infinity solution of the theory. The model that we obtain is different from previous proposals, which are proven to be incorrect. Our matrix model contains the Penner and Kontsevich cases, and we study its quantum effective action. The simplicity of our model leads to an encouraging interpretation in the context of background-independent noncritical string field theory.
The Kontsevich-Penner model, an Airy matrix model with a logarithmic potential, may be derived from a simple Gaussian two-matrix model through a duality. In this dual version the Fourier transforms of the n-point correlation functions can be computed in closed form. Using Virasoro constraints, we find that in addition to the parameters $t_n$, which appears in the KdV hierarchies, one needs to introduce here half-integer indices $t_{n/2}$ . The free energy as a function of those parameters may be obtained from these Virasoro constraints. The large N limit follows from the solution to an integra…
We construct a Hermitian matrix model for the total descendant potential of a simple singularity of type D similar to the Kontsevich matrix model for the generating function of intersection numbers on the Deligne--Mumford moduli spaces $\overline{\mathcal{M}}_{g,n}$.
This thesis studies matrix field theories, which are a special type of matrix models. First, the different types of applications are pointed out, from (noncommutative) quantum field theory over 2-dimensional quantum gravity up to algebraic geometry with explicit computation of intersection numbers on the moduli space of complex curves. The Kontsevich model, which has proved the Witten conjecture, is the simplest example of a matrix field theory. Generalisations of this model will be studied, where different potentials and the spectral dimension (defined by the asymptotics of the external matri…
The same complex matrix model calculates both tachyon scattering for the c=1 non-critical string at the self-dual radius and certain correlation functions of half-BPS operators in N=4 super-Yang-Mills. It is dual to another complex matrix model where the couplings of the first model are encoded in the Kontsevich-like variables of the second. The duality between the theories is mirrored by the duality of their Feynman diagrams. Analogously to the Hermitian Kontsevich-Penner model, the correlation functions of the second model can be written as sums over discrete points in subspaces of the modul…
The review is devoted to the integrable properties of the Generalized Kontsevich Model which is supposed to be an universal matrix model to describe the conformal field theories with $c<1$. It is shown that the deformations of the "monomial" phase to "polynomial" one have the natural interpretation in context of so-called equivalent hierarchies. The dynamical transition between equivalent integrable systems is exactly along the flows of the dispersionless Kadomtsev-Petviashvili hierarchy; the coefficients of the potential are shown to be directly related with the flat (quasiclassical) times ar…
The main goal of the paper is to present a new approach via Hurwitz numbers to Kontsevich's combinatorial/matrix model for the intersection theory of the moduli space of curves. A secondary goal is to present an exposition of the circle of ideas involved: Hurwitz numbers, Gromov-Witten theory of the projective line, matrix integrals, and the theory of random trees. Further topics will be treated in a sequel.
Riemann surfaces with nodes can be described by introducing simple composite operators in matrix models. In the case of the Kontsevich model, it is sufficient to add the quadratic, but ``non-propagating'', term (tr[X])^2 to the Lagrangian. The corresponding Jenkins-Strebel differentials have pairwise identified simple poles. The result is in agreement with a conjecture formulated by Kontsevich and recently investigated by Arbarello and Cornalba that the set ${\cal M}_{m*,s}$ of ribbon graphs with s faces and $m*=(m_0,m_1,\ldots,m_j,\ldots)$ vertices of valencies $(1,3,\ldots,2j+1,\ldots)$ ``ca…
The generalized Kontsevich model (GKM) is a one-matrix model with arbitrary potential. Its partition function belongs to the KP hierarchy. When the potential is monomial, it is an $r$-reduced tau-function that governs the $r$-spin intersection numbers. In this paper, we present an ordered exponential representation of monomial GKM in terms of the $W_{1+\infty}$ operators that preserves the KP integrability. In fact, this representation is naturally the solution of a $W_{1+\infty}$ constraint that uniquely determines the tau-function. Furthermore, we show that, for the cases of Kontsevich-Witte…
We investigate the NBI matrix model with the potential $XΛ+X^{-1}+(2η+1)\log X$ recently proposed to describe IIB superstrings. With the proper normalization, using Virasoro constraints, we prove the equivalence of this model and the Kontsevich matrix model for $η\ne0$ and find the explicit transformation between the two models.
We investigate the two-logarithm matrix model with the potential $XΛ+α\log(1+X)+β\log(1-X)$ related to an exactly solvable Kazakov-Migdal model. In the proper normalization, using Virasoro constraints, we prove the equivalence of this model and the Kontsevich-Penner matrix model and construct the 1/N-expansion solution of this model.
The exact FZZT brane partition function for topological gravity with matter is computed using the dual two-matrix model. We show how the effective theory of open strings on a stack of FZZT branes is described by the generalized Kontsevich matrix integral, extending the earlier result for pure topological gravity. Using the well-known relation between the Kontsevich integral and a certain shift in the closed-string background, we conclude that these models exhibit open/closed string duality explicitly. Just as in pure topological gravity, the unphysical sheets of the classical FZZT moduli space…
We introduce a new 1-matrix model with arbitrary potential and the matrix-valued background field. Its partition function is a $τ$-function of KP-hierarchy, subjected to a kind of ${\cal L}_{-1}$-constraint. Moreover, partition function behaves smoothly in the limit of infinitely large matrices. If the potential is equal to $X^{K+1}$, this partition function becomes a $τ$-function of $K$-reduced KP-hierarchy, obeying a set of ${\cal W} _K$-algebra constraints identical to those conjectured in \cite{FKN91} for double-scaling continuum limit of $(K-1)$-matrix model. In the case of $K=2$ the stat…
We examine the Kac-Schwarz problem of specification of point in Grassmannian in the restricted case of gap-one first-order differential Kac-Schwarz operators. While the pair of constraints satisfying $[{\cal K}_1,W] = 1$ always leads to Kontsevich type models, in the case of $[{\cal K}_1,W] = W$ the corresponding KP $τ$-functions are represented as more sophisticated matrix integrals.
