Kontsevich Matrix Model Research Articles

A Note on BKP for the Kontsevich Matrix Model with Arbitrary Potential

We exhibit the Kontsevich matrix model with arbitrary potential as a BKP tau-function with respect to polynomial deformations of the potential. The result can be equivalently formulated in terms of Cartan-Plücker relations of certain averages of Schur $Q$-function. The extension of a Pfaffian integration identity of de Bruijn to singular kernels is instrumental in the derivation of the result.

Matrix model for the total descendant potential of a simple singularity of type D

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 $\overline{\mathcal{M}}_{g,n}$.

Superintegrability of Kontsevich matrix model

Many eigenvalue matrix models possess a peculiar basis of observables that 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. As a peculiarity of matrix models, the relevant basis is formed by the Schur polynomials (characters) and their generalizations, and superintegrability looks like a property langle characterrangle ,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 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.

Matrix models for stationary Gromov–Witten invariants of the Riemann sphere

Inspired by recent formulæ of Dubrovin, Yang, and Zagier, we interpret the tau function enumerating stationary Gromov–Witten invariants of as an isomonodromic tau function associated with a difference equation. As a byproduct we obtain an analogue of the Kontsevich matrix model for this tau function. A connection with the Charlier ensemble is also considered.

KP integrability of triple Hodge integrals. II. Generalized Kontsevich matrix model

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.

Topological open/closed string dualities: matrix models and wave functions

We sharpen the duality between open and closed topological string partition functions for topological gravity coupled to matter. The closed string partition function is a generalized Kontsevich matrix model in the large dimension limit. We integrate out off-diagonal degrees of freedom associated to one source eigenvalue, and find an open/closed topological string partition function, thus proving open/closed duality. We match the resulting open partition function to the generating function of intersection numbers on moduli spaces of Riemann surfaces with boundaries and boundary insertions. Moreover, we connect our work to the literature on a wave function of the KP integrable hierarchy and clarify the role of the extended Virasoro generators that include all time variables as well as the coupling to the open string observable.

Universality of the matrix Airy and Bessel functions at spectral edges of unitary ensembles

This paper deals with products and ratios of average characteristic polynomials for unitary ensembles. We prove universality at the soft edge of the limiting eigenvalues’ density, and write the universal limit in function of the Kontsevich matrix model (“matrix Airy function”, as originally named by Kontsevich). For the case of the hard edge, universality is already known. We show that also in this case the universal limit can be expressed as a matrix integral (“matrix Bessel function”) known in the literature as generalized Kontsevich matrix model.

From r-spin intersection numbers to Hodge integrals

Generalized Kontsevich Matrix Model (GKMM) with a certain given potential is the partition function of $r$-spin intersection numbers. We represent this GKMM in terms of fermions and expand it in terms of the Schur polynomials by boson-fermion correspondence, and link it with a Hurwitz partition function and a Hodge partition by operators in a $\widehat{GL}(\infty)$ group. Then, from a $W_{1+\infty}$ constraint of the partition function of $r$-spin intersection numbers, we get a $W_{1+\infty}$ constraint for the Hodge partition function. The $W_{1+\infty}$ constraint completely determines the Schur polynomials expansion of the Hodge partition function.

From free fields to AdS. III

In previous work, we have shown that large $N$ field theory amplitudes, in Schwinger parametrized form, can be organized into integrals over the stringy moduli space ${\mathcal{M}}_{g,n}\ifmmode\times\else\texttimes\fi{}{R}_{+}^{n}$. Here we flesh this out into a concrete implementation of open-closed string duality. In particular, we propose that the closed string world sheet is reconstructed from the unique Strebel quadratic differential that can be associated to (the dual of) a field theory skeleton graph. We are led, in the process, to identify the inverse Schwinger proper times (${\ensuremath{\sigma}}_{i}=1/{\ensuremath{\tau}}_{i}$) with the lengths of edges of the critical graph of the Strebel differential. Kontsevich's matrix model derivation of the intersection numbers in moduli space provides a concrete example of this identification. It also exhibits how closed string correlators emerge very naturally from the Schwinger parameter integrals. Finally, to illustrate the utility of our approach to open-closed string duality, we outline a method by which a world sheet operator product expansion can be directly extracted from the field theory expressions. Limits of the Strebel differential for the four punctured sphere play a key role.

Matrix Models: Geometry of Moduli Spaces and Exact Solutions

We study the connection between characteristics of moduli spaces of Riemann surfaces with marked points and matrix models. The Kontsevich matrix model describes intersection indices on continuous moduli spaces, and the Kontsevich–Penner matrix model describes intersection indices on discretized moduli spaces. Analyzing the constraint algebras satisfied by various generalized Kontsevich matrix models, we derive time transformations that establish exact relations between different models appearing in mathematical physics. We solve the Hermitian one-matrix model using the moment technique in the genus expansion and construct a recursive procedure for solving this model in the double scaling limit.

The NBI matrix model of IIB superstrings

We investigate the NBI matrix model with the potential $X\Lambda+X^{-1}+(2\eta+1)\log X$ 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 $\eta\ne0$ and find the explicit transformation between the two models.

RG equations from Whitham hierarchy

The second derivatives of the prepotential with respect to Whitham time variables in the Seiberg-Witten theory are expressed in terms of Riemann theta-functions. These formulas give a direct transcendental generalization of the algebraic ones for the Kontsevich matrix model. In particular case they provide an explicit derivation of the renormalization group (RG) equation proposed recently in papers on Donaldson theory.