Related papers: Open intersection numbers, matrix models and MKP h…
The theory of Q-Cartier divisors on the space of n-pointed, genus 0, stable maps to projective space is considered. Generators and Picard numbers are computed. A recursive algorithm computing all top intersection products of Q-Divisors is…
In a recent work of Duke, Imamo\={g}lu, and T\'{o}th, the linking number of certain links on the space $\text{SL}(2,\mathbb{Z})\backslash\text{SL}(2,\mathbb{R})$ is investigated. This linking number has an alternative interpretation as the…
In our previous two papers, we constructed an $r$-spin theory in genus zero for Riemann surfaces with boundary and fully determined the corresponding intersection numbers, providing an analogue of Witten's $r$-spin conjecture in genus zero…
Okounkov [36] proved a remarkable formula relating $n$-point GUE (Gaussian unitary ensemble) correlators of a fixed genus to Witten's intersection numbers of the same genus. The partition function of GUE correlators is a tau-function for…
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…
Kontsevitch's work on Airy matrix integrals has led to explicit results for the intersection numbers of the moduli space of curves. In a subsequent work Okounkov rederived these results from the edge behavior of a Gaussian matrix integral.…
In [5] I.P. Goulden, D.M. Jackson, and R. Vakil formulated a conjecture relating certain Hurwitz numbers (enumerating ramified coverings of the sphere) to the intersection theory on a conjectural Picard variety. We are going to use their…
We present a family of solvable multi-matrix models associated with an arbitrary embedded graph $\Gamma$ with a single vertex. The graph with $n$ edges is equipped with $2n$ corner matrices. The partition function of each member of the…
An explicit construction is presented of homotopy-invariant iterated integrals on a Riemann surface of arbitrary genus in terms of a flat connection valued in a freely generated Lie algebra. The integration kernels consist of modular…
We identify the formulas of Buryak and Okounkov for the n-point functions of the intersection numbers of psi-classes on the moduli spaces of curves. This allows us to combine the earlier known results and this one into a principally new…
We review progress on the generalized Witten conjecture and some of its major ingredients. This conjecture states that certain intersection numbers on the moduli space of higher spin curves assemble into the logarithm of the tau function of…
M.Kazarian and S.Lando found a 1-parametric interpolation between Kontsevich and Hurwitz partition functions, which entirely lies within the space of KP tau-functions. V.Bouchard and M.Marino suggested that this interpolation satisfies some…
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…
Okounkov's generating function of the double Hurwitz numbers of the Riemann sphere is a hypergeometric tau function of the 2D Toda hierarchy in the sense of Orlov and Scherbin. This tau function turns into a tau function of the lattice KP…
We present the multi-matrix models that are the generating functions for branched covers of the complex projective line ramified over $n$ fixed points $z_i$, $i=1,\dots,n$, (generalized Grotendieck's dessins d'enfants) of fixed genus,…
The connection between Riemann surfaces with boundaries and the theory of vertex operator algebras is discussed in the framework of conformal field theories defined by Kontsevich and Segal and in the framework of their generalizations in…
The paper contains some new results and a review of recent achievements, concerning the multisupport solutions to matrix models. In the leading order of the 't Hooft expansion for matrix integral, these solutions are described by…
The partition function for unitary two matrix models is known to be a double KP tau-function, as well as providing solutions to the two dimensional Toda hierarchy. It is shown how it may also be viewed as a Borel sum regularization of…
The main objects under consideration in this thesis are called maps, a certain class of graphs embedded on surfaces. Our problems have a powerful relatively recent tool in common, the so-called topological recursion (TR) introduced by…
Inspired by recent formul\ae\ of Dubrovin, Yang, and Zagier, we interpret the tau function enumerating stationary Gromov-Witten invariants of $\mathbb{P}^1$ as an isomonodromic tau function associated with a difference equation. As a…