Partition Functions of Matrix Models as the First Special Functions of String Theory. II. Kontsevich Model

TL;DR

This paper advances the understanding of matrix-model tau-functions, focusing on the Kontsevich and GKM models, by deriving explicit resolvent formulas, analyzing phases, and exploring dualities relevant to string theory.

Contribution

It provides explicit calculations of resolvents in the Kontsevich model, extends results to the GKM, and discusses dualities and phases, enriching the mathematical framework of string theory.

Findings

Explicit resolvent formulas in the Gaussian phase

Genus zero and one resolvent expressions

Analysis of p-q duality and Riemann surfaces

Abstract

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 surprising integral formula, and the expressions for the resolvents in the genus zero and one are especially simple (in particular, we generalize the known genus zero result to genus one). We also discuss various features of generic phases of the Kontsevich model, in particular, a counterpart of the unambiguous Gaussian solution in the generic case, the solution called Dijkgraaf-Vafa (DV) solution. Further, we extend the results to the GKM and, in particular, discuss the p-q duality in terms of resolvents and corresponding Riemann surfaces in the example of dualities between (2,3) and (3,2) models.