Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

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

A. Alexandrov, A. Mironov, A. Morozov, P. Putrov

TL;DR

The work extends the program of turning matrix-model tau-functions into first-class special functions by providing a comprehensive treatment of the Kontsevich model and its Generalized version. It derives loop equations and Virasoro-like constraints for Kontsevich/GKM, constructs resolvents and their genus expansions, and highlights the Gaussian branch where resolvents admit elegant integral representations (via Okounkov’s Laplace transform) and a DV-type KdV solution. The paper also develops the generalized framework, including p-q duality, and applies it to the p=3 case, exposing both dual spectral curves and explicit densities across reduced constraint sectors. Together, these results deepen the bridge between matrix models, spectral curves, and string-theoretic tau-functions, with explicit low-genus densities and duality maps that illuminate the structure of the Kontsevich family.

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.

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

TL;DR

The work extends the program of turning matrix-model tau-functions into first-class special functions by providing a comprehensive treatment of the Kontsevich model and its Generalized version. It derives loop equations and Virasoro-like constraints for Kontsevich/GKM, constructs resolvents and their genus expansions, and highlights the Gaussian branch where resolvents admit elegant integral representations (via Okounkov’s Laplace transform) and a DV-type KdV solution. The paper also develops the generalized framework, including p-q duality, and applies it to the p=3 case, exposing both dual spectral curves and explicit densities across reduced constraint sectors. Together, these results deepen the bridge between matrix models, spectral curves, and string-theoretic tau-functions, with explicit low-genus densities and duality maps that illuminate the structure of the Kontsevich family.

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.

Paper Structure

This paper contains 49 sections, 1 theorem, 299 equations.

Table of Contents

  1. Introduction
  2. Aim of the paper
  3. Correlators in matrix models
  4. IZK integral
  5. Kontsevich model
  6. Solving loop equations
  7. Kontsevich model: definitions
  8. The loop equation and recursion relations on the multi-resolvents
  9. Solving the reduced Virasoro constraints
  10. Example of transition: ${N=2 \rightarrow N=1}$
  11. Resolvents
  12. CFT representation
  13. Gaussian branch of Kontsevich model
  14. Specific of the Gaussian branch
  15. Important formula:
  16. ...and 34 more sections

Key Result

Lemma A.1

There exist the following formulae for $T_k$ and $\frac{\partial F^{(0)}}{\partial T_k}:$

Theorems & Definitions (1)

  • Lemma A.1