Combinatorics of the Modular Group II: the Kontsevich integrals

TL;DR

This paper explores the algebraic properties of Kontsevich integrals, linking them to intersection theory on moduli space and deriving important constraints like Virasoro and KdV, with implications for mathematical physics and algebraic geometry.

Contribution

It provides a detailed analysis of Kontsevich integrals, including their expansion, relation to Grassmannians, and generalizations to higher potentials, advancing understanding of their algebraic structure.

Findings

Derivation of Virasoro and KdV constraints for Kontsevich integrals

Expansion of the integral in terms of characters and Schur functions

Connection between Kontsevich integrals and Painlevé equations

Abstract

We study algebraic aspects of Kontsevich integrals as generating functions for intersection theory over moduli space and review the derivation of Virasoro and KdV constraints. 1. Intersection numbers 2. The Kontsevich integral 2.1. The main theorem 2.2 Expansion of Z on characters and Schur functions 2.3 Proof of the first part of the Theorem 3. From Grassmannians to KdV 4. Matrix Airy equation and Virasoro highest weight conditions 5. Genus expansion 6. Singular behaviour and Painlev'e equation. 7. Generalization to higher degree potentials