Polynomial averages in the Kontsevich model

TL;DR

This paper derives explicit formulas for polynomial averages in the Kontsevich model, connecting them to derivatives of the partition function and introducing new invariant polynomials related to Schur functions.

Contribution

It provides a closed-form solution for polynomial averages and proves Witten's conjecture linking these averages to derivatives of the partition function.

Findings

Explicit formulas for polynomial averages in the Kontsevich model

Proof of Witten's conjecture relating averages to derivatives of the partition function

Introduction of new invariant polynomials related to Schur functions

Abstract

We obtain in closed form averages of polynomials, taken over hermitian matrices with the Gaussian measure involved in the Kontsevich integral, and prove a conjecture of Witten enabling one to express analogous averages with the full (cubic potential) measure, as derivatives of the partition function with respect to traces of inverse odd powers of the external argument. The proofs are based on elementary algebraic identities involving a new set of invariant polynomials of the linear group, closely related to the general Schur functions.