KdV integrability in GUE correlators

TL;DR

This paper presents a new proof of the Witten--Kontsevich theorem by connecting GUE correlators, intersection numbers, and the KdV hierarchy through integrability and tau-functions.

Contribution

It establishes a novel proof linking GUE correlators and Witten's intersection numbers via the KdV integrable hierarchy, building on known results about tau-functions and Toda lattice.

Findings

GUE correlators relate to Witten's intersection numbers

Partition function of GUE is a tau-function for Toda hierarchy

New proof of Witten--Kontsevich theorem connecting intersection numbers and KdV hierarchy

Abstract

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 the Toda lattice hierarchy. In this note, based on the knowledge of these two statements we give a new proof of the Witten--Kontsevich theorem, that relates Witten's intersection numbers to the KdV (Korteweg--de Vries) integrable hierarchy.