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KdV integrability in GUE correlators

Di Yang

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.

KdV integrability in GUE correlators

Abstract

Okounkov [36] proved a remarkable formula relating -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.

Paper Structure

This paper contains 5 sections, 6 theorems, 56 equations.

Table of Contents

  1. Introduction
  2. Review of Toda lattice hierarchy and GUE partition function
  3. Tau-function for the Toda lattice hierarchy
  4. The GUE partition function
  5. A new proof of Witten's conjecture

Key Result

Lemma 1

There exists a unique $2\times 2$ matrix series satisfying the equation along with the normalization conditions

Theorems & Definitions (11)

  • Lemma 1: DY17Y20
  • Lemma 2: DY17Y20
  • Lemma 3
  • proof
  • Lemma 4
  • proof
  • Remark 1
  • Lemma 5
  • proof
  • Theorem 1: Kontsevich Ko92
  • ...and 1 more