Schur Q-Polynomials and Kontsevich-Witten Tau Function

TL;DR

This paper proves a formula representing the Kontsevich-Witten tau-function as a Schur Q-polynomial expansion, demonstrating it satisfies Virasoro constraints and confirming its status as a hypergeometric tau-function of the BKP hierarchy.

Contribution

It provides a direct proof of the Schur Q-polynomial expansion formula for the tau-function without relying on matrix models, and confirms Alexandrov's conjecture.

Findings

Q-polynomial expansion satisfies Virasoro constraints

Tau-function is a hypergeometric BKP tau-function after re-scaling

Proof of the formula without matrix model approach

Abstract

Using matrix model, Mironov and Morozov recently gave a formula which represents Kontsevich-Witten tau-function as a linear expansion of Schur Q-polynomials. In this paper, we will show directly that the Q-polynomial expansion in this formula satisfies the Virasoro constraints, and consequently obtain a proof of this formula without using matrix model. We also give a proof for Alexandrov's conjecture that Kontsevich-Witten tau-function is a hypergeometric tau-function of the BKP hierarchy after re-scaling.