Schur Q-Polynomials and Kontsevich-Witten Tau Function

Abstract

In this paper, we prove a conjecture of Mironov and Morozov which expresses Kontsevich-Witten tau-function as a linear combination of Schur Q-polynomials. We also prove Alexandrov's conjecture that Kontsevich-Witten tau-function is a hypergeometric tau-function of the BKP hierarchy after re-scaling.

Action of Virasoro operators on Hall-Littlewood polynomials

“…Schur Q-functions were also used to study Kontsevich-Witten and Brezin-Gross-Witten tau functions which are generating functions of certain intersection numbers on moduli spaces of stable curves (c.f. [MM20], [A20], [A21], [LY1], [LY2]). Recently Mironov and Morozov proposed to use Hall-Littlewood functions specialized at roots of unity to study generalized Kontsevich matrix models (c.f.…”

BKP Hierarchy, Affine Coordinates, and a Formula for Connected Bosonic $N$-Point Functions

“…Now we discuss some applications of the above results. Using the recent results [24,25] of X. Liu and the second author, we obtain automatically the affine coordinates of the Witten-Kontsevich tau-function and the Brézin-Gross-Witten tau-function (as tau-functions of the BKP hierarchy), then we are able to apply Theorem 4.1 to compute the free energies. In general, one can write down the affine coordinates of a hypergeometric tau-function [31] of the BKP hierarchy and compute the connected n-point functions.…”

“…Then is automatically a tau-function of the BKP hierarchy due to [3]. The following Schur Q-expansion formula was first proposed by Mironov-Morozov [27] (see also [2], and see [24] for a proof by Virasoro constraints):…”

“…The discussions in this work may hopefully provide some new understandings from the point of view of the BKP hierarchy. This is clearly inspired by the works [2,4,24,25,27], in which the Witten-Kontsvich tau-function was related to Schur Q-functions; and the work of Zhou [43], in which the KP-affine coordinates and boson-fermion correspondence are applied to compute the free energies.…”

“…Using some of the results in [2,4,24,25,27], we are able to write down the explicit expressions of the BKP-affine coordinates of the Witten-Kontsevich tau-function and the BGW tau-function, then we may apply Theorem 1.1 to compute their free energies. The generating series of the BKP-affine coordinates of these two taufunctions have simple expressions (in terms of the first two basis vectors of the corresponding elements in the Sato-Grassmannian):…”