From Hurwitz numbers to Kontsevich-Witten tau-function: a connection by Virasoro operators

TL;DR

This paper conjectures a simple Virasoro group element connecting the Hurwitz and Kontsevich-Witten tau-functions, potentially enabling new derivations of Virasoro constraints and integrable descriptions.

Contribution

It proposes a conjectural formula linking the two tau-functions via a Virasoro group element, advancing understanding of their relationship.

Findings

Conjectural connection between Hurwitz and Kontsevich-Witten tau-functions.

Potential derivation of Virasoro constraints for Hurwitz tau-function.

Framework for an integrable operator description of the tau-functions.

Abstract

In this letter,we present our conjecture on the connection between the Kontsevich--Witten and the Hurwitz tau-functions. The conjectural formula connects these two tau-functions by means of the $GL(\infty)$ group element. An important feature of this group element is its simplicity: this is a group element of the Virasoro subalgebra of $gl(\infty)$. If proved, this conjecture would allow to derive the Virasoro constraints for the Hurwitz tau-function, which remain unknown in spite of existence of several matrix model representations, as well as to give an integrable operator description of the Kontsevich--Witten tau-function.