Topological recursion on the Bessel curve

TL;DR

This paper extends topological recursion to the Bessel curve, demonstrating its partition function is a KdV tau-function satisfying Virasoro constraints and quantum curve equations, thus broadening the understanding of spectral curves in integrable systems.

Contribution

It introduces the Bessel curve as a new spectral curve for topological recursion and proves its associated partition function is a KdV tau-function with Virasoro constraints.

Findings

Partition function satisfies Virasoro constraints

Partition function obeys a cut-and-join recursion

Partition function fulfills a quantum curve equation

Abstract

The Witten-Kontsevich theorem states that a certain generating function for intersection numbers on the moduli space of stable curves is a tau-function for the KdV integrable hierarchy. This generating function can be recovered via the topological recursion applied to the Airy curve $x=\frac{1}{2}y^2$. In this paper, we consider the topological recursion applied to the irregular spectral curve $xy^2=\frac{1}{2}$, which we call the Bessel curve. We prove that the associated partition function is also a KdV tau-function, which satisfies Virasoro constraints, a cut-and-join type recursion, and a quantum curve equation. Together, the Airy and Bessel curves govern the local behaviour of all spectral curves with simple branch points.