Topological recursion on the Bessel curve

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.