Communications in Number Theory and Physics

Volume 12 (2018)

Number 1

Topological recursion on the Bessel curve

Pages: 53 – 73



Norman Do (School of Mathematical Sciences, Monash University, Melbourne, Australia)

Paul Norbury (School of Mathematics and Statistics, University of Melbourne, Australia)


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.

Received 7 September 2016

Accepted 20 September 2017

Published 27 April 2018