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 $x{y}^2=\frac{1}{2}$, 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.

12 pages

References in corpus (7)

• Hermitean matrix model free energy: Feynman graph technique for all genera
• Correlation functions of the KdV hierarchy and applications to intersection numbers over ${\overline{\mathcal{M}}}_{g,n}$
• The Eynard-Orantin recursion and equivariant mirror symmetry for the projective line
• Topological recursion for irregular spectral curves
• Intersection numbers on Deligne-Mumford moduli spaces and quantum Airy curve
• The Harer--Zagier recursion for an irregular spectral curve
• Formal pseudodifferential operators and Witten's r-spin numbers

Cited by in corpus (3)

• Integrable differential systems of topological type and reconstruction by the topological recursion
• Whittaker vectors for $\mathcal{W}$-algebras from topological recursion
• Voros Coefficients for the Hypergeometric Differential Equations and Eynard-Orantin's Topological Recursion - Part II : For the Confluent Family of Hypergeometric Equations