Topological recursion for irregular spectral curves

Abstract

We study topological recursion on the irregular spectral curve $x{y}^2-xy+1=0$, which produces a weighted count of dessins d'enfant. This analysis is then applied to topological recursion on the spectral curve $x{y}^2=1$, which takes the place of the Airy curve $x=y^2$ to describe asymptotic behaviour of enumerative problems associated to irregular spectral curves. In particular, we calculate all one-point invariants of the spectral curve $x{y}^2=1$ via a new three-term recursion for the number of dessins d'enfant with one face.

28 pages

References in corpus (2)

• Raney distributions and random matrix theory
• Combinatorics of loop equations for branched covers of sphere

Cited by in corpus (16)

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• Gromov-Witten invariants of ${\mathbb{P}}^1$ coupled to a KdV tau function
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• Triply mixed coverings of arbitrary base curves: Quasimodularity, quantum curves and a mysterious topological recursions
• Integration on complex Grassmannians, deformed monotone Hurwitz numbers, and interlacing phenomena
• The factorial growth of topological recursion
• Generalisations of the Harer-Zagier recursion for 1-point functions