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arxiv: 1406.5976 · v3 · pith:MOW5KIHYnew · submitted 2014-06-23 · 🧮 math.CO

Virasoro constraints and topological recursion for Grothendieck's dessin counting

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keywords constraintsgiveninftynumbersrecursiontopologicalvirasorocompute
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We compute the number of coverings of ${\mathbb{C}}P^1\setminus\{0, 1, \infty\}$ with a given monodromy type over $\infty$ and given numbers of preimages of 0 and 1. We show that the generating function for these numbers enjoys several remarkable integrability properties: it obeys the Virasoro constraints, an evolution equation, the KP (Kadomtsev-Petviashvili) hierarchy and satisfies a topological recursion in the sense of Eynard-Orantin.

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