Virasoro Constraints and Topological Recursion for Grothendieck’s Dessin Counting


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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Kazarian, M., Zograf, P. Virasoro Constraints and Topological Recursion for Grothendieck’s Dessin Counting. Lett Math Phys 105, 1057–1084 (2015).

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Mathematics Subject Classification

  • 37K10
  • 05C30


  • Grothendieck’s “dessins d’enfants”
  • ribbon graphs
  • Virasoro constraints
  • topological recursion