Combinatorics of loop equations for branched covers of sphere

Abstract

We prove, in a purely combinatorial way, the spectral curve topological recursion for the problem of enumeration of bi-colored maps, which are dual objects to dessins d'enfant. Furthermore, we give a proof of the quantum spectral curve equation for this problem. Then we consider the generalized case of 4-colored maps and outline the idea of the proof of the corresponding spectral curve topological recursion.

23 pages; clarifications added

References in corpus (2)

• On KP-integrable Hurwitz functions
• The matrix model for hypergeometric Hurwitz numbers

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