The matrix model for dessins d'enfants

We present the matrix models that are the generating functions for branched covers of the complex projective line ramified over 0, 1, and $\infty$ (Grotendieck’s dessins d’enfants) of fixed genus, degree, and the ramification profile at infinity. For general ramifications at other points, the model is the two-logarithm matrix model with the external field studied previously by one of the authors (L.Ch.) and K.Palamarchuk. It lies in the class of the generalised Kontsevich models (GKM) thus being the Kadomtsev–Petviashvili (KP) hierarchy tau function and, upon the shift of times, this model is equivalent to a Hermitian one-matrix model with a general potential whose coefficients are related to the KP times by a Miwa-type transformation. The original model therefore enjoys a topological recursion and can be solved in terms of shifted moments of the standard Hermitian one-matrix model at all genera of the topological expansion. We also derive the matrix model for clean Belyi morphisms, which turns out to be the Kontsevich–Penner model introduced by the authors and Yu. Makeenko. Its partition function is also a KP hierarchy tau function, and this model is in turn equivalent to a Hermitian one-matrix model with a general potential. Finally we prove that the generating function for general two-profile Belyi morphisms is a GKM thus proving that it is also a KP hierarchy tau function in proper times.

Финансовая поддержка	Номер гранта
European Research Council	291092
Independent Research Fund Denmark
Canada Research Chair
Российский фонд фундаментальных исследований	14-01-00860-a 13-01-12405-ofi-m
Российская академия наук - Федеральное агентство научных организаций
The authors acknowledge support from the ERC Advance Grant 291092 “Exploring the Quantum Universe” (EQU). J.A. acknowledges support of the FNU, the Free Danish Research Council, from the grant “Quantum gravity and the role of black holes.” The work of J.A. was supported in part by Perimeter Institute of Theoretical Physics. Research of Perimeter Institute is supported by the Government of Canada through Industry Canada and by Province of Ontario through the Ministry of Economic Development and Innovation. The work of L.Ch. was supported by the Russian Foundation for Basic Research (Grant Nos. 14-01-00860-a and 13-01-12405-ofi-m) and by the Program Mathematical Methods for Nonlinear Dynamics.