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The matrix model for dessins d’enfants. (English) Zbl 1304.81130
Summary: 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 the second author and K. Palamarchuk [“Two logarithm matrix model with an external field”, Mod. Phys. Lett. A 14, 2229–2243 (1999)]. 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, C. F. Kristjansen and Yu. Makeenko [Nucl. Phys., B 404, No. 1–2, 127–172 (1995; Zbl 1043.81636); erratum ibid. 449, No. 3, 681 (1995)]. 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.

81T30 String and superstring theories; other extended objects (e.g., branes) in quantum field theory
05A15 Exact enumeration problems, generating functions
15B52 Random matrices (algebraic aspects)
14H57 Dessins d’enfants theory
37K10 Completely integrable infinite-dimensional Hamiltonian and Lagrangian systems, integration methods, integrability tests, integrable hierarchies (KdV, KP, Toda, etc.)
11G32 Arithmetic aspects of dessins d’enfants, Belyĭ theory
14H70 Relationships between algebraic curves and integrable systems
Full Text: DOI
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