The matrix model for hypergeometric Hurwitz numbers

Abstract

We present the multi-matrix models that are the generating functions for branched covers of the complex projective line ramified over $n$ fixed points $z_i$, $i=1,\dots ,n$, (generalized Grotendieck's dessins d'enfants) of fixed genus, degree, and the ramification profiles at two points, $z_1$ and $z_n$. We take a sum over all possible ramifications at other $n-2$ points with the fixed length of the profile at $z_2$ and with the fixed total length of profiles at the remaining $n-3$ points. All these models belong to a class of hypergeometric Hurwitz models thus being tau functions of the Kadomtsev--Petviashvili (KP) hierarchy. In the case described above, we can present the obtained model as a chain of matrices with a (nonstandard) nearest-neighbor interaction of the type $\tr M_iM_{i+1}^{-1}$. We describe the technique for evaluating spectral curves of such models, which opens the possibility of applying the topological recursion for developing $1/N^2$-expansions of these model. These spectral curves turn out to be of an algebraic type.

12 pages, 2 figures in LaTeX, contribution to the volume of TMPh celebrating the 75th birthday of A A Slavnov

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Cited by in corpus (20)

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• Generating functions for weighted Hurwitz numbers
• Weighted Hurwitz numbers and topological recursion
• Combinatorics of loop equations for branched covers of sphere
• Quantum Hurwitz numbers and Macdonald polynomials
• Hurwitz numbers from Feynman diagrams
• Rationally weighted Hurwitz numbers, Meijer $G$-functions and matrix integrals
• Schwinger-Dyson and loop equations for a product of square Ginibre random matrices
• The Harer--Zagier recursion for an irregular spectral curve
• Hurwitz numbers and matrix integrals labeled with chord diagrams
• Matrix integrals and Hurwitz numbers
• Grothendieck's Dessins d'Enfants in a Web of Dualities. III
• Spectral curves for hypergeometric Hurwitz numbers
• Generating weighted Hurwitz numbers
• Product matrix processes for coupled multi-matrix models and their hard edge scaling limits
• Matrix model generating function for quantum weighted Hurwitz numbers
• From Hurwitz numbers to Feynman diagrams: counting rooted trees in log gravity
• Asymptotics of quantum weighted Hurwitz numbers
• Notes about KP/BKP correspondence
• 2D Toda $\tau$ Functions, Weighted Hurwitz Numbers and the Cayley Graph: Determinant Representation and Recursion Formula