Around spin Hurwitz numbers

Abstract We study ћ expansion of the KP hierarchy following Takasaki-Takebe [1] considering several examples of matrix model τ-functions with natural genus expansion. Among the examples there are solutions of KP equations of special interest, such as generating function for simple Hurwitz numbers, Hermitian matrix model, Kontsevich model and Brezin-Gross-Witten model. We show that all these models with parameter ћ are τ-functions of the ћ-KP hierarchy and the expansion in ћ for the ћ-KP coincides with the genus expansion for these models. Furthermore, we show a connection of recent papers considering the ћ-formulation of the KP hierarchy [2, 3] with original Takasaki-Takebe approach. We find that in this approach the recovery of enumerative geometric meaning of τ-functions is straightforward and algorithmic.

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Generating functions for weighted Hurwitz numbers: enumeration of branched coverings

Tau Functions and their Applications ◽

10.1017/9781108610902.015 ◽

2021 ◽

pp. 369-387

Keyword(s):

Generating Functions ◽

Hurwitz Numbers ◽

Branched Coverings

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Jacobi Ensemble, Hurwitz Numbers and Wilson Polynomials

Letters in Mathematical Physics ◽

10.1007/s11005-021-01396-z ◽

2021 ◽

Vol 111 (3) ◽

Author(s):

Massimo Gisonni ◽

Tamara Grava ◽

Giulio Ruzza

Keyword(s):

Generating Functions ◽

Combinatorial Interpretation ◽

Hurwitz Numbers ◽

Jacobi Ensemble ◽

Matrix Ensembles ◽

Wilson Polynomials ◽

Unitary Ensemble ◽

Topological Expansion

AbstractWe express the topological expansion of the Jacobi Unitary Ensemble in terms of triple monotone Hurwitz numbers. This completes the combinatorial interpretation of the topological expansion of the classical unitary invariant matrix ensembles. We also provide effective formulæ for generating functions of multipoint correlators of the Jacobi Unitary Ensemble in terms of Wilson polynomials, generalizing the known relations between one point correlators and Wilson polynomials.

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