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arXiv · 2002.07935

Weighted Hurwitz numbers, $τ$-functions and matrix integrals

J. Harnad
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Abstract

The basis elements spanning the Sato Grassmannian element corresponding to the KP $τ$-function that serves as generating function for rationally weighted Hurwitz numbers are shown to be Meijer $G$-functions. Using their Mellin-Barnes integral representation the $τ$-function, evaluated at the trace invariants of an externally coupled matrix, is expressed as a matrix integral. Using the Mellin-Barnes integral transform of an infinite product of $Γ$ functions, a similar matrix integral representation is given for the KP $τ$-function that serves as generating function for quantum weighted Hurwitz numbers.

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J. Harnad. 2020-02-19. Weighted Hurwitz numbers, $τ$-functions and matrix integrals. https://doi.org/10.1007/978-3-030-55777-5_7

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