Weighted Hurwitz Numbers,
τ-Functions, and Matrix Integrals
J. Harnad
Dedicated to Prof. Decio Levi on the occasion of his 70th
birthday
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 MellinBarnes 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.
Keywords Hurwitz numbers · τ -functions
1 Hurwitz Numbers: Classical and Weighted
The fact that KP and 2D-Toda τ -functions of hypergeometric type serve as generating functions for weighted Hurwitz numbers was shown in [3–6], generalizing the
case of simple (single and double) Hurwitz numbers [8, 9]. Sections 1.1 and 1.2
below, and Sect. 2 give a brief review of this theory, together with two illustrative
examples: rational and quantum weighted Hurwitz numbers. In Sect. 3, it is shown
how evaluation of such τ -functions at the trace invariants of a finite matrix may be
expressed either as a Wronskian determinant or as a matrix integral. The content of
J. Harnad ()
Department of Mathematics and Statistics, Concordia University, Montréal, QC, Canada
Centre de recherches mathématiques, Université de Montréal, Montréal, QC, Canada
e-mail: harnad@crm.umontreal.ca
© Springer Nature Switzerland AG 2021
M. B. Paranjape et al. (eds.), Quantum Theory and Symmetries, CRM Series in
Mathematical Physics, https://doi.org/10.1007/978-3-030-55777-5_7
77
τ-Functions, and Matrix Integrals
J. Harnad
Dedicated to Prof. Decio Levi on the occasion of his 70th
birthday
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 MellinBarnes 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.
Keywords Hurwitz numbers · τ -functions
1 Hurwitz Numbers: Classical and Weighted
The fact that KP and 2D-Toda τ -functions of hypergeometric type serve as generating functions for weighted Hurwitz numbers was shown in [3–6], generalizing the
case of simple (single and double) Hurwitz numbers [8, 9]. Sections 1.1 and 1.2
below, and Sect. 2 give a brief review of this theory, together with two illustrative
examples: rational and quantum weighted Hurwitz numbers. In Sect. 3, it is shown
how evaluation of such τ -functions at the trace invariants of a finite matrix may be
expressed either as a Wronskian determinant or as a matrix integral. The content of
J. Harnad ()
Department of Mathematics and Statistics, Concordia University, Montréal, QC, Canada
Centre de recherches mathématiques, Université de Montréal, Montréal, QC, Canada
e-mail: harnad@crm.umontreal.ca
© Springer Nature Switzerland AG 2021
M. B. Paranjape et al. (eds.), Quantum Theory and Symmetries, CRM Series in
Mathematical Physics, https://doi.org/10.1007/978-3-030-55777-5_7
77