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On KP-integrable Hurwitz functions

By Germany Freiburg D-79104 Albertstraße 19 University of Freiburg A.Freiburg Institute for Advanced Studies (FRIAS) Alexandrov, Russia) 117218 Moscow Bol. Cheremushkinskaya 25 A.(ITEP Mironov, Russia) 117218 Moscow Bol. Cheremushkinskaya 25 A.(ITEP Morozov and Russia) Moscow Vavilova st. 7a S.(National Research University Higher School of Economics Natanzon


There is now a renewed interest [1]–[4] to a Hurwitz τ -function, counting the isomorphism classes of Belyi pairs, arising in the study of equilateral triangulations and Grothiendicks’s dessins d ’ enfant . It is distinguished by belonging to a particular family of Hurwitz τ -functions, possessing conventional Toda/KP integrability properties. We explain how the variety of recent observations about this function fits into the general theory of matrix model τ -functions. All such quantities possess a number of different descriptions, related in a standard way: these include Toda/KP integrability, several kinds of W -representations (we describe four), two kinds of integral (multi-matrix model) descriptions (of Hermitian and Kontsevich types), Virasoro constraints, character expansion, embedding into generic set of Hurwitz τ -functions and relation to knot theory. When approached in this way, the family of models in the literature has a natural extension, and additional integrability with respect to associated new time-variables. Another member of this extended family is the Itsykson-Zuber integral

Year: 2014
DOI identifier: 10.1007/JHEP11(2014)080
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