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Regular version of the site

Article

On KP-integrable Hurwitz functions

Journal of High Energy Physics. 2014. Vol. 11. No. 80. P. 1-31.
Alexandrov A., Mironov A., Morozov A., Natanzon S.

There is now a renewed interest  to a Hurwitz tau-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 tau-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 tau-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.