Abstract
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.
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Alexandrov, A., Mironov, A., Morozov, A. et al. On KP-integrable Hurwitz functions. J. High Energ. Phys. 2014, 80 (2014). https://doi.org/10.1007/JHEP11(2014)080
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DOI: https://doi.org/10.1007/JHEP11(2014)080