Hurwitz numbers and integrable hierarchy of Volterra type

Abstract

A generating function of the single Hurwitz numbers of the Riemann sphere ${\mathbb{CP}}^1$ is a tau function of the lattice KP hierarchy. The associated Lax operator $L$ turns out to be expressed as $L=e^{\mathfrak{L}}$, where $\mathfrak{L}$ is a difference-differential operator of the form $\mathfrak{L}={\partial }_s-v{e}^{-{\partial }_s}$. $\mathfrak{L}$ satisfies a set of Lax equations that form a continuum version of the Bogoyavlensky-Itoh (aka hungry Lotka-Volterra) hierarchies. Emergence of this underlying integrable structure is further explained in the language of generalized string equations for the Lax and Orlov-Schulman operators of the 2D Toda hierarchy. This leads to logarithmic string equations, which are confirmed with the help of a factorization problem of operators.

12 pages, no figure; (v2) typos in eqs. (14), (15) etc. are corrected; (v3) typos are corrected, final version for publication

References in corpus (4)

• On KP-integrable Hurwitz functions
• Matrix Models for Random Partitions
• Virasoro constraints for Kontsevich-Hurwitz partition function
• Fractional Volterra Hierarchy

Cited by in corpus (2)

• Bogoyavlensky lattices and generalized Catalan numbers
• Cubic Hodge integrals and integrable hierarchies of Volterra type