Hurwitz numbers and integrable hierarchy of Volterra type

A generating function of the single Hurwitz numbers of the Riemann sphere is a tau function of the lattice KP hierarchy. The associated Lax operator L turns out to be expressed as , where is a difference-differential operator of the form . 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. (letter)