Generalized string equations for double Hurwitz numbers

Details Generalized string equations for double Hurwitz numbers Generalized string equations for double Hurwitz numbers

2010-12-27

arxiv.org/abs/1012.5554v2

J. Geom. Phys. 62 (2012), 1135-1156

Physics

Mathematical Physics

latex2e using packages amsmath,amssymb,amsthm, 41 pages, no figure; (v2) sections are fully reorganized, a proof of hbar-expan

Scientific paper

The generating function of double Hurwitz numbers is known to become a tau function of the Toda hierarchy. The associated Lax and Orlov-Schulman operators turn out to satisfy a set of generalized string equations. These generalized string equations resemble those of $c = 1$ string theory except that the Orlov-Schulman operators are contained therein in an exponentiated form. These equations are derived from a set of intertwining relations for fermiom bilinears in a two-dimensional free fermion system. The intertwiner is constructed from a fermionic counterpart of the cut-and-join operator. A classical limit of these generalized string equations is also obtained. The so called Lambert curve emerges in a specialization of its solution. This seems to be another way to derive the spectral curve of the random matrix approach to Hurwitz numbers.

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