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Generalized string equations for double Hurwitz numbers

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.