Generalized string equations for double Hurwitz numbers

Abstract

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.

latex2e using packages amsmath,amssymb,amsthm, 41 pages, no figure; (v2) sections are fully reorganized, a proof of hbar-expansion of the tau function is added, many typos are corrected

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Cited by in corpus (11)

• Free fermions and tau-functions
• Toda hierarchies and their applications
• Fermionic approach to weighted Hurwitz numbers and topological recursion
• Laplacian growth in a channel and Hurwitz numbers
• Hurwitz numbers from Feynman diagrams
• Thermodynamic limit of random partitions and dispersionless Toda hierarchy
• Hurwitz numbers and integrable hierarchy of Volterra type
• Symmetric solutions to dispersionless 2D Toda hierarchy, Hurwitz numbers and conformal dynamics
• $q$-Difference Kac-Schwarz Operators in Topological String Theory
• Symmetric solutions of the dispersionless Toda hierarchy and associated conformal dynamics
• Old and New Reductions of Dispersionless Toda Hierarchy