Virasoro constraints for Kontsevich-Hurwitz partition function

Abstract

In [1, 2] M.Kazarian and found a 1-parametric interpolation between Kontsevich and Hurwitz partition functions, which entirely lies within the space of KP τ-functions. In [3] V.Bouchard and M.Marino suggested that this interpolation satisfies some deformed Virasoro constraints. However, they described the constraints in a somewhat sophisticated form of AMM-Eynard equations [4-7] for the rather involved Lambert spectral curve. Here we present the relevant family of Virasoro constraints explicitly. They differ from the conventional continuous Virasoro constraints in the simplest possible way: by a constant shift u 2/24 of the -1 operator, where u is an interpolation parameter between Kontsevich and Hurwitz models. This trivial modification of the string equation gives rise to the entire deformation which is a conjugation of the Virasoro constraints mm-1 and ''twists'' the partition function, KH = Z K. The conjugation = exp{(u 2/3)( 1- 1)+O(u 6)} = exp{(u 2/12)(∑ kT k/T k+1-(g 2/2) 2/T 02)+O(u 6)} is expressed through the previously unnoticed operators like 1 = ∑ k(k+1) 2T k/T k+1 which annihilate the quasiclassical (planar) free energy F K(0) of the Kontsevich model, but do not belong to the symmetry group GL() of the universal Grassmannian. © 2009 SISSA.