Virasoro constraints for Kontsevich-Hurwitz partition function

Abstract

In [1, 2] M.Kazarian and S.Lando 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 u2/24 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 m→m−1 and ``twists'' the partition function, KH = ZK. The conjugation = exp{(u2/3)(1−1)+O(u6)} = exp{(u2/12)(∑kTk∂/∂Tk+1−(g2/2) ∂2/∂T02)+O(u6)} is expressed through the previously unnoticed operators like 1 = ∑k(k+1)2Tk∂/∂Tk+1 which annihilate the quasiclassical (planar) free energy FK(0) of the Kontsevich model, but do not belong to the symmetry group GL(∞) of the universal Grassmannian.