Virasoro constraints for Kontsevich-Hurwitz partition function

TL;DR

This paper explicitly derives a family of Virasoro constraints for the Kontsevich-Hurwitz partition function, revealing a simple shift in the constraints that deforms the string equation and relates the models through conjugation.

Contribution

It provides an explicit form of the Virasoro constraints for the interpolating partition function, showing how they differ from conventional constraints by a constant shift and a conjugation.

Findings

Derived explicit Virasoro constraints for the Kontsevich-Hurwitz model.

Identified a simple shift in the L_{-1} operator as the deformation.

Connected the deformation to conjugation by operators annihilating the planar free energy.

Abstract

M.Kazarian and S.Lando found a 1-parametric interpolation between Kontsevich and Hurwitz partition functions, which entirely lies within the space of KP tau-functions. 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 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 L_{-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 L_m -> U L_m U^{-1} and "twists" the partition function, Z_{KH}= U Z_K. The conjugation U is expressed through the previously unnoticed operators which annihilate the quasiclassical (planar) free energy of the Kontsevich model, but do not belong to the symmetry group GL(\infty) of the universal Grassmannian.