Virasoro constraints for Kontsevich-Hurwitz partition function

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, 5, 6, 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 L̂−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 L̂m → ÛL̂mÛ −1 and ”twists” the partition function, ZKH = ÛZK . The conjugation Û = exp { u 3 (N̂1 − L̂1) } = exp { u 12 ( ∑ k Tk∂/∂Tk+1 − g 2 ∂/∂T 2 0 )} is expressed through the previously