Symplectic duality for topological recursion

Joint work with B. Bychkov, P. Dunun-Barkovski, and S. Shadrin, work in progress.
There are many enumerative problems whose answers are encoded in the Taylor coefficients of a sequence of the so-called m-point correlator functions. The topological recursion (due to Chekhov-Eunard-Orantin) is an inductive procedure for explicit computation of these functions in a closed form starting from a relative small amount of initial data. A small suspension of the problem leads to a collection of (m,n)-point correlator functions such that the original ones correspond to the case n=0. It proves out that the sequence of (0,n) functions also satisfies its own topological recursion with its own initial data. This fact was known before for the two-matrix model related to the problem of enumeration of (hyper)maps. The two recursions are related in this case by the x-y duality which is well studied in a general formalism of topological recursion. We generalize this fact to the case of enumeration of generalized Hurwitz numbers. The former x-y duality does not hold literally in this case; its analogue for the generalized Hurwitz numbers is exactly what we mean by the symplectic duality.