Virasoro contraints for simple maps and free probability

Symplectic invariance is a known feature of topological recursion at the theoretical level. An explicit instance of such phenomenon has been recently found and proved by Borot and Garcia-Failde: two enumerative geometric problems satisfying topological recursion, whose spectral curves are related by the swap of x and y — The enumeration of usual maps and the enumeration of maps with an extra combinatorial condition, named “simple”. The transition coefficients between their correlators is given by monotone Hurwitz numbers, another enumerative problem satisfying topological recursion. Since the Fock space operators for this problem is known, we can link the two partition functions involved in the symplectic invariance, and compute the Virasoro algebra of the simple maps from the usual maps one. This has interesting applications in the context of free probability, in particular towards the computation of higher order free cumulants and towards the introduction of the the concept of genus for such cumulants (from a joint work in progress w/ G.Borot and E.Garcia Failde)