Non-orientable branched coverings, b-Hurwitz numbers, and positivity for multiparametric Jack expansions (extended abstract)

We introduce a one-parameter deformation of the 2-Toda tau-function of maps (or more generally, constellations), obtained by deforming Schur functions into Jack symmetric functions. We show that its coefficients are polynomials in the deformation parameter b with nonnegative integer coefficients. These coefficients count generalized constellations on an arbitrary surface, orientable or not, with an appropriate b-weighting that “measures” in some sense their non-orientability. The particular case of bipartite maps gives the best progress so far towards the “b-conjecture” of Goulden and Jackson from 1996. Our proof consists in showing that the partition function satisfies an infinite set of PDEs. These PDEs have two definitions, one given by Lax equations, the other one following an explicit combinatorial decomposition.