Non-orientable branched coverings, b-Hurwitz numbers, and positivity for multiparametric Jack expansions

We introduce a one-parameter deformation of the 2-Toda tau-function of (weighted) Hurwitz numbers, 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 branched coverings of the sphere by an arbitrary surface, orientable or not, with an appropriate b-weighting that "measures" in some sense their non-orientability. Notable special cases include non-orientable dessins d'enfants for which we prove the most general result so far towards the Matching-Jack conjecture and the "b-conjecture" of Goulden and Jackson from 1996, expansions of the beta-ensemble matrix model, deformations of the HCIZ integral, and b-Hurwitz numbers that we introduce here and that are b-deformations of classical (single or double) Hurwitz numbers obtained for b = 0. A key role in our proof is played by a combinatorial model of non-orientable constellations equipped with a suitable b-weighting, whose partition function satisfies an infinite set of PDEs. These PDEs have two definitions, one given by Lax