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

Abstract

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 equations, the other one following an explicit combinatorial decomposition.