Topological Recursion for Orlov–Scherbin Tau Functions, and Constellations with Internal Faces

We study the correlators Wg,n\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$W_{g,n}$$\end{document} arising from Orlov–Scherbin 2-Toda tau functions with rational content-weight G(z), at arbitrary values of the two sets of time parameters. Combinatorially, they correspond to generating functions of weighted Hurwitz numbers and (m, r)-factorisations of permutations. When the weight function is polynomial, they are generating functions of constellations on surfaces in which two full sets of degrees (black/white) are entirely controlled, and in which internal faces are allowed in addition to boundaries. We give the spectral curve (the “disk” function W0,1\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$W_{0,1}$$\end{document}, and the “cylinder” function W0,2\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$W_{0,2}$$\end{document}) for this model, generalising Eynard’s solution of the 2-matrix model which corresponds to G(z)=1+z\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$G(z)=1+z$$\end{document}, by the addition of arbitrarily many free parameters. Our method relies both on the Albenque–Bouttier combinatorial proof of Eynard’s result by slice decompositions, which is strong enough to handle the polynomial case, and on algebraic arguments. Building on this, we establish the topological recursion (TR) for the model. Our proof relies on the fact that TR is already known at time zero (or, combinatorially, when the underlying graphs have only boundaries, and no internal faces) by work of Bychkov–Dunin-Barkowski–Kazarian–Shadrin (or Alexandrov–Chapuy–Eynard–Harnad for the polynomial case), and on the general idea of deformation of spectral curves due to Eynard and Orantin, which we make explicit in this case. As a result of TR, we obtain strong structure results for all fixed-genus generating functions. Our techniques also cover the case where G(z) is a rational function times an exponential (containing in particular the case of classical Hurwitz numbers).