String Theory Journal Club

Weighted Hurwitz numbers were introduced by Harnad and Guay-Paquet as objects covering a broad class of Hurwitz numbers of various types. A robust property of Hurwitz numbers is that they are governed by the celebrated topological recursion (TR) of Chekhov--Eynard--Orantin: a universal algorithm that allows computation of them recursively with respect to their topology. The program of understanding how TR can be used to compute different types of Hurwitz numbers was carried out over the last two decades by considering each case separately, and finally, the general case of rationally-weighted Hurwitz numbers was recently proved by Bychkov--Dunin-Barkowski--Kazarian--Shadrin. We will discuss a more general case of weighted $b$-Hurwitz numbers, which is a one-parameter deformation of Hurwitz numbers. We show that their generating function is annihilated by an explicit system of PDE that has an elegant combinatorial description and can be obtained from a representation of $W$-algebras. We also explain how to use this description to prove refined topological recursion for various interesting models. Our result gives a new explanation of the remarkable enumerative properties of Hurwitz numbers and extends it to the $b$-deformed case. This is joint work with Nitin Chidambaram and Kento Osuga based on 2401.12814 and 2412.17502.