Elements of spin Hurwitz theory: closed algebraic formulas, blobbed topological recursion, and a proof of the Giacchetto-Kramer-Lewanski conjecture

TL;DR

This paper develops algebraic formulas and topological recursion methods for spin Hurwitz numbers, proving a generalized conjecture and connecting generating functions to integrable hierarchies.

Contribution

It introduces closed algebraic formulas for correlation functions and proves blobbed topological recursion for spin Hurwitz numbers, extending the Giacchetto--Kramer--Lewański conjecture.

Findings

Derived algebraic formulas for correlation functions.

Proved blobbed topological recursion for spin Hurwitz numbers.

Established a version of topological recursion for generalized spin Hurwitz numbers.

Abstract

In this paper, we discuss the properties of the generating functions of spin Hurwitz numbers. In particular, for spin Hurwitz numbers with arbitrary ramification profiles, we construct the weighed sums which are given by Orlov's hypergeometric solutions of the 2-component BKP hierarchy. We derive the closed algebraic formulas for the correlation functions associated with these tau-functions, and under reasonable analytical assumptions we prove the loop equations (the blobbed topological recursion). Finally, we prove a version of topological recursion for the spin Hurwitz numbers with the spin completed cycles (a generalized version of the Giacchetto--Kramer--Lewa\'nski conjecture).