A new family of weighted double Hurwitz numbers and a new ELSV-type formula with $\Omega$-classes

TL;DR

This paper introduces a new family of weighted double Hurwitz numbers, develops techniques linking hypergeometric KP tau functions with intersection theory, and derives an explicit ELSV-type formula involving $\,\Omega$-classes.

Contribution

It presents a novel family of weighted double Hurwitz numbers and a new explicit ELSV-type formula with $\,\Omega$-classes, advancing the understanding of their intersection theory.

Findings

Demonstrates the interaction of hypergeometric KP tau functions with moduli space intersection theory.

Refines techniques for deriving ELSV-type formulas in this context.

Derives a new explicit ELSV-type formula involving $\,\Omega$-classes.

Abstract

We analyze a new family of weighted double Hurwitz numbers that was introduced as a notable example in the context of the $x-y$ duality for logarithmic topological recursion. We use this family to systematically demonstrate, refine and develop techniques that play a crucial role in the interaction of hypergeometric (Orlov--Scherbin) KP tau functions and intersection theory of moduli spaces of curves. In particular, we discuss the subtleties related to the derivation of the ELSV-type formulas in this context and derive a new, explicit ELSV-type formula in terms of the so-called $\Omega$-classes.