On the structure of Goulden-Jackson-Vakil formula

TL;DR

This paper analyzes the Goulden-Jackson-Vakil formula, providing explicit generating functions for related intersection numbers and proving they satisfy Hirota equations, thus advancing understanding of Hurwitz numbers and their geometric connections.

Contribution

It offers explicit formulas for generating functions of intersection numbers and proves they satisfy Hirota equations, simplifying previous results and deepening the geometric understanding.

Findings

Explicit formulas for generating functions of intersection numbers

Proof that these functions satisfy Hirota equations

Generalization and simplification of earlier results

Abstract

We study the structure of the Goulden-Jackson-Vakil formula that relates Hurwitz numbers to some conjectural "intersection numbers" on a conjectural family of varieties $X_{g,n}$ of dimension $4g-3+n$. We give explicit formulas for the properly arranged generating function for these "intersection numbers", and prove that it satisfies Hirota equations. This generalizes and substantially simplifies our earlier results with Zvonkine in arXiv:math/0602457.