A short proof of the lambda_g-conjecture without Gromov-Witten theory: Hurwitz theory and the moduli of curves

TL;DR

This paper presents a concise proof of the lambda_g-conjecture using Hurwitz theory and the polynomiality of Hurwitz numbers, avoiding Gromov-Witten theory, and discusses potential extensions to other intersection number conjectures.

Contribution

It provides a novel, direct proof of the lambda_g-conjecture leveraging Hurwitz theory and polynomiality, independent of Gromov-Witten theory.

Findings

Proof of the lambda_g-conjecture without Gromov-Witten theory

Application of the Ekedahl-Lando-Shapiro-Vainshtein theorem

Potential extension of the approach to other intersection number conjectures

Abstract

We give a short and direct proof of the $\lambda_g$-Conjecture. The approach is through the Ekedahl-Lando-Shapiro-Vainshtein theorem, which establishes the ``polynomiality'' of Hurwitz numbers, from which we pick off the lowest degree terms. The proof is independent of Gromov-Witten theory. We briefly describe the philosophy behind our general approach to intersection numbers and how it may be extended to other intersection number conjectures.