Faber's socle intersection numbers via Gromov--Witten theory of elliptic curve

TL;DR

This paper provides a new proof of Faber's formula for socle intersection numbers in the tautological ring of moduli spaces of curves, utilizing Gromov--Witten theory of elliptic curves and double ramification cycles.

Contribution

It introduces a novel tautological relation derived from Gromov--Witten theory and double ramification cycles, offering an alternative proof of Faber's formula.

Findings

New tautological relation from Gromov--Witten theory

Alternative proof of Faber's formula

Connection to KdV hierarchy recursion

Abstract

The goal of this very short note is to give a new proof of Faber's formula for the socle intersection numbers in the tautological ring of $\mathcal{M}_g$. This new proof exhibits a new beautiful tautological relation that stems from the recent work of Oberdieck--Pixton on the Gromov--Witten theory of the elliptic curve via a refinement of their argument, and some straightforward computation with the double ramification cycles that enters the recursion relations for the Hamiltonians of the KdV hierarchy.