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

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.