Relations on $\overline M_{g,n}$ via equivariant Gromov-Witten theory of $\mathbb P^1$

TL;DR

This paper proves Pixton's generalized Faber-Zagier relations in the tautological Chow ring of  M_{g,n} using equivariant Gromov-Witten theory of    , extending known cohomological results to Chow via virtual localization.

Contribution

It extends the proof of Pixton's relations from cohomology to Chow by applying equivariant Gromov-Witten theory and the Givental-Teleman classification in Chow.

Findings

Pixton's relations hold in the Chow ring of  M_{g,n}.

Equivariant Gromov-Witten theory of  can be used to derive tautological relations.

Givental-Teleman classification applies in Chow via virtual localization.

Abstract

We give a proof of Pixton's generalized Faber-Zagier relations in the tautological Chow ring of $\overline M_{g,n}$. The strategy is very similar to the work of Pandharipande-Pixton-Zvonkine, who have given a proof of the same result in cohomology. The main tool is the Givental-Teleman classification of semisimple cohomological field theories, which, while in general only known in cohomology, via virtual localization can be shown to be also valid in Chow for the equivariant Gromov-Witten theory of the projective line. We obtain the relations just from this theory.