Relations on Mbar_{g,n} via 3-spin structures

TL;DR

This paper constructs explicit tautological relations on the moduli space of curves using 3-spin structures and CohFT classification, confirming Pixton's conjecture and advancing understanding of Witten's class.

Contribution

It provides an explicit formula for Witten's class in the tautological ring and constructs tautological relations that match Pixton's conjecture, using semisimple CohFT classification.

Findings

Derived explicit formula for Witten's class in tautological ring

Constructed tautological relations matching Pixton's conjecture

Proved relations in cohomology via CohFT classification

Abstract

Witten's class on the moduli space of 3-spin curves defines a (non-semisimple) cohomological field theory. After a canonical modification, we construct an associated semisimple CohFT with a non-trivial vanishing property obtained from the homogeneity of Witten's class. Using the classification of semisimple CohFTs by Givental-Teleman, we derive two main results. The first is an explicit formula in the tautological ring of Mbar_{g,n} for Witten's class. The second, using the vanishing property, is the construction of relations in the tautological ring of Mbar_{g,n}. Pixton has previously conjectured a system of tautological relations on Mbar_{g,n} (which extends the established Faber-Zagier relations on M_g). Our 3-spin construction exactly yields Pixton's conjectured relations. As the classification of CohFTs is a topological result depending upon the Madsen-Weiss theorem (Mumford's conjecture), our construction proves relations in cohomology. The study of Witten's class and the associated tautological relations for r-spin curves via a parallel strategy will be taken up in a following paper.