Relations on $\overline {\mathcal {M}}_{g,n}$ via $3$-spin structures

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 $\overline {\mathcal {M}}_{g,n}$ for Witten’s class. The second, using the vanishing property, is the construction of relations in the tautological ring of $\overline {\mathcal {M}}_{g,n}$.

Pixton has previously conjectured a system of tautological relations on $\overline {\mathcal {M}}_{g,n}$ (which extends the established Faber-Zagier relations on $\mathcal {M}_{g}$). 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.