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

Source 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 M ¯ g , n \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 M ¯ g , n \overline {\mathcal {M}}_{g,n} . Pixton has previously conjectured a system of tautological relations on M ¯ g , n \overline {\mathcal {M}}_{g,n} (which extends the established Faber-Zagier relations on M g \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 r -spin curves via a parallel strategy will be taken up in a following paper.