Relations on Mbar_{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 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.

45 pages, 2 figures, more typos fixed (parity factor in leg term)

References in corpus (4)

• Relations in the tautological ring of the moduli space of curves
• Conjectural relations in the tautological ring of ${\bar{M}}_{g,n}$
• On almost duality for Frobenius manifolds
• Witten's top Chern class via K-theory

Cited by in corpus (14)

• Dubrovin's superpotential as a global spectral curve
• Equivalence of ELSV and Bouchard-Mariño conjectures for $r$-spin Hurwitz numbers
• Relations on ${\overline{M}}_{g,n}$ via equivariant Gromov-Witten theory of ${\mathbb{P}}^1$
• Emergent Geometry and Mirror Symmetry of A Point
• Structure of Higher Genus Gromov-Witten Invariants of Quintic 3-folds
• Tautological relations in moduli spaces of weighted pointed curves
• The hypergeometric functions of the Faber-Zagier and Pixton relations
• Comparing tautological relations from the equivariant Gromov-Witten theory of projective spaces and spin structures
• The Chern character of the Verlinde bundle over the moduli space of stable curves
• Half-Spin Tautological Relations and Faber's Proportionalities of Kappa Classes
• Cycles on curves and Jacobians: a tale of two tautological rings
• Les Houches lecture notes on moduli spaces of Riemann surfaces
• Relations on Mbar_{g,n} via orbifold stable maps
• Teleman's classification of 2D semisimple cohomological field theories