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

Pandharipande, Rahul¹ ; Pixton, Aaron² ; Zvonkine, Dimitri³

¹ Departement Mathematik, ETH Zürich 8092, Switzerland
² Department of Mathematics, Princeton University, Princeton, New Jersey 08544
³ CNRS, Institut Mathématique de Jussieu, 4 place Jussieu 75005 Paris, France

Journal of the American Mathematical Society, Tome 28 (2015) no. 1, pp. 279-309

Résumé

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.

Détail

DOI : 10.1090/S0894-0347-2014-00808-0

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