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

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- by Rahul Pandharipande, Aaron Pixton and Dimitri Zvonkine PDF
- J. Amer. Math. Soc.
**28**(2015), 279-309 Request permission

## 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.

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## Additional Information

**Rahul Pandharipande**- Affiliation: Departement Mathematik, ETH Zürich 8092, Switzerland
- MR Author ID: 357813
- Email: rahul@math.ethz.ch
**Aaron Pixton**- Affiliation: Department of Mathematics, Princeton University, Princeton, New Jersey 08544
- Email: apixton@math.princeton.edu
**Dimitri Zvonkine**- Affiliation: CNRS, Institut Mathématique de Jussieu, 4 place Jussieu 75005 Paris, France
- MR Author ID: 621483
- Email: dimitri.zvonkine@imj-prg.fr
- Received by editor(s): July 3, 2013
- Received by editor(s) in revised form: February 4, 2014
- Published electronically: May 28, 2014
- © Copyright 2014 American Mathematical Society
- Journal: J. Amer. Math. Soc.
**28**(2015), 279-309 - MSC (2010): Primary 14H10; Secondary 14N35
- DOI: https://doi.org/10.1090/S0894-0347-2014-00808-0
- MathSciNet review: 3264769