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


Authors: Rahul Pandharipande, Aaron Pixton and Dimitri Zvonkine
Journal: J. Amer. Math. Soc. 28 (2015), 279-309
MSC (2010): Primary 14H10; Secondary 14N35
Published electronically: May 28, 2014
MathSciNet review: 3264769
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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
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
Email: dimitri.zvonkine@imj-prg.fr

DOI: https://doi.org/10.1090/S0894-0347-2014-00808-0
Received by editor(s): July 3, 2013
Received by editor(s) in revised form: February 4, 2014
Published electronically: May 28, 2014
Article copyright: © Copyright 2014 American Mathematical Society