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Journal of the American Mathematical Society

Published by the American Mathematical Society, the Journal of the American Mathematical Society (JAMS) is devoted to research articles of the highest quality in all areas of mathematics.

ISSN 1088-6834 (online) ISSN 0894-0347 (print)

The 2020 MCQ for Journal of the American Mathematical Society is 4.83.

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Relations on $\overline {\mathcal {M}}_{g,n}$ via $3$-spin structures
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by Rahul Pandharipande, Aaron Pixton and Dimitri Zvonkine
J. Amer. Math. Soc. 28 (2015), 279-309
Published electronically: May 28, 2014


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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Bibliographic Information
  • Rahul Pandharipande
  • Affiliation: Departement Mathematik, ETH Zürich 8092, Switzerland
  • MR Author ID: 357813
  • Email:
  • Aaron Pixton
  • Affiliation: Department of Mathematics, Princeton University, Princeton, New Jersey 08544
  • Email:
  • Dimitri Zvonkine
  • Affiliation: CNRS, Institut Mathématique de Jussieu, 4 place Jussieu 75005 Paris, France
  • MR Author ID: 621483
  • Email:
  • 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:
  • MathSciNet review: 3264769