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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
DOI: https://doi.org/10.1090/S0894-0347-2014-00808-0
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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  • [1] Pavel Belorousski and Rahul Pandharipande, A descendent relation in genus 2, Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4) 29 (2000), no. 1, 171-191. MR 1765541 (2001g:14044)
  • [2] Alessandro Chiodo, The Witten top Chern class via $ K$-theory, J. Algebraic Geom. 15 (2006), no. 4, 681-707. MR 2237266 (2007f:14021), https://doi.org/10.1090/S1056-3911-06-00444-9
  • [3] B. Dubrovin, Geometry of 2d topological field theories, available at arXiv:hep-th/9407018.
  • [4] Boris Dubrovin, On almost duality for Frobenius manifolds, Geometry, topology, and mathematical physics, Amer. Math. Soc. Transl. Ser. 2, vol. 212, Amer. Math. Soc., Providence, RI, 2004, pp. 75-132. MR 2070050 (2005e:53142)
  • [5] E. Getzler, Intersection theory on $ \overline {\mathcal {M}}_{1,4}$ and elliptic Gromov-Witten invariants, J. Amer. Math. Soc. 10 (1997), no. 4, 973-998. MR 1451505 (98f:14018), https://doi.org/10.1090/S0894-0347-97-00246-4
  • [6] Alexander B. Givental, Gromov-Witten invariants and quantization of quadratic Hamiltonians, Mosc. Math. J. 1 (2001), no. 4, 551-568, 645 (English, with English and Russian summaries). Dedicated to the memory of I. G. Petrovskii on the occasion of his 100th anniversary. MR 1901075 (2003j:53138)
  • [7] Alexander B. Givental, Semisimple Frobenius structures at higher genus, Internat. Math. Res. Notices 23 (2001), 1265-1286. MR 1866444 (2003b:53092), https://doi.org/10.1155/S1073792801000605
  • [8] T. Graber and R. Pandharipande, Constructions of nontautological classes on moduli spaces of curves, Michigan Math. J. 51 (2003), no. 1, 93-109. MR 1960923 (2004e:14043), https://doi.org/10.1307/mmj/1049832895
  • [9] Huijun Fan, Tyler Jarvis, and Yongbin Ruan, The Witten equation, mirror symmetry, and quantum singularity theory, Ann. of Math. (2) 178 (2013), no. 1, 1-106. MR 3043578, https://doi.org/10.4007/annals.2013.178.1.1
  • [10] Eleny-Nicoleta Ionel, Relations in the tautological ring of $ \mathcal {M}_g$, Duke Math. J. 129 (2005), no. 1, 157-186. MR 2155060 (2006c:14040), https://doi.org/10.1215/S0012-7094-04-12916-1
  • [11] M. Kontsevich and Yu. Manin, Gromov-Witten classes, quantum cohomology, and enumerative geometry [ MR1291244 (95i:14049)], Mirror symmetry, II, AMS/IP Stud. Adv. Math., vol. 1, Amer. Math. Soc., Providence, RI, 1997, pp. 607-653. MR 1416351
  • [12] Takuro Mochizuki, The virtual class of the moduli stack of stable $ r$-spin curves, Comm. Math. Phys. 264 (2006), no. 1, 1-40. MR 2211733 (2007e:14043), https://doi.org/10.1007/s00220-006-1538-3
  • [13] David Mumford, Towards an enumerative geometry of the moduli space of curves, Arithmetic and geometry, Vol. II, Progr. Math., vol. 36, Birkhäuser Boston, Boston, MA, 1983, pp. 271-328. MR 717614 (85j:14046)
  • [14] R. Pandharipande and A. Pixton, Relations in the tautological ring of the moduli space of curves, available at arXiv:1301.4561.
  • [15] R. Pandharipande, A. Pixton, and D. Zvonkine. in preparation.
  • [16] A. Pixton, Conjectural relations in the tautological ring of $ \overline {\mathcal {M}}_{g,n}$, available at arXiv:1207.1918.
  • [17] Alexander Polishchuk and Arkady Vaintrob, Algebraic construction of Witten's top Chern class, Advances in algebraic geometry motivated by physics (Lowell, MA, 2000), Contemp. Math., vol. 276, Amer. Math. Soc., Providence, RI, 2001, pp. 229-249. MR 1837120 (2002d:14012), https://doi.org/10.1090/conm/276/04523
  • [18] Alexander Polishchuk, Witten's top Chern class on the moduli space of higher spin curves, Frobenius manifolds, Aspects Math., E36, Vieweg, Wiesbaden, 2004, pp. 253-264. MR 2115773 (2006d:14006)
  • [19] Sergey Shadrin, BCOV theory via Givental group action on cohomological fields theories, Mosc. Math. J. 9 (2009), no. 2, 411-429, back matter (English, with English and Russian summaries). MR 2568443 (2010h:53140)
  • [20] Constantin Teleman, The structure of 2D semi-simple field theories, Invent. Math. 188 (2012), no. 3, 525-588. MR 2917177, https://doi.org/10.1007/s00222-011-0352-5
  • [21] Edward Witten, Algebraic geometry associated with matrix models of two-dimensional gravity, Topological methods in modern mathematics (Stony Brook, NY, 1991) Publish or Perish, Houston, TX, 1993, pp. 235-269. MR 1215968 (94c:32012)

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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

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