## 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 - DOI: https://doi.org/10.1090/S0894-0347-2014-00808-0
- Published electronically: May 28, 2014
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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.

## References

- 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** - Alessandro Chiodo,
*The Witten top Chern class via $K$-theory*, J. Algebraic Geom.**15**(2006), no. 4, 681–707. MR**2237266**, DOI 10.1090/S1056-3911-06-00444-9 - B. Dubrovin,
*Geometry of 2d topological field theories*, available at arXiv:hep-th/9407018., DOI 10.1007/BFb0094793 - 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**, DOI 10.1090/trans2/212/05 - E. Getzler,
*Intersection theory on $\overline {\scr M}_{1,4}$ and elliptic Gromov-Witten invariants*, J. Amer. Math. Soc.**10**(1997), no. 4, 973–998. MR**1451505**, DOI 10.1090/S0894-0347-97-00246-4 - 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**, DOI 10.17323/1609-4514-2001-1-4-551-568 - Alexander B. Givental,
*Semisimple Frobenius structures at higher genus*, Internat. Math. Res. Notices**23**(2001), 1265–1286. MR**1866444**, DOI 10.1155/S1073792801000605 - 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**, DOI 10.1307/mmj/1049832895 - 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**, DOI 10.4007/annals.2013.178.1.1 - Eleny-Nicoleta Ionel,
*Relations in the tautological ring of $\scr M_g$*, Duke Math. J.**129**(2005), no. 1, 157–186. MR**2155060**, DOI 10.1215/S0012-7094-04-12916-1 - 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**, DOI 10.1090/amsip/001/23 - 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**, DOI 10.1007/s00220-006-1538-3 - 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** - R. Pandharipande and A. Pixton,
*Relations in the tautological ring of the moduli space of curves*, available at arXiv:1301.4561. - R. Pandharipande, A. Pixton, and D. Zvonkine. in preparation.
- A. Pixton,
*Conjectural relations in the tautological ring of $\overline {\mathcal {M}}_{g,n}$*, available at arXiv:1207.1918. - 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**, DOI 10.1090/conm/276/04523 - Alexander Polishchuk,
*Witten’s top Chern class on the moduli space of higher spin curves*, Frobenius manifolds, Aspects Math., E36, Friedr. Vieweg, Wiesbaden, 2004, pp. 253–264. MR**2115773** - 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**, DOI 10.17323/1609-4514-2009-9-2-411-429 - Constantin Teleman,
*The structure of 2D semi-simple field theories*, Invent. Math.**188**(2012), no. 3, 525–588. MR**2917177**, DOI 10.1007/s00222-011-0352-5 - 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**

## Bibliographic 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