Tautological relations via $r$-spin structures
Authors:
R. Pandharipande, A. Pixton and D. Zvonkine
Journal:
J. Algebraic Geom. 28 (2019), 439-496
DOI:
https://doi.org/10.1090/jag/736
Published electronically:
March 29, 2019
MathSciNet review:
3959068
Full-text PDF
Abstract |
References |
Additional Information
Abstract:
Relations among tautological classes on $\overline {\mathcal {M}}_{g,n}$ are obtained via the study of Witten’s $r$-spin theory for higher $r$. In order to calculate the quantum product, a new formula relating the $r$-spin correlators in genus 0 to the representation theory of ${\mathsf {sl}}_2(\mathbb {C})$ is proven. The Givental-Teleman classification of CohFT (cohomological field theory) is used at two special semisimple points of the associated Frobenius manifold. At the first semisimple point, the $R$-matrix is exactly solved in terms of hypergeometric series. As a result, an explicit formula for Witten’s $r$-spin class is obtained (along with tautological relations in higher degrees). As an application, the $r=4$ relations are used to bound the Betti numbers of $R^*(\mathcal {M}_g)$. At the second semisimple point, the form of the $R$-matrix implies a polynomiality property in $r$ of Witten’s $r$-spin class.
In Appendix A (with F. Janda), a conjecture relating the $r=0$ limit of Witten’s $r$-spin class to the class of the moduli space of holomorphic differentials is presented.
References
- Matt Bainbridge, Dawei Chen, Quentin Gendron, Samuel Grushevsky, and Martin Möller, Compactification of strata of Abelian differentials, Duke Math. J. 167 (2018), no. 12, 2347–2416. MR 3848392, DOI https://doi.org/10.1215/00127094-2018-0012
- 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
- Dawei Chen, Strata of abelian differentials and the Teichmüller dynamics, J. Mod. Dyn. 7 (2013), no. 1, 135–152. MR 3071469, DOI https://doi.org/10.3934/jmd.2013.7.135
- Alessandro Chiodo, The Witten top Chern class via $K$-theory, J. Algebraic Geom. 15 (2006), no. 4, 681–707. MR 2237266, DOI https://doi.org/10.1090/S1056-3911-06-00444-9
- Carel Faber, A conjectural description of the tautological ring of the moduli space of curves, Moduli of curves and abelian varieties, Aspects Math., E33, Friedr. Vieweg, Braunschweig, 1999, pp. 109–129. MR 1722541
- C. Faber and R. Pandharipande, Logarithmic series and Hodge integrals in the tautological ring, Michigan Math. J. 48 (2000), 215–252. With an appendix by Don Zagier; Dedicated to William Fulton on the occasion of his 60th birthday. MR 1786488, DOI https://doi.org/10.1307/mmj/1030132716
- C. Faber and R. Pandharipande, Tautological and non-tautological cohomology of the moduli space of curves, Handbook of moduli. Vol. I, Adv. Lect. Math. (ALM), vol. 24, Int. Press, Somerville, MA, 2013, pp. 293–330. MR 3184167
- Gavril Farkas and Rahul Pandharipande, The moduli space of twisted canonical divisors, J. Inst. Math. Jussieu 17 (2018), no. 3, 615–672. MR 3789183, DOI https://doi.org/10.1017/S1474748016000128
- William Fulton, Intersection theory, Ergebnisse der Mathematik und ihrer Grenzgebiete (3) [Results in Mathematics and Related Areas (3)], vol. 2, Springer-Verlag, Berlin, 1984. MR 732620
- William Fulton and Joe Harris, Representation theory, Graduate Texts in Mathematics, vol. 129, Springer-Verlag, New York, 1991. A first course; Readings in Mathematics. MR 1153249
- 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 https://doi.org/10.17323/1609-4514-2001-1-4-551-568
- 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 https://doi.org/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 https://doi.org/10.4007/annals.2013.178.1.1
- F. Janda, Comparing tautological relations from the equivariant Gromov-Witten theory of projective spaces and spin structures, arXiv:1407.4778, 2014.
- Felix Janda, Frobenius manifolds near the discriminant and relations in the tautological ring, Lett. Math. Phys. 108 (2018), no. 7, 1649–1675. MR 3802725, DOI https://doi.org/10.1007/s11005-018-1047-2
- Felix Janda, Relations on $\overline M_{g,n}$ via equivariant Gromov-Witten theory of $\Bbb P^1$, Algebr. Geom. 4 (2017), no. 3, 311–336. MR 3652083, DOI https://doi.org/10.14231/AG-2017-018
- F. Janda, R. Pandharipande, A. Pixton, and D. Zvonkine, Double ramification cycles on the moduli spaces of curves, Publ. Math. Inst. Hautes Études Sci. 125 (2017), 221–266. MR 3668650, DOI https://doi.org/10.1007/s10240-017-0088-x
- M. Kontsevich and Yu. Manin, Gromov-Witten classes, quantum cohomology, and enumerative geometry, Comm. Math. Phys. 164 (1994), no. 3, 525–562. MR 1291244
- Eduard Looijenga, On the tautological ring of ${\scr M}_g$, Invent. Math. 121 (1995), no. 2, 411–419. MR 1346214, DOI https://doi.org/10.1007/BF01884306
- 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 https://doi.org/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
- Rahul Pandharipande, The $\varkappa $ ring of the moduli of curves of compact type, Acta Math. 208 (2012), no. 2, 335–388. MR 2931383, DOI https://doi.org/10.1007/s11511-012-0078-2
- Rahul Pandharipande, A calculus for the moduli space of curves, Algebraic geometry: Salt Lake City 2015, Proc. Sympos. Pure Math., vol. 97, Amer. Math. Soc., Providence, RI, 2018, pp. 459–487. MR 3821159, DOI https://doi.org/10.4310/pamq.2015.v11.n4.a3
- R. Pandharipande, Cohomological field theory calculations, Proceedings of the ICM (Rio de Janeiro 2018), Vol. 1, pp 869–898.
- Rahul Pandharipande, Aaron Pixton, and Dimitri Zvonkine, Relations on $\overline {\scr M}_{g,n}$ via $3$-spin structures, J. Amer. Math. Soc. 28 (2015), no. 1, 279–309. MR 3264769, DOI https://doi.org/10.1090/S0894-0347-2014-00808-0
- A. Pixton, Conjectural relations in the tautological ring of $\overline {\mathcal {M}}_{g,n}$, arXiv:1207.1918, 2012.
- Aaron Pixton, The tautological ring of the moduli space of curves, ProQuest LLC, Ann Arbor, MI, 2013. Thesis (Ph.D.)–Princeton University. MR 3153424
- 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 https://doi.org/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
- A. Polishchuk, Moduli spaces of curves with effective $r$-spin structures, Gromov-Witten theory of spin curves and orbifolds, Contemp. Math., vol. 403, Amer. Math. Soc., Providence, RI, 2006, pp. 1–20. MR 2234882, DOI https://doi.org/10.1090/conm/403/07592
- A. Sauvaget, Cohomology classes of strata of differentials, arXiv:1701.07867, 2017.
- Constantin Teleman, The structure of 2D semi-simple field theories, Invent. Math. 188 (2012), no. 3, 525–588. MR 2917177, DOI https://doi.org/10.1007/s00222-011-0352-5
- Erik Verlinde, Fusion rules and modular transformations in $2$D conformal field theory, Nuclear Phys. B 300 (1988), no. 3, 360–376. MR 954762, DOI https://doi.org/10.1016/0550-3213%2888%2990603-7
- 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
References
- Matt Bainbridge, Dawei Chen, Quentin Gendron, Samuel Grushevsky, and Martin Möller, Compactification of strata of Abelian differentials, Duke Math. J. 167 (2018), no. 12, 2347–2416. MR 3848392, DOI https://doi.org/10.1215/00127094-2018-0012
- 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
- Dawei Chen, Strata of abelian differentials and the Teichmüller dynamics, J. Mod. Dyn. 7 (2013), no. 1, 135–152. MR 3071469, DOI https://doi.org/10.3934/jmd.2013.7.135
- Alessandro Chiodo, The Witten top Chern class via $K$-theory, J. Algebraic Geom. 15 (2006), no. 4, 681–707. MR 2237266, DOI https://doi.org/10.1090/S1056-3911-06-00444-9
- Carel Faber, A conjectural description of the tautological ring of the moduli space of curves, Moduli of curves and abelian varieties, Aspects Math., E33, Friedr. Vieweg, Braunschweig, 1999, pp. 109–129. MR 1722541
- C. Faber and R. Pandharipande, Logarithmic series and Hodge integrals in the tautological ring, with an appendix by Don Zagier, dedicated to William Fulton on the occasion of his 60th birthday, Michigan Math. J. 48 (2000), 215–252. MR 1786488, DOI https://doi.org/10.1307/mmj/1030132716
- C. Faber and R. Pandharipande, Tautological and non-tautological cohomology of the moduli space of curves, Handbook of moduli. Vol. I, Adv. Lect. Math. (ALM), vol. 24, Int. Press, Somerville, MA, 2013, pp. 293–330. MR 3184167
- Gavril Farkas and Rahul Pandharipande, The moduli space of twisted canonical divisors, J. Inst. Math. Jussieu 17 (2018), no. 3, 615–672. MR 3789183, DOI https://doi.org/10.1017/S1474748016000128
- William Fulton, Intersection theory, Ergebnisse der Mathematik und ihrer Grenzgebiete (3) [Results in Mathematics and Related Areas (3)], vol. 2, Springer-Verlag, Berlin, 1984. MR 732620
- William Fulton and Joe Harris, Representation theory, Graduate Texts in Mathematics, vol. 129, Springer-Verlag, New York, 1991. A first course; Readings in Mathematics. MR 1153249
- Alexander B. Givental, Gromov-Witten invariants and quantization of quadratic Hamiltonians, dedicated to the memory of I. G. Petrovskii on the occasion of his 100th anniversary, Mosc. Math. J. 1 (2001), no. 4, 551–568, 645 (English, with English and Russian summaries). MR 1901075
- 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 https://doi.org/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 https://doi.org/10.4007/annals.2013.178.1.1
- F. Janda, Comparing tautological relations from the equivariant Gromov-Witten theory of projective spaces and spin structures, arXiv:1407.4778, 2014.
- Felix Janda, Frobenius manifolds near the discriminant and relations in the tautological ring, Lett. Math. Phys. 108 (2018), no. 7, 1649–1675. MR 3802725, DOI https://doi.org/10.1007/s11005-018-1047-2
- Felix Janda, Relations on $\overline M_{g,n}$ via equivariant Gromov-Witten theory of $\mathbb {P}^1$, Algebr. Geom. 4 (2017), no. 3, 311–336. MR 3652083, DOI https://doi.org/10.14231/AG-2017-018
- F. Janda, R. Pandharipande, A. Pixton, and D. Zvonkine, Double ramification cycles on the moduli spaces of curves, Publ. Math. Inst. Hautes Études Sci. 125 (2017), 221–266. MR 3668650, DOI https://doi.org/10.1007/s10240-017-0088-x
- M. Kontsevich and Yu. Manin, Gromov-Witten classes, quantum cohomology, and enumerative geometry, Comm. Math. Phys. 164 (1994), no. 3, 525–562. MR 1291244
- Eduard Looijenga, On the tautological ring of ${\mathcal {M}}_g$, Invent. Math. 121 (1995), no. 2, 411–419. MR 1346214, DOI https://doi.org/10.1007/BF01884306
- 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 https://doi.org/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
- Rahul Pandharipande, The $\varkappa$ ring of the moduli of curves of compact type, Acta Math. 208 (2012), no. 2, 335–388. MR 2931383, DOI https://doi.org/10.1007/s11511-012-0078-2
- Rahul Pandharipande, A calculus for the moduli space of curves, Algebraic geometry: Salt Lake City 2015, Proc. Sympos. Pure Math., vol. 97, Amer. Math. Soc., Providence, RI, 2018, pp. 459–487. MR 3821159
- R. Pandharipande, Cohomological field theory calculations, Proceedings of the ICM (Rio de Janeiro 2018), Vol. 1, pp 869–898.
- Rahul Pandharipande, Aaron Pixton, and Dimitri Zvonkine, Relations on $\overline {\mathcal {M}}_{g,n}$ via $3$-spin structures, J. Amer. Math. Soc. 28 (2015), no. 1, 279–309. MR 3264769, DOI https://doi.org/10.1090/S0894-0347-2014-00808-0
- A. Pixton, Conjectural relations in the tautological ring of $\overline {\mathcal {M}}_{g,n}$, arXiv:1207.1918, 2012.
- Aaron Pixton, The tautological ring of the moduli space of curves, Thesis (Ph.D.)–Princeton University, 2013, ProQuest LLC, Ann Arbor, MI. MR 3153424
- 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 https://doi.org/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
- A. Polishchuk, Moduli spaces of curves with effective $r$-spin structures, Gromov-Witten theory of spin curves and orbifolds, Contemp. Math., vol. 403, Amer. Math. Soc., Providence, RI, 2006, pp. 1–20. MR 2234882, DOI https://doi.org/10.1090/conm/403/07592
- A. Sauvaget, Cohomology classes of strata of differentials, arXiv:1701.07867, 2017.
- Constantin Teleman, The structure of 2D semi-simple field theories, Invent. Math. 188 (2012), no. 3, 525–588. MR 2917177, DOI https://doi.org/10.1007/s00222-011-0352-5
- Erik Verlinde, Fusion rules and modular transformations in $2$D conformal field theory, Nuclear Phys. B 300 (1988), no. 3, 360–376. MR 954762, DOI https://doi.org/10.1016/0550-3213%2888%2990603-7
- 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
Additional Information
R. Pandharipande
Affiliation:
Departement Mathematik, ETH Zürich, Ramistrasse 101, 8092 Zurich, Switzerland
MR Author ID:
357813
Email:
rahul@math.ethz.ch
A. Pixton
Affiliation:
Department of Mathematics, Massachusetts Institute of Technology, Cambridge, Massachusetts 02139
MR Author ID:
818035
Email:
apixton@mit.edu
D. Zvonkine
Affiliation:
CNRS, Institut Mathématique de Jussieu, 75013 Paris, France
MR Author ID:
621483
Email:
dimitri.zvonkine@ump-prg.fr
Received by editor(s):
October 11, 2016
Received by editor(s) in revised form:
September 20, 2018
Published electronically:
March 29, 2019
Additional Notes:
The first author was partially supported by SNF-200021143274, SNF-200020162928, ERC-2012-AdG-320368-MCSK, ERC-2017-AdG-786580-MACI, SwissMAP, and the Einstein Stiftung. The second author was supported by a fellowship from the Clay Mathematics Institute. The third author was supported by the grants ANR-09-JCJC-0104-01 and ANR-18-CE40-0009 ENUMGEOM. This project received funding from the European Research Council (ERC) under the European Union Horizon 2020 research and innovation program (grant agreement No. 786580).
Article copyright:
© Copyright 2019
University Press, Inc.
