Symmetry, Integrability and Geometry: Methods and Applications (SIGMA)

SIGMA 15 (2019), 080, 27 pages      arXiv:1902.02742      https://doi.org/10.3842/SIGMA.2019.080
Contribution to the Special Issue on Integrability, Geometry, Moduli in honor of Motohico Mulase for his 65th birthday

Half-Spin Tautological Relations and Faber's Proportionalities of Kappa Classes

Elba Garcia-Failde ^a, Reinier Kramer ^b, Danilo Lewański ^b and Sergey Shadrin ^c
a) Institute de Physique Théorique, CEA Paris-Saclay, Orme des Merisiers, 91191 Gif-sur-Yvette, France
^b) Max Planck Institut für Mathematik, Vivatsgasse 7, 53111 Bonn, Germany
^c) Korteweg-de Vries Instituut voor Wiskunde, Universiteit van Amsterdam, Postbus 94248, 1090GE Amsterdam, The Netherlands

Received June 19, 2019, in final form October 14, 2019; Published online October 18, 2019

Abstract
We employ the $1/2$-spin tautological relations to provide a particular combinatorial identity. We show that this identity is a statement equivalent to Faber's formula for proportionalities of kappa-classes on $\mathcal{M}_g$, $g\geq 2$. We then prove several cases of the combinatorial identity, providing a new proof of Faber's formula for those cases.

Key words: tautological ring; tautological relations; moduli spaces of curves; Faber intersection number conjecture; odd-even binomial coefficients.

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References

1. Buryak A., Shadrin S., A new proof of Faber's intersection number conjecture, Adv. Math. 228 (2011), 22-42, arXiv:0912.5115.
2. Clader E., Janda F., Wang X., Zakharov D., Topological recursion relations from Pixton's formula, arXiv:1704.02011.
3. Faber C., A non-vanishing result for the tautological ring of $\mathcal{M}_g$, arXiv:math.AG/9711219.
4. Faber C., A conjectural description of the tautological ring of the moduli space of curves, in Moduli of Curves and Abelian Varieties, Aspects Math., Vol. E33, Friedr. Vieweg, Braunschweig, 1999, 109-129, arXiv:math.AG/9711218.
5. Faber C., Pandharipande R., Relative maps and tautological classes, J. Eur. Math. Soc. (JEMS) 7 (2005), 13-49, arXiv:math.AG/0304485.
6. Getzler E., Pandharipande R., Virasoro constraints and the Chern classes of the Hodge bundle, Nuclear Phys. B 530 (1998), 701-714, arXiv:math.AG/9805114.
7. Givental A.B., Gromov-Witten invariants and quantization of quadratic Hamiltonians, Mosc. Math. J. 1 (2001), 551-568, arXiv:math.AG/0108100.
8. Goulden I.P., Jackson D.M., Vakil R., The moduli space of curves, double Hurwitz numbers, and Faber's intersection number conjecture, Ann. Comb. 15 (2011), 381-436, arXiv:math.AG/0611659.
9. Kramer R., Labib F., Lewanski D., Shadrin S., The tautological ring of ${\mathcal M}_{g,n}$ via Pandharipande-Pixton-Zvonkine $r$-spin relations, Algebr. Geom. 5 (2018), 703-727, arXiv:1703.00681.
10. Lin Y.J., Zhou J., Topological recursion relations from Pixton relations, Acta Math. Sin. (Engl. Ser.) 33 (2017), 470-494.
11. Liu K., Xu H., A proof of the Faber intersection number conjecture, J. Differential Geom. 83 (2009), 313-335, arXiv:0803.2204.
12. Looijenga E., On the tautological ring of ${\mathcal M}_g$, Invent. Math. 121 (1995), 411-419, arXiv:alg-geom/9501010.
13. Mumford D., Towards an enumerative geometry of the moduli space of curves, in Arithmetic and Geometry, Vol. II, Progr. Math., Vol. 36, Birkhäuser Boston, Boston, MA, 1983, 271-328.
14. Pandharipande R., A calculus for the moduli space of curves, in Algebraic Geometry: Salt Lake City 2015, Proc. Sympos. Pure Math., Vol. 97, Amer. Math. Soc., Providence, RI, 2018, 459-487, arXiv:1603.05151.
15. Pandharipande R., Pixton A., Zvonkine D., Relations on $\overline{\mathcal M}_{g,n}$ via $3$-spin structures, J. Amer. Math. Soc. 28 (2015), 279-309, arXiv:1303.1043.
16. Pandharipande R., Pixton A., Zvonkine D., Tautological relations via $r$-spin structures, J. Algebraic Geom. 28 (2019), 439-496, arXiv:1607.00978.
17. Pixton A., The tautological ring of the moduli space of curves, Princeton University, Ph.D. Thesis, 2013.
18. Polishchuk A., Vaintrob A., Algebraic construction of Witten's top Chern class, in Advances in Algebraic Geometry Motivated by Physics (Lowell, MA, 2000), Contemp. Math., Vol. 276, Amer. Math. Soc., Providence, RI, 2001, 229-249, arXiv:math.AG/0011032.
19. Tavakol M., The moduli space of curves and its invariants, arXiv:1610.09589.
20. Vakil R., The moduli space of curves and its tautological ring, Notices Amer. Math. Soc. 50 (2003), 647-658, available at https://www.ams.org/notices/200306/fea-vakil.pdf.
21. Witten E., Algebraic geometry associated with matrix models of two-dimensional gravity, in Topological Methods in Modern Mathematics (Stony Brook, NY, 1991), Publish or Perish, Houston, TX, 1993, 235-269, available at https://www.sns.ias.edu/content/algebraic-geometry-associated-matrix-models-two-dimensional-gravity.
22. Zvonkine D., An introduction to moduli spaces of curves and their intersection theory, in Handbook of Teichmüller Theory, Vol. III, IRMA Lect. Math. Theor. Phys., Vol. 17, Eur. Math. Soc., Zürich, 2012, 667-716.