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


SIGMA 16 (2020), 086, 13 pages      arXiv:2004.02749      https://doi.org/10.3842/SIGMA.2020.086
Contribution to the Special Issue on Algebra, Topology, and Dynamics in Interaction in honor of Dmitry Fuchs

Uniform Lower Bound for Intersection Numbers of $\psi$-Classes

Vincent Delecroix a, Élise Goujard b, Peter Zograf cd and Anton Zorich ef
a) LaBRI, Domaine universitaire, 351 cours de la Libération, 33405 Talence, France
b) Institut de Mathématiques de Bordeaux, Université de Bordeaux,351 cours de la Libération, 33405 Talence, France
c) Steklov Mathematical Institute, Fontanka 27, St. Petersburg 191023, Russia
d) Chebyshev Laboratory, St. Petersburg State University,14th Line V.O. 29B, St. Petersburg, 199178, Russia
e) Center for Advanced Studies, Skoltech, Russia
f) Institut de Mathématiques de Jussieu - Paris Rive Gauche, B^atiment Sophie Germain,Case 7012, 8 Place Aurélie Nemours, 75205 PARIS Cedex 13, France

Received April 09, 2020, in final form August 21, 2020; Published online August 26, 2020

Abstract
We approximate intersection numbers $\big\langle \psi_1^{d_1}\cdots \psi_n^{d_n}\big\rangle_{g,n}$ on Deligne-Mumford's moduli space $\overline{\mathcal M}_{g,n}$ of genus $g$ stable complex curves with $n$ marked points by certain closed-form expressions in $d_1,\dots,d_n$. Conjecturally, these approximations become asymptotically exact uniformly in $d_i$ when $g\to\infty$ and $n$ remains bounded or grows slowly. In this note we prove a lower bound for the intersection numbers in terms of the above-mentioned approximatingexpressions multiplied by an explicit factor $\lambda(g,n)$, which tends to $1$ when $g\to\infty$ and $d_1+\dots+d_{n-2}=o(g)$.

Key words: intersection numbers; $\psi$-classes; Witten-Kontsevich correlators; moduli space of curves; large genus asymptotics.

pdf (419 kb)   tex (19 kb)  

References

  1. Aggarwal A., Large genus asymptotics for Siegel-Veech constants, Geom. Funct. Anal. 29 (2019), 1295-1324, arXiv:1810.05227.
  2. Aggarwal A., Large genus asymptotics for volumes of strata of abelian differentials, J. Amer. Math. Soc., to appear, arXiv:1804.05431.
  3. Aggarwal A., Large genus asymptotics for intersection numbers and principal strata volumes of quadratic differentials, arXiv:2004.05042.
  4. Aggarwal A., Delecroix V., Goujard É., Zograf P., Zorich A., Conjectural large genus asymptotics of Masur-Veech volumes and of area Siegel-Veech constants of strata of quadratic differentials, Arnold Math. J. 6 (2020), 149-161, arXiv:1912.11702.
  5. Chen D., Möller M., Sauvaget A., Masur-Veech volumes and intersection theory: the principal strata of quadratic differentials (with an appendix by G. Borot, A. Giacchetto, D. Lewanski), arXiv:1912.02267.
  6. Chen D., Möller M., Sauvaget A., Zagier D., Masur-Veech volumes and intersection theory on moduli spaces of abelian differentials, Invent. Math., to appear, arXiv:1901.01785.
  7. Delecroix V., Goujard É., Zograf P., Zorich A., Masur-Veech volumes, frequencies of simple closed geodesics and intersection numbers of moduli spaces of curves, arXiv:1908.08611.
  8. Delecroix V., Goujard É., Zograf P., Zorich A., Large genus asymptotic geometry of random square-tiled surfaces and of random multicurves, arXiv:2007.04740.
  9. Dijkgraaf R., Intersection theory, integrable hierarchies and topological field theory, in New Symmetry Principles in Quantum Field Theory (Cargèse, 1991), NATO Adv. Sci. Inst. Ser. B Phys., Vol. 295, Plenum, New York, 1992, 95-158, arXiv:hep-th/9201003.
  10. Dijkgraaf R., Verlinde H., Verlinde E., Topological strings in $d<1$, HREF="https://doi.org/10.1016/0550-3213(91)90129-L" Nuclear Phys. B 352 (1991), 59-86.
  11. Eskin A., Okounkov A., Pillowcases and quasimodular forms, in Algebraic Geometry and Number Theory, Progr. Math., Vol. 253, Birkhäuser Boston, Boston, MA, 2006, 1-25, arXiv:math.DS/0505545.
  12. Eskin A., Zorich A., Volumes of strata of Abelian differentials and Siegel-Veech constants in large genera, Arnold Math. J. 1 (2015), 481-488, arXiv:1507.05296.
  13. Goujard É., Volumes of strata of moduli spaces of quadratic differentials: getting explicit values, Ann. Inst. Fourier (Grenoble) 66 (2016), 2203-2251, arXiv:1501.01611.
  14. Kazarian M., Recursion for Masur-Veech volumes of moduli spaces of quadratic differentials, J. Inst. Math. Jussieu, to appear, arXiv:1912.10422.
  15. Kazarian M., Lando S., An algebro-geometric proof of Witten's conjecture, J. Amer. Math. Soc. 20 (2007), 1079-1089, arXiv:math.AG/0601760.
  16. Kontsevich M., Intersection theory on the moduli space of curves and the matrix Airy function, Comm. Math. Phys. 147 (1992), 1-23.
  17. Liu K., Xu H., A remark on Mirzakhani's asymptotic formulae, Asian J. Math. 18 (2014), 29-52, arXiv:1103.5136.
  18. Mirzakhani M., Simple geodesics and Weil-Petersson volumes of moduli spaces of bordered Riemann surfaces, Invent. Math. 167 (2007), 179-222.
  19. Mirzakhani M., Weil-Petersson volumes and intersection theory on the moduli space of curves, J. Amer. Math. Soc. 20 (2007), 1-23.
  20. Okounkov A., Pandharipande R., Gromov-Witten theory, Hurwitz numbers, and matrix models, in Algebraic Geometry - Seattle 2005, Part 1, Proc. Sympos. Pure Math., Vol. 80, Amer. Math. Soc., Providence, RI, 2009, 325-414, arXiv:math.AG/0101147.
  21. Witten E., Two-dimensional gravity and intersection theory on moduli space, in Surveys in Differential Geometry (Cambridge, MA, 1990), Lehigh University, Bethlehem, PA, 1991, 243-310.
  22. Yang D., Zagier D., Zhang Y., Masur-Veech volumes of quadratic differentials and their asymptotics, arXiv:2005.02275.
  23. Zograf P.G., An explicit formula for Witten's 2-correlators, J. Math. Sci. 240 (2019), 535-538.

Previous article  Next article  Contents of Volume 16 (2020)