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.

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