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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