# Local topological recursion governs the enumeration of lattice points in $\overline{\mathcal M}_{g,n}$

## TL;DR

This paper proves that the enumeration of lattice points in the Deligne-Mumford compactifications of moduli spaces of curves is governed by local topological recursion, revealing new connections between enumerative geometry and topological recursion.

## Contribution

The authors demonstrate that lattice point enumeration in $ar{ ext{M}}_{g,n}$ satisfies local topological recursion with modified initial data, linking it to algebro-geometric invariants.

## Key findings

- Enumeration satisfies local topological recursion
- Modified spectral curve data is required
- Intermediate polynomial coefficients encode geometric information

## Abstract

The second author and Norbury initiated the enumeration of lattice points in the Deligne-Mumford compactifications of moduli spaces of curves. They showed that the enumeration may be expressed in terms of polynomials, whose top and bottom degree coefficients store psi-class intersection numbers and orbifold Euler characteristics of $\overline{\mathcal M}_{g,n}$, respectively. Furthermore, they ask whether the enumeration is governed by the topological recursion and whether the intermediate coefficients also store algebro-geometric information. In this paper, we prove that the enumeration does indeed satisfy the topological recursion, although with a modification to the initial spectral curve data. Thus, one can consider this to be one of the first known instances of a natural enumerative problem governed by the so-called local topological recursion. Combining the present work with the known relation between local topological recursion and cohomological field theory should uncover the geometric meaning of the intermediate coefficients of the aforementioned polynomials.