Topological 1D Gravity, KP Hierarchy, and Orbifold Euler Characteristics of $\overline{\mathcal M}_{g,n}$

TL;DR

This paper explores the tau-function of topological 1D gravity within the KP hierarchy and introduces algorithms based on topological recursion and affine coordinates to compute orbifold Euler characteristics of moduli spaces.

Contribution

It develops two novel algorithms leveraging topological recursion and Sato Grassmannian coordinates for calculating orbifold Euler characteristics.

Findings

Algorithms effectively compute orbifold Euler characteristics.

Topological recursion formulas derived from Virasoro constraints.

Connection established between KP tau-functions and moduli space invariants.

Abstract

In this work we study the tau-function $Z^{1D}$ of the KP hierarchy specified by the topological 1D gravity. As an application, we present two types of algorithms to compute the orbifold Euler characteristics of $\overline{\mathcal M}_{g,n}$. The first is to use (fat or thin) topological recursion formulas emerging from the Virasoro constraints for $Z^{1D}$; and the second is to use a formula for the connected $n$-point functions of a KP tau-function in terms of its affine coordinates on the Sato Grassmannian. This is a sequel to an earlier work.