Categorical enumerative invariants of the ground field

TL;DR

This paper introduces a new operad construction called Feynman compactification for $S^1$-framed modular operads, linking it to the homology of moduli spaces and relating categorical invariants to Gromov-Witten invariants of a point.

Contribution

It defines the Feynman compactification for $S^1$-framed modular operads and establishes an isomorphism with the homology of the Deligne-Mumford operad, connecting categorical invariants to classical Gromov-Witten invariants.

Findings

Homology of Feynman compactification matches the homology of the Deligne-Mumford operad.

Explicit formula for the fundamental class of moduli space quotients.

Categorical invariants of the ground field coincide with Gromov-Witten invariants of a point.

Abstract

For an $S^1$-framed modular operad $P$, we introduce its "Feynman compactification" denoted by $FP$ which is a modular operad. Let $\{\mathbb{M}^{\sf fr}(g,n)\}_{(g,n)}$ be the $S^1$-framed modular operad defined using moduli spaces of smooth curves with framings along punctures. We prove that the homology operad of $F\mathbb{M}^{\sf fr}$ is isomorphic to $H_*(\overline{M})$, the homology operad of the Deligne-Mumford operad. Using this isomorphism, we obtain an explicit formula of the fundamental class of $[\overline{M}_{g,n}/S_n]$ in terms of Sen-Zwiebach's string vertices. As an immediate application, under mild assumptions, we prove that Costello's categorical enumerative invariants of the ground field match with the Gromov-Witten invariants of a point.