Categorical enumerative invariants of the ground field

Abstract

For an $S^1$-framed modular operad $P$, we introduce its "Feynman compactification" denoted by $FP$ which is a modular operad. Let $\{{\mathbb{M}}^{\mathsf{fr}}(g,n){\}}_{(g,n)}$ 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 $F{\mathbb{M}}^{\mathsf{fr}}$ is isomorphic to $H_{\ast }(\overline{M})$, 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.

41 pages, revised and some parts rewritten