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 ${\mathbb{M}^{\sf fr}(g,n)}{(g,n)}$ 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.