Moduli spaces and formal operads

Abstract

Let ${\stackrel{̲}{\mathcal{M}}}_{g,l}$ be the moduli space of stable algebraic curves of genus $g$ with $l$ marked points. With the operations that relate the different moduli spaces identifying marked points, the family ${({\stackrel{̲}{\mathcal{M}}}_{g,l})}_{g,l}$ is a modular operad of projective smooth Deligne-Mumford stacks $\stackrel{̲}{\mathcal{M}}$. In this paper, we prove that the modular operad of singular chains $S_{*}(\stackrel{̲}{\mathcal{M}};\mathbb{Q})$ is formal, so it is weakly equivalent to the modular operad of its homology $H_{*}(\stackrel{̲}{\mathcal{M}};\mathbb{Q})$. As a consequence, the up-to-homotopy algebras of these two operads are the same. To obtain this result, we prove a formality theorem for operads analogous to the Deligne-Griffiths-Morgan-Sullivan formality theorem, the existence of minimal models of modular operads, and a characterization of formality for operads which shows that formality is independent of the ground field.