Moduli spaces and formal operads

Abstract

Let M̲g,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 (M̲g,l)g,l is a modular operad of projective smooth Deligne-Mumford stacks M̲. In this paper, we prove that the modular operad of singular chains S*(M̲;Q) is formal, so it is weakly equivalent to the modular operad of its homology H*(M̲;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.