The Deligne-Mumford operad as a trivialization of the circle action
Alexandru Oancea, Dmitry Vaintrob
TL;DR
This work proves that the tree-like Deligne–Mumford operad $\text{DM}^{\text{tree}}$ furnishes a homotopical model for the trivialization of the circle action in the higher-genus framed little disks operad by comparing a homotopy pushout diagram $\text{pt} \leftarrow \widetilde{\text{Ann}} \rightarrow \widetilde{\text{Fr}}_{\partial}$ with $\text{DM}^{\text{tree}}$. The authors introduce the operad of framed surfaces $\text{Fr}_{\partial}$ and its suboperads $\text{Ann}$ and nodal generalizations, and they develop the formalism of topological moduli problems and Segal operads to handle automorphisms and orbifold structure. A geometric pushout theorem and a sequence of W-constructions provide cofibrant-model-level control, enabling a genus-0 proof that recovers Drummond-Cole, and a higher-genus argument via topological moduli problems that extends the result to arbitrary genus. The corollaries connect chain-level data to derived $S^1$-trivializations and Hodge-to-de Rham degeneration, with implications for homological mirror-symmetry contexts. Overall, the paper deepens the link between moduli of Riemann surfaces, operadic pushouts, and algebraic structures arising in enumerative geometry and mirror symmetry.
Abstract
We prove that the tree-like Deligne-Mumford operad is a homotopical model for the trivialization of the circle in the higher-genus framed little discs operad. Our proof is based on a geometric argument involving nodal annuli. We use as a model for the higher-genus framed little discs an operad of Riemann surfaces with analytically parametrized boundary. We develop the formalism of topological moduli problems as a framework to accommodate the orbifold nature of the Deligne-Mumford operad.