The Deligne–Mumford operad as a trivialization of the circle action

Oancea, Alexandru; Vaintrob, Dmitry

doi:10.2140/agt.2026.26.1229

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The Deligne–Mumford operad as a trivialization of the circle action

Alexandru Oancea and Dmitry Vaintrob

Algebraic & Geometric Topology 26 (2026) 1229–1291

DOI: 10.2140/agt.2026.26.1229

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.

Keywords

moduli space of curves, Deligne–Mumford compactification, framed little discs operad, categorical mirror symmetry, topological moduli problems

Mathematical Subject Classification

Primary: 18M75, 53D37

Secondary: 18G85, 55P48

References

Publication

Received: 8 April 2020

Revised: 11 April 2025

Accepted: 5 May 2025

Published: 25 April 2026

Authors

Alexandru Oancea
Institut de Recherche Mathématique Avancée Université de Strasbourg Strasbourg France

Dmitry Vaintrob
Institut des Hautes Études Scientifiques Bures-sur-Yvette France

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Volume 26, issue 4 (2026)