A cohomological framework for homotopy moment maps

TL;DR

This paper introduces a cohomological approach to homotopy moment maps, simplifying their computation and exploring their properties, including relations to equivariant cohomology and obstruction theory.

Contribution

It provides a new cohomological framework for homotopy moment maps, extending previous results and enabling easier analysis of their properties.

Findings

Homotopy moment maps are described as coboundaries in a specific complex.

The framework simplifies computations of homotopy moment maps.

The paper explores their relation to equivariant cohomology and obstruction theory.

Abstract

Given a Lie group acting on a manifold $M$ preserving a closed $n+1$-form $\omega$, the notion of homotopy moment map for this action was introduced in Callies-Fregier-Rogers-Zambon [6], in terms of $L_{\infty}$-algebra morphisms. In this note we describe homotopy moment maps as coboundaries of a certain complex. This description simplifies greatly computations, and we use it to study various properties of homotopy moment maps: their relation to equivariant cohomology, their obstruction theory, how they induce new ones on mapping spaces, and their equivalences. The results we obtain extend some of the results of [6].