Homotopy moment maps

Abstract

Associated to any manifold equipped with a closed form of degree >1 is an ‘L∞-algebra of observables’ which acts as a higher/homotopy analog of the Poisson algebra of functions on a symplectic manifold. In order to study Lie group actions on these manifolds, we introduce a theory of homotopy moment maps. Such a map is a L∞-morphism from the Lie algebra of the group into the observables which lifts the infinitesimal action. We establish the relationship between homotopy moment maps and equivariant de Rham cohomology, and analyze the obstruction theory for the existence of such maps. This allows us to easily and explicitly construct a large number of examples. These include results concerning group actions on loop spaces and moduli spaces of flat connections. Relationships are also established with previous work by others in classical field theory, algebroid theory, and dg geometry. Furthermore, we use our theory to geometrically construct various L∞-algebras as higher central extensions of Lie algebras, in analogy with Kostant's quantization theory. In particular, the so-called ‘string Lie 2-algebra’ arises this way.