Associated to any manifold equipped with a closed form of degree >1 is an
`L-infinity 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-infinity 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-infinity 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.Comment: Final version will appear in Advances in Mathematics. Results
concerning equivariant cohomology strengthened. In particular, we exhibit the
explicit relationship between equivariant de Rham cocycles of arbitrary
degree and homotopy moment maps. 62 pages. Comments are welcome. arXiv admin
note: text overlap with arXiv:1402.0144 by other author
