Group Actions on DG-Manifolds and Exact Courant Algebroids

Abstract

Let G be a Lie group acting by diffeomorphisms on a manifold M and consider the image of T[1]G and T[1]M, of G and M respectively, in the category of differential graded manifolds. We show that the obstruction to lift the action of T[1]G on T[1]M to an action on a \({\mathbb{R}[n]}\)-bundle over T[1]M is measured by the G equivariant cohomology of M. We explicitly calculate the differential graded Lie algebra of the symmetries of the \({\mathbb{R}[n]}\)-bundle over T[1]M and we use this differential graded Lie algebra to understand which actions are hamiltonian. We show how split Exact Courant algebroids could be obtained as the derived Leibniz algebra of the symmetries of \({\mathbb{R}[2]}\)-bundles over T[1]M, and we use this construction to propose that the infinitesimal symmetries of a split Exact Courant algebroid should be encoded in the differential graded Lie algebra of symmetries of a \({\mathbb{R}[2]}\)-bundle over T[1]M. With this setup at hand, we propose a definition for an action of a Lie group on an Exact Courant algebroid and we propose conditions for the action to be hamiltonian.