Eugenia Boffo

• We explore a framework for gauge and gravity theories, based on a combination of methods from graded symplectic and generalized geometry. We reformulate some gravitational theories in this language and establish a relation to gauge theories. The models of gravity we consider range from the type II effective string action for the NS-NS bosonic fields and other actions with T-dual fluxes, to General Relativity in the Palatini formulation with frame fields. A sketch of the technique for reconstructing non-abelian gauge theories is also given. The idea is the following: On the symplectic geometry side, we implement a grading up to degree 2 that enlarges the set of coordinates, so as to naturally support the geometric data of a metric tensor. This is essential since the metric is the fundamental field for a gravity theory. Then we implement interactions with gauge and other fields by deforming the Poisson brackets of the graded phase space coordinates. We do not deform the Hamiltonian but rather retain the free one. The relation to gauge theory is obtained via a graded version of Moser lemma: The deformation can be undone by a change of local phase space coordinates. It is carried by graded diffeomorphisms parametrized by a non-unique gauge field. The freedom is a gauge symmetry. Differential graded manifolds are classified by higher algebraic structures, Lie and Courant algebroids, that encode the symmetries of the bundle of generalized geometry. The correspondence stems from derived brackets with the Hamiltonian vector field. Here the latter is left unchanged by the deformation and all the novelty in the algebroid can be tracked back to the deformed Poisson brackets. Furthermore, we present a new formulation of generalized differential geometry that together with the algebroid brackets, enable us to characterize an affine connection, and torsion and curvature tensors on the generalized bundle. The gravity actions are obtained almost as Hilbert-Einstein actions.