TT-duality and exceptional generalized geometry through symmetries of dg-manifolds

We study dg-manifolds which are R[2]R[2]-bundles over R[1]R[1]-bundles over manifolds, we calculate its symmetries, its derived symmetries and we introduce the concept of TT-dual dg-manifolds. Within this framework, we construct the TT-duality map as a degree −1 map between the cohomologies of the TT-dual dg-manifolds and we show an explicit isomorphism between the differential graded algebra of the symmetries of the TT-dual dg-manifolds. We, furthermore, show how the algebraic structure underlying BnBn generalized geometry could be recovered as derived dg-Leibniz algebra of the fixed points of the TT-dual automorphism acting on the symmetries of a self TT-dual dg-manifold, and we show how other types of algebraic structures underlying exceptional generalized geometry could be obtained as derived symmetries of certain dg-manifolds.