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

Abstract

We study dg-manifolds which are R[2]-bundles over R[1]-bundles over manifolds, we calculate its symmetries, its derived symmetries and we introduce the concept of T-dual dg-manifolds. Within this framework, we construct the T-duality map as a degree -1 map between the cohomologies of the T-dual dg-manifolds and we show an explicit isomorphism between the differential graded algebra of the symmetries of the T-dual dg-manifolds. We, furthermore, show how the algebraic structure underlying Bn generalized geometry could be recovered as derived dg-Leibniz algebra of the fixed points of the T-dual automorphism acting on the symmetries of a self T-dual dgmanifold, and we show how other types of algebraic structures underlying exceptional generalized geometry could be obtained as derived symmetries of certain dg-manifolds.2010 Mathematics Subject Classification. 53D18 53D17 (primary), 55R65, 57R22 (secondary).

Brane Wess–Zumino terms from AKSZ and exceptional generalised geometry as an $L_\infty$-algebroid

“…Tensor hierarchies (different from the EFT one considered here) have been connected to L ∞ -algebras before, by at least two different groups [43,44]. There have been a number of papers proposing derived bracket structures for DFT [21,45,46], two for EGG [17] [16] and one for EFT [19]; in the last two one also finds the EGG/EFT L ∞ -algebra structure respectively in the M-theory case (although none exhibit the L ∞ -algebroid or dg-symplectic structure which is the main point of this paper). On topological sigma models in the context of extended geometries we mention the works [46][47][48] in which generalisations of the Courant sigma model are used to study non-geometry in string theory (which strongly suggest appropriate generalisations of the theories developed in this paper would be relevant for non-geometry in M-theory), and also [49] wherein WZ terms for various branes (including the M5) are found from L ∞algebraic generalisations of the super-Poincaré algebra (in a manner reminiscent of [50] and references therein).…”

“…Algebraically speaking the Dorfman bracket is thus not a Lie algebra bracket. It has been known for a while in the mathematical literature [16,17] that the algebraic structure of the exceptional Dorfman bracket is instead that of an L ∞ -algebra [18], and while this paper was being written the L ∞ -algebra was first related to the "tensor hierarchy" of EFT in the physics literature [19]. (For the Courant bracket, relevant for the usual O(d, d) generalised geometry, the L ∞ structure has been known for decades [20], see also [21,22].…”

Dissections and automorphisms of regular Courant algebroids

“…In this last section we detail three examples to illustrate the results obtained in the two previous sections. For an approach based on dg-manifolds, see [Uri13] and [LCU14]. ).…”