L∞ algebras and tensor hierarchies in Exceptional Field Theory and Gauged Supergravity

Abstract

We show how the gauge and field structure of the tensor hierarchies in Double and E 7(7)Exceptional Field Theory fits into L ∞ algebras. Special attention is paid to redefinitions, the role of covariantly constrained fields and intertwiners. The results are connected to Gauged Supergravities through generalized Scherk-Schwarz reductions. We find that certain gaugingdependent parameters generate trivial gauge transformations, giving rise to novel symmetries for symmetries that are absent in their ungauged counterparts.

Fluxes in exceptional field theory and threebrane sigma-models

“…The same expressions are also interpreted as the conditions for gauge invariance and closure of gauge transformations for a topological threebrane sigmamodel, where the fluxes appear as generalized Wess-Zumino terms. It would be interesting to understand better the geometric features of the algebroid structure defined by our SL(5) projection of the Lie algebroid up to homotopy on T M ⊕ 2 T * M. Given that the higher Courant algebroid structure on this extended bundle has a well-known realization as an L ∞ -algebra, see for example [18,19,59], it is an interesting problem to see how our specific Lie algebroid up to homotopy fits into recent discussions of the L ∞ -algebra structure of gauge symmetries underlying the tensor hierarchy in exceptional field theory, see for instance [60][61][62].…”

Tensor hierarchies and Leibniz algebras

“…Given that in supergravity models, the Bianchi identities induce a L ∞ -algebra structure on the (shifted) tensor hierarchy [17], it would seem natural to understand how one passes from the tensor hierarchy algebra structure on T = h ⊕ T ′ to a L ∞ -algebra on T ′ [−1]. This is all the more important since L ∞ algebras have recently drawn much interests in supergravity theories [3,5,[13][14][15]. This topic is indeed important because these L ∞ structures encode the field strengths of the theory and their corresponding Bianchi identities.…”

“…In the last few years, a renewal of interest in L ∞ -algebras has soared in the supergravity community, in relation with the formalism of tensor hierarchies [3,5,[13][14][15]17]. These L ∞algebras would in some sense encode the field strength of the model.…”

The L∞-algebra of the S-matrix

“…The recent resurgence of physics interest in L ∞ -algebras (e.g. [27][28][29][30][31][32][33][34][35][36][37][38]) mostly centres on gauge symmetries of classical theories. (Most relevant here is [28], which articulates the lore that classical BV master actions have canonical associated L ∞ -algebras [16,[39][40][41][42][43][44][45][46][47][48][49][50]).…”