Extended geometries

Abstract

We present a unified and completely general formulation of extended geometry, characterised by a Kac-Moody algebra and a highest weight coordinate module. Generalised diffeomorphisms are constructed, as well as solutions to the section constraint. Generically, additional ("ancillary") gauge transformations are present, and we give a concrete criterion determining when they appear. A universal form of the (pseudo-)action determines the dynamics in all cases without ancillary transformations, and also for a restricted set of cases based on the adjoint representation of a finite-dimensional simple Lie group. Our construction reproduces (the internal sector of) all previously considered cases of double and exceptional field theories.

On supersymmetric E11 exceptional field theory

“…This section constraint restricts the dependence of all fields and parameters on the extra coordinates such that they can depend at most on n independent coordinates on the neighbourhood of each point. For n ≥ 8, the algebra of generalised diffeomorphism needs to be extended to include not only diffeomorphism preserving the E n structure, but also additional gauge transformations that involve constrained parameters [4,60,61]. The formulation of E n exceptional field theory extends that of gauged maximal supergravity.…”

“…The gauge parameter ξ M of the generalised Lie derivative transforms also in R(Λ 1 ) just as the coordinates z M . Note that additional gauge transformations with constrained gauge parameters are required for the closure of the algebra of generalised Lie derivatives for e n with n ≥ 8 [4,60,61,66]. We shall not check the closure of the algebra of generalised Lie derivatives for e 11 , but we will comment on these additional transformations in Section 3.6.…”

Pre‐NQ Manifolds and Correspondence Spaces: the Nilmanifold Example

“…Studying derived brackets, algebraic conditions as (56) might give a more systematic insight into dualities of such theories (U-duality). Extensions of spacetime which parameterize diffeomorphisms as well as gauge transformations of the corresponding fluxes in M-theory (exceptional field theories [58][59][60][61][62][63][64][65][66][67]) in their generalized diffeomorphisms are an interesting playground to generalize the constructions reviewed in this article. Finally, the derived bracket form of Courant-and C-brackets by Poisson brackets on symplectic graded manifolds suggests that they arise as classical limits of some quantized versions.…”

Supergravity Fluxes and Generalised Geometry

“…These are of course the continuous forms of the relevant T duality and U duality groups that appear in compactifications on a torus. The fibre of the generalised tangent space forms a representation of , whose Dynkin label follows a schematic pattern described in [] (see also [] for further discussion).…”