Leibniz algebroids, twistings and exceptional generalized geometry

Abstract

We investigate a class of Leibniz algebroids which are invariant under diffeomorphisms and symmetries involving collections of closed forms. Under appropriate assumptions we arrive at a classification which in particular gives a construction starting from graded Lie algebras. In this case the Leibniz bracket is a derived bracket and there are higher derived brackets resulting in an $L_\infty$-structure. The algebroids can be twisted by a non-abelian cohomology class and we prove that the twisting class is described by a Maurer-Cartan equation. For compact manifolds we construct a Kuranishi moduli space of this equation which is shown to be affine algebraic. We explain how these results are related to exceptional generalized geometry.Comment: 58 page

Generalised geometry for string corrections

“…The resulting extended generalised tangent space with the corresponding generalised Lie derivative is known as a "transitive Courant algebroid". That such algebroids can be constructed by reduction was first observed by Severa [39] (see also [38] for a discussion) and first discussed in the generalised geometric context in [35,36] and [37]. It was specifically applied to the heterotic theory in [30,31].…”

“…4 More generally one can reduce on a general group manifold and obtain the non-Abelian versions of (1.1). A closely related notion, B n generalised geometry on a bundle T M ⊕ R ⊕ T * M is discussed in [35][36][37]. Incorporation of the non-Abelian degrees of freedom requires a new extension of the generalised tangent bundle.…”

“…We find a generalised geometry whose Bianchi identity is sourced by the first Pontryagin class of two U(1) bundles. Similarly, one can also discuss the Bianchi identity with only one U(1) bundle, giving rise to B n generalised geometry on a bundle T M ⊕ R ⊕ T * M [35][36][37]. In section 3.1 we will discuss non-Abelian gauge groups in generalised geometry by generalizing the bundle construction employed here to the non-Abelian case.…”

Spheres, Generalised Parallelisability and Consistent Truncations

“…However, this is only one of family of possible generalised geometries where one considers structures on different generalised tangent spaces [19][20][21][22]. These capture the bosonic degrees of freedom of the bosonic fields of other supergravity theories, in particular those of type II and eleven-dimensional supergravity.…”

Extended geometries

“…However, the classifications in refs. [50,51] rely on the absence of such transformations, so it is relevant to reconsider the general setting.…”