Leibniz algebroids, twistings and exceptional generalized geometry

TL;DR

This paper classifies Leibniz algebroids invariant under diffeomorphisms and symmetries, explores their derived bracket structures, and connects them to exceptional generalized geometry through twistings and moduli spaces.

Contribution

It provides a classification of Leibniz algebroids using graded Lie algebras, introduces a non-abelian twisting framework, and relates these structures to exceptional generalized geometry.

Findings

Leibniz algebroids can be constructed from graded Lie algebras.

Twisting by non-abelian cohomology is governed by a Maurer-Cartan equation.

A Kuranishi moduli space for the twisting class is constructed and shown to be affine algebraic.

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.