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

In this paper, we use reduction by extended actions to give a construction of transitive Courant algebroids from string classes. We prove that T-duality commutes with the reductions and thereby determine global conditions for the existence of T-duals in heterotic string theory. In particular we find that T-duality exchanges string structures and gives an isomorphism of transitive Courant algebroids. Consequently we derive the T-duality transformation for generalised metrics and show that the heterotic Einstein equations are preserved. The presence of string structures significantly extends the domain of applicability of T-duality and this is illustrated by several classes of examples.

Abstract. We introduce a new approach for computing the monodromy of the Hitchin map and use this to completely determine the monodromy for the moduli spaces of L-twisted G-Higgs bundles, for the groups G = GL(2, C), SL(2, C) and P SL(2, C). We also determine the twisted Chern class of the regular locus, which obstructs the existence of a section of the moduli space of L-twisted Higgs bundles of rank 2 and degree deg(L) + 1. By counting orbits of the monodromy action with Z 2 -coefficients, we obtain in a unified manner the number of components of the character varieties for the real groups G = GL(2, R), SL(2, R), P GL(2, R), P SL(2, R), as well as the number of components of the Sp(4, R)-character variety with maximal Toledo invariant. We also use our results for GL(2, R) to compute the monodromy of the SO(2, 2) Hitchin map and determine the components of the SO(2, 2) character variety.

Abstract. We construct triples of commuting real structures on the moduli space of Higgs bundles, whose fixed loci are branes of type (B, A, A), (A, B, A) and (A, A, B). We study the real points through the associated spectral data and describe the topological invariants involved using KO, KR and equivariant K-theory.