Abstract. This work revisits the notions of connection and curvature in generalized geometry, with emphasis on torsion-free generalized connections on a transitive Courant algebroid. As an application, we provide a mathematical derivation of the equations of motion of heterotic supergravity in terms of the Ricci tensor of a generalized metric, inspired by the work of Coimbra, Strickland-Constable and Waldram.

We study equations on a principal bundle over a compact complex manifold coupling a connection on the bundle with a Kähler structure on the base. These equations generalize the conditions of constant scalar curvature for a Kähler metric and Hermite-Yang-Mills for a connection. We provide a moment map interpretation of the equations and study obstructions for the existence of solutions, generalizing the Futaki invariant, the Mabuchi K-energy and geodesic stability. We finish by giving some examples of solutions. = 0, where Ω p,q J (ad E) denotes the space of (ad E)-valued smooth (p, q)-forms with respect to J and F 0,2 A is the projection of F A into Ω 0,2 J (ad E). This space is in bijection with the space of holomorphic structures on the principal G c -bundle E c over the complex manifold (X, J) (see [52]). Definition 1.1. A connection A ∈ A 1,1 J is called Hermitian-Yang-Mills if it satisfies the Hermitian-Yang-Mills equation ΛF A = z.(1.7)Remark 1.2. The element z ∈ z in the right-hand side of (1.7) is determined by the cohomology class Ω := [ω] ∈ H 2 (X) and the topology of the principal bundle E. This follows after applying (z j , ·) to (1.7), for an orthonormal basis {z j } of z ⊂ g, and then integrating over X, we obtain z = j

Abstract. We construct the space of infinitesimal variations for the Strominger system and an obstruction space to integrability, using elliptic operator theory. We initiate the study of the geometry of the moduli space, describing the infinitesimal structure of a natural foliation on this space. The associated leaves are related to generalized geometry and correspond to moduli spaces of solutions of suitable Killing spinor equations on a Courant algebroid. As an application, we propose a unifying framework for metrics with holonomy SU(3) and solutions of the Strominger system.