Heterotic supersymmetric backgrounds with compact holonomy revisited

We simplify the classification of supersymmetric solutions with compact holonomy of the Killing spinor equations of heterotic supergravity using the field equations and the additional assumption that the 3-form flux is closed. We determine all the fractions of supersymmetry that the solutions preserve and find that there is a restriction on the number of supersymmetries which depends on the isometry group of the background. We examine the geometry of spacetime in all cases. We find that the supersymmetric solutions of heterotic supergravity are associated with a large number of geometric structures which include 7-dimensional manifolds with $G_2$ structure, 6-dimensional complex and almost complex manifolds, and 4-dimensional hyper-K\"ahler, K\"ahler and anti-self-dual Weyl manifolds.