Supersymmetric heterotic string backgrounds

We present the main features of the solution of the gravitino and dilatino Killing spinor equations derived in hep-th/0510176 and hep-th/0703143 which have led to the classification of geometric types of all type I backgrounds. We then apply these results to the supersymmetric backgrounds of the heterotic string. In particular, we solve the gaugino Killing spinor equation together with the other two Killing spinor equations of the theory. We also use our results to classify all supersymmetry conditions of ten-dimensional gauge theory. The effective theory of the heterotic string can be described by a type I supergravity which includes higher curvature corrections that are organized in an α and gs expansion. The α corrections can be computed by a sigma model loop calculation and modify the field equations of the theory. In addition, the anomaly cancelation mechanism modifies the Bianchi identity of the three-form field strength. Up to and including two-loops in the sigma model computation, the Killing spinor equations of the effective theory [1, 2, 3] can be written as Dǫ = ∇̂ǫ+O(α) = 0 , Aǫ = (dΦ− 1 2 H)ǫ+O(α) = 0 , Fǫ = Fǫ+O(α) = 0 , (1) where we have suppressed the spacetime indices and the gamma matrix dependence. The first equation is the gravitino Killing spinor equation for a metric connection ∇̂ with torsion the three-form field strength H . The second equation is the dilatino Killing spinor equation, where Φ is the dilaton, and the last is the gaugino Killing spinor equation, where F is the gauge field strength. It is clear that these Killing spinor equations are as those expected from type I supergravity to the order indicated. The only difference is that H is not closed. In particular it is modified at one-loop due to the Green-Schwarz anomaly cancellation mechanism as