We analyze the necessary and sufficient conditions for the preservation of supersymmetry for bosonic geometries of the form $R^{1,9-d}\times M_d,$ in the common Neveu-Schwarz–Neveu-Schwarz (NS-NS) sector of type II string theory and also type I or heterotic string theory. The results are phrased in terms of the intrinsic torsion of G structures and provide a comprehensive classification of static supersymmetric backgrounds in these theories. Generalized calibrations naturally appear since the geometries can always arise as solutions describing NS or type I or heterotic fivebranes wrapping calibrated cycles. Some new solutions are presented. In particular we find $d=6\mathrm{}$ examples with a fibered structure which preserve $\mathcal{N}=1,2,3\mathrm{}$ supersymmetry in type II and include compact type I or heterotic geometries.

• Received 11 June 2003

DOI:https://doi.org/10.1103/PhysRevD.69.086002