Supersymmetric backgrounds from generalized Calabi‐Yau manifolds

We show that the supersymmetry transformations for type II string theories onsix‐manifolds can be written as differential conditions on a pair of purespinors, the exponentiated Kähler form e iJ and the holomorphic formO. The equations are explicitly symmetric under exchange of thetwo pure spinors and a choice of even or odd‐rank RR field.This is mirror symmetry formanifolds with torsion. Moreover, RR fluxes affect only one of the twoequations: e iJ is closed under the action of the twisted exteriorderivative in IIA theory, and similarly O is closed in IIB. This means thatsupersymmetric SU(3)‐structure manifolds are always complex in IIB while they are twisted symplecticin IIA. Modulo adifferent action of the B ‐field, these are all generalized Calabi‐Yau manifolds, asdefined by Hitchin.