On the supersymmetric solutions of the Heterotic Superstring effective action

We consider the effective action of the Heterotic Superstring to first order in alpha ' and derive the necessary and sufficient conditions that a field configuration has to satisfy in order to admit at least one Killing spinor using the spinor bilinear method in an arbitrary spinorial basis and corresponding arbitrary gamma matrices. As a previous step in this derivation, we compute the complete spinor bilinear algebra using the Fierz identities, obtaining as a by-product the algebra satisfied by the Spin(7) structure contained in the bilinears in an arbitrary basis. We find the off-shell relations existing between the bosonic equations of motion evaluated on supersymmetric field configurations using the Killing Spinor Identities instead of the (far more complicated) integrability conditions of the Killing Spinor Equations as it is common in the literature. We show how to include the Kalb-Ramond's Bianchi identity in the Killing Spinor Identities.