Kappa symmetry , generalized calibrations and spinorial geometry

We extend the spinorial geometry techniques developed for the solution of su-pergravity Killing spinor equations to the kappa symmetry condition for supersym-metric brane probe configurations in any supergravity background. In particular, we construct the linear systems associated with the kappa symmetry projector of M-and type II branes acting on any Killing spinor. As an example, we show that static supersymmetric M2-brane configurations which admit a Killing spinor representing the SU (5) orbit of Spin(10, 1) are generalized almost hermitian calibrations and the embedding map is pseudo-holomorphic. We also present a bound for the Euclidean action of M-and type II branes embedded in a supersymmetric background with non-vanishing fluxes. This leads to an extension of the definition of generalized calibrations which allows for the presence of non-trivial Born-Infeld type of fields in the brane actions.