Kappa symmetry, generalized calibrations and spinorial geometry

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We extend the spinorial geometry techniques developed for the solution of supergravity Killing spinor equations to the kappa symmetry condition for supersymmetric 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.