Supersymmetric D-branes and calibrations on general Script N = 1 backgrounds

We study the conditions to have supersymmetric D-branes on general {\cal N}=1backgrounds with Ramond-Ramond fluxes. These conditions can be written in termsof the two pure spinors associated to the SU(3)\times SU(3) structure onT_M\oplus T^\star_M, and can be split into two parts each involving a differentpure spinor. The first involves the integrable pure spinor and requires theD-brane to wrap a generalised complex submanifold with respect to thegeneralised complex structure associated to it. The second contains thenon-integrable pure spinor and is related to the stability of the brane. Thetwo conditions can be rephrased as a generalised calibration condition for thebrane. The results preserve the generalised mirror symmetry relating the typeIIA and IIB backgrounds considered, giving further evidence for this duality.Comment: 23 pages. Some improvements and clarifications, typos corrected and references added. v3: Version published in JHE