D-brane networks in flux vacua, generalized cycles and calibrations

We consider chains of genralized submanifolds, as defined by Gualtieri in the context of generalized complex geometry, and define a boundary operator that acts on them. This allows us to define generalized cycles and the corresponding homology theory. Gauge invariance demands that D-brane networks on flux acua must wrap these generalized cycles, while deformations of genralized cycles inside of a certain homology class describe physical processes such as the dissolution of D-branes in higher-dimensional D-branes and MMS-like instantonic transitions. We introduce calibrations that identify the supersymmetric D-brane networks, which minimize their energy inside of the corresponding homology class of generalized cycles. Such a calibration is explicitly presented for type II N=1 flux compactifications to foru dimensions. In particualr networks of walls and strings in compactifications on warped Calabi-Yasu's are treated, wich explicit examples on a toroidal orientifold vacuum and on the Klebanov-Strassler gemoertry.