The N=1N=1 effective actions of D-branes in Type IIA and IIB orientifolds

We discuss the four-dimensional N=1N=1 effective actions of single space–time filling DpDp-branes in general Type IIA and Type IIB Calabi–Yau orientifold compactifications. The effective actions depend on an infinite number of normal deformations and gauge connection modes. For D6-branes the N=1N=1 Kähler potential, the gauge-coupling function, the superpotential and the D-terms are determined as functions of these fields. They can be expressed as integrals over chains which end on the D-brane cycle and a reference cycle. The infinite deformation space will reduce to a finite dimensional moduli space of special Lagrangian submanifolds upon imposing F- and D-term supersymmetry conditions. We show that the Type IIA moduli space geometry is captured by three real functionals encoding the deformations of special Lagrangian submanifolds, holomorphic three-forms and Kähler two-forms of Calabi–Yau manifolds. These elegantly combine in the N=1N=1 Kähler potential, which reduces after applying mirror symmetry to the results previously determined for space–time filling D3-, D5- and D7-branes. We also propose general chain integral expressions for the Kähler potentials of Type IIB D-branes.