The D5-brane effective action and superpotential in [formula omitted] compactifications

Abstract

The four-dimensional effective action for D5-branes in generic compact Calabi–Yau orientifolds is computed by performing a Kaluza–Klein reduction. The N = 1 Kähler potential, the superpotential, the gauge–kinetic coupling function and the D-terms are derived in terms of the geometric data of the internal space and of the two-cycle wrapped by the D5-brane. In particular, we obtain the D5-brane and flux superpotential by integrating out four-dimensional three-forms which couple via the Chern–Simons action. Also the infinitesimal complex structure deformations of the two-cycle induced by the deformations of the ambient space contribute to the F-terms. The superpotential can be expressed in terms of relative periods depending on both the open and closed moduli. To analyze this dependence we blow up along the two-cycle and obtain a rigid divisor in an auxiliary compact threefold with negative first Chern class. The variation of the mixed Hodge structure on this blown-up geometry is equivalent to the original deformation problem and can be analyzed by Picard–Fuchs equations. We exemplify the blow-up procedure for a non-compact Calabi–Yau threefold given by the canonical bundle over del Pezzo surfaces.