The D5-brane effective action and superpotential in N=1 compactifications

TL;DR

This paper derives the effective four-dimensional action for D5-branes in Calabi-Yau orientifolds, including superpotentials and moduli dependence, using geometric and Hodge theory techniques.

Contribution

It provides a detailed computation of the D5-brane superpotential and effective action incorporating open and closed string moduli via geometric blow-up and Picard-Fuchs equations.

Findings

Derived explicit superpotential expressions involving open and closed moduli.

Connected deformation problems to variations of mixed Hodge structures.

Applied the method to a non-compact Calabi-Yau example.

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 Kaehler 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.