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

Thomas W. Grimm, Tae-Won Ha, Albrecht Klemm, Denis Klevers

TL;DR

This work computes the four-dimensional $ il$ effective action for D5-branes in generic Calabi–Yau orientifolds via KK reduction of the bulk and brane actions, yielding explicit $ il$ data including a brane-corrected Kähler potential and a superpotential that combines flux and brane contributions. The brane superpotential, expressed as $W= frac1{ } obreak oldsymbol{∫}_Y F_3\,Ω+oldsymbol μ_5 obreak obreak\int_{oldsymbol Σ_+} zeta eg Ω$, depends on both open and closed moduli through relative periods; to analyze this dependence, the authors blow up the brane curve along $oldsymbol Σ$ to a rigid divisor $D$ in a blown-up manifold $ ilde Y$ and study the variation of mixed Hodge structure via Picard–Fuchs equations. They develop two complementary approaches to PF equations, provide explicit formulas for the $ il$ coordinates, Kähler potential, and gauge-kinetic functions, and illustrate the blow-up construction with a non-compact Calabi–Yau example over del Pezzo surfaces. The results lay groundwork for open–closed moduli stabilization and open-string phenomenology in compact Type IIB setups, offering a framework to compute the full $ il$ potential and its dependence on brane moduli.

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.

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

TL;DR

This work computes the four-dimensional effective action for D5-branes in generic Calabi–Yau orientifolds via KK reduction of the bulk and brane actions, yielding explicit data including a brane-corrected Kähler potential and a superpotential that combines flux and brane contributions. The brane superpotential, expressed as , depends on both open and closed moduli through relative periods; to analyze this dependence, the authors blow up the brane curve along to a rigid divisor in a blown-up manifold and study the variation of mixed Hodge structure via Picard–Fuchs equations. They develop two complementary approaches to PF equations, provide explicit formulas for the coordinates, Kähler potential, and gauge-kinetic functions, and illustrate the blow-up construction with a non-compact Calabi–Yau example over del Pezzo surfaces. The results lay groundwork for open–closed moduli stabilization and open-string phenomenology in compact Type IIB setups, offering a framework to compute the full potential and its dependence on brane moduli.

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.

Paper Structure

This paper contains 26 sections, 162 equations, 1 figure, 2 tables.

Table of Contents

  1. Introduction
  2. The D5-brane action
  3. The four-dimensional spectrum
  4. The closed string spectrum
  5. The open string spectrum
  6. Special relations on the $\mathcal{N}=1$ moduli space
  7. Reduction of the D5-brane action
  8. Dirac-Born-Infeld action and tadpole cancellation
  9. Chern-Simons action
  10. The scalar potential
  11. The $\mathcal{N}=1$ characteristic data
  12. The Kähler potential and $\mathcal{N}=1$ coordinates
  13. The superpotential
  14. The gauge-kinetic function, gaugings and D-term potential
  15. The structure of the $\mathcal{N}=1$ open-closed field space
  16. ...and 11 more sections

Figures (1)

  • Figure 1: Polyhedron for $\mathcal{B}_3$.