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The effective action of D7-branes in N=1 Calabi-Yau orientifolds

Hans Jockers, Jan Louis

TL;DR

The authors compute the four-dimensional $ ext{N}=1$ effective action for a D7-brane wrapped on a four-cycle in a generic Calabi–Yau orientifold by KK reducing the DBI and CS actions. They derive the bulk-brane couplings, identify the holomorphic gauge kinetic functions, and show how a Green–Schwarz mechanism induces a D-term, all within a framework that avoids fixing a particular orientifold. A careful choice of chiral coordinates reveals a Kähler potential that encapsulates brane fluctuations, Wilson lines, and bulk moduli, with W=0 at this order. The geometry of the combined moduli space is clarified using relative cohomology and a generalized variation of Hodge structure, connecting to $ ext{N}=1$ special geometry and providing a platform for exploring moduli stabilization and SUSY breaking in brane-influenced setups. The results offer a foundation for incorporating nonperturbative effects (e.g., gaugino condensation, Euclidean D3-instantons) and for extending to multiple D7-branes, with implications for de Sitter constructions and phenomenological model building in string theory.

Abstract

Using a Kaluza-Klein reduction of the Dirac-Born-Infeld and Chern-Simons action we compute the four dimensional N=1 effective action for the massless modes of a D7-brane which is wrapped on a four-cycle of a compact Calabi-Yau orientifold. We do not consider a specific orientifold but instead determine the Kahler potential, the gauge kinetic functions and the scalar potential in terms of geometrical data of a generic orientifold and its wrapped four-cycle. In particular we derive the couplings of the D-brane excitations to the bulk moduli of the orientifold as they are important for the study of soft supersymmetry breaking terms. We relate the resulting Kahler geometry to the N=1 special geometry of Lerche, Mayr and Warner. Finally we comment on the structure of the D-term which is induced by a Green-Schwarz term in the Chern-Simons action.

The effective action of D7-branes in N=1 Calabi-Yau orientifolds

TL;DR

The authors compute the four-dimensional effective action for a D7-brane wrapped on a four-cycle in a generic Calabi–Yau orientifold by KK reducing the DBI and CS actions. They derive the bulk-brane couplings, identify the holomorphic gauge kinetic functions, and show how a Green–Schwarz mechanism induces a D-term, all within a framework that avoids fixing a particular orientifold. A careful choice of chiral coordinates reveals a Kähler potential that encapsulates brane fluctuations, Wilson lines, and bulk moduli, with W=0 at this order. The geometry of the combined moduli space is clarified using relative cohomology and a generalized variation of Hodge structure, connecting to special geometry and providing a platform for exploring moduli stabilization and SUSY breaking in brane-influenced setups. The results offer a foundation for incorporating nonperturbative effects (e.g., gaugino condensation, Euclidean D3-instantons) and for extending to multiple D7-branes, with implications for de Sitter constructions and phenomenological model building in string theory.

Abstract

Using a Kaluza-Klein reduction of the Dirac-Born-Infeld and Chern-Simons action we compute the four dimensional N=1 effective action for the massless modes of a D7-brane which is wrapped on a four-cycle of a compact Calabi-Yau orientifold. We do not consider a specific orientifold but instead determine the Kahler potential, the gauge kinetic functions and the scalar potential in terms of geometrical data of a generic orientifold and its wrapped four-cycle. In particular we derive the couplings of the D-brane excitations to the bulk moduli of the orientifold as they are important for the study of soft supersymmetry breaking terms. We relate the resulting Kahler geometry to the N=1 special geometry of Lerche, Mayr and Warner. Finally we comment on the structure of the D-term which is induced by a Green-Schwarz term in the Chern-Simons action.

Paper Structure

This paper contains 21 sections, 124 equations, 1 figure, 5 tables.

Table of Contents

  1. Introduction
  2. Type IIB compactified on Calabi-Yau orientifolds
  3. Type IIB orientifold spectrum with O3/O7 planes
  4. Democratic low energy effective bulk action
  5. D7-brane: Spectrum and effective action
  6. D7-brane spectrum
  7. Dirac-Born-Infeld action and brane fluxes
  8. Supersymmetry and calibration condition
  9. Reduction of the D7-brane action
  10. Bulk and D7-brane: Effective theory
  11. Branes and orientifolds: Tadpole cancellation
  12. Bulk and brane effective action
  13. $\mathcal{N}=1$ effective action in chiral coordinates
  14. Geometry of the $\mathcal{N}=1$ moduli space
  15. Relative cohomology and relative homology
  16. ...and 6 more sections

Figures (1)

  • Figure 5.1: Variation of Hodge structure of $H^3_-(Y,S^\Lambda)$.