Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Special Geometry and Mirror Symmetry for Open String Backgrounds with N=1 Supersymmetry

Wolfgang Lerche

TL;DR

The work extends mirror symmetry to open/closed Type II backgrounds with D-branes and fluxes by formulating a relative cohomology framework that unifies flux- and brane-induced superpotentials as relative-period data. It develops a flat open/closed moduli space and a relative Picard–Fuchs system, enabling exact, non-perturbative computations of $N=1$ superpotentials via chain and period integrals. A concrete non-compact example demonstrates the method and reveals how disk instantons and boundary conditions shape the brane sector, including framing and phase dependencies. This approach provides a principled way to study the quantum geometry of D-branes and their interplay with bulk fluxes, with potential connections to large-N transitions and fourfold dualities.

Abstract

We review an approach for computing non-perturbative, exact superpotentials for Type II strings compactified on Calabi-Yau manifolds, with extra fluxes and D-branes on top. The method is based on an open string generalization of mirror symmetry, and takes care of the relevant sphere and disk instanton contributions. We formulate a framework based on relative (co)homology that uniformly treats the flux and brane sectors on a similar footing. However, one important difference is that the brane induced potentials are of much larger functional diversity than the flux induced ones, which have a hidden N=2 structure and depend only on the bulk geometry. This introductory lecture is meant for an audience unfamiliar with mirror symmetry.

Special Geometry and Mirror Symmetry for Open String Backgrounds with N=1 Supersymmetry

TL;DR

The work extends mirror symmetry to open/closed Type II backgrounds with D-branes and fluxes by formulating a relative cohomology framework that unifies flux- and brane-induced superpotentials as relative-period data. It develops a flat open/closed moduli space and a relative Picard–Fuchs system, enabling exact, non-perturbative computations of superpotentials via chain and period integrals. A concrete non-compact example demonstrates the method and reveals how disk instantons and boundary conditions shape the brane sector, including framing and phase dependencies. This approach provides a principled way to study the quantum geometry of D-branes and their interplay with bulk fluxes, with potential connections to large-N transitions and fourfold dualities.

Abstract

We review an approach for computing non-perturbative, exact superpotentials for Type II strings compactified on Calabi-Yau manifolds, with extra fluxes and D-branes on top. The method is based on an open string generalization of mirror symmetry, and takes care of the relevant sphere and disk instanton contributions. We formulate a framework based on relative (co)homology that uniformly treats the flux and brane sectors on a similar footing. However, one important difference is that the brane induced potentials are of much larger functional diversity than the flux induced ones, which have a hidden N=2 structure and depend only on the bulk geometry. This introductory lecture is meant for an audience unfamiliar with mirror symmetry.

Paper Structure

This paper contains 12 sections, 54 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Recapitulation of $N=2$ "closed string" mirror symmetry
  3. Type II strings on Calabi-Yau manifolds
  4. The Special Geometry of the N=2 vector-multiplet moduli space
  5. Open/closed string backgrounds with $N=1$ supersymmetry
  6. Adding background fluxes
  7. Adding $D$-branes
  8. Mirror symmetry of $D$-brane configurations
  9. Relative homology
  10. The Special Geometry underlying the $N=1$ superpotential
  11. An explicit example
  12. Conclusions

Figures (2)

  • Figure 1: Sketch of the mirror pair of the $D$-brane configurations we consider. In the $A$-model on the left side, disk and $S^1$ instantons deform the quantum geometry of the SL 3-cycle $\gamma^{(3)}$, around which a $D6$-brane is partially wrapped. On the other hand, for the $B$-model on the right side the brane geometry remains uncorrected. Note that this is a simplified picture, in that for the concrete physical models under consideration, the Calabi-Yau manifolds and the relevant cycles are non-compact.
  • Figure 2: There are many viewpoints from which one can tackle string vacua with $N=1$ supersymmetry, each of which has its own scope, merits and limitations. What we have covered in these lectures is a small neighborhood of "$N=1$ Special Geometry".