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D-Brane Stability and Monodromy

Paul S. Aspinwall, Michael R. Douglas

TL;DR

This work develops a stability framework for B-type D-branes on Calabi–Yau manifolds using $\Pi$-stability within the derived category $\mathbf{D}(X)$, highlighting the octahedral axiom as essential for coherent decays and stability across moduli. It applies this machinery to the quintic Calabi–Yau, deriving exact central charges from periods and mapping lines of marginal stability for various branes, including D4-branes and exotic constructions, with detailed monodromy analyses. The paper proves Kontsevich’s conjecture on conifold monodromy as a Fourier–Mukai transform and discusses the broader implications for autoequivalences and the structure of the brane category, including the Gepner-point and orbifold regimes. Finally, it sketches an application to supersymmetry breaking in string compactifications, arguing that global SUSY-breaking requires a global stability landscape across Teichmüller space and identifying obstacles and possible routes via brane configurations. Overall, the work connects derived-category stability, explicit quintic data, and monodromy to physical questions about D-brane spectra and SUSY breaking in string theory.

Abstract

We review the idea of Pi-stability for B-type D-branes on a Calabi-Yau manifold. It is shown that the octahedral axiom from the theory of derived categories is an essential ingredient in the study of stability. Various examples in the context of the quintic Calabi-Yau threefold are studied and we plot the lines of marginal stability in several cases. We derive the conjecture of Kontsevich, Horja and Morrison for the derived category version of monodromy around a "conifold" point. Finally, we propose an application of these ideas to the study of supersymmetry breaking.

D-Brane Stability and Monodromy

TL;DR

This work develops a stability framework for B-type D-branes on Calabi–Yau manifolds using -stability within the derived category , highlighting the octahedral axiom as essential for coherent decays and stability across moduli. It applies this machinery to the quintic Calabi–Yau, deriving exact central charges from periods and mapping lines of marginal stability for various branes, including D4-branes and exotic constructions, with detailed monodromy analyses. The paper proves Kontsevich’s conjecture on conifold monodromy as a Fourier–Mukai transform and discusses the broader implications for autoequivalences and the structure of the brane category, including the Gepner-point and orbifold regimes. Finally, it sketches an application to supersymmetry breaking in string compactifications, arguing that global SUSY-breaking requires a global stability landscape across Teichmüller space and identifying obstacles and possible routes via brane configurations. Overall, the work connects derived-category stability, explicit quintic data, and monodromy to physical questions about D-brane spectra and SUSY breaking in string theory.

Abstract

We review the idea of Pi-stability for B-type D-branes on a Calabi-Yau manifold. It is shown that the octahedral axiom from the theory of derived categories is an essential ingredient in the study of stability. Various examples in the context of the quintic Calabi-Yau threefold are studied and we plot the lines of marginal stability in several cases. We derive the conjecture of Kontsevich, Horja and Morrison for the derived category version of monodromy around a "conifold" point. Finally, we propose an application of these ideas to the study of supersymmetry breaking.

Paper Structure

This paper contains 18 sections, 1 theorem, 47 equations, 8 figures.

Table of Contents

  1. Introduction
  2. $\Pi$-Stability
  3. Topological D-Branes
  4. Cones and Triangles
  5. Physical Branes
  6. Stability and Octahedra
  7. Stable Sheaves and Vector Bundles
  8. Abelian subcategories and the orbifold point
  9. Examples on the Quintic
  10. Basics of the moduli space and periods
  11. The exact computation of ${\varphi}$
  12. $N$ D4-branes
  13. Some exotic D-branes
  14. The peculiar case of ${\Scr X}_2$
  15. Monodromy
  16. ...and 3 more sections

Key Result

Theorem 1

An bundle is stable if and only if it admits an irreducible Hermitian-Yang-Mills connection. This connection is unique.

Figures (8)

  • Figure 1: Competing lines of stability.
  • Figure 2: The octahedron for $C=A+B$ and $B=E+F$.
  • Figure 3: Stability of $N$ D4-branes in the $t$-plane.
  • Figure 4: Stability of the exotic objects ${\Scr X}_N$ in the $t$-plane.
  • Figure 5: Lines of stability for the case of ${\Scr X}_2$.
  • ...and 3 more figures

Theorems & Definitions (1)

  • Theorem 1