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D-Branes on Calabi-Yau Manifolds

Paul S. Aspinwall

TL;DR

The work presents a comprehensive bridge between D-brane physics on Calabi–Yau threefolds and advanced algebraic geometry. It demonstrates that B-branes are captured by the derived category of coherent sheaves, with Pi-stability encoding physical stability and open-string spectra described by Ext groups. The text develops the Fukaya category for A-branes, explains the need for a triangulated, derived-category perspective to realize homological mirror symmetry, and applies the framework to Quintic, flops, and orbifolds, uncovering rich stability structures and monodromies. This synthesis provides a powerful, mathematically guided lens for understanding D-brane dynamics and mirror symmetry in string theory, with central tools including periods, Picard–Fuchs equations, and Fourier–Mukai transforms.

Abstract

In this review we study BPS D-branes on Calabi-Yau threefolds. Such D-branes naturally divide into two sets called A-branes and B-branes which are most easily understood from topological field theory. The main aim of this paper is to provide a self-contained guide to the derived category approach to B-branes and the idea of Pi-stability. We argue that this mathematical machinery is hard to avoid for a proper understanding of B-branes. A-branes and B-branes are related in a very complicated and interesting way which ties in with the ``homological mirror symmetry'' conjecture of Kontsevich. We motivate and exploit this form of mirror symmetry. The examples of the quintic 3-fold, flops and orbifolds are discussed at some length. In the latter case we describe the role of McKay quivers in the context of D-branes. These notes are to be submitted to the proceedings of TASI03.

D-Branes on Calabi-Yau Manifolds

TL;DR

The work presents a comprehensive bridge between D-brane physics on Calabi–Yau threefolds and advanced algebraic geometry. It demonstrates that B-branes are captured by the derived category of coherent sheaves, with Pi-stability encoding physical stability and open-string spectra described by Ext groups. The text develops the Fukaya category for A-branes, explains the need for a triangulated, derived-category perspective to realize homological mirror symmetry, and applies the framework to Quintic, flops, and orbifolds, uncovering rich stability structures and monodromies. This synthesis provides a powerful, mathematically guided lens for understanding D-brane dynamics and mirror symmetry in string theory, with central tools including periods, Picard–Fuchs equations, and Fourier–Mukai transforms.

Abstract

In this review we study BPS D-branes on Calabi-Yau threefolds. Such D-branes naturally divide into two sets called A-branes and B-branes which are most easily understood from topological field theory. The main aim of this paper is to provide a self-contained guide to the derived category approach to B-branes and the idea of Pi-stability. We argue that this mathematical machinery is hard to avoid for a proper understanding of B-branes. A-branes and B-branes are related in a very complicated and interesting way which ties in with the ``homological mirror symmetry'' conjecture of Kontsevich. We motivate and exploit this form of mirror symmetry. The examples of the quintic 3-fold, flops and orbifolds are discussed at some length. In the latter case we describe the role of McKay quivers in the context of D-branes. These notes are to be submitted to the proceedings of TASI03.

Paper Structure

This paper contains 62 sections, 5 theorems, 216 equations, 14 figures, 2 tables.

Table of Contents

  1. Introduction
  2. Worldsheet Models of Closed Strings
  3. The $N=(2,2)$ non-linear $\sigma$-model
  4. The A model
  5. The B model
  6. Mirror Symmetry
  7. Boundaries
  8. The A-model
  9. A-branes
  10. Open strings for one A-brane
  11. Open strings for many A-branes
  12. The B-model
  13. B-branes
  14. Open strings for B-branes
  15. A failure of mirror symmetry
  16. ...and 47 more sections

Key Result

Theorem 1

Given any maps $f:A\to B$, $g:A\to I$ in an abelian category with I an injective object, a map $g'$ can be constructed to make the following commutative: \xymatrix{ A\ar[r]^f\ar[d]^g&B\ar@{-->}[dl]^{g'}\\ I& }so long as $g\ker(f)=0$.

Figures (14)

  • Figure 1: Five fundamental regions for the moduli space of the quintic.
  • Figure 2: Loops which do and do not give an anomaly.
  • Figure 3: Composition of morphisms.
  • Figure 4: Instanton Tunneling.
  • Figure 5: Disk instanton associated to three-point functions.
  • ...and 9 more figures

Theorems & Definitions (15)

  • Definition 1
  • Definition 2
  • Definition 3
  • Definition 4
  • Definition 5
  • Definition 6
  • Definition 7
  • Definition 8
  • Definition 9
  • Theorem 1
  • ...and 5 more