TL;DR
This paper reviews the mathematical and physical understanding of BPS D-branes on Calabi-Yau threefolds, emphasizing the derived category approach, Pi-stability, and the homological mirror symmetry conjecture, with detailed examples including the quintic, flops, and orbifolds.
Contribution
It provides a comprehensive, self-contained guide to the derived category framework and Pi-stability for B-branes, highlighting their relation to A-branes and mirror symmetry.
Findings
Derived category approach is essential for understanding B-branes.
Homological mirror symmetry relates A-branes and B-branes in complex ways.
McKay quivers play a role in D-branes on orbifolds.
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.
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