Dirichlet branes, homological mirror symmetry, and stability

TL;DR

This paper explores the mathematical structures arising from Dirichlet branes in superstring theory, linking stability conditions, homological mirror symmetry, and derived categories in the context of Calabi-Yau compactifications.

Contribution

It introduces a notion of stability for objects in derived categories, connecting physical concepts with mathematical conjectures like Kontsevich's homological mirror symmetry.

Findings

Formulated stability conditions for derived category objects.

Connected stability with homological mirror symmetry.

Provided physics-inspired proofs for conjectures.

Abstract

We discuss some mathematical conjectures which have come out of the Dirichlet branes in superstring theory, focusing on the case of supersymmetric branes in Calabi-Yau compactification. This has led to the formulation of a notion of stability for objects in a derived category, contact with Kontsevich's homological mirror symmetry conjecture, and "physics proofs" for many of the subsequent conjectures based on it, such as the representation of Calabi-Yau monodromy by autoequivalences of the derived category.