Dirichlet branes, homological mirror symmetry, and stability

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.

15 pp., amstex using special .sty (included). To appear in the 2002 ICM proceedings

References in corpus (6)

• Derived Categories and Zero-Brane Stability
• D-Brane Stability and Monodromy
• Enhanced D-Brane Categories from String Field Theory
• A Point's Point of View of Stringy Geometry
• Derived Category Automorphisms from Mirror Symmetry
• D-equivalence and K-equivalence

Cited by in corpus (72)

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• The Landau-Ginzburg to Calabi-Yau Dictionary for D-Branes
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• Bridgeland Stability conditions on threefolds I: Bogomolov-Gieseker type inequalities
• The Calabi homomorphism, Lagrangian paths and special Lagrangians
• Symmetries of Lagrangian fibrations
• Quantum hypermultiplet moduli spaces in N=2 string vacua: a review
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• Dynamical systems and categories
• Stability conditions on $A_n$-singularities
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• Derived category of coherent sheaves and counting invariants
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• Birational Calabi-Yau 3-folds and BPS state counting
• Automorphisms and autoequivalences of generic analytic K3 surfaces
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• Stability conditions on curves
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• Stability and Applications
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• Homological mirror symmetry of Fermat polynomials
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• Stability conditions and extremal contractions
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• Phases of Lagrangian-invariant objects in the derived category of an abelian variety
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• Mirror symmetry and K3 surfaces
• A mathematical theory of D-string world-sheet instantons, I: Compactness of the stack of $Z$-semistable Fourier-Mukai transforms from a compact family of nodal curves to a projective Calabi-Yau 3-fold
• Topologies on a triangulated category
• Stability conditions and the braid group
• Gluing stability conditions
• Characteristic classes and stability conditions for projective Kleinian orbisurfaces
• Bridgeland stability conditions on the category of holomorphic triples over curves
• Sheaves of t-structures and valuative criteria for stable complexes
• Degenerations and Lagrangian fibrations of Calabi-Yau manifolds
• Stability Conditions Under the Fourier-Mukai Transforms on Abelian Threefolds
• Mirror stability conditions and SYZ conjecture for Fermat polynomials
• Quintic periods and stability conditions via homological mirror symmetry
• The space of stability conditions for quivers with two vertices
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• From Algebraic Geometry to Machine Learning
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• On Homological and Homotopical Algebra of Supersymmetries and Integrability in String Theory