A-branes and Noncommutative Geometry

Abstract

We argue that for a certain class of symplectic manifolds the category of A-branes (which includes the Fukaya category as a full subcategory) is equivalent to a noncommutative deformation of the category of B-branes (which is equivalent to the derived category of coherent sheaves) on the same manifold. This equivalence is different from Mirror Symmetry and arises from the Seiberg-Witten transform which relates gauge theories on commutative and noncommutative spaces. More generally, we argue that for certain generalized complex manifolds the category of generalized complex branes is equivalent to a noncommutative deformation of the derived category of coherent sheaves on the same manifold. We perform a simple test of our proposal in the case when the manifold in question is a symplectic torus.

15 pages, latex

References in corpus (13)

• Deformation quantization of Poisson manifolds, I
• Remarks on A-branes, Mirror Symmetry, and the Fukaya category
• Topological strings on noncommutative manifolds
• Categories of holomorphic vector bundles on noncommutative two-tori
• Topological sigma-models with H-flux and twisted generalized complex manifolds
• Mirror Symmetry and Generalized Complex Manifolds
• Generalized complex geometry, generalized branes and the Hitchin sigma model
• Generalized Kahler geometry and manifest N=(2,2) supersymmetric nonlinear sigma-models
• Mirror symmetry for topological sigma models with generalized Kahler geometry
• Coisotropic Branes, Noncommutativity, and the Mirror Correspondence
• Generalized complex geometry and the Poisson Sigma Model
• Mirror symmetry and deformation quantization
• Open String BRST Cohomology for Generalized Complex Branes

Cited by in corpus (24)

• The Omega Deformation, Branes, Integrability, and Liouville Theory
• Boundaries, Mirror Symmetry, and Symplectic Duality in 3d $\mathcal{N}=4$ Gauge Theory
• Chaos and entanglement spreading in a non-commutative gauge theory
• Topological Chern-Simons/Matter Theories
• A stable infinity-category of Lagrangian cobordisms
• String states, loops and effective actions in noncommutative field theory and matrix models
• Dynamical tachyons on fuzzy spheres
• A Note on Quantum Geometric Langlands Duality, Gauge Theory, and Quantization of the Moduli Space of Flat Connections
• Microlocal branes are constructible sheaves
• Holomorphic Lagrangian branes correspond to perverse sheaves
• Geometric Transitions on non-Kaehler Manifolds
• A new mechanism for non-locality from string theory: UV-IR quantum entanglement and its imprints on the CMB
• Time dependence of entanglement entropy on the fuzzy sphere
• On higher rank coisotropic A-branes
• D-branes and Azumaya/matrix noncommutative differential geometry,II: Azumaya/matrix supermanifolds and differentiable maps therefrom -- with a view toward dynamical fermionic D-branes in string theory
• Batalin--Vilkovisky quantization and supersymmetric twists
• Quantum chaos in a weakly-coupled field theory with nonlocality
• Branes and DAHA Representations
• Two Categories of Dirac Manifolds
• Strings and vortex rings
• Field Theory with Coordinate Dependent Noncommutativity
• Generalized Complex and Dirac Structures on Homogeneous Spaces
• D-branes, actions effectives et symétrie miroir
• From Surface Operators to Non-Abelian Volume Operators in Puff Field Theory