Remarks on A-branes, mirror symmetry, and the Fukaya category

By A. Kapustin, Dmitri Orlov. 2001

206 citations recorded by the source corpus.

We discuss D-branes of the topological A-model (A-branes), which are believed to be closely related to the Fukaya category. We give string theory arguments which show that A-branes are not necessarily Lagrangian submanifolds in the Calabi–Yau: more general coisotropic branes are also allowed, if the line bundle on the brane is not flat. We show that a coisotropic A-brane has a natural structure of a foliated manifold with a transverse holomorphic structure. We argue that the Fukaya category must be enlarged with such objects for the Homological Mirror Symmetry Conjecture to be true.

Subfields: Geometry and Topology, Mathematical Physics, Nuclear and High Energy Physics

View Paper: https://www.semanticscholar.org/paper/a5354e9d26cc15d52aa8e4c784a01890cd61cafd

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Citations in the corpus

Electric-Magnetic Duality And The Geometric Langlands Program (2006, background): “Using the fact that ω and F are closed 2-forms, one can show that IF is automatically integrable [32].”
A new look at the path integral of quantum mechanics (2010, unknown): “So whenever we study the A-model of a symplectic manifold X – such as X = M̂ – endowed with a symplectic form ω that is the real part of a holomorphic two-form Ω, in some complex structure J , there is a canonical way to obey the KapustinOrlov conditions.”
Homological mirror symmetry for the 4-torus (2009, unknown): “In another direction, Kapustin and Orlov [ KO04 ] argued that the Fukaya category of a high-dimensional torus did not seem “large enough” (from the point of view of K-theory) to be mirror to the derived category of sheaves of an abelian variety, and suggested that one should augment the Fukaya categ”
Generalized geometry and the Hodge decomposition (2004, background): “lds [3]. These sub-objects correspond exactly with the physicists’ notion of topological D-branes; in particular, one recovers, in the symplectic case, the co-isotropic A-branes of Kapustin and Orlov [6]. There is also a natural notion of generalized holomorphic bundle supported on a generalized com”
The geometry of supersymmetric partition functions (2013, background): “The supercharge QA = Q+ Q̃ reduces to the twodimensional BRST charge of the A-model.”
Branes and Quantization (2008, methodology): “of L enables us, following [ 5 ] and [6], to define a coisotropic A-brane Bcc, whose support is”
Generalized Complex Geometry (2004, methodology): “These are precisely the co-isotropic A-branes discovered by Kapustin and Orlov [ 22 ].”
The omega deformation, branes, integrability and Liouville theory (2010, background): “The condition for a rank 1 brane whose support is precisely X and whose Chan-Paton bundle has curvature F to be an A-brane is [10] that the linear transformation of the tangent bundle defined by I = ωF should obey I2 = −1.”
Orientifolds and mirror symmetry (2003, background): “neral index formula, with the replacement Aˆ →td, is given by IB τΩ (E,τE) = Z Xτ ch(2E)εB eB/π td(Xτ) ch(∧NXτ ) . (3.18) 1There are non-Lagrangian A-branes with non-ﬂat connection or non-zero B-ﬁeld [43,58]. We do not include this in our paper. It would be interesting to see the consistency of pari”
D branes in Landau-Ginzburg models and algebraic geometry (2002, unknown): “ar X iv :h ep -t h/ 02 10 29 6v 2 2 7 N ov 2 00”
Gauge Theory, Ramification, And The Geometric Langlands Program (2006, background): “ection whose curvature form is F= (Imτ)ωJ. This formula ensures that (ω)−1F is a complex structure, indeed the one that we have called I. This is the condition for a coisotropic brane in the sense of [72]. We call this brane the c.c. or canonical coisotropic brane, and denote it as Bc.c.. For Bc.c. ”
D-branes on general Script N = 1 backgrounds: superpotentials and D-terms (2006, background): “In the SU(3)-structure cases, where the internal manifold is either complex (IIB) or symplectic (IIA), this requirement is completely equivalent [7] to require that the D-brane must be holomorphically embedded with F(2,0) = 0 in Type IIB, and must wrap Lagrangian or more general coisotropic [39] gen”
Branes on Poisson varieties (2007, background): “As shown in [12], in the complex case (and for the trivial gerbe), generalized complex submanifolds correspond to holomorphic submanifolds equipped with unitary holomorphic line bundles, whereas in the symplectic case they correspond to Lagrangian submanifolds equipped with flat line bundles or the ”
Phases Of N=2 Theories In 1+1 Dimensions With Boundary (2008, background): “1 There are more general boundary conditions which eventually lead to coisotropic branes [68].”
Supersymmetric D-branes and calibrations on general Script N = 1 backgrounds (2005, background): “The SU(3) structure subcase provides a clear example where this condition means that the brane must wrap a holomorphic cycle with (1, 1) field strength F in type IIB and a coisotropic cycle of the kind discussed in [4, 27] in type IIA.”