Fukaya Category Research Articles

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

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.

Vertex Algebras, Mirror Symmetry, and D-Branes: The Case of Complex Tori

A vertex algebra is an algebraic counterpart of a two-dimensional conformal field theory. We give a new definition of a vertex algebra which includes chiral algebras as a special case, but allows for fields which are neither meromorphic nor anti-meromorphic. To any complex torus equipped with a flat Kahler metric and a closed 2-form we associate an N=2 superconformal vertex algebra (N=2 SCVA) in the sense of our definition. We find a criterion for two different tori to produce isomorphic N=2 SCVA's. We show that for algebraic tori isomorphism of N=2 SCVA's implies the equivalence of the derived categories of coherent sheaves corresponding to the tori or their noncommutative generalizations (Azumaya algebras over tori). We also find a criterion for two different tori to produce N=2 SCVA's related by a mirror morphism. If the 2-form is of type (1,1), this condition is identical to the one proposed by Golyshev, Lunts, and Orlov, who used an entirely different approach inspired by the Homological Mirror Symmetry Conjecture of Kontsevich. Our results suggest that Kontsevich's conjecture must be modified: coherent sheaves must be replaced with modules over Azumaya algebras, and the Fukaya category must be ``twisted'' by a closed 2-form. We also describe the implications of our results for BPS D-branes on Calabi-Yau manifolds.

Graded lagrangians, exotic topological D-branes and enhanced triangulated categories

I point out that (BPS saturated) A-type D-branes in superstring compactification on Calabi-Yau threefolds correspond to graded special lagrangian submanifolds, a particular case of the graded lagrangian submanifolds considered by M. Kontsevich and P. Seidel. Combining this with the categorical formulation of cubic string field theory in the presence of D-branes, I consider a collection oftopological D-branes wrapped over the same lagrangian cycle and derive its string field action from first string-theoretic principles. The result is a -graded version of super-Chern-Simons field theory living on the lagrangian cycle, whose relevant string field is a degree one superconnection in a -graded superbundle, in the sense previously considered in mathematical work of J.M. Bismutt and J. Lott. This gives a refined (and modified) version of a proposal previously made by C. Vafa. I analyze the vacuum deformations of this theory and relate them to topological D-brane composite formation, upon using the general formalism developed in a previous paper. This allows me to identify a large class of topological D-brane composites (generalized, or ``exotic'' topological D-branes) which do not admit a traditional description. Among these are objects which correspond to the ``covariantly constant sequences of flat bundles'' considered by Bismut and Lott, as well as more general structures, which are related to the enhanced triangulated categories of Bondal and Kapranov. I also give a rough sketch of the relation between this construction and the large radius limit of a certain version of the ``derived category of Fukaya's category''. This paper forms part of a joint project with Prof. S. Popescu, a brief announcement of which can be found in the second part of the note hep-th/0102183. The paralel B-model realization, as well as the relation with the enhanced triangulated categories of Bondal and Kapranov, was recently discussed by D.E. Diaconescu in the paper hep-th/0104200, upon using the observations contained in that announcement.

Categorial Mirror Symmetry for K3 Surfaces

We study the structure of a modified Fukaya category ${\frak F}(X)$ associated with a K3 surface $X$, and prove that whenever $X$ is an elliptic K3 surface with a section, the derived category of $\fF(X)$ is equivalent to a subcategory of the derived category ${\bold D}(\hat X)$ of coherent sheaves on the mirror K3 surface $\hat X$.

The Hall algebras of annuli

We refine and prove the central conjecture of our first paper for annuli with at least two marked intervals on each boundary component by computing the derived Hall algebras of their Fukaya categories.

Stability conditions and the braid group

We find stability conditions [6, 3] on some derived categories of differential graded modules over a graded algebra studied in [12, 10]. This category arises in both derived Fukaya categories and derived categories of coherent sheaves. This gives the first examples of stability conditions on the A-model side of mirror symmetry, where the triangulated category is not naturally the derived category of an abelian category. The existence of stability conditions, however, gives many such abelian categories, as predicted by mirror symmetry. In our examples in 2 dimensions, we completely describe a connected component of the space of stability conditions. It is the universal cover of the configuration space C k+1 of k + 1 points in C with centre of mass zero, with deck transformations the braid group action of [10, 15]. This gives a geometric origin for these braid group actions and their faithfulness, and axiomatises the proposal for stability of Lagrangians in [18] and the example proved by mean curvature flow in [19].

On Hochschild invariants of Landau–Ginzburg orbifolds

We develop an approach to calculating the cup and cap products on Hochschild cohomology and homology of curved algebras associated with polynomials and their finite abelian symmetry groups. For polynomials with isolated critical points, the approach yields a complete description of the products. We also reformulate the result for the corresponding categories of equivariant matrix factorizations. In an Appendix written jointly with Alexey Basalaev, we apply the formulas to calculate the Hochschild cohomology of a simple but non-trivial class of so-called invertible LG orbifold models. The resulting algebras turn out to be isomorphic to what has already appeared in the literature on LG mirror symmetry under the name of twisted or orbifolded Milnor/Jacobian algebras. We conjecture that this holds true for all invertible LG models. In the second part of the Appendix, the formulas are applied to a different class of LG orbifolds which have appeared in the context of homological mirror symmetry for varieties of general type as mirror partners of surfaces of genus 2 and higher. In combination with a homological mirror symmetry theorem for the surfaces, our calculation yields a new proof of the fact that the Hochschild cohomology of the Fukaya category of a surface is isomorphic, as an algebra, to the cohomology of the surface.

Mirror of Atiyah flop in symplectic geometry and stability conditions

We study the mirror operation of the Atiyah flop in symplectic geometry. We formulate the operation for a symplectic manifold with a Lagrangian fibration. Furthermore we construct geometric stability conditions on the derived Fukaya category of the deformed conifold and study the action of the mirror Atiyah flop on these stability conditions.