Fukaya category

Fukaya category
Name	Fukaya category
Field	Symplectic geometry
Introduced	1990s
Introduced by	Kenji Fukaya

Fukaya category The Fukaya category is a central invariant in symplectic geometry connecting Kenji Fukaya, American Mathematical Society, Maxim Kontsevich, Paul Seidel, Mohammed Abouzaid, and Yong-Geun Oh. It encodes intersection and holomorphic curve data of Lagrangian submanifolds in a symplectic manifold and plays a crucial role in relations between Mirror symmetry, Homological mirror symmetry, Floer homology, Gromov–Witten invariants, and categorical aspects of Derived category of coherent sheaves.

Introduction

The Fukaya category arose from attempts by Kenji Fukaya and collaborators to formalize Lagrangian intersection theory using techniques developed by Andreas Floer and later extended by Dusa McDuff, Dietmar Salamon, Yasha Eliashberg, and Paul Seidel. It sits alongside invariants such as Quantum cohomology and Symplectic cohomology and interacts with structures studied by Maxim Kontsevich in the formulation of Homological mirror symmetry. Contributions from researchers like Mohammed Abouzaid, Peter Seidel, Paul Kronheimer, and Jake Solomon shaped foundational aspects and technical machinery for transversality, bounding cochains, and obstruction theories.

Definitions and Construction

A Fukaya category is typically associated to a compact or noncompact symplectic manifold (examples include Calabi–Yau manifold, K3 surface, Cotangent bundle, Symplectic torus). Objects are suitably decorated Lagrangian submanifolds often equipped with brane structures introduced in analogy to constructions related to Dirichlet brane, Strominger–Yau–Zaslow conjecture, and choices inspired by Homological mirror symmetry proposals. Morphism spaces are versions of Lagrangian Floer cohomology groups as developed by Andreas Floer and further formalized by Fukaya–Oh–Ohta–Ono techniques. The composition maps count pseudo-holomorphic polygons constrained by boundary conditions in the style of Gromov compactness and Pseudoholomorphic curve theory developed by Mikhail Gromov and expanded by Cliff Taubes and Richard Hamilton-type analytic frameworks. Technical foundations invoke virtual cycles, Kuranishi structures, and polyfold theory advanced by Fukaya, Kenji Ono, Helmut Hofer, and Kai Cieliebak for regularization.

A∞-Structures and Homological Algebra

Fukaya categories are A∞-categories whose higher composition maps μ^k satisfy A∞-relations introduced by Jim Stasheff. The algebraic formalism connects to Derived category of coherent sheaves techniques used by Alexander Beilinson and Joseph Bernstein. Homological algebra tools from Bernard Keller and concepts like triangulated envelopes, split-closures, and idempotent completion are essential. Bounding cochains and Maurer–Cartan elements resolve curvature issues akin to constructions in Deformation theory studied by Maxim Kontsevich and Pierre Deligne. The categorical framework relates to invariants used by Michael Kontsevich in noncommutative geometry and to structures appearing in work of Tom Bridgeland on stability conditions and by Paul Aspinwall in string-theoretic contexts.

Examples and Computations

Computations of Fukaya categories for concrete symplectic manifolds have been carried out for Symplectic torus, Cotangent bundle of a sphere, Riemann surface, K3 surface, and simple Weinstein manifolds studied by Yakov Eliashberg and Aaron Weinstein. For the two-torus explicit descriptions link to Theta functions and aspects of Elliptic curve mirror symmetry studied by Dori Bejleri and Maxim Kontsevich. The wrapped Fukaya category for noncompact examples was developed by Mohammed Abouzaid and Paul Seidel and computed in cases related to Plumbings of cotangent bundles and links to Symplectic field theory as developed by Eliashberg–Givental–Hofer. Calculations often invoke surgery exact triangles, generation criteria established by Aaron Weinstein, generation results of Mohammed Abouzaid, and functoriality under Lagrangian correspondences studied by Alexander Weinstein and Konstanze Rietsch.

Relationship to Mirror Symmetry

The Fukaya category is the symplectic side of Homological mirror symmetry, conjectured by Maxim Kontsevich to be equivalent to the derived category of coherent sheaves on a mirror complex variety such as Calabi–Yau manifold mirrors for Quintic threefold examples explored by Philip Candelas and Borisov–Cox. Works by Paul Seidel, Denis Auroux, Ludmil Katzarkov, Paul Aspinwall, and Klaus Hulek produced verifications in examples including Quartic surface and punctured Riemann surfaces. Mirror symmetry applications connect Fukaya categories to enumerative predictions involving Gromov–Witten invariants and physical constructions in String theory frameworks used by Edward Witten and Cumrun Vafa.

Variants and Generalizations

Variants include the wrapped Fukaya category, monotone Fukaya category, and relative Fukaya categories adapted for pairs (symplectic manifold, divisor) studied in contexts by Paul Seidel, Mohammed Abouzaid, Fukaya–Oh–Ohta–Ono, and Denis Auroux. Generalizations incorporate equivariant versions with group actions relevant to Hamiltonian dynamics studied by Alan Weinstein and equivariant Floer theory by Kenji Fukaya collaborators, and categorical enhancements interacting with Symplectic cohomology and Topological quantum field theory frameworks developed by Graeme Segal and Kevin Costello. Ongoing developments link Fukaya categories with advances in polyfold theory of Helmut Hofer and with categorical dualities in noncommutative geometry studied by Maxim Kontsevich and Alexander Rosenberg.

Category:Symplectic geometry