Wrapped Fukaya CategoryEdit

Wrapped Fukaya category

The wrapped Fukaya category is a deep construction in symplectic geometry that organizes certain non-compact Lagrangian submanifolds of a Liouville manifold into an algebraic object known as an A-infinity category. It extends the classical Fukaya category, which is defined for compact symplectic manifolds, to settings where Lagrangians extend to infinity. The key idea is to use a Hamiltonian flow that “wraps” Lagrangians at infinity, producing finite and well-behaved morphism spaces via wrapped Floer cohomology. These morphisms, together with higher composition maps defined by counts of pseudo-holomorphic discs, assemble into an A-infinity structure that encodes interactions among Lagrangians. The wrapped Fukaya category plays a central role in modern approaches to homological mirror symmetry and to the study of symplectic invariants of non-compact spaces such as Liouville domains and their completions.

Overview and definitions

  • The ambient setting: the wrapped Fukaya category is defined for a Liouville manifold (or Liouville domain with completion). The Liouville structure provides a natural way to speak about behavior at infinity, which is essential when dealing with non-compact Lagrangian submanifolds. Objects in the category are suitably brane-decorated Lagrangian submanifolds that are exact or have appropriate wrapping-compatible structures. See Lagrangian submanifold and Liouville domain for foundational concepts; the broader context sits inside symplectic geometry.
  • Objects and morphisms: objects are (properly embedded) Lagrangian branes, and morphisms between two objects are generated by chords of a chosen Hamiltonian that grows toward infinity, yielding the wrapped Floer cochain complex. The cohomology of this complex gives HW*(L0,L1), the wrapped Floer cohomology groups. This construction generalizes the usual Floer cohomology to the non-compact setting. See Floer cohomology and A-infinity category for the algebraic framework.
  • Higher compositions and A-infinity structure: the A-infinity structure maps μ^k are defined by counting rigid pseudo-holomorphic discs with boundary on the sequence of Lagrangian branes, in the spirit of the compact Fukaya category, but with the wrapping allowing discs to interact with the infinity in controlled ways. The resulting algebraic relations satisfy the A-infinity equations, providing a robust categorical object. See A-infinity category for the abstract algebraic notion.
  • Invariance and stops: the wrapped Fukaya category depends on choices such as the Hamiltonian used to wrap and auxiliary stop data that control the behavior at infinity; however, under suitable hypotheses (e.g., appropriate stops and isotopies), the category is invariant up to quasi-equivalence. This mirrors, in the non-compact setting, a version of functoriality in Viterbo theory. See Viterbo functoriality and stop (symplectic geometry) for related ideas.

Foundations, constructions, and technical landscape

  • From compact to wrapped: the classical Fukaya category associates to a compact symplectic manifold a category whose objects are Lagrangian branes and whose morphisms come from Floer cohomology, with higher A-infinity operations determined by counts of holomorphic discs. The wrapped variant adapts this to non-compact manifolds by introducing a Hamiltonian that enforces wrapping and by working with a directed system of Floer complexes whose colimit yields HW*(L0,L1). See Fukaya category and Floer cohomology.
  • Generating objects and generation criteria: a central theme is identifying a (often finite) set of Lagrangian branes that generate the wrapped Fukaya category, in the sense that every object is obtained (up to isomorphism) from these generators under cones and shifts in the A-infinity sense. Abouzaid’s generation criterion provides practical tools for proving generation in a wide class of examples. See Abouzaid and generation criterion (as general concepts linked to A-infinity category).
  • Connections with mirrors: in the landscape of homological mirror symmetry, the wrapped Fukaya category of a non-compact symplectic manifold is conjectured (and in many cases proved) to correspond to a derived category of coherent sheaves (or related categories) on a mirror object. This bridge has powered dramatic advances in understanding how “symplectic” data translates into “algebraic” data on the mirror side. See mirror symmetry and derived category of coherent sheaves.
  • Examples and computations: the wrapped Fukaya category of the cotangent bundle T*Q, for a smooth manifold Q, serves as a guiding example. Here one shows that the category is generated by the cotangent fiber, and the endomorphism algebra recovers information about the based loop space of Q. This connects to broader themes in geometric representation theory and algebraic topology. See cotangent bundle and based loop space for related concepts.

Key results and major themes

  • Relationships to symplectic cohomology: wrapped Floer cohomology groups HW*(L,L) for a Lagrangian L relate to broader invariants of the ambient Liouville manifold, and the wrapped Fukaya category sits in a web of structures that include symplectic cohomology and Viterbo-type functorialities. See symplectic cohomology.
  • Stop removal and invariance theorems: techniques for proving invariance under changes of stop data and Hamiltonians have matured, with several strategies (including microlocal sheaf-theoretic perspectives) contributing a robust understanding of when the wrapped category remains equivalent. See stop (symplectic geometry) and microlocal sheaf theory (in connection with symplectic methods).
  • Generation and HMS programs: generation results, particularly Abouzaid’s work, have turned wrapped Fukaya categories into practical computational tools for verifying instances of non-compact HMS. The broader program analyzes when a collection of Lagrangians suffices to reconstruct the entire category, and how these results reflect on the geometry of the Liouville domain. See Abouzaid and Homological mirror symmetry.
  • Non-compact geometries and stops: the non-compact setting invites a refined toolkit, including the notion of stops, stops’ role in controlling infinity, and the interplay with wrapped dynamics. This has sharpened the understanding of which geometric features influence the algebraic structure of the category. See stop (symplectic geometry) and wrap (as part of the wrapping construction).

Controversies and ongoing debates

  • Foundations and transversality: the precise foundations of the wrapped Fukaya category involve delicate transversality issues for moduli spaces of holomorphic discs in non-compact settings. Different approaches (e.g., perturbative methods, polyfold theory, or Kuranishi structures) have been developed to address these technicalities, leading to debates about the most transparent and flexible framework. See discussions surrounding polyfold theory and Kuranishi structure as general approaches to virtual perturbations.
  • Invariance and stops: while many invariance results exist, there is ongoing work to understand the limits of stop removal and to characterize when the wrapped category remains unchanged under broader kinds of deformations. The precise dependence on auxiliary data can be subtle, prompting careful formulations of hypotheses and proofs. See Viterbo functoriality and stop (symplectic geometry) for the core ideas and current research.
  • Generators and computational practicality: while generation criteria provide powerful benchmarks, there are families of Liouville manifolds for which determining a finite generating set is challenging, and for which the full structure of the wrapped category remains difficult to compute explicitly. This has spurred developments in both abstract theory and computational techniques, tying into broader questions about the scope of HMS in non-compact settings. See Abouzaid and Homological mirror symmetry.
  • Relation to other models: multiple models and perspectives on the wrapped Fukaya category (geometric, sheaf-theoretic, and algebraic) have proven complementary but at times technically distinct. Debates about the most natural or efficient framework persist, particularly when comparing with alternative categorical approaches to non-compact symplectic geometry. See microlocal sheaf theory and Fukaya category for related viewpoints.

Examples and connections to broader theory

  • The cotangent bundle example: for a manifold Q, the wrapped Fukaya category of T*Q is generated by the cotangent fiber, and the endomorphism algebra recovers information connected to the based loop space of Q. This instance illuminates the deep ties between symplectic geometry and algebraic topology, and it provides a concrete laboratory for exploring HMS in non-compact settings. See cotangent bundle and based loop space.
  • Mirror symmetry and non-compact geometries: in many non-compact Calabi–Yau or Landau–Ginzburg models, the wrapped Fukaya category corresponds to a derived (or triangulated) category of coherent sheaves on a mirror object with a superpotential, reflecting a duality between symplectic and algebraic data in a non-compact environment. See mirror symmetry and Landau–Ginzburg model.
  • Interactions with categorical dynamics: the A-infinity structure of the wrapped Fukaya category enables one to study functors that arise from exact Lagrangian correspondences, leading to a richer understanding of how symplectic maps induce algebraic actions on wrapped categories. See A-infinity category and Fukaya category.

See also