Fukaya category

Definition

The Fukaya category is a sophisticated structure in mathematics that arises from symplectic geometry, focusing on the study of Lagrangian submanifolds and their intersections. It serves as a framework for understanding the relationships between these submanifolds through objects and morphisms, making it an essential tool in current research trends within homological algebra. The category encapsulates deep geometrical and topological information about the underlying symplectic manifold and its Lagrangian submanifolds.

5 Must Know Facts For Your Next Test

1. The Fukaya category is constructed from the data of Lagrangian submanifolds and their intersection points, where objects correspond to Lagrangians and morphisms to intersection data.
2. The Fukaya category is enriched over the A-infinity category, allowing for the inclusion of higher homotopical structures which provide a richer framework for homological algebra.
3. In practice, Fukaya categories have applications in mirror symmetry, relating symplectic geometry to algebraic geometry through derived categories.
4. The formalism of the Fukaya category also allows for the definition of Floer cohomology, which links holomorphic curves to intersection theory in Lagrangian manifolds.
5. Current research trends focus on expanding the Fukaya category's applications, particularly in relation to deformation theory and categorical structures arising from physical theories like string theory.

Review Questions

• How does the Fukaya category utilize Lagrangian submanifolds to establish connections within symplectic geometry?
  • The Fukaya category utilizes Lagrangian submanifolds by defining objects as these submanifolds and morphisms based on their intersection properties. This approach allows mathematicians to study the relationships between different Lagrangians through their intersections, leading to insights into both geometric properties and algebraic structures. By considering how Lagrangians interact with each other within a symplectic manifold, researchers can uncover deeper relationships that bridge various areas of mathematics.
• Discuss the significance of A-infinity structures in the context of Fukaya categories and their application in homological algebra.
  • A-infinity structures are crucial for Fukaya categories as they enable a flexible framework where morphism compositions are defined up to homotopy rather than strictly. This flexibility is essential for capturing the complexities that arise from intersections of Lagrangian submanifolds. In homological algebra, this leads to powerful tools for understanding derived categories and has profound implications for developments in areas like mirror symmetry and deformation theory.
• Evaluate how current research trends involving Fukaya categories might influence future developments in symplectic geometry and related fields.
  • Current research trends involving Fukaya categories are expanding their applications into new areas such as mirror symmetry and categorical structures arising from theoretical physics. As these categories become more intertwined with modern mathematical theories, they could lead to breakthroughs that enhance our understanding of both symplectic geometry and its connections with other disciplines like algebraic geometry. Future developments may also explore further generalizations of Fukaya categories or their relations to other topological constructs, pushing the boundaries of what is known in contemporary mathematics.

Related terms

Lagrangian submanifold: A type of submanifold in symplectic geometry that is half-dimensional, meaning its dimension is half of the ambient symplectic manifold.

Symplectic geometry: A branch of differential geometry that studies symplectic manifolds, which are smooth manifolds equipped with a closed non-degenerate 2-form.

A-infinity category: A generalization of categories in which the composition of morphisms is not strictly associative but defined up to higher homotopies, playing a key role in the formulation of Fukaya categories.