Fukaya Category

Definition

The Fukaya category is a mathematical structure that arises in the study of symplectic geometry, particularly in relation to Lagrangian submanifolds. It organizes these submanifolds and their associated morphisms into a category, allowing for the exploration of the geometric and topological properties of symplectic manifolds. The Fukaya category has deep connections to both algebraic geometry and mirror symmetry, revealing profound links between seemingly disparate areas of mathematics.

5 Must Know Facts For Your Next Test

1. The Fukaya category was introduced by Kenji Fukaya in the 1990s as a tool to study symplectic manifolds using methods from algebraic geometry and category theory.
2. Objects in the Fukaya category are typically Lagrangian submanifolds, while morphisms between these objects are given by Floer homology, which captures intersection data.
3. The Fukaya category can be enhanced to include more structures, such as A-infinity (A∞) structures, which allow for more flexible algebraic manipulations.
4. Understanding the Fukaya category can lead to insights about the topology of symplectic manifolds and their relationships with complex varieties through mirror symmetry.
5. The Fukaya category plays a crucial role in modern mathematical physics, particularly in string theory and in understanding D-branes' geometry.

Review Questions

• How does the Fukaya category relate to Lagrangian submanifolds and their morphisms?
  • The Fukaya category is fundamentally built on Lagrangian submanifolds, which serve as its objects. Morphisms in this category are derived from Floer homology, reflecting how these submanifolds intersect within a symplectic manifold. This relationship allows mathematicians to study complex geometrical features and provide insights into the topology of these structures.
• Discuss the significance of Floer homology in defining morphisms within the Fukaya category.
  • Floer homology provides a robust framework for defining morphisms between Lagrangian submanifolds in the Fukaya category. By examining the moduli space of pseudo-holomorphic curves that pass through intersection points of these submanifolds, Floer homology captures essential topological information about them. This approach not only enriches the structure of the Fukaya category but also connects it deeply to both symplectic geometry and algebraic topology.
• Evaluate how the Fukaya category contributes to our understanding of mirror symmetry and its implications for algebraic geometry.
  • The Fukaya category serves as a critical bridge between symplectic geometry and mirror symmetry, illuminating how geometrical properties translate across dual spaces. By providing a framework that relates Lagrangian submanifolds in one context to complex structures in another, it enhances our understanding of these dualities. This interplay has significant implications for algebraic geometry as it suggests new ways to approach problems involving complex varieties and provides tools for deriving invariants that are consistent across different geometrical settings.

Related terms

Lagrangian Submanifold: A submanifold of a symplectic manifold where the symplectic form restricts to zero, often used as the objects in the Fukaya category.

Mirror Symmetry: A duality between certain pairs of geometrical structures that suggests a correspondence between symplectic and complex geometric properties.

Homological Algebra: A branch of mathematics that studies homology in a general algebraic setting, which is foundational for understanding morphisms in categories like the Fukaya category.