Fukaya Categories

Definition

Fukaya categories are mathematical structures that encode the relationships between Lagrangian submanifolds in symplectic geometry, focusing on their intersection theory and their morphisms. These categories are essential for understanding how Lagrangian submanifolds behave under deformations and play a pivotal role in mirror symmetry, providing a way to study both geometric and algebraic aspects of these objects. They allow for the categorification of Floer homology and serve as a bridge between symplectic geometry and algebraic geometry.

5 Must Know Facts For Your Next Test

1. Fukaya categories are defined over a field, often the field of complex numbers, which allows for the computation of their homological properties.
2. The objects in a Fukaya category correspond to Lagrangian submanifolds, while morphisms are constructed using Floer cohomology, reflecting the intersections of these submanifolds.
3. They can be enhanced with additional structure, such as A-infinity structures, which allows for more sophisticated algebraic manipulations.
4. Fukaya categories can vary depending on the choice of the ambient symplectic manifold and the specific Lagrangians considered, leading to rich and diverse mathematical landscapes.
5. These categories have deep implications in both mathematics and theoretical physics, particularly in areas like mirror symmetry and string theory, linking geometry with topology.

Review Questions

• How do Fukaya categories relate to the study of Lagrangian submanifolds and their intersections?
  • Fukaya categories provide a systematic framework to study Lagrangian submanifolds by associating each Lagrangian with an object in the category. The morphisms between these objects are derived from Floer cohomology, which captures information about how Lagrangians intersect and evolve under deformation. This relationship is crucial for understanding the geometric structure and behavior of Lagrangian submanifolds within symplectic manifolds.
• Discuss the role of Fukaya categories in establishing connections between symplectic geometry and mirror symmetry.
  • Fukaya categories play a central role in mirror symmetry by offering a way to relate two different geometrical setups—often a symplectic manifold and its mirror algebraic counterpart. Through these categories, one can define invariants that allow mathematicians to translate problems in symplectic geometry into algebraic terms, facilitating deep connections between seemingly disparate fields. This interplay has led to significant insights in both theoretical physics and pure mathematics.
• Evaluate the impact of Fukaya categories on modern mathematical research, particularly in fields such as algebraic geometry and theoretical physics.
  • The introduction of Fukaya categories has transformed modern mathematical research by providing powerful tools for analyzing complex relationships between Lagrangian submanifolds and their algebraic counterparts. In algebraic geometry, they help establish rigorous connections through mirror symmetry, leading to new insights into both geometric structures and homological algebra. In theoretical physics, Fukaya categories have implications for string theory and quantum field theory, suggesting that they are key components in understanding fundamental concepts such as duality and topological invariance.

Related terms

Lagrangian Submanifolds: A type of submanifold in symplectic geometry where the symplectic form restricts to zero, often studied in relation to their intersections and properties.

Floer Homology: A type of homology theory that arises from studying the intersection of Lagrangian submanifolds, providing invariants that capture their geometric properties.

Mirror Symmetry: A phenomenon in string theory and algebraic geometry where two seemingly different geometrical objects have equivalent physical and mathematical properties, often involving Fukaya categories.