What is Fukaya Algebra?

A Fukaya Algebra is not a single, rigidly defined algebraic object but rather a family of algebraic structures associated with a symplectic manifold. It is the foundational algebraic structure that emerges from Fukaya Categories. Think of it this way: in a Fukaya Category, the objects are Lagrangian submanifolds, and the "spaces of morphisms" between them are Floer chain complexes. A Fukaya Algebra is what you get when you focus on a single Lagrangian submanifold and study the algebraic structure on its Floer cohomology. This structure is an A_∞-algebra (A-infinity algebra). In essence, the Fukaya algebra of a Lagrangian submanifold is an A_∞-algebra that encodes the "self-interaction" of that Lagrangian via pseudo-holomorphic curves.

The Building Blocks: The Stage and the Actors

To understand Fukaya algebras, we need the stage and the actors:

A Symplectic Manifold (M, ω) is the "stage." It's a smooth manifold equipped with a symplectic form ω, a closed, non-degenerate 2-form. This structure is the mathematical formulation for the phase space of classical mechanics.

A Lagrangian Submanifold (L) is our "actor." It's a submanifold of half the dimension of M on which the symplectic form vanishes completely. Lagrangian submanifolds are the symplectic analogue of lines or planes. A classic example is the zero-section of a cotangent bundle.

The Plot: "Counting" Curves with Floer Theory

The central idea, pioneered by Andreas Floer, is to study the intersections of Lagrangian submanifolds. Given two Lagrangians L₁ and L₂, their Floer cohomology HF(L₁, L₂) is a homology theory whose chain complex is generated by their intersection points.

The differential in this complex is defined by counting pseudo-holomorphic curves. Specifically, one counts the number of certain maps from a disk into the symplectic manifold M such that the boundary of the disk lies on the Lagrangians and connects two intersection points.

Focusing on a Single Object

Now, let's focus on a single Lagrangian L. We want to understand its "self-homology" HF(L, L). The chain complex leading to this is CF(L, L). On this complex, we don't just have a differential. We have an infinite tower of operations that together form an A_∞-algebra.

The Algebraic Structure: A_∞-Algebras

On the Floer chain complex CF(L, L), we define a series of operations for k = 1, 2, 3, ...:

m_k : CF(L, L)^{⊗ k} → CF(L, L)

The operation m₁ is the Floer differential itself. It satisfies m₁ ∘ m₁ = 0, which allows us to take homology.

The operation m₂ is a product operation, like a multiplication. However, it is not associative at the chain level.

The operations m₃, m₄, ... are higher homotopies that precisely measure the failure of associativity of m₂. The operation m₃ provides a specific way in which (xy)z is related to x(yz).

This entire package of operations {m₁, m₂, m₃, ...} satisfies a specific, intricate set of equations called the A_∞-relations. A vector space equipped with such operations is called an A_∞-algebra.

The Fukaya algebra of L is this A_∞-algebra structure on CF(L, L).

Geometrically, the operation m_k is defined by counting pseudo-holomorphic disks with (k+1) marked points on the boundary, all mapped into L.

Key Properties and Refinements

The Fukaya algebra is a tool for studying deformations of the Lagrangian L. Deforming the Lagrangian corresponds to deforming the A_∞-algebra structure. This is central to the homological mirror symmetry program.

A Lagrangian is called unobstructed if its Fukaya algebra admits a solution to the Maurer-Cartan equation. Such a solution, called a bounding cochain, is an element of degree 1 that "fixes" the non-associativity and allows one to define a deformed, honest associative multiplication. Only unobstructed Lagrangians with a choice of bounding cochain give well-defined objects in the Fukaya category.

For full technical power, the Fukaya algebra is often defined over a Novikov field, which keeps track of the areas of the holomorphic disks, and can be equipped with a grading.

Significance and Applications

The Fukaya algebra of L is the endomorphism algebra of the object L in the Fukaya category of M. So, understanding these algebras is synonymous with understanding the Fukaya category itself.

This is most famously applied in Homological Mirror Symmetry (HMS). HMS conjectures that the Fukaya category of one symplectic manifold is equivalent to the derived category of coherent sheaves on a "mirror" complex manifold. Under this equivalence, the Fukaya algebra of a Lagrangian should correspond to the endomorphism algebra of a sheaf on the mirror, providing a powerful bridge between symplectic geometry and algebraic geometry.

Finally, the Fukaya algebra provides a deep algebraic invariant to distinguish Lagrangians that might be topologically identical but situated differently in the symplectic sense.