Fukaya Algebra and the Role of Riemann & Complex Analysis
How complex geometry provides the foundation for symplectic algebraic structures
The roles of Riemann surfaces and Complex Analysis are absolutely fundamental and constitutive in the Fukaya model. They are not just incidental tools; they are the very source of the algebraic structure itself.
The Core Idea: Geometry Becomes Algebra via Complex Analysis
The entire premise of the Fukaya category is to take geometric and topological data about Lagrangians in a symplectic manifold and translate it into an algebraic structure. The engine for this translation is Complex Analysis, specifically the theory of pseudo-holomorphic curves.
The Role of Riemann Surfaces: The "Shape" of Interactions
In the Fukaya model, Riemann surfaces serve as the domains for the maps that define the algebraic operations. The choice of domain is crucial and depends on the operation:
For the Differential (m₁)
The domain is the infinite cylinder ℝ × S¹, which is conformally equivalent to the punctured plane. This is viewed as a "strip" where we count maps that connect two intersection points of a Lagrangian with itself.
For the Product (m₂) and Higher Operations (mₖ)
The domain is the unit disk in the complex plane D², with (k+1) marked points on its boundary. This disk is a Riemann surface with boundary. The positions of these marked points are fixed to specific points to break the symmetry and define the inputs and output of the operation mₖ.
Geometrically: You have a disk. k of the boundary points are labeled as "inputs," and 1 is labeled as the "output." The operation mₖ(x₁, x₂, ..., xₖ) is defined by counting pseudo-holomorphic maps from this disk into the symplectic manifold M such that the boundary of the disk is mapped into the Lagrangian submanifold L, and the map converges to the intersection points x₁, x₂, ..., xₖ and the output point at the respective marked points.
Why this shape? The disk is the simplest Riemann surface with a boundary, and the boundary condition is where the Lagrangian L lives. The different mₖ operations correspond to disks with different numbers of "corners" or "insertion points."
The Role of Complex Analysis: Defining the "Counting"
The heart of the construction is the type of map we consider from our Riemann surface domain into the symplectic manifold. We use pseudo-holomorphic curves, a generalization of holomorphic maps.
Holomorphic Map (Complex Analysis)
A map f : Σ → X between complex manifolds is holomorphic if its derivative is complex-linear. This is equivalent to satisfying the Cauchy-Riemann equations:
where J is the complex structure (multiplication by i).
Pseudo-holomorphic Curve (Symplectic Geometry)
A map u : Σ → M from a Riemann surface Σ to a symplectic manifold M is pseudo-holomorphic if it satisfies a generalized Cauchy-Riemann equation:
Here:
- • j is the complex structure on the Riemann surface Σ
- • J is an almost complex structure on the symplectic manifold M, compatible with the symplectic form ω
- • du is the differential of the map u
Interpretation: This equation means that the map u sends the complex structure on its domain (Σ) to the almost complex structure J on its target (M). In other words, the map "preserves the geometric structure" in a generalized sense. For a fixed J, these equations become a system of elliptic partial differential equations.
How It All Fits Together: The Analytic Foundation of the A∞-Algebra
The link between the complex analysis and the Fukaya algebra is provided by the moduli space of these pseudo-holomorphic curves.
Step 1: Define the Moduli Space
For a given Riemann surface (D², j) with marked points and a Lagrangian L, we consider the space of all pseudo-holomorphic maps u : (D², ∂D²) → (M, L) with the specified boundary conditions.
Step 2: Transversality and Compactness (The Hard Analysis)
This is the most technically challenging part. One must prove that this moduli space is:
- • Transverse (or "regular"): This ensures the moduli space is a smooth manifold. It often requires perturbing the equation.
- • Compact: This ensures that when we "count" the number of curves, we get a finite number. The Gromov compactness theorem is essential here.
Step 3: "Counting" Defines the Operation
Once we have a well-behaved, zero-dimensional moduli space (a finite set of points), we "count" the number of such pseudo-holomorphic disks. This count becomes the coefficient in the definition of the operation mₖ(x₁, ..., xₖ).
Step 4: The A∞-Relations Emerge from Gluing
Why does this infinite tower of operations {m₁, m₂, m₃, ...} satisfy the precise A∞-relations? The answer is geometric and comes from the boundary of the moduli spaces.
The A∞-relation for m₁ and m₂ looks like:
This relation is proven by considering the one-dimensional moduli space of disks with three boundary points. The boundary of this compactified one-dimensional space consists of pairs of disks that have "glued together" along a node.
One type of boundary corresponds to m₁(m₂(x,y)) (a disk for m₂ with a "bubble" for m₁ attached). Another corresponds to m₂(m₁(x), y), and so on. Since the total boundary of a compact one-dimensional manifold is a set of points with a net count of zero, these algebraic terms must sum to zero. This is exactly the A∞-relation.
In summary: The intricate algebraic laws governing the Fukaya algebra are a direct consequence of the analytic compactness and gluing theorems for the moduli spaces of pseudo-holomorphic curves.
The Big Picture: A Synthesis of Geometries
The Fukaya model is a profound synthesis of three great fields of geometry:
| Field | Role in the Fukaya Model |
|---|---|
| Complex Analysis / Riemann Surfaces | Provides the domain and the defining equation (pseudo-holomorphic curve equation) for the maps. |
| Symplectic Geometry | Provides the target space (M, ω) and the objects of study (Lagrangians L). The symplectic form ω is used to control the energy of the curves. |
| Algebra | The output. The geometric data of counting curves is packaged into the algebraic structure of an A∞-algebra and the Fukaya category. |
Therefore, without the theory of Riemann surfaces and the complex-analytic machinery of pseudo-holomorphic curves, the Fukaya algebra would have no definition. The algebra is a shadow cast by the geometry, and complex analysis is the light that creates it.
No comments:
Post a Comment