Homological Mirror Symmetry - Core Definitions

Homological mirror symmetry (HMS) is Kontsevich's profound categorical reformulation of mirror symmetry, positing an equivalence between derived categories of coherent sheaves and Fukaya categories. This framework provides a mathematically rigorous foundation for mirror symmetry.

DefinitionHomological Mirror Symmetry Conjecture

For a mirror pair $(X, X^\vee)$ of Calabi-Yau manifolds, there exists an equivalence of triangulated categories: $D^b(Coh(X)) \simeq D\mathcal{F}uk(X^\vee)$

where $D^b(Coh(X))$ is the bounded derived category of coherent sheaves on $X$ , and $D\mathcal{F}uk(X^\vee)$ is the derived Fukaya category of $X^\vee$ .

This conjecture unifies various aspects of mirror symmetry: the A-model corresponds to the Fukaya side (symplectic geometry), while the B-model corresponds to the coherent sheaf side (complex geometry). The categorical equivalence implies all previously observed mirror phenomena.

DefinitionDerived Category of Coherent Sheaves

The derived category $D^b(Coh(X))$ is constructed from:

The abelian category $Coh(X)$ of coherent sheaves on $X$
Complexes of coherent sheaves: $\mathcal{F}^\bullet = (\cdots \to \mathcal{F}^i \to \mathcal{F}^{i+1} \to \cdots)$
Quasi-isomorphisms inverted to form the derived category

Objects are complexes, morphisms are chain maps up to homotopy, and triangles come from mapping cones.

The bounded derived category contains complexes with finitely many non-zero cohomology sheaves. Key examples include:

Structure sheaf: $\mathcal{O}_X$ (skyscraper at a point)
Ideal sheaves: $\mathcal{I}_Z$ for subvarieties $Z \subset X$
Complexes from resolutions: $0 \to \mathcal{O}_X \to \mathcal{O}_X(D) \to 0$

DefinitionFukaya Category

The Fukaya category $\mathcal{F}uk(X)$ of a symplectic manifold $X$ has:

Objects: Lagrangian submanifolds $L \subset X$ equipped with flat $U(1)$ connections
Morphisms: $\text{Hom}(L_0, L_1) = HF^*(L_0, L_1)$ , the Floer cohomology
Composition: Defined by counting pseudo-holomorphic triangles

The Floer cohomology $HF^*(L_0, L_1)$ is computed as the homology of the complex generated by intersection points $L_0 \cap L_1$ , with differential counting pseudo-holomorphic strips.

The $A_\infty$ structure on the Fukaya category arises from higher compositions involving pseudo-holomorphic polygons. This structure is essential for making the category well-defined.

Remark

The Fukaya category depends on choices: almost complex structure $J$ , symplectic form $\omega$ , and perturbation data. However, the derived category $D\mathcal{F}uk(X)$ is expected to be independent of these choices, providing an intrinsic invariant of the symplectic manifold.

ExampleElliptic Curves

For an elliptic curve $E$ viewed as a complex torus $\mathbb{C}/\Lambda$ :

B-side: $D^b(Coh(E))$ is generated by $\mathcal{O}_E$ and skyscraper sheaves
A-side: $D\mathcal{F}uk(E)$ is generated by Lagrangian circles

The HMS equivalence matches line bundles to Lagrangian tori and point sheaves to Lagrangian circles.

Homological mirror symmetry elevates mirror symmetry from numerical coincidences to categorical equivalences, providing a framework that explains and generalizes classical mirror symmetry phenomena.