Homological Mirror Symmetry - Key Properties

The categorical equivalence in homological mirror symmetry has profound structural consequences, connecting invariants and structures on both sides. Understanding these properties reveals the deep mathematics underlying mirror symmetry.

Derived Equivalences and K-Theory

The HMS equivalence induces isomorphisms on K-theory and Hochschild (co)homology.

DefinitionK-Theory Isomorphism

The derived equivalence $D^b(Coh(X)) \simeq D\mathcal{F}uk(X^\vee)$ induces an isomorphism: $K_0(X) \cong HH_0(\mathcal{F}uk(X^\vee))$

where $K_0(X)$ is the Grothendieck group and $HH_0$ is the zeroth Hochschild homology.

For Calabi-Yau threefolds, this explains the swap of Hodge numbers: $h^{1,1}(X)$ (dimension of $K_0$ modulo numerical equivalence) equals $h^{2,1}(X^\vee)$ and vice versa.

Remark

The Hochschild cohomology $HH^*(D^b(Coh(X)))$ is isomorphic to $H^*(\Lambda^* TX)$ by the Hochschild-Kostant-Rosenberg theorem. Under HMS, this should match the quantum cohomology of $X^\vee$ , explaining the A-model/B-model correspondence.

Serre Functors and Canonical Bundles

Both categories possess Serre functors reflecting the geometric canonical bundle.

DefinitionSerre Functor

A Serre functor $S: \mathcal{C} \to \mathcal{C}$ on a triangulated category satisfies: $\text{Hom}(A, B)^* \cong \text{Hom}(B, S(A))$

functorially in $A$ and $B$ .

For $D^b(Coh(X))$ , the Serre functor is $S = - \otimes K_X [n]$ where $K_X$ is the canonical bundle and $[n]$ is shift by dimension. For Calabi-Yau manifolds with $K_X \cong \mathcal{O}_X$ , the Serre functor is $S = [n]$ , the shift functor.

On the Fukaya side, the Serre functor corresponds to the antipodal map or symplectic rotation, reflecting the duality in Floer theory.

Stability Conditions and BridgelandSpaces

The space of stability conditions on a derived category carries important geometric information.

DefinitionBridgeland Stability

A stability condition on a triangulated category $\mathcal{C}$ consists of:

A heart $\mathcal{A} \subset \mathcal{C}$ (abelian subcategory)
A central charge $Z: K_0(\mathcal{C}) \to \mathbb{C}$

satisfying compatibility: $\arg(Z(E)) > \arg(Z(F))$ for subobjects $F \subset E$ in $\mathcal{A}$ .

The space $\text{Stab}(\mathcal{C})$ of stability conditions is conjectured to be a complex manifold. For Calabi-Yau manifolds, this space connects to:

Stringy Kähler moduli: The space of stability conditions mirrors the complexified Kähler cone
Mirror symmetry: Stability conditions on $D^b(Coh(X))$ parametrize special Lagrangians in $X^\vee$

Categorical Entropy and Dynamics

The auto-equivalences of derived categories encode geometric transformations.

Remark

The auto-equivalence group $\text{Aut}(D^b(Coh(X)))$ contains:

Tensor product with line bundles: $- \otimes \mathcal{L}$
Shift functors: $[k]$ for $k \in \mathbb{Z}$
Spherical twists: Associated to spherical objects

These generate a large group connecting to birational geometry and mirror symmetry dualities.

Hodge Structures from Categories

The categorical data recovers Hodge-theoretic information.

DefinitionCategorical Hodge Structure

The categorical Hodge structure is defined using: $H^k_{\text{cat}}(X) = \bigoplus_{i-j=k} \text{Ext}^i(E, E \otimes K_X^j)$

for suitable objects $E$ . This recovers the geometric Hodge structure under appropriate conditions.

For Calabi-Yau manifolds, the periodic cyclic homology of the derived category carries a mixed Hodge structure isomorphic to the geometric one via the Hochschild-Kostant-Rosenberg isomorphism.

ExampleHMS for Tori

For complex tori $T$ and dual torus $\hat{T}$ : $D^b(Coh(T)) \simeq D\mathcal{F}uk(\hat{T})$

The equivalence exchanges:

Line bundles $\leftrightarrow$ Lagrangian tori (fibers of moment map)
Skyscraper sheaves $\leftrightarrow$ Special Lagrangian sections

This classical example illustrates all key features of HMS in a computable setting.

These properties demonstrate how HMS unifies symplectic and complex geometry through category theory, providing structural explanations for mirror symmetry phenomena.