Homological Mirror Symmetry - Applications

Homological mirror symmetry has profound applications extending beyond verifying classical mirror predictions. The categorical framework provides new tools for studying both symplectic and algebraic geometry, with implications for physics and representation theory.

Bridgeland Stability and Moduli Spaces

HMS connects stability conditions on derived categories to geometric structures on moduli spaces.

TheoremStability Conditions and Special Lagrangians

For a mirror pair $(X, X^\vee)$ , stability conditions on $D^b(Coh(X))$ correspond to families of special Lagrangian submanifolds in $X^\vee$ via HMS: $\text{Stab}(D^b(Coh(X))) \cong \text{SLag}(X^\vee)/\text{Ham}$

where $\text{SLag}(X^\vee)$ is the space of special Lagrangian submanifolds modulo Hamiltonian isotopy.

This identifies the stringy Kähler moduli space with the space of stability conditions.

This deep connection allows transfer of techniques: algebraic methods for stability conditions inform symplectic geometry, while geometric intuition guides categorical constructions.

Derived Categories and Birational Geometry

HMS provides new perspectives on birational transformations.

Remark

Flops and other birational modifications induce auto-equivalences of derived categories. For Calabi-Yau threefolds, the derived category is invariant under flops: $D^b(Coh(X)) \simeq D^b(Coh(X'))$

even when $X$ and $X'$ are not isomorphic. HMS suggests this reflects the symplectic geometry of the mirror remaining unchanged.

Knot Homology and Categorification

HMS techniques extend to knot theory via categorical constructions.

TheoremHMS and Khovanov Homology

The Khovanov homology of links can be understood via HMS for certain symplectic manifolds (cotangent bundles). This connection provides:

Geometric interpretations of knot invariants
Spectral sequences relating different homology theories
Categorification of quantum invariants

The Fukaya category of $T^*S^3$ relates to the category of Soergel bimodules, which categorifies Hecke algebras.

Physics Applications

In string theory, HMS explains dualities and provides computational frameworks.

Remark

Open string mirror symmetry computes D-brane charges and central charges:

B-branes (coherent sheaves on $X$ ) $\leftrightarrow$ A-branes (Lagrangians in $X^\vee$ )
K-theory charges $\leftrightarrow$ Floer homology dimensions

HMS predicts that BPS spectra coincide, allowing calculations on whichever side is more tractable.

Enumerative Geometry

HMS recovers and extends Gromov-Witten theory predictions.

TheoremOpen Gromov-Witten Invariants via HMS

For a Lagrangian $L \subset X$ , open Gromov-Witten invariants counting holomorphic discs with boundary on $L$ equal structure constants in: $HF^*(L, L) = \text{Hom}_{\mathcal{F}uk}(L, L)$

Under HMS, these match Ext groups: $\text{Ext}^*(\mathcal{F}, \mathcal{F})$

for the corresponding sheaf $\mathcal{F} \in D^b(Coh(X^\vee))$ , providing algebraic methods for computing disc invariants.

Non-Commutative Geometry

HMS extends to non-commutative spaces via derived categories of non-commutative algebras.

ExampleNon-Commutative Tori

For non-commutative 2-tori $\mathbb{T}_\theta^2$ (deformation quantizations), HMS takes the form: $D^b(\text{Mod}(\mathbb{T}_\theta^2)) \simeq \mathcal{F}uk(\mathbb{T}^2, B)$

where $B$ is a B-field (closed 2-form). This generalizes classical HMS to non-commutative geometry, relevant for string theory with B-fields.

Representation Theory

Categorical techniques from HMS illuminate representation theory.

Remark

For flag varieties $G/P$ , the derived category $D^b(Coh(G/P))$ has exceptional collections related to Weyl group combinatorics. HMS perspectives suggest these should correspond to Lagrangian skeleta in mirror Landau-Ginzburg models, connecting representation theory to symplectic topology.

These applications demonstrate HMS as a unifying framework connecting diverse mathematical fields through categorical equivalences.