Let $E = \mathbb{C}/\Lambda$ be an elliptic curve and $\hat{E} = \mathbb{C}/\Lambda^\vee$ its dual torus. We construct an equivalence $\Phi: D^b(Coh(E)) \to D\mathcal{F}uk(\hat{E})$.

Step 1: Generators

The derived category $D^b(Coh(E))$ is generated by:

• Structure sheaf $\mathcal{O}_E$
• Skyscraper sheaf $\mathcal{O}_p$ at a point $p \in E$

Every coherent sheaf on $E$ can be built from these via exact triangles and shifts.

Step 2: Lagrangian Generators

The Fukaya category $D\mathcal{F}uk(\hat{E})$ is generated by:

• Zero section $L_0 \subset \hat{E}$
• Fiber $F$ of the projection $\hat{E} = \mathbb{C}/\Lambda^\vee \to S^1$

These Lagrangian circles generate via products and taking cones.

Step 3: Define Functor on Generators

Set: $\Phi(\mathcal{O}_E) = L_0, \quad \Phi(\mathcal{O}_p) = F$

This assignment must preserve triangulated structure and Ext groups.

Step 4: Verify Ext Groups

We need: $\text{Ext}^i(\mathcal{O}_E, \mathcal{O}_p) \cong HF^i(L_0, F)$

Left side: By Serre duality and Riemann-Roch: $\text{Ext}^0(\mathcal{O}_E, \mathcal{O}_p) = \mathbb{C}$ $\text{Ext}^1(\mathcal{O}_E, \mathcal{O}_p) = 0$

Right side: The Lagrangians $L_0$ and $F$ intersect transversely at one point, so: $HF^0(L_0, F) = \mathbb{C}, \quad HF^1(L_0, F) = 0$

The differential in Floer complex vanishes since there are no holomorphic strips between single intersection points.

Step 5: Self-Exts and $A_\infty$ Structure

For $\text{Ext}^*(\mathcal{O}_p, \mathcal{O}_p)$: $\text{Ext}^0(\mathcal{O}_p, \mathcal{O}_p) = \mathbb{C}, \quad \text{Ext}^1(\mathcal{O}_p, \mathcal{O}_p) = \mathbb{C}^2$

This matches: $HF^*(F, F) = H^*(S^1) = \mathbb{C} \oplus \mathbb{C}^2$

The $A_\infty$ products on both sides (Yoneda product vs. holomorphic polygon counts) must match. This requires detailed analysis of holomorphic strips in $\hat{E}$.

Step 6: Line Bundles and Lagrangian Tori

Line bundles $\mathcal{L}_\alpha$ parametrized by $\alpha \in \hat{E}$ correspond to Lagrangian tori $T_\alpha \subset \hat{E}$ via: $\Phi(\mathcal{L}_\alpha) = T_\alpha$

where $T_\alpha$ is the translate of $L_0$ by $\alpha$. The Ext groups: $\text{Ext}^i(\mathcal{L}_\alpha, \mathcal{L}_\beta) \cong \begin{cases} \mathbb{C} & i=0, \alpha=\beta \\ \mathbb{C} & i=1, \alpha=\beta \\ 0 & \text{otherwise} \end{cases}$

match the Floer cohomology $HF^*(T_\alpha, T_\beta)$ computed from parallel Lagrangian tori.

Step 7: Fully Faithful and Essentially Surjective

Having verified:

1. $\Phi$ sends generators to generators
2. All Ext/HF groups match
3. $A_\infty$ products agree

We conclude $\Phi$ is fully faithful. Essential surjectivity follows from generation: every object in both categories is built from generators.

Conclusion: $\Phi$ is an equivalence of triangulated categories.