Let $L \subset X$ be a compact special Lagrangian $n$-submanifold in a Calabi-Yau $n$-fold $(X, \omega, \Omega)$.

Step 1: Deformation Complex

Special Lagrangian deformations are infinitesimally described by normal vector fields $\nu \in \Gamma(NL)$ satisfying: $\text{Im}(\Omega)(-, J\nu) = 0$

where $NL$ is the normal bundle and $J$ is the complex structure.

The tangent space to the moduli space is: $T_L\mathcal{M}_{\text{SLag}} = \{\nu : \text{div}_L(\nu) = 0, \, \text{Im}(\Omega)(-, J\nu) = 0\}$

Step 2: Identification with Cohomology

Using the Calabi-Yau structure, normal vectors can be identified with 1-forms on $L$ via: $\nu \mapsto \alpha = \iota_\nu(\omega|_L)$

The special Lagrangian condition $\text{Im}(\Omega)|_L = 0$ implies: $d\alpha = 0 \quad \text{(harmonicity)}$

Thus: $T_L\mathcal{M}_{\text{SLag}} \cong H^1(L, \mathbb{R})$

Step 3: Obstruction Theory

Obstructions to extending infinitesimal deformations lie in: $H^2(L, \mathbb{R})$

For a 3-torus $L = T^3$ in a Calabi-Yau threefold, we have: $H^2(T^3, \mathbb{R}) = \mathbb{R}^3$

However, the special Lagrangian equation is elliptic, and McLean shows obstructions vanish.

Step 4: Elliptic PDE Analysis

The special Lagrangian equation for a graph over $L$ with height function $u$ is: $\text{div}\left(\frac{Du}{\sqrt{1 + |Du|^2}}\right) = H(u)$

where $H(u)$ depends on the embedding. This is a fully nonlinear elliptic PDE.

The linearization at $u=0$ is: $\Delta_L v = 0$

The Fredholm theory for this operator gives:

• Index = $\dim H^1(L) - \dim H^2(L) = b_1(L) - b_2(L)$
• For $T^3$: index $= 3 - 3 = 0$

Step 5: Smoothness via Implicit Function Theorem

The nonlinear operator: $F: C^{k,\alpha}(L) \to C^{k-2,\alpha}(L), \quad u \mapsto \text{special Lagrangian equation}(u)$

is smooth and has surjective linearization $DF|_0 = \Delta_L$. By the implicit function theorem for Banach spaces: $F^{-1}(0) \cong \ker(DF|_0) = H^1(L, \mathbb{R})$

near $L$.

Step 6: Unobstructedness

The key observation is that for Calabi-Yau threefolds with $L = T^3$: $H^2(L, \mathbb{R}) \neq 0 \quad \text{but obstructions vanish}$

This follows from the fact that the obstruction map: $\mathcal{O}: H^1(L) \otimes H^1(L) \to H^2(L)$

vanishes identically for special Lagrangians in Calabi-Yau manifolds.

Conclusion: The moduli space $\mathcal{M}_{\text{SLag}}$ is smooth near $L$ with: $\dim \mathcal{M}_{\text{SLag}} = h^1(L) = 3$

for $L = T^3$.