The HYM equation famously fits into a formal geometric invariant theory (GIT) framework. For this, we slightly change viewpoint, and consider the bundle $E$ equipped with a fixed Hermitian structure, while the $(0,1)$-connection $\bar{\partial}_{E}$ encoding the holomorphic structure is allowed to vary. Each $\bar{\partial}_{E}$ uniquely determines the Chern connection $\nabla_{E}$. The group of complex gauge transformations $GL(E,\mathbb{C})$ acts on the space of integrable $(0,1)$-connections, via $\bar{\partial}_{E}\mapsto g\circ\bar{\partial}_{E}\circ g^{-1}$. This action is analogous to a complex reductive group action on a finite dimensional Kähler manifold. The subgroup $U(E)$ of unitary gauge transformations is analogous to the maximal compact subgroup. The space $GL(E,\mathbb{C})/U(E)$ can be formally identified with the space of Hermitian metrics on $E$. The HYM equation arises naturally from considerations of the moment map, and the Donaldson-Uhlenbeck-Yau theorem can be formally motivated from this picture [27].

This kind of infinite dimensional GIT framework has successfully suggested the answer in many problems within Kähler geometry, so it is only natural that many people have attempted to find an analogue suitable for special Lagrangian geometry. One such attempt is as follows. Thomas [65, section 3] considered the space

$$\mathcal{Z}=\{(L,A):L\subset X\text{ is Lagrangian},A\text{ is a flat $U(1)$-connection on }L\}$$

(not up to gauge equivalence!). The tangent space is $Z^{1}(L)\oplus Z^{1}(L)$, where $Z^{1}(L)$ is the space of closed 1-forms on $L$. This suggests an almost complex structure on $\mathcal{Z}$

$$J=\begin{pmatrix}0&1\\ -1&0\end{pmatrix}.$$

With some hesitation,²⁷²⁷ 27 Thomas was aware of the possible objections, and did not use the GIT analogy as the principal basis of his proposal. Thomas attempted to complexify the Hamiltonian group action into a complex infinite dimensional group action. Unfortunately, there seems to be no natural way to do this in general, and $J$ is not quite an integrable complex structure. On the other hand, the moduli space of special Lagrangians with $U(1)$-local systems does have a natural complex structure induced from $J$, which is however naïve in the sense that the complex structure of the moduli space of Lagrangian branes is subject to further quantum corrections due to holomorphic curves, known also as ‘worldsheet instantons’ [8].

There seems to be no agreed interpretation, but in the author’s view, this suggests the infinite dimensional GIT framework is itself inadequate for the purpose of special Lagrangian geometry.²⁸²⁸ 28 Thomas’s suggestion is not the only possible way to achieve a GIT analogy. However, the other proposals [28][57] do not exhibit the mirror analogy in the same intuitively plausible way. As we explained in section 2.4, the main predictions of mirror symmetry are quantum in nature, and one should be cautious about taking an overly classical perspective. From our perspective, one underlying reason why the infinite dimensional GIT framework is successful in Kähler geometry, is that on the B-side one does not see the worldsheet instanton effect directly. On the A-side we have no such luxury.

As a word of console, although GIT is very good at suggesting the correct stability conditions for a PDE problem in Kähler geometry, it is almost never involved in the actual proofs of existence and uniqueness results for PDEs.