first of all, I need to confess my ignorance concerning physics since I'm a mathematician. I'm interested in the physical intuition of the Langlands program, therefore I need to understand what physicists think about homological mirror symmetry. This question is related to this other one Intuition for S-duality.
Mirror Symmetry
As I have heard, the mirror symmetry can be derived from S-duality by picking a topological sector using an element of the super Lie algebra of the Lorentz group $Q$, such that things commute with $Q$, $Q^2 = 0$ and some other properties that I don't understand. Then to construct this $Q$, one would need to recover the action of $\text{Spin}(6)$ (because dimension 4 is a reduction of a 10-dimensional theory? is this correct?) and there are different ways of doing this. Anyway, passing through all the details, this is a twisting of the theory giving families of topological field theory parametrized by $\mathbb{P}^1$.
Compactifying this $M_4 = \Sigma \times X$ gives us a topological $\sigma$-model with values in Hitchin moduli space (that is hyperkähler). The Hitchin moduli space roughly can be described as semi-stable flat $G$ bundles or vector bundles with a Higgs field. However since the Hitchin moduli are kähler, there will be just two $\ sigma$ models: A-models and B-models. I don't want to write more details, so, briefly, there is an equivalence between symplectic structures and complex structures (for more details see http://arxiv.org/pdf/0906.2747v1.pdf).
So the main point is that Lagrangian submanifolds (of a Kähler-Einstein manifold) with a unitary local system should be dual to flat bundles.
But what's the physical interpretation of a Lagrangian submanifold with a unitary local system?
What's the physical intuition for A-models and B-models (or exchanging "models" by "branes")?
What's the physical interpretation of this interplay between complex structures and symplectic ones (coming from the former one)?
Thanks in advance.
