Intuition for Homological Mirror Symmetry

Question

first of all, I need to confess my ignorance with respect to any 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.

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 commutes with $Q$, $Q^2 = 0$ and some other properties that I actually don't understand. Then to construct this $Q$, one would need to recover the action of $\text{Spin}(6)$ (because the 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 a 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 is 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 sympletic 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.

1) But what's the physical interpretation of a Lagrangian submanifold with a unitary local system?

2) What's the physical intuition for A-models and B-models (or exchanging "models" by "branes")?

3) What's the physical interpretation of this interplay between complex structures and sympletic ones (coming from the former one)?

Thanks in advance.

This post imported from StackExchange Physics at 2015-03-17 04:42 (UTC), posted by SE-user user40276

Comments

Re 'And, then the theory would be non-perturbative, since it would be defined "for all" τ, because amplitudes are computed with an expansion in power series in τ':

Actually, to a physicist, such a power-series expansion is the hallmark of (the outcome of) a perturbative theory: Such power series typically correspond to some perturbation calculated to (arbitrarily) high order. A base in the coefficient corresponds to a physical coupling constant and causes such approaches to become invalid for large (e.g. unity) coupling constants as the power series no longer converges.

Please take this comment with a grain of salt: I am myself from a foreign field (to theoretical physics) as I am a mere quantumoptics experimental physicist curious about expanding my mental horizon. This is just my first "that's usually like this" association.

---

The answer to this question would easily take thousands of pages. A first thousand is the Clay math book "Mirror Symmetry".