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.

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 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.

  • $\begingroup$ so, if necessary, I will split this post into several ones. please do! As ACuriousMind states, the two headings could easily be separated into two questions. $\endgroup$ – Kyle Kanos Mar 6 '15 at 13:37
  • $\begingroup$ @KyleKanos Done! $\endgroup$ – user40276 Mar 6 '15 at 15:51

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