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, Q2=0 and some other properties that I actually don't understand. Then to construct this Q, one would need to recover the action of 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 P1.
Compactifying this M4=Σ×X gives us a topological σ-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 σ-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.
No comments:
Post a Comment