There has been a significant amount of work in recent years by many mathematical physicists viz. Witten, Kapustin etc. on understanding the geometric Langlands and its relation to non-perturbative effects string theory and gauge theory.
My questions are as follows:
a) What is the physical interpretation of the various conjectures in the original Langlands programme?
b) The geometric Langlands has an interpretation in terms of the S-duality of the gauge theory and also can be used to study mirror symmetries of $2d$ and $4d$ gauge theories. (Witten, Kapustin) Is this $2d$ - $4d$ gauge theory appearance a coincidence or is it related to exact non-perturbative symmetries in which a $6d$ theory is reduced to a $2d$ or a $4d$ theory in various limits? In other words, what can the geometric Langlands tell us about AGT and exact symmetries? In particular, what theories are related by mirror symmetry in this case?
c) What can one learn about non-Abelian gauge theories from the Geometric Langlands?
This post imported from StackExchange Physics at 2017-04-23 15:20 (UTC), posted by SE-user Eh-whaaa