Electric-magnetic duality and the geometric Langlands program

Abstract

The geometric Langlands program can be described in a natural way by compactifying on a Riemann surface C a twisted version of N = 4 super Yang-Mills theory in four dimensions. The key ingredients are electric-magnetic duality of gauge theory, mirror symmetry of sigma-models, branes, Wilson and 't Hooft operators, and topological field theory. Seemingly esoteric notions of the geometric Langlands program, such as Hecke eigensheaves and D-modules, arise naturally from the physics.

Localized modes at a D-brane—O-plane intersection and heterotic Alice strings

“…This result agrees with the transformation for the N = 4 supersymmetries and fermions derived in [50] by other means. Also, observe that under S 2 = −½ we have ψ → iψ.…”

“…Fortunately, there is an ambiguity in the SL(2, ) transformation of the fermions. As was pointed out in [50], the transformation is not unique, but rather can be combined with an automorphism of the supersymmetry algebra. The Montonen-Olive conjecture states that SL(2, ) should commute with the Poincare symmetries, but there is still the global SU (4) R symmetry.…”

“…It is not physically equivalent to a shift by π, or −π/2, etc. A shift by π, for example, would be generated by the element J 2 in SU (4)R. But, as is also discussed in [50], this corresponds to a 2 fermion number transformation (−1) F . And this is the 2 element that is already used to extend SL(2, ) to its double cover, under which the fermions transform.…”

BPS string webs in the 6-dim (2,0) theories

“…(3.29) of Ref. [17] which discusses the geometric Langlands program. One can show the above equations imply that the BPS configurations satisfy the field equations.…”

Mirror Symmetry, Hitchin's Equations, and Langlands Duality

“…This rather surprising fact is natural from the point of view of generalized complex geometry, see Hitchin (2003), and has been explained from that point of view in section 4.6 of Gualtieri (2003), as a general statement about complex symplectic manifolds. In Kapustin & Witten (2007), sections 5.2 and 11.3, it was shown that quantum geometric Langlands is naturally understood in precisely this setting.…”