  We construct and study a closed, two-dimensional, quasi-topological (0,2)
gauged sigma model with target space a smooth G-manifold, where G is any
compact and connected Lie group. When the target space is a flag manifold of
simple G, and the gauge group is a Cartan subgroup thereof, the perturbative
model describes, purely physically, the recently formulated mathematical theory
of "Twisted Chiral Differential Operators". This paves the way, via a
generalized T-duality, for a natural physical interpretation of the geometric
Langlands correspondence for simply-connected, simple, complex Lie groups. In
particular, the Hecke eigensheaves and Hecke operators can be described in
terms of the correlation functions of certain operators that underlie the
infinite-dimensional chiral algebra of the flag manifold model. Nevertheless,
nonperturbative worldsheet twisted-instantons can, in some situations,
trivialize the chiral algebra completely. This leads to a spontaneous breaking
of supersymmetry whilst implying certain delicate conditions for the existence
of Beilinson-Drinfeld D-modules. Via supersymmetric gauged quantum mechanics on
loop space, these conditions can be understood to be intimately related to a
conjecture by Hohn-Stolz [1] regarding the vanishing of the Witten genus on
string manifolds with positive Ricci curvature. An interesting connection to
Chern-Simons theory is also uncovered, whence we would be able to (i) relate
general knot invariants of three-manifolds and Khovanov homology to "quantum"
ramified D-modules and Lagrangian intersection Floer homology; (ii) furnish
physical proofs of mathematical conjectures by Seidel-Smith [2] and Gaitsgory
[3, 4] which relate knots to symplectic geometry and Langlands duality,
respectively.
