Quasi-topological gauged sigma models, the geometric Langlands program, and knots

We construct and study a closed, 2-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 $\mathcal{D}$-modules. Via supersymmetric gauged quantum mechanics on loop space, these conditions can be understood to be intimately related to a conjecture by Höhn-Stolz 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 $\mathcal{D}$-modules and Lagrangian intersection Floer homology; (ii) furnish physical proofs of mathematical conjectures by Seidel-Smith and Gaitsgory which relate knots to symplectic geometry and Langlands duality, respectively.

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Published 23 March 2015