[Paper Review] Quasi-Topological Gauged Sigma Models, The Geometric Langlands Program, And Knots

Meng-Chwan Tan|arXiv (Cornell University)|Nov 2, 2011

Black Holes and Theoretical Physics65 references3 citations

TL;DR

This paper constructs a quasi-topological (0,2) gauged sigma model on a flag manifold with a Cartan subgroup gauge group, showing its chiral algebra realizes Twisted Chiral Differential Operators (TCDOs). Using T-duality, it provides a physical derivation of the geometric Langlands correspondence for simply-connected, simple, complex Lie groups, linking Hecke eigensheaves to correlation functions and connecting knot invariants to quantum D-modules and Langlands duality.

ABSTRACT

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.

Motivation & Objective

• To construct a (0,2) quasi-topological gauged sigma model on a G-manifold with a Cartan subgroup gauge group, realizing the mathematical theory of Twisted Chiral Differential Operators (TCDOs) in a physical framework.
• To establish a physical interpretation of the geometric Langlands correspondence for simply-connected, simple, complex Lie groups via T-duality and chiral algebra structures.
• To relate nonperturbative worldsheet instantons to spontaneous supersymmetry breaking and conditions for Beilinson-Drinfeld D-modules.
• To connect knot invariants in three-manifolds and Khovanov homology to quantum ramified D-modules and Lagrangian Floer homology via Chern-Simons theory.
• To provide a physical proof of mathematical conjectures by Seidel-Smith and Gaitsgory relating knots, symplectic geometry, and Langlands duality.

Proposed method

• Construct a non-dynamically gauged (0,2) twisted sigma model on a flag manifold with a Cartan subgroup as the gauge group, deriving its chiral algebra from the perturbative sector.
• Identify the chiral algebra of the model with the sheaf of TCDOs on the flag manifold, establishing a physical realization of this mathematical structure.
• Apply generalized T-duality to relate the chiral algebra of the original model to that of the Langlands dual group, mapping W-algebras and affine Kac-Moody algebras at critical level.
• Analyze the infinite-volume limit of the model to recover Chern-Simons theory, linking correlation functions to knot invariants and quantum group representations.
• Use supersymmetric gauged quantum mechanics on loop space to interpret nonperturbative instanton effects as obstructions to chiral algebra structure, linking to the Witten genus vanishing conjecture.
• Relate correlation functions of chiral operators to Hecke eigensheaves and Hecke modifications, showing that T-duality maps the geometric Langlands correspondence to a classical limit (c → 0).

Experimental results

Research questions

• RQ1 How can a (0,2) quasi-topological gauged sigma model realize the mathematical structure of Twisted Chiral Differential Operators (TCDOs) in a physical framework?
• RQ2 What is the role of T-duality in connecting the chiral algebra of the model to the geometric Langlands correspondence for simply-connected, simple, complex Lie groups?
• RQ3 How do nonperturbative worldsheet instantons affect the chiral algebra and what is their implication for the existence of Beilinson-Drinfeld D-modules?
• RQ4 Can knot invariants in three-manifolds and Khovanov homology be physically interpreted via the model’s correlation functions and Chern-Simons theory?
• RQ5 Does the model provide a physical proof of Gaitsgory’s conjecture relating the classical limit of the geometric Langlands correspondence to representations of the Langlands dual group?

Key findings

• The perturbative chiral algebra of the (0,2) gauged sigma model on a flag manifold with Cartan subgroup gauge group is isomorphic to the sheaf of Twisted Chiral Differential Operators (TCDOs) on the flag manifold.
• T-duality maps the chiral algebra of the model to that of the Langlands dual group, realizing the geometric Langlands correspondence as a duality between chiral algebras and correlation functions.
• In the R → 0 limit, the model's correlation functions correspond to classical ramified D-modules on the moduli space of holomorphic bundles, establishing the classical geometric Langlands correspondence.
• Nonperturbative worldsheet twisted-instantons can trivialize the chiral algebra, leading to spontaneous supersymmetry breaking and imposing delicate conditions on the existence of Beilinson-Drinfeld D-modules.
• The model links knot invariants of three-manifolds to quantum ramified D-modules and Lagrangian intersection Floer homology, with correlation functions realizing quantum group representations via Uq(LgC).
• The paper provides a physical proof of Gaitsgory’s conjecture that the classical limit (c → 0) of the geometric Langlands correspondence corresponds to representations of the Langlands dual group LGC.