Gauging Spacetime Symmetries on the Worldsheet and the Geometric Langlands Program

TL;DR

This paper explores the connection between two-dimensional twisted sigma-models on complex flag manifolds and the geometric Langlands program, offering a new physical perspective on the correspondence for G=SL(N,C).

Contribution

It establishes an equivalence at the level of holomorphic chiral algebra between a bosonic string on G/B and a B-gauged version on G, linking to the geometric Langlands correspondence.

Findings

Identifies an isomorphism of classical W-algebras underlying the Langlands correspondence.

Provides an alternative physical interpretation of Hecke operators and eigensheaves.

Proposes a framework for a quantum geometric Langlands correspondence.

Abstract

We study the two-dimensional twisted (0,2) sigma-model on various smooth complex flag manifolds G/B, and explore its relevance to the geometric Langlands program. We find that an equivalence - at the level of the holomorphic chiral algebra - between a bosonic string on G/B and a B-gauged version of itself on G, will imply an isomorphism of classical W-algebras and a level relation which underlie a geometric Langlands correspondence for G=SL(N,C). This furnishes an alternative physical interpretation of the geometric Langlands correspondence for G=SL(N,C), to that demonstrated earlier by Kapustin and Witten via an electric-magnetic duality of four-dimensional gauge theory. Likewise, the Hecke operators and Hecke eigensheaves will have an alternative physical interpretation in terms of the correlation functions of local operators in the holomorphic chiral algebra of a quasi-topological sigma-model without boundaries. A forthcoming paper will investigate the interpretation of a ``quantum'' geometric Langlands correspondence for G=SL(N,C) in a similar setting, albeit with fluxes of the sigma-model moduli which induce a ``quantum'' deformation of the relevant classical algebras turned on.