Gauging Spacetime Symmetries On The Worldsheet And The Geometric Langlands Program -- II

TL;DR

This paper extends the physical understanding of the geometric Langlands program by relating holomorphic chiral algebras of bosonic strings on coset manifolds to W-algebras, generalizing previous results beyond SL(N,C).

Contribution

It demonstrates an equivalence between certain bosonic string models and their gauged versions, establishing a new physical interpretation of the geometric Langlands correspondence for complex ADE groups.

Findings

Isomorphism of classical W-algebras for ADE groups

Level relations underpinning the geometric Langlands correspondence

Physical interpretation of Hecke operators via chiral algebra correlations

Abstract

We generalise the analysis carried out in [arXiv:0710.5796], and find that our previous results can be extended beyond the case of SL(N,C). In particular, we show that an equivalence--at the level of the holomorphic chiral algebra--between a bosonic string on a smooth coset manifold 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 the simply-laced, complex ADE-groups. In addition, as opposed to line operators and branes of an open topological sigma-model, the Hecke operators and Hecke eigensheaves, can, instead, be physically interpreted in terms of the correlation functions of local operators in the holomorphic chiral algebra of a closed, quasi-topological sigma-model. Our present results thus serve as an alternative physical interpretation--to that of an electric-magnetic duality of four-dimensional gauge theory demonstrated earlier by Kapustin and Witten in [arXiv:hep-th/0604151]--of the geometric Langlands correspondence for complex ADE-groups. The cases with tame and mild "ramifications" are also discussed.