Two-parameter Langlands Correspondence

TL;DR

This paper surveys the authors' previous work on a q-deformation of the Langlands correspondence for simply-laced Lie algebras, connecting conformal blocks, quantum affine algebras, and quantum K-theory, and proposes an extension to non-simply laced cases.

Contribution

It extends the quantum Langlands correspondence to non-simply laced Lie algebras, proposing a new duality framework.

Findings

Established a canonical isomorphism for simply-laced Lie algebras

Linked the isomorphism to 6d little string theory duality

Proposed extension to non-simply laced Lie algebras

Abstract

In our paper arXiv:1701.03146 we established, for every simply-laced Lie algebra g, a canonical isomorphism between the spaces of deformed conformal blocks of the deformed W-algebra and the quantum affine algebra corresponding to g, which we view as a q-deformation of the quantum Langlands correspondence. This was done by realizing the deformed conformal blocks of these algebras via the quantum K-theory of the Nakajima quiver varieties. We also linked this isomorphism to a duality emerging from the 6d little string theory. Here, we give a brief survey of these results and propose an extension to the non-simply laced case, which exhibits a Langlands-type duality.