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Two-parameter Langlands Correspondence

Mina Aganagic, Edward Frenkel, Andrei Okounkov

TL;DR

This work introduces a two-parameter deformation of the geometric Langlands correspondence by equating deformed conformal blocks of the $W_{q,t}(g)$ algebra with those of a quantum affine algebra, for simply-laced $g$, via the quantum K-theory of Nakajima quiver varieties. The central technical bridge is the realization of both block spaces as vertex function solutions to $q$-difference equations, enabling a canonical identification mediated by elliptic stable envelopes and the geometry of the quiver variety $X$. The authors connect this algebraic correspondence to six-dimensional string theory dualities and propose a non-simply laced extension, where the electric side undergoes a more subtle deformation and a folded/modified $q$-deformation of KZ-type equations is anticipated. Overall, the paper provides a concrete, geometrically anchored formulation of a quantum ($q$-)Langlands program and links representation theory to enumerative geometry and string-duality structures, with clear directions for extending the framework beyond simply-laced cases.

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.

Two-parameter Langlands Correspondence

TL;DR

This work introduces a two-parameter deformation of the geometric Langlands correspondence by equating deformed conformal blocks of the algebra with those of a quantum affine algebra, for simply-laced , via the quantum K-theory of Nakajima quiver varieties. The central technical bridge is the realization of both block spaces as vertex function solutions to -difference equations, enabling a canonical identification mediated by elliptic stable envelopes and the geometry of the quiver variety . The authors connect this algebraic correspondence to six-dimensional string theory dualities and propose a non-simply laced extension, where the electric side undergoes a more subtle deformation and a folded/modified -deformation of KZ-type equations is anticipated. Overall, the paper provides a concrete, geometrically anchored formulation of a quantum (-)Langlands program and links representation theory to enumerative geometry and string-duality structures, with clear directions for extending the framework beyond simply-laced cases.

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.
Paper Structure (14 sections, 1 theorem, 22 equations, 1 figure)

This paper contains 14 sections, 1 theorem, 22 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Statement of the correspondence
  3. Electric side
  4. Magnetic side
  5. Main result
  6. Geometry behind the correspondence
  7. Nakajima quiver variety
  8. Equivariant variables
  9. Vertex functions
  10. Outline of the proof of the main theorem
  11. Electric side
  12. Magnetic side
  13. The match of deformed conformal blocks
  14. Non-simply laced case

Key Result

Theorem 1

Let ${\mathfrak{g}}$ be a simply-laced simple Lie algebra. The deformed conformal blocks of $U_{\hbar}({{\widehat{^L\mathfrak{g}}}})_{^Lk}$ in electric and the deformed conformal blocks of ${\mathcal{W}}_{q,t}({\mathfrak{g}})$ in magnetic, whose parameters are generic and are related by equation zer

Figures (1)

  • Figure 1: The cylinder ${\cal C}$ with the insertions of vertex operators corresponding to finite-dimensional $U_{\hbar}({{\widehat{^L\mathfrak{g}}}})_{^Lk}$-modules $^L\!\rho_i$ at the points $a_i\in {\cal C}$. Boundary conditions at infinity are the highest weight vectors $\langle \lambda'|$ and $|\lambda \rangle$.

Theorems & Definitions (3)

  • Remark 1.1
  • Remark 1.2
  • Theorem 1