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Lectures on the Langlands Program and Conformal Field Theory

Edward Frenkel

TL;DR

The notes give a comprehensive tour of the Langlands program from its number-field origins to its geometric reformulation and a conformal-field-theory perspective. They articulate how local systems on curves correspond to Hecke eigensheaves on Bun_G, with Beilinson–Drinfeld providing a concrete CFT-based construction via opers and the critical-level center. A central thread is the geometric Satake correspondence, the affine Grassmannian, and the Hitchin system, which together realize a non-abelian Fourier–Mukai-like duality between D-modules on Bun_G and O-modules on Loc_{^L G}. The treatise also develops the theory of ramification, parabolic structures, and deformations, highlighting deep links to S-duality, D-branes, and Hitchin integrable systems, and laying groundwork for a full geometric Langlands program for general G.

Abstract

These lecture notes give an overview of recent results in geometric Langlands correspondence which may yield applications to quantum field theory. We start with a motivated introduction to the Langlands Program, including its geometric reformulation, addressed primarily to physicists. I tried to make it as self-contained as possible, requiring very little mathematical background. Next, we describe the connections between the Langlands Program and two-dimensional conformal field theory that have been found in the last few years. These connections give us important insights into the physical implications of the Langlands duality.

Lectures on the Langlands Program and Conformal Field Theory

TL;DR

The notes give a comprehensive tour of the Langlands program from its number-field origins to its geometric reformulation and a conformal-field-theory perspective. They articulate how local systems on curves correspond to Hecke eigensheaves on Bun_G, with Beilinson–Drinfeld providing a concrete CFT-based construction via opers and the critical-level center. A central thread is the geometric Satake correspondence, the affine Grassmannian, and the Hitchin system, which together realize a non-abelian Fourier–Mukai-like duality between D-modules on Bun_G and O-modules on Loc_{^L G}. The treatise also develops the theory of ramification, parabolic structures, and deformations, highlighting deep links to S-duality, D-branes, and Hitchin integrable systems, and laying groundwork for a full geometric Langlands program for general G.

Abstract

These lecture notes give an overview of recent results in geometric Langlands correspondence which may yield applications to quantum field theory. We start with a motivated introduction to the Langlands Program, including its geometric reformulation, addressed primarily to physicists. I tried to make it as self-contained as possible, requiring very little mathematical background. Next, we describe the connections between the Langlands Program and two-dimensional conformal field theory that have been found in the last few years. These connections give us important insights into the physical implications of the Langlands duality.

Paper Structure

This paper contains 66 sections, 10 theorems, 267 equations.

Table of Contents

  1. The origins of the Langlands Program
  2. The Langlands correspondence over number fields
  3. Galois group
  4. Abelian class field theory
  5. Frobenius automorphisms
  6. Rigidifying ACFT
  7. Non-abelian generalization?
  8. Automorphic representations of $GL_2({\mathbb A}_{\mathbb Q})$ and modular forms
  9. Elliptic curves and Galois representations
  10. From number fields to function fields
  11. Function fields
  12. Galois representations
  13. Automorphic representations
  14. The Langlands correspondence
  15. The geometric Langlands Program
  16. ...and 51 more sections

Key Result

Theorem 1

There is a bijection between the sets ${\mathcal{G}}_n$ and ${\mathcal{A}}_n$ defined above which satisfies the following matching condition. If $\sigma \in {\mathcal{G}}_n$ corresponds to $\pi \in {\mathcal{A}}_n$, then the sets of points where they are unramified are the same, and for each $x$ fro up to permutation.

Theorems & Definitions (11)

  • Theorem 1
  • Lemma 2
  • Theorem 3
  • Theorem 4
  • Theorem 5
  • Theorem 6
  • Conjecture 1
  • Theorem 7
  • Theorem 8
  • Theorem 9
  • ...and 1 more