Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Quantum Hall Effect and Langlands Program

Kazuki Ikeda

Abstract

Recent advances in the Langlands program shed light on a vast area of modern mathematics from an unconventional viewpoint, including number theory, gauge theory, representation, knot theory and etc. By applying to physics, these novel perspectives endow with a unified account of the (integer/ fractional) quantum Hall effect. The plateaus of the Hall conductance are described by Hecke eigensheaves of the geometric Langlands correspondence. Especially, the particle-vortex duality, which is explained by S-duality of Chern-Simons theory, corresponds to the Langlands duality in Wilson and Hecke operators. Moreover the Langlands duality in the quantum group associated with the Hamiltonian describes fractal energy spectrum structure, know as Hofstadter's butterfly. These results suggest that the Langlands program has many physically realistic meanings.

Quantum Hall Effect and Langlands Program

Abstract

Recent advances in the Langlands program shed light on a vast area of modern mathematics from an unconventional viewpoint, including number theory, gauge theory, representation, knot theory and etc. By applying to physics, these novel perspectives endow with a unified account of the (integer/ fractional) quantum Hall effect. The plateaus of the Hall conductance are described by Hecke eigensheaves of the geometric Langlands correspondence. Especially, the particle-vortex duality, which is explained by S-duality of Chern-Simons theory, corresponds to the Langlands duality in Wilson and Hecke operators. Moreover the Langlands duality in the quantum group associated with the Hamiltonian describes fractal energy spectrum structure, know as Hofstadter's butterfly. These results suggest that the Langlands program has many physically realistic meanings.

Paper Structure

This paper contains 10 sections, 29 equations, 4 figures.

Table of Contents

  1. Introduction
  2. Integer Quantum Hall Effect and Langlands Program
  3. General Setup
  4. Hall Conductance
  5. Geometric Langlands Correspondence
  6. Hecke Eigensheaf, Landau Level, and Anderson Localization
  7. Duality, Hecke operator, and K-theoretic view
  8. Chern-Simons Theory, $S$-duality and Mirror Symmetry
  9. Hofstadter's Butterfly and Langlands Duality of Quantum Groups
  10. Finale

Figures (4)

  • Figure 1: Integer dependency of the Hall conductance
  • Figure 2: Landau level (left) and sheaf (right)
  • Figure 3: Hall conductance and Hecke eigensheaves
  • Figure 4: Fractal energy spectrum, called Hofstadter's butterfly