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Hofstadter's Butterfly and Langlands Duality

Kazuki Ikeda

Abstract

We dig out a deeper mathematical structure of the quantum Hall system from a perspective of the Langlands program. An algebraic expression of the Hamiltonian with the quantum group is a cornerstone. The Langlands duality of the quantum group sheds light on the fractal structure of Hofstadter's butterfly. This would imply a "quantum Langlands duality".

Hofstadter's Butterfly and Langlands Duality

Abstract

We dig out a deeper mathematical structure of the quantum Hall system from a perspective of the Langlands program. An algebraic expression of the Hamiltonian with the quantum group is a cornerstone. The Langlands duality of the quantum group sheds light on the fractal structure of Hofstadter's butterfly. This would imply a "quantum Langlands duality".

Paper Structure

This paper contains 5 sections, 16 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Hofstadter's Butterfly and Lanlands Duality
  3. Preliminary
  4. Langlands Duality of Quantum Groups
  5. Discussion

Figures (1)

  • Figure 1: Fractal energy spectrum, called Hofstadter's butterfly

Theorems & Definitions (1)

  • Remark 2.1