Quantum Hall Effect and Langlands Program

TL;DR

This paper explores how the Langlands program provides a unifying mathematical framework for understanding various phenomena in quantum physics, including the quantum Hall effect and fractal energy spectra.

Contribution

It connects the Langlands program with quantum physics, offering new insights into the quantum Hall effect, particle-vortex duality, and fractal spectra through geometric and quantum group dualities.

Findings

Hall conductance plateaus linked to Hecke eigensheaves

Particle-vortex duality explained by Langlands duality

Fractal energy spectrum related to Langlands duality in quantum groups

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.