Quantum Hall Effect and Langlands Program
arXiv:1708.00419 · doi:10.1016/j.aop.2018.08.002
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.
arXiv comment21 pages, 4 figures
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Cited by in corpus (9)
- Hyperbolic Band Theory under Magnetic Field and Dirac Cones on a Higher Genus Surface
- A slow review of the AGT correspondence
- Selberg trace formula in hyperbolic band theory
- Hyperbolic band theory through Higgs bundles
- Hofstadter's Butterfly and Langlands Duality
- Fractal defect states in the Hofstadter butterfly
- Algebra of Hyperbolic Band Theory under Magnetic Field
- Topological Aspects of Matters and Langlands Program
- Universal Computation with Quantum Fields