Discussion of the integer quantum Hall effect via a Brillouin torus with noncommutative geometry and using the Connes-Chern character:
Generalization of BvESB94 to the fractional quantum Hall effect:
See also exposition in:
Discussion of the fractional quantum Hall effect via abelian but noncommutative (matrix model-)Chern-Simons theory:
Leonard Susskind: The Quantum Hall Fluid and Non-Commutative Chern Simons Theory [arXiv:hep-th/0101029]
Simeon Hellerman, Leonard Susskind: Realizing the Quantum Hall System in String Theory [arXiv:hep-th/0107200]
(relating this to M5-branes via the BFSS matrix model)
Alexios P. Polychronakos: Quantum Hall states as matrix Chern-Simons theory, JHEP 0104:011 (2001) [doi:10.1088/1126-6708/2001/04/011, arXiv:hep-th/0103013]
Simeon Hellerman, Mark Van Raamsdonk: Quantum Hall Physics = Noncommutative Field Theory, JHEP 0110:039 (2001) [doi:10.1088/1126-6708/2001/10/039, arXiv:hep-th/0103179]
Eduardo Fradkin, Vishnu Jejjala, Robert G. Leigh: Non-commutative Chern-Simons for the Quantum Hall System and Duality, Nucl. Phys. B 642 (2002) 483-500 [doi:10.1016/S0550-3213(02)00616-8, arXiv:cond-mat/0205653]
Andrea Cappelli, Ivan D. Rodriguez: Matrix Effective Theories of the Fractional Quantum Hall effect, J. Phys. A 42 (2009) 304006 [doi:10.1088/1751-8113/42/30/304006, arXiv:0902.0765]
Zhihuan Dong, T. Senthil: Non-commutative field theory and composite Fermi Liquids in some quantum Hall systems, Phys. Rev. B 102 (2020) 205126 [doi:10.1103/PhysRevB.102.205126, arXiv:2006.01282]
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