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Abelian Group Theory by D. Arnold, R. Hunter, E. Walker

By D. Arnold, R. Hunter, E. Walker

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Noncommutative field theory such as Eq. , 1985; Laughlin, 1985). See also Fradkin (1991). In this picture, the original quasiparticles are vortex solutions, and their fractional statistics is reproduced by an Abelian Chern-Simons term in the action S CS ϭ im 2␲ ͵⑀ ijk A i‫ ץ‬jA k . (127) With this term, a vortex with unit magnetic charge will also carry electric charge 1/m, so the Aharonov-Bohm effect will lead to fractional statistics (Wilczek, 1982). Recently Susskind (2001) has proposed that noncommutative Chern-Simons theory is a better description of fractional quantum Hall states, which can reproduce the detailed properties of these quasiparticles.

Vol. 73, No. 4, October 2001 FIG. 6. T duality to an anisotropic torus. wind about the x 2 dimension will also become light and must be included in the action; their winding number w 2 becomes a new component of momentum in 2ϩ1-dimensional gauge theory. In this anisotropic geometry, the two ends of a winding open string will have different locations in x 1 , with separation ␪ w 2 . Thus the fundamental objects turn out to be dipoles in exactly the sense described in Sec. 2, with the corresponding noncommutative interactions.

In particular, the quasiparticles are well-defined excitations of the noncommutative gauge field; for example, the quasihole is rather similar to the fluxon Eq. (83). We refer to the cited references for more details. Rev. Mod. , Vol. 73, No. 4, October 2001 VI. MATHEMATICAL ASPECTS As we mentioned in the Introduction, noncommutative gauge theory was first clearly formulated by mathematicians to address questions in noncommutative geometry. Limitations on length would not permit more than the most cursory introduction to this subject here, and since so many introductions are already available, starting with the excellent Connes (1994), much of which is quite readable by physicists, and including Connes (1995, 2000a, 2000b), Douglas (1999), Gracia-Bondia et al.

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