Operator Algebra in Chern-Simons Theory on a Torus.pdf

Operator Algebra in Chern-Simons Theory on a Torus.pdf

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Operator Algebra in Chern-Simons Theory on a Torus

a r X i v : h e p - t h / 9 2 1 0 1 0 3 v 1 1 9 O c t 1 9 9 2 Theoretical Physics Institute University of Minnesota September 1992 TPI-MINN-92/41-T UMN-TH-1105/92 Operator Algebra in Chern-Simons Theory on a Torus Choon-Lin Ho Theoretical Physics Institute, University of Minnesota, Minneapolis, MN 55455, U.S.A., and Department of Physics, Tamkang University, Tamsui, Taiwan 25137, R.O.C.? and Yutaka Hosotani School of Physics and Astronomy, University of Minnesota, Minneapolis, MN 55455, U.S.A. Abstract We consider Chern-Simons gauge theory on a torus with both nonrela- tivistic and relativistic matter. It is shown that the Hamiltonian and two total momenta commute among themselves only in the physical Hilbert space. We also discuss relations among degenerate physical states, degenerate vacua, and the existence of multicomponent Schro?dinger wavefunctions. ? Permanent address Chern-Simons field theory with matter coupling have attracted intense interest in recent years, owing to its relevance to condensed matter systems such as quantum Hall systems, and possibly high Tc superconductors [1]. Setting aside such physical applica- bility, Chern-Simons theory is in itself very interesting in view of its rich and beautiful mathematical structures. As such, various aspects of this theory deserve a careful study. Much of recent efforts have been directed towards the issue of consistent quantisation of the theory [2,3]. While the majority of works in the field is concerned with planar systems, Chern- Simons field theory on compact Riemann surfaces has also captured considerable interests [4–14]. It has even richer structures, which, being topological in nature, are absent in planar system. Among them are the multicomponent structure of many-body wavefunctions [9,11,14] and the degeneracy of physical states [5]. Moreover, the analysis on a torus is mathematically rigorous, being free from infrared divergences and ambiguity in boundary conditions at space infinity on a pla

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