On fermion gauge groups, current algebras and Kac-Moody algebras

Abstract

Representations of groups of loops in U(N), SO(N) and various subgroups are studied. The representations are defined on fermion Fock spaces, and may be regarded as local gauge groups in the context of the two-dimensional many-particle Dirac theory for charged or neutral particles with rest mass m/> 0. For m = 0, the representations are shown to give rise to type I® factors, while for m > 0 hyperfinite, type III~ factors arise. A key point in the structure analysis is a convergence result: We prove that suitably rescaled representers of certain nonzero winding number loops converge to the free Dirac fields. We also present applications to cyclicity and irreducibility questions concerning the Dirac currents, and to the representation theory of a class of Kac-Moody Lie algebras.