Free fields and new cosets of current algebras

Abstract

We introduce a new free field representation of current algebras by considering the affine compact and non-compact groups G k = SU( N + M) k , SU( N, M) k and supergroups SU( N M )k using cosets of the form G k ( SU(N) k + ηM× SU(M) ηk + ηN) , where η = ± for group/supergroup respectively. The subgroup H = SU( N)×SU( M) does not include a U(1) factor. Because of the subgroup levels k + ηM, ( k + N) η these cosets differ from GKO cosets of the type G k H k . We discuss simultaneously compact, non-compact and supergroup current algebras all in the same formalism. Borrowing ideas from induced representation theory of Lie groups we provide a basis in which we split the currents into “orbital” and “intrinsic spin” parts. The “orbital” part is constructed from NM canonical pairs of complex free fields (analogous to position and momentum) classified in G (H × U(1)) . These provide a new generalization of Wakimoto's SU (2) β- γ system. There is also a single free scalar field φ in a background charge which is associated with the remaining (twisted) U(1). The “intrinsic spin” part corresponds to currents in H = SU( N) × SU( M). The resulting expressions for the currents are simple and elegant and they are reminiscent of Wigner's construction of the Poincaré group generators in terms of orbital and intrinsic spin variables. The Sugawara stress tensor splits into four commuting parts T G = T [ G H×U(1) ] + T U(1) + T H where the first two termse are constructed only from the free fields ( β- γ), φ respectively, while T H = T SU( N) + T SU( M) is the Sugawara stress tensor for the “intrinsic spin” currents belonging to H. By iterating our G H method, the “intrinsic spin” part H may, in turn, be written in terms of new free fields, thus reducing the entire current algebra of G to a free field theory.