Abstract
We show how the Feigin-Fuchs Coulomb-gas construction, with two free Gaussian bosons, can be used to derive the representation theory of the SU(2) parafermion models. We identify the generators of the chiral algebra within the bosonic Fock space and derive the chiral algebra of the finitely reducible models, which correspond to the SU(2) and SU(1, 1) parafermion algebras. We then focus on the SU(2) highest-weight modules in the remainder of the paper. Unitarity of the modules requires that the states of the parafermion theory be independent of the zero modes of two fermionic vertex operators of the bosonic theory. The expressions for the Virasoro highest weights of the models are doubly degenerate in the bosonic Fock space. We formulate the correlation functions of these operators in the parafermion Hilbert space, and in particular, the fusion rules for the Virasoro highest weights are derived in an elegant way. Finally, the irreducible parafermion characters are derived. We discuss the connection between our analysis and previous work on representation theory based on BRST cohomology.
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