Operator Quantization of Constrained Wznw Theories and Coset Constructions

Operator quantization of the WZNW theory for Lie algebra g with constrained affine Lie algebrâ h, h ⊂ g, is performed.The extended phase space and effective first class constraints are constructed which determine a class of g/h coset primary fields and currents. In a recent article [1] for generic g/h coset conformal field theory [2, 3] a class of Virasoro currents and primary fields was constructed.These fields are explicitly expressed in terms of two Wess-Zumino-Novikov-Witten (WZNW) theories for Lie algebras g and h ⊂ g. The object of this paper is to present the quantization scheme which leads from the WZNW theory for g to the g/h coset primary fields and currents of ref. [1]. To this end we perform operator quantization of the WZNW theory for g with constrained affine Lie algebrâ h. We follow ideas of the generalized canonical quantization method [4]. However our approach contains some new elements.It is different trom the BRST quantization of gauged WZNW actions [5, 6].In particular, we do not introduce ghost fields.Operator quantization of the constrained WZNW theories which are Abelian g/u(1) d cosets was performed in [7]. The phase space of the WZNW theory is extended by introducing broken affine primary fields and an auxiliary WZNW theory for h. The original constraints are replaced by effective ones in two steps. First we replace the original constraints by some new ones and demand that all the fields of the reduced theory commute with the new constraints.In the operator product algebra of these fields there is the subalgebra which can be treated as an algebra of effective first class constraints and used for further reduction. All the fields are invariant * [email protected]