Free Field Realization of WBC n and WG 2 algebras

We study the BRST-cohomology in the quantum hamiltonian reduction of affine Lie algebras of non-simply laced type. We obtain the free field realization of the Wg-algebra for g = B2, B3, C3 and G2. The WC3 algebra is shown to be equal to the WB3 algebra at the quantum level by duality transformation. W -algebra symmetry [1, 2] plays an important role in the classification of rational conformal field theory and integrable systems such as Toda field theory and 2d gravity. The classical Wg-algebra associated with a simple Lie algebra g can be realized as the second hamiltonian structure of the generalized KdV hierarchy [3]. The classical hamiltonian reduction of affine Lie algebra provides a systematic method to the construction of generalized KdV hierarchy associated with any Lie algebra[4]. In order to study the quantum W algebra it is necessary to calculate operator product expansions of normalordered composite operators. The most general form of the W -algebra can be determined by consistency conditions such as the Jacobi identity. Such an approach has been done systematically for the W -algebras with two and three generators [5, 6]. However, it is technically difficult to extend this approach to general W -algebra because of the lack of Lie algebraic viewpoint. The free field realization of theWg algebra is a crucial step to understand the representation of the algebra, correlation functions through screening operators, which is defined as the commutant of the W -algebra. So far the free field realization is well-understood for the Wg-algebra for simply laced Lie algebras g = An and Dn[7]. The Wg-algebra based on a simply laced Lie algebra has self-dual property in the torus, on which free bosons are compactified. Self-duality allows us to generalize the classical Miura transformation to the quantum one by taking normal ordered product of the scalar Lax operator and replacing the level k of an affine Lie algebra by a parameter α0 = a− 1/a , where a = √ k + h and h is the dual Coxeter number. Concerning the quantumWg algebra associated with a non-simply laced Lie algebra g, it gets non-trivial quantum correction due to the lack of self-duality. In fact, the classical Miura transformation for non-simply laced Lie algebra does not work in the quantum case. The purpose of the present article is to study the free field realization of the quantum W -algebra associated with non-simply laced Lie algebras. We will use the method of quantum hamiltonian reduction [8, 9, 10] based on the BRST quantization. We examine the BRST cohomology explicitly for non-simply laced Lie algebras B2, B3, C3 and G2. An interesting properties in non-simply laced Lie algebra is the duality relation between Lie algebras Bn and Cn. This duality predicts a unique quantum W algebra for BC type Lie algebra[11]. We will confirm this duality for BC3 case.