Lie Superalgebra and Extended Topological Conformal Symmetry in Non - critical W 3 Strings

We obtain a new free field realization of N = 2 super W3 algebra using the technique of quantum hamiltonian reduction. The construction is based on a particular choice of the simple root system of the affine Lie superalgebra sl(3|2)(1) associated with a non-standard sl(2) embedding. After twisting and a similarity transformation, this W algebra can be identified as the extended topological conformal algebra of non-critical W3 string theory. Many attempts have been done to construct non-critical string theories withW algebra symmetry (non-critical W strings)[1], which may be defined beyond the c = 1 barrier of non-critical string theory. Physical states of non-criticalW -strings can be characterized by the BRST cohomology. It is known that the BRST current has quite non-trivial structure due to non-linearity of W -algebra [2]. It is recently understood that the BRST algebra in non-critical string theory may be enlarged to the twisted N = 2 superconformal algebra i.e. topological conformal algebra[3]. Bershadsky, Lerche, Nemeschansky and Warner[4] found the topological W -symmetry in the non-critical W3-string. This fact would be universal in a class of non-critical W -string theories and essential for investigation of their properties as topological strings, which is a clue to a non-perturbative formulation of string theories in higher dimensions. Practical computations involving (topological) W algebra often become complicated due to its non-linearity. In this paper we regard the topological WN symmetry as a result of the quantum Hamiltonian reduction of an affine Lie superalgebra sl(N |N − 1). We believe that the viewpoint of Lie superalgebra is helpful in more systematic understanding of the algebraic structure of topological W symmetry, which is obscured by the non-linearity. For example, the existence of the BRST current which has completely vanishing nilpotent OPE relation is most clearly understood from the hidden symmetry of sl(N |N − 1) [4]. The Lie superalgebra may also explain the origin of the screening operators which play an essential role in investigating the physical spectrum, especially the problem ofW gravitational dressing. This implies the Lie superalgebra sl(N |N− 1) is important for geometrical aspects of the theory, such as W moduli. In a previous paper[5], we have shown the relation between the topological conformal algebra and the Lie superalgebra sl(2|1)(1) using the hamiltonian reduction. In this article we will examine the quantum hamiltonian reduction of an affine Lie superalgebra sl(3|2)(1) and study a free field realization of the N = 2 super-W3 algebra [6] relevant to the non-critical W3-string theory. This reduction at classical level has been discussed in [4]. However, the fermionic ghosts that they employed are not free fields due to ghost number violation term in the U(1) current, which cannot be expected from the standard hamiltonian reduction. We will show that by a similarity transformation the fermionic ghosts of Bershadsky et.al. are related to the genuine free fields which naturally appear