W - symmetries on the Homogeneous Space G / U ( 1 ) r

A construction of W-symmetries is given only in terms of the nonlocal fields (parafermions ψ α), which take values on the homogeneous space G/U (1) r , where G is a simply connected compact Lie group manifold (its accompanying Lie algebra G is a simple one of rank r). Only certain restriction of the root set of Lie algebra on which the parafermionic fields take values are satisfied, then a consistent and non-trivial extension of the stress momentum tensor may exist. For arbitrary simple-laced algebras, i.e. the A − D − E cases, a more detailed discussion is given. The OPE of spin three primary field are calculated, in which a primary field with spin four is emerging. The discovery of the AdS d+1 /CF T d correspondence [24, 18, 32] brings the role of conformal field theory to a special stage. The type IIB string theory on the AdS 5 ⊗S 5 are equivalent to N = 4 super Yang-Mills in Minkowski space-time. However, the calculation of the correlation functions of physical quantities are limited by our knowledge, except for the CF T 2 , saying 2d conformal field theory (CFT) case. With the help of infinitely dimensional symmetries of 2d CFT, much more information can be obtained. For example, Seiberg et al. considered the duality between AdS 3 and CF T 2 [16, 23]. As a special case of this duality, a type IIB string 1