The grading Becchi-Rouet-Stora-Tyutin (BRST) method gives a way to construct the integer W 2,s strings, where the BRST charge is written as Q B = Q 0 + Q 1 . Using this method, we reconstruct the nilpotent BRST charges Q 0 for the integer W 2,s strings and the half-integer W 2,s strings. Then we construct the exact grading BRST charge with spinor fields and give the new realizations of the half-integer W 2,s strings for the cases of s = 3/2, 5/2, and 7/2.
Contents1. Introduction 1 2. Review of the grading BRST method 2 3. Construct BRST charges Q 0 for the integer and the half-integer W 2,s strings 4 3.1 Solutions for the case s = integer 5 3.2 Solutions for the case s = half-integer 6 4. Spinor field realizations of the half-integer W 2,s strings 8 4.1 Spinor field realizations of the W 2,3/2 string 8 4.2 Spinor field realizations of the W 2,5/2 string 8 4.3 Spinor field realizations of the W 2,7/2 string 10 5. Conclusion 11were obtained [24,25,26,27]. Recently, we constructed the nilpotent BRST charges of the spinor non-critical W 2,s (s=3,4) strings by taking account of the property of spinor field [28]. It was shown that certain integer W 2,s (s=3,4) algebras can be linearized as the W 1,2,s algebras by the inclusion of a spin-1 current. This provides a way of obtaining new realizations of the integer W 2,s strings. Based on the linear W 1,2,s algebras, the ghost field realizations and the spinor realizations of the spinor non-critical W 2,s strings were given in Refs. [29,30]. The half-integer W 2,s algebra with s = 3/2 is known as the N = 1 superconformal algebra or SW (3/2, 2) superconformal algebra. Recently, several constructions of the algebra were obtained in Refs. [31,32] . Much research had been done also on N = 2 superconformal algebra, in which another s = 3/2 current is added [33,34,35,36]. Correspondingly, the N = 1 and N = 2 superstrings had been investigated in [37,38]. In addition, the case of s = 5/2 was investigated in Ref. [39]. It was found that this algebra can not be closed with spin-2 current T and spin-5/2 current G only, and so there is no realization for this algebra. But by introducing other spin currents, this algebra may be linearized [40], which implies that the realizations of the algebra could be found. The W 2,s (s = 7/2, 9/2, 11/2, 13/2, 15/2) algebras were also studied in Ref. [41], in which the values of central charges for these algebras were given.Since up to now there is no work focused on the research of spinor field realizations of the half-integer W 2,s strings, we will construct the nilpotent BRST charges of spinor halfinteger W 2,s strings for the first time by using the grading BRST method. This method provides a more easy way to construct W 2,s algebras and strings. And the physical states can be obtained by the grading form of BRST operators for these strings. Firstly, we reconstruct the BRST charges Q 0 for integer W 2,s strings and the half-integer W 2,s strings. For each case we obtain four solutions. All these solutions indicate that we can construct the BRST...
