Explicit Construction of the BRST Charge for W4

We give the explicite form of the BRST charge Q for the algebra W4 = WA3 in the basis where the spin-3 and the spin-4 field are primary as well as for a basis where the algebra closes quadratically. ∗e-mail: [email protected]; 31890::HORNFECK For the construction of W-strings [1], especially its physical states, it is important to know the cohomology of the BRST charge [2, 3, 4, 5, 6, 7, 8, 9]. Unfortunately, for the W-strings based on the algebras Wn = WAn−1 only the one for the simplest algebra, the Zamolodchikokv W3-algebra [10] has been constructed previously [11]. Since so far there is no simplifying ansatz of the terms appearing in Qfor a given (general) W-algebra 1, we constructed the BRST charge for the W4-algebra by taking into account all possible terms of even parity (where the matter field W3 and the ghost fields c3 and b3 have parity odd, all other fields have parity even) and determining the coefficients by demanding that Q is nilpotent. However, contrary to the case of the W3-algebra, one has to be careful about two points: We have the freedom of redefining the spin-4 field W4 (that is usually taken to be quasi-primary) by transforming W4 → W4+κΛ, Λ being the spin-4 quasi-primary TT−3/10T ′′ (all products of fields are considered to be normal ordered in the standard way whenever necessary). There are two obvious choices: Either W4 is such that the algebra closes quadratically or else W4 is taken to be primary, leading also to cubic terms (in the Virasoro operator T ) in the algebra. Whereas in the latter case the operator product expansions (OPEs) (and therefore the commutators) are unique (apart from a sign-ambiguity due to the transformation W4 → −W4, there exist two (equivalent) possibilities for the algebra that closes quadratically. The structure constants of W4 can be found in refs. [13, 14]. To the matter part (T,W3,W4) we have to introduce ghost and anti-ghost fields ({c2, b2}, {c3, b3}, {c4, b4}), obeying the OPEs ci ⋆ cj = bi ⋆ bj = 0 ci ⋆ bj = δij 1 (1) and the BRST charge will be of the form Q = ∮ dw ( T c2 +W3 c3 + (W4 + κ1 TT + κ2 T ′′) c4 ) (w) + (2) contributions containing anti-ghost fields bi We introduced κ1 and κ2 to include different bases for the field W4. as it exists for a certain sub-class of W-algebras that close quadratically [12].