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We investigate whether there are unitary families of W-algebras with spin onefields in the natural example of the Feigin-Semikhatov W^(2)_n-algebra. Thisalgebra is conjecturally a quantum Hamiltonian reduction corresponding to anon-principal nilpotent element. We conjecture that this algebra admits aunitary real form for even n. Our main result is that this conjecture isconsistent with the known part of the operator product algebra, and especiallyit is true for n=2 and n=4. Moreover, we find certain ranges of allowed levelswhere a positive definite inner product is possible. We also find a unitaryconformal field theory for every even n at the special level k+n=(n+1)/(n-1).At these points, the W^(2)_n-algebra is nothing but a compactified free boson.This family of W-algebras admits an 't Hooft limit that is similar to theoriginal minimal model 't Hooft limit. Further, in the case of n=4, wereproduce the algebra from the higher spin gravity point of view. In general,gravity computations allow us to reproduce some leading coefficients of theoperator product.

Unitary W-algebras and three-dimensional higher spin gravities with spin one symmetry