Unitary W-algebras and three-dimensional higher spin gravities with spin   one symmetry
  We investigate whether there are unitary families of W-algebras with spin one fields in the natural example of the Feigin-Semikhatov W^(2)_n-algebra. This algebra is conjecturally a quantum Hamiltonian reduction corresponding to a non-principal nilpotent element. We conjecture that this algebra admits a unitary real form for even n. Our main result is that this conjecture is consistent with the known part of the operator product algebra, and especially it is true for n=2 and n=4. Moreover, we find certain ranges of allowed levels where a positive definite inner product is possible. We also find a unitary conformal 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 the original minimal model 't Hooft limit. Further, in the case of n=4, we reproduce the algebra from the higher spin gravity point of view. In general, gravity computations allow us to reproduce some leading coefficients of the operator product. 
Hamid Afshar, Thomas Creutzig, Daniel Grumiller, Yasuaki Hikida and   Peter B. Ronne
hep-th