Abstract
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.
Highlights
A universal property of all non-principal embeddings is the presence of at least one singlet in the wedge algebra, which translates into a current algebra as part of the asymptotic symmetry algebra
Our main technical result is that the Wn(2)-algebra of level k of Feigin and Semikhatov seems to allow for a unitary real form if n is even
The main result of this work is that we conjectured that the Wn(2) algebra for even n allows a unitary real form, and that for a certain range of levels it even can have a positive definite inner product
Summary
Our main technical result is that the Wn(2)-algebra of level k of Feigin and Semikhatov seems to allow for a unitary real form if n is even. This algebra is constructed in [17] as both a coset of supergroup type and as the chiral algebra of a conformal field theory associated to a set of screening charges of supergroup type inside a lattice super algebra Both constructions allow, in principle, for a computation of the operator algebra, though this is very tedious and Feigin and Semikhatov provide the leading contributions of important operator products. The cases where λn−2(n, k) vanishes are exactly the discrete levels investigated in [18] They lead to unitary real forms of the Wn(2) algebra provided that the central charge is nonnegative. If we compare the expression for Λ with the one of the previous section, we see that contributions of type ∂3J4(x) and J4(w)J4(w)∂J4(w) disappear This is essential for closure of our unitary form with real coefficients, and that these terms vanish is a non-trivial computation.
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