W-algebras and superalgebras from constrained WZW models: A group theoretical classification

Abstract

We present a classification of W algebras and superalgebras arising in Abelian as well as non Abelian Toda theories. Each model, obtained from a constrained WZW action, is related with an Sl(2) subalgebra (resp. OSp(1|2) superalgebra) of a simple Lie algebra (resp. superalgebra) G. However, the determination of an U(1) Y factor, commuting with Sl(2) (resp. OSp(1|2)), appears, when it exists, particularly useful to characterize the corresponding W algebra. The (super) conformal spin contents of each W (super)algebra is performed. The class of all the superconformal algebras (i.e. with conformal spins s ≤ 2) is easily obtained as a byproduct of our general results.

TWISTED YANGIANS AND FOLDED ${\mathcal W}$ -ALGEBRAS

“…Looking at the decomposition of the fundamental of gl(Np) with respect to the principal embedding of sl (2) in N.sl(p) (see [19] and [12] for the technic used here) one shows that the subalgebra N.sl(p), generated by the J aa jm 's, is folded into a n.sl(p) (resp. n.sl(p) ⊕ so(k), where p = 2k + 1 is chosen odd to get N odd) when N = 2np (resp.…”

Semi-classical spectrum of the homogeneous sine-Gordon theories

“…However, Λ ± are regular elements of g 0 0 and, therefore, all this implies that α ± β cannot be a root of g for any α and β in ∆. In other words, h ι provide a soliton solution only if ι(su (2)) is the principal su(2) subalgebra of some regular A 1 ⊕ · · · ⊕ A 1 subalgebra of g (for a nice review about su(2) embeddings see [34] and references therein) and, in this case, h ι is the product of the soliton solutions associated with all the roots in ∆:…”

-superalgebras as truncations of super-Yangians

“…However, a simple counting (using the method given in [19]) of the generators shows that it is the twisted super-Yangians that have to be considered.…”