Folding the W Algebras

TL;DR

This paper explores how folding Dynkin diagrams and symmetry algebras can derive W-algebras for non-simply laced and superalgebra cases from unitary algebra cases, aiding in understanding fusion rules.

Contribution

It introduces a method to obtain W-algebras for B, C, D series and super W-algebras from unitary W-algebras via diagram folding and symmetry considerations.

Findings

W algebras for B, C, D series derived from A series

Fusion rules for non-simply laced algebras obtained from A case

Super W-algebras related to orthosymplectic superalgebras deduced from A(m,n) series

Abstract

In the same way the folding of the Dynkin diagram of A_{2n} (resp. A_{2n-1}) produces the B_n (resp. C_n) Dynkin diagram, the symmetry algebra W of a Toda model based on B_n (resp. C_n) can be seen as resulting from the folding of a W-algebra based on A_{2n} (resp. A_{2n-1}). More generally, W algebras related to the B-C-D algebra series can appear from W algebras related to the unitary ones. Such an approach is in particular well adapted to obtain fusion rules of W algebras based on non simply laced algebras from fusion rules corresponding to the A_n case. Anagously, super W algebras associated to orthosymplectic superalgebras are deduced from those relative to the unitary A(m,n) series.