Families of vertex algebras associated to nilpotent elements of simply-laced
Lie algebras are constructed. These algebras are close cousins of logarithmic
W-algebras of Feigin and Tipunin and they are also obtained as modifications of
semiclassical limits of vertex algebras appearing in the context of S-duality
for four-dimensional gauge theories. In the case of type A and principal
nilpotent element the character agrees precisely with the Schur-Index formula
for corresponding Argyres-Douglas theories with irregular singularities. For
other nilpotent elements they are identified with Schur-indices of type IV
Argyres-Douglas theories. Further, there is a conformal embedding pattern of
these vertex operator algebras that nicely matches the RG-flow of
Argyres-Douglas theories as discussed by Buican and Nishinaka.Comment: Comments are welcom