Logarithmic W-algebras and Argyres-Douglas theories at higher rank

Understand

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.