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

TL;DR

This paper constructs new vertex algebras linked to nilpotent elements of Lie algebras, connecting them to Argyres-Douglas theories and their RG flows, revealing deep algebraic and physical correspondences.

Contribution

It introduces families of vertex algebras associated to nilpotent elements, relating them to logarithmic W-algebras and Argyres-Douglas theories, expanding the understanding of their algebraic structures and dualities.

Findings

Vertex algebras match Schur-Index formulas for Argyres-Douglas theories.

Identification of these algebras with Schur-indices of type IV Argyres-Douglas theories.

Conformal embedding patterns reflect RG-flow of Argyres-Douglas theories.

Abstract

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.