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

TL;DR

The paper constructs vertex operator algebras whose graded characters reproduce the Schur indices of higher-rank Argyres-Douglas theories, linking VOA structure to gauge-theory data. It introduces the $\mathcal{B}(n)_Q$ algebras as higher-rank logarithmic W-algebras and proves that for $Q=A_{N-1}$ their characters equal the Schur indices $\mathcal{I}(N,n)$ with a compatible central charge, while also providing a second, S-duality–based construction of related VOAs and their Euler-Poincaré characters. The work emphasizes conformal embeddings to model RG flows and discusses automorphism groups and potential generalizations via quantum Hamiltonian reductions, braid-reversed equivalences, and Heisenberg cosets. Overall, it provides a VOA framework that mirrors Beem-Nishinaka indices and RG flows for Argyres-Douglas theories, enriching the bridge between 4d N=2 physics and logarithmic CFTs. The results have implications for modular structures, representation theory, and potential quasi-lisse properties of the new algebras.

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.