4d $\mathcal{N}=2$ SCFTs and lisse W-algebras

We continue our studies of the correspondence between 4d $$ \mathcal{N} $$ = 2 SCFTs and 2d W-algebras. The purpose of this paper is to study the relationship between 2d lisse W-algebras and their 4d SCFT partners. The lisse W-algebra is the W-algebra whose associated Zhu’s C2 algebra is finite dimensional. As the associated variety of Zhu’s C2 algebra is identified with the Higgs branch in the 4d/2d correspondence, the lisse condition is equivalent to the absence of the Higgs branch on the 4d side. We classify 4d $$ \mathcal{N} $$ = 2 SCFTs which do not admit Higgs branch, then these theories would give lisse W-algebras through the 4d/2d correspondence. In particular, we predict the existence of a large class of new non-admissible lisse W-algebras, which have not been studied before. The 4d theories corresponding to lisse W-algebra can appear in the Higgs branches of generic 4d $$ \mathcal{N} $$ = 2 SCFTs, therefore they are crucial to understand the Higgs branches of $$ \mathcal{N} $$ = 2 SCFTs.