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

Abstract

We continue our studies of the correspondence between 4d $$ \mathcal{N} $$ 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 $$ \mathcal{N} $$ 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 $$ \mathcal{N} $$ N = 2 SCFTs, therefore they are crucial to understand the Higgs branches of $$ \mathcal{N} $$ N = 2 SCFTs.

3d mirror for Argyres-Douglas theories

“…The 3d mirror of AD theory is quite useful in studying the properties of 4d AD theory: a): it confirms the S duality conjectures of these theories [22,23]: the S duality could be found by decomposing the 3d mirror into various pieces corresponding to the mirror of AD matter, see [13] for how to use 3d mirror to find S duality of AD theory; b): it predicts the Higgs branch of 4d theories: the Coulomb branch of the mirror quiver can be computed and they agree with the Higgs branch result of AD theories found in [24]. It also has interesting applications for the study of 3d N = 4 SCFTs: we find a large class of new interesting theories and some of them can be useful to construct new Chern-Simons matter theory.…”

“…The Higgs branch of 4d AD matter theory is in general a mixed branch, i.e. the generic point of the Higgs branch consists of free hypermultiplets and an interacting AD theory which has no Higgs branch [24]. The Higgs branch part are described by following variety:…”

The Hitchhiker's Guide to 4d $\mathcal{N}=2$ Superconformal Field Theories

“…The chiral algebra sector is a subset of the local operators in the 4d SCFT that closes under OPE restricted to a 2d plane (known as the chiral algebra plane) [8]. A lot of progress has been made to identify the chiral algebra sectors of AD SCFTs and the corresponding associated varieties [62,63,199,[213][214][215][216][217][218][219] including for Class S theories with twisted irregular punctures [45], which we will come to shortly. For AD SCFTs where both the 3d mirror and the chiral algebra (and its associated variety) are available, the two approaches discussed here have been demonstrated to produce the same Higgs branch, providing nontrivial evidence for the conjectured relation between associated varieties of 2d chiral algebras and 4d Higgs branches (see for example [199,213]).…”

Mixed Hodge structure and $\mathcal{N}=2$ Coulomb branch solution

“…Associated with a local singularity, one has an interacting 4d N = 2 theories. If the local singularity is quasi-homogenenous, the 4d theory is superconformal and a lot of studies for them were performed in [12,20,29], in particular the spectral numbers play a crucial role there. The study of 4d theories associated with general local singularity will be given in [53].…”

“…, r s ) (unitarity bound requires these numbers to be larger than 1). The Higgs branch operators also form a ring, which is in general quite complicated, see [20] for recent development.…”