Boundary vertex algebras for 3d $\mathcal{N}=4$ rank-0 SCFTs

TL;DR

This work leverages the HT twist as a practical handle to access boundary VOAs for topologically twisted 3d ${\cal N}=4$ rank-0 SCFTs. For ${\cal T}_{\text{min}}$ it identifies the boundary algebra after the $B$ twist as the rational $L_1(\mathfrak{osp}(1|2))$, with a level-rank-like relation to the Virasoro minimal model $M(2,5)$; it further furnishes a complete picture of modules, fusion, and modular properties. Extending to the family ${\cal T}_r$, the paper provides strong evidence that the boundary VOA is $L_r(\mathfrak{osp}(1|2))$, supported by half-indices and modular data, and connects these to the minimal models $M(2,2r+3)$ via a coset/decomposition framework. The results illuminate how 3d bulk twists encode 2d rational CFT data, reveal a braid-reversed duality with minimal models, and lay groundwork for a systematic boundary-VOA classification of rank-0 ${\cal N}=4$ theories.

Abstract

We initiate the study of boundary Vertex Operator Algebras (VOAs) of topologically twisted 3d $\mathcal{N}=4$ rank-0 SCFTs. This is a recently introduced class of $\mathcal{N}=4$ SCFTs that by definition have zero-dimensional Higgs and Coulomb branches. We briefly explain why it is reasonable to obtain rational VOAs at the boundary of their topological twists. When a rank-0 SCFT is realized as the IR fixed point of a $\mathcal{N}=2$ Lagrangian theory, we propose a technique for the explicit construction of its topological twists and boundary VOAs based on deformations of the holomorphic-topological twist of the $\mathcal{N}=2$ microscopic description. We apply this technique to the $B$ twist of a newly discovered family of 3d $\mathcal{N}=4$ rank-0 SCFTs ${\mathcal T}_r$ and argue that they admit the simple affine VOAs $L_r(\mathfrak{osp}(1|2))$ at their boundary. In the simplest case, this leads to a novel level-rank duality between $L_1(\mathfrak{osp}(1|2))$ and the minimal model $M(2,5)$. As an aside, we present a TQFT obtained by twisting a 3d $\mathcal{N}=2$ QFT that admits the $M(3,4)$ minimal model as a boundary VOA and briefly comment on the classical freeness of VOAs at the boundary of 3d TQFTs.