Three-Dimensional Topological Field Theories and Non-Unitary Minimal Models

TL;DR

The paper builds a bridge between 3d non-unitary topological field theories derived from rank-0 ${\cal N}=4$ SCFTs and non-unitary Virasoro minimal models $M(2,2r+3)$. It develops a UV ${\cal N}=2$ description of rank-0 theories, shows IR SUSY enhancement to ${\cal N}=4$, and constructs two topological twists whose A-twisted sector captures the minimal-model data via half-indices and Seifert-manifold partition functions. The authors demonstrate fermionic-sum representations for the minimal-model characters and align the modular data with the corresponding 2d theories, identifying a precise boundary-operator realization through Wilson loops. This work suggests a general correspondence between rank-0 3d SCFTs and 2d non-unitary RCFTs and outlines future directions for classifying boundary algebras and broader rank-0 constructions.

Abstract

We find an intriguing relation between a class of 3-dimensional non-unitary topological field theories (TFTs) and Virasoro minimal models $M(2,2r+3)$ with $r \geq 1$. The TFTs are constructed by topologically twisting $3d$ ${\mathcal N}=4$ superconformal field theories (SCFTs) of rank-0, i.e. having zero-dimensional Coulomb and Higgs branches. We present ultraviolet (UV) field theory descriptions of the SCFTs with manifest ${\mathcal N}=2$ supersymmetry, which we argue is enhanced to ${\mathcal N}=4$ in the infrared. From the UV description, we compute various partition functions of the TFTs and reproduce some basic properties of the minimal models, such as their characters and modular matrices. We expect more general correspondence between topologically twisted $3d$ ${\mathcal N}=4$ rank-0 SCFTs and $2d$ non-unitary rational conformal field theories.