QFT Realization of Non-Unitary $\mathfrak{sl}(2,\mathbb{C})$ WRT Invariants and Their Galois Conjugations

TL;DR

This paper introduces a field theoretic model for non-unitary $rak{sl}(2,b C)$ WRT invariants, connecting them to twisted 3D $b N=4$ theories and clarifying their modular properties.

Contribution

It proposes a novel field theoretic realization of non-unitary WRT TQFTs at non-principal roots of unity using topological twists of 3D $b N=4$ theories.

Findings

Constructed modular matrices matching WRT TQFTs.

Established relation between parameters of the theories.

Identified a decoupled unitary TQFT component.

Abstract

We propose a field theoretic realization of the non-unitary $\mathfrak{sl}(2,\mathbb{C})$ Witten-Reshetikhin-Turaev Topological Quantum Field Theory(WRT TQFT). The WRT TQFT at the principal root of unity is unitary. It is known to be realized by $\mathrm{SU}(2)$ Chern-Simons theory. However, the WRT TQFT at a non-principal root of unity is non-unitary. Its field theoretic realization has remained unclear. We propose that such a non-unitary TQFT arises from the topological twist of the 3-dimensional $\mathcal{N}=4$ rank-0 theory constructed by joining multiple $T[\mathrm{SU}(2)]$ theories. We construct its modular matrices and identify them with those of the WRT TQFT, establishing a concrete relation between the parameters, up to a decoupled unitary TQFT.