QFT realization of non-unitary 𝔰𝔩(2, ℂ) WRT invariants and their galois conjugations

A bstract We propose a field theoretic realization of the non-unitary 𝔰𝔩(2, 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 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 𝒩 = 4 rank-0 theory constructed by joining multiple T [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.