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

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.

17+27 pages, 8 figures. V2: comment about exceptional cases added, see footnote 6 on page 28, reference added, typo fixed. V3: sections reorganized, section 2.2 added to provide physical motivation, reference added, published version

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