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

TL;DR

The work provides a concrete QFT construction for the non-unitary $\mathfrak{sl}(2,\mathbb{C})$ WRT TQFT by topologically twisting a 3d $\mathcal{N}=4$ rank-0 theory built from gluing multiple $T[\mathrm{SU}(2)]$ blocks, and shows that the A-twisted theory factorizes into a non-unitary $\mathcal{D}(p,q)|_A$ part and a decoupled unitary TFT$[\vec{k}]$, with a precise mapping between $(k,m)$ and $(p,q)$ that reproduces the WRT modular data. It provides explicit HF data and modular-matrix constructions for both odd and even $p$, demonstrating a factorized S and T structure in the odd case and partial sector identifications in the even case, while discussing refined IR equivalence across different UV data and the role of the decoupled TFT. The results establish a tangible field-theoretic realization of non-unitary WRT TQFTs and illuminate how Galois-conjugate TQFTs emerge from topological twists of higher-rank 3d theories, with open questions about fully identifying the UV decoupled sector. These insights bridge bulk TQFT data with QFT realizations and deepen the understanding of non-unitary topological phases in three dimensions.

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.

Figures (8)

• Figure 1: The problem we are going to deal with in this paper.
• Figure 2: Quiver diagram associated with the $T[\mathrm{SU}(2)]$ theory.
• Figure 3: Cartoon of the $T[\mathrm{SU}(2)]$ theory.
• Figure 4: $\mathrm{SL}(2,\mathbb{Z})$ action for the $\mathrm{SU}(2)$ flavor symmetry. $[k]$ under the flavor symmetry box indicates the supersymmetric Chern-Simons term of level $k$.
• Figure 5: Construction of the $D(\vec{k})$ theory. $2^{1,n}$ indicate flavor $\mathrm{SU}(2)^{1,n}$. Note that the $D(\vec{k})$ theory is analogous to the $\mathrm{U}(1)$ Chern-Simons effective description for the fractional quantum Hall effect(FQHE). After replacing whole $\mathrm{SU}(2)$ to the $\mathrm{U}(1)$ and regarding the $S$ transformation as Witten's $S$ transformationWitten:2003ya, we get the dynamical part of the theory describes the $\sigma=-\frac{e^2}{2\pi \hbar}\times\frac{q}{p}$ FQHEBurgess:2001.