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.

arXiv comment17+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

References in corpus (25)

• Edge states and topological invariants of non-Hermitian systems
• Observation of non-Hermitian bulk-boundary correspondence in quantum dynamics
• Exact Results for Wilson Loops in Superconformal Chern-Simons Theories with Matter
• S-Duality of Boundary Conditions In N=4 Super Yang-Mills Theory
• The Exact Superconformal R-Symmetry Extremizes Z
• SUSY Gauge Theories on Squashed Three-Spheres
• Notes on SUSY Gauge Theories on Three-Sphere
• New topological invariants in non-Hermitian systems
• Localization techniques in quantum field theories
• A topologically twisted index for three-dimensional supersymmetric theories
• Comments on One-Form Global Symmetries and Their Gauging in 3d and 4d
• N=2 supersymmetric theories on squashed three-sphere
• Comments on twisted indices in 3d supersymmetric gauge theories
• SL(2,R) Chern-Simons, Liouville, and Gauge Theory on Duality Walls
• A theory of 2+1D bosonic topological orders
• Supersymmetric Index Of Three-Dimensional Gauge Theory
• Particle-Vortex Duality and the Modular Group: Applications to the Quantum Hall Effect and Other 2-D Systems
• Bethe/Gauge correspondence on curved spaces
• Solving 3d Gravity with Virasoro TQFT
• AGT on the S-duality Wall
• Comments on nonunitary conformal field theories
• Three-dimensional $\mathcal{N}=2$ supersymmetric gauge theories and partition functions on Seifert manifolds: A review
• Three-dimensional gauge theories with supersymmetry enhancement
• Non-unitary TQFTs from 3D $\mathcal{N}=4$ rank 0 SCFTs
• Non-hyperbolic 3-manifolds and 3D field theories for 2D Virasoro minimal models