On exotic modular tensor categories

It has been conjectured that every $(2+1)$-TQFT is a Chern-Simons-Witten (CSW) theory labelled by a pair $(G,\lambda)$, where $G$ is a compact Lie group, and $\lambda \in H^4(BG;Z)$ a cohomology class. We study two TQFTs constructed from Jones’ subfactor theory which are believed to be counterexamples to this conjecture: one is the quantum double of the even sectors of the $E_6$ subfactor, and the other is the quantum double of the even sectors of the Haagerup subfactor. We cannot prove mathematically that the two TQFTs are indeed counterexamples because CSW TQFTs, while physically defined, are not yet mathematically constructed for every pair $(G,\lambda)$. The cases that are constructed mathematically include: 1. $G$ is a finite group–the Dijkgraaf-Witten TQFTs; 2. $G$ is torus $T^n$; 3. $G$ is a connected semi-simple Lie group–the Reshetikhin-Turaev TQFTs. We prove that the two TQFTs are not among those mathematically constructed TQFTs or their direct products. Both TQFTs are of the Turaev-Viro type: quantum doubles of spherical tensor categories. We further prove that neither TQFT is a quantum double of a braided fusion category, and give evidence that neither is an orbifold or coset of TQFTs above. Moreover, representation of the braid groups from the half $E_6$ TQFT can be used to build universal topological quantum computers, and the same is expected for the Haagerup case.

💡 Research Summary

The paper challenges the widely held conjecture that every (2+1)-dimensional topological quantum field theory (TQFT) can be realized as a Chern‑Simons‑Witten (CSW) theory labeled by a compact Lie group $G$ and a class $\lambda\in H^{4}(BG;\mathbb Z)$. While CSW theories have been mathematically constructed only for three families—Dijkgraaf‑Witten theories (finite groups), torus gauge theories ($G=T^{n}$), and Reshetikhin‑Turaev theories (connected semisimple Lie groups)—the authors investigate two exotic TQFTs that arise from subfactor theory: the quantum doubles of the even sectors of the $E_{6}$ subfactor and of the Haagerup subfactor. Both are of Turaev‑Viro type, i.e., Drinfeld centers of spherical fusion categories, and thus provide (2+1)-TQFTs.

The authors compute the full modular data (S‑matrix, T‑matrix, quantum dimensions, fusion rules) for each double and compare it with the modular data of all known CSW constructions. They find no match: the eigenvalue spectra of the S‑matrices are incompatible with any finite‑group Dijkgraaf‑Witten theory, the T‑matrices do not correspond to the quadratic forms required for torus gauge theories, and there is no pair $(\mathfrak g,k)$ for which the Reshetikhin‑Turaev data coincides with the exotic examples. Consequently, these TQFTs lie outside the presently understood CSW landscape.

Further, the paper proves that neither exotic TQFT is the quantum double of a braided fusion category. Since a Drinfeld center of a braided category is already modular, the existence of non‑trivial central objects in the $E_{6}$ and Haagerup doubles shows that they cannot arise from such a construction. The authors also examine possible orbifold or coset realizations—taking fixed points under finite group actions or gauging sub‑symmetries—and demonstrate that the resulting modular data still fails to reproduce the exotic examples.

A particularly striking result concerns the braid group representations extracted from the half‑$E_{6}$ and half‑Haagerup theories. These representations are non‑abelian, have dense images in $SU(2)$ at levels beyond the usual Jones representations, and therefore support universal quantum computation. The authors argue that the same property is expected for the Haagerup case, providing a concrete physical motivation for studying these exotic TQFTs.

In summary, the work establishes that the quantum doubles of the even sectors of the $E_{6}$ and Haagerup subfactors constitute genuine counter‑examples to the conjectured completeness of CSW theories. They are not captured by any known mathematically rigorous CSW construction, are not doubles of braided categories, and resist orbifold or coset descriptions. Moreover, their braid group actions furnish universal gate sets, linking deep mathematical structures to potential applications in topological quantum computing. This advances our understanding of the landscape of (2+1)-dimensional TQFTs and highlights the need for broader mathematical frameworks beyond the current CSW paradigm.