On exotic modular tensor categories
Seung-moon Hong, Eric Rowell, Zhenghan Wang
TL;DR
This work investigates two purported exotic (non-CSW) 2+1D TQFTs arising as quantum doubles of the even sectors of the E6 and Haagerup subfactors. It provides detailed modular data (S/T matrices, fusion rules) for the doubles $\mathcal{Z}(\mathcal{E})$ and $\mathcal{Z}(\mathcal{H})$, shows they are not of finite or torus-Lie quantum-group type, and proves they are not doubles of braided fusion categories or orbifold/coset constructions. The paper also analyzes SL(2,Z) representations and braid-group actions, finding finite SL(2,Z) images and irreducible, potentially universal braid representations in the E6 case, with the Haagerup case remaining open for braid irreducibility. Collectively, the results supply substantial evidence that these two MTCs are exotic and not derivable from conventional CSW, quantum-group, orbifold, or coset frameworks, while highlighting their potential relevance to topological quantum computation. Overall, the work advances the classification of (2+1)D TQFTs by identifying concrete, highly structured examples that lie outside established CSW constructions.
Abstract
It has been conjectured that every $(2+1)$-TQFT is a Chern-Simons-Witten (CSW) theory labelled by a pair $(G,λ)$, where $G$ is a compact Lie group, and $λ\in H^4(BG;Z)$ 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 $(G,λ)$. 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.