The goal of the paper is an exposition of the simplest $(2+1)$-TQFTs in a sense following a pictorial approach. In the end, we fell short on details in the later sections where new results are stated and proofs are outlined. Comments are welcome and should be sent to the 4th author.
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.
