  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.
