Mathematics – Geometric Topology
Scientific paper
2007-10-30
Mathematics
Geometric Topology
Minor changes and addition of a few references. To appear in the special volume of CCM dedicated to Xiao-Song Lin
Scientific paper
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.
Hong Seung-moon
Rowell Eric
Wang Zhenghan
No associations
LandOfFree
On exotic modular tensor categories does not yet have a rating. At this time, there are no reviews or comments for this scientific paper.
If you have personal experience with On exotic modular tensor categories, we encourage you to share that experience with our LandOfFree.com community. Your opinion is very important and On exotic modular tensor categories will most certainly appreciate the feedback.
Profile ID: LFWR-SCP-O-13960