Narrow Results

In recent joint work with Jones and Shlyakhtenko, we have given a diagrammatic description of Popa's symmetric enveloping inclusion for planar algebra subfactors. In this paper, we give a diagrammatic construction of the associated Jones tower, in the case that the planar algebra is finite-depth. We then use this construction to describe the planar algebra of the symmetric enveloping inclusion, which is known to be isomorphic to the planar algebra of Ocneanu's asymptotic inclusion by a result of Popa. As an application we give a planar algebraic computation of the (reduced) fusion algebra of the asymptotic inclusion, recovering some well-known results of Ocneanu and Evans–Kawahigashi.

We introduce fusion, contragradient and braiding of Hilbert affine representations of a subfactor planar algebra P (not necessarily having finite depth). We prove that if N ⊂ M is a subfactor realization of P, then the Drinfeld center of the N–N-bimodule category generated by _NL²(M)_M, is equivalent to the category of Hilbert affine representations of P satisfying certain finiteness criterion. As a consequence, we prove Kevin Walker's conjecture for planar algebras.

It has been conjectured that every (2 + 1)-TQFT is a Chern-Simons-Witten (CSW) theory labeled by a pair (G, λ), where G is a compact Lie group, and λ ∈ H⁴(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 E₆ 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 Tⁿ; (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 E₆ TQFT can be used to build universal topological quantum computers, and the same is expected for the Haagerup case.