group L G . † The abovementioned conjectures are known to hold when the gauge group is abelian or of type A 1 . Our answer to the second question is bicommutant categories. The latter are higher categorical analogs of von Neumann algebras: They are tensor categories that are equivalent to their bicommutant inside Bim ( R ) , the category of bimodules over a hyperfinite 𝐼𝐼𝐼 1 factor. We prove that, modulo certain conjectures, the category of representations of the based loop group is a bicommutant category. The relevant conjectures are known to hold when the gauge group is abelian or of type A n . The Chern–Simons theories are certain 3 D topological quantum field theories introduced by Witten ( 1 ). They are parameterized by a compact Lie group G known as the gauge group and by a cohomology class k ∈ H 4 ( B G , ℤ ) known as the level of the theory ( 2 – 4 ). The Chern–Simons action S = 1 4 π ∫ M 3 ⟨ A ∧ d A ⟩ k + 1 3 ⟨ A ∧ [ A ∧ A ] ⟩ k   ( mod 2 π ) [1] is a functional of G bundles with connections over compact 3 manifolds. Here, A is the connection form, ⟨   ⟩ k : 𝔤 ⊗ 𝔤 → ℝ is a certain metric constructed from the level, and the integral is taken over a global section of the principal bundle. ‡ We point out that not every level k ∈ H 4 ( B G , ℤ ) yields a quantum field theory. For example, it is important that ⟨   ⟩ k be nondegenerate. In this paper, we deal only with those levels k that satisfy the following positivity condition: Definition 1. Let G be a compact Lie group. A level k ∈ H 4 ( B G , ℤ ) is positive if its image under the Chern–Weil homomorphism H 4 ( B G ) → Sym 2 ( 𝔤 ∗ ) G is a positive definite symmetric bilinear form ⟨   ⟩ k : 𝔤 ⊗ 𝔤 → ℝ . We write C S G , k for the Chern–Simons theory associated to the gauge group G and the level k . In the case of finite gauge groups, Chern–Simons theory is also known as Dijkgraaf–Witten theory. It is well known that a G bundle with connection is a critical point of the Chern–Simons action functional (a classi