Tortile Category and Philosophy of Non-Commutative Geometry

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In terms of Feynman diagrams, a quantum field theory is nothing but a finitely generated subcategory QFT of the ribbon category Rep(G). A ribbon category (also called a tortile category) is a braided pivotal category, or equivalently a balanced autonomous category, which satisfies  θ*=θ, where  θ is the twist. This is a kind of category with duals. QFT is generated by the fundamental particles (irreducible representations) and all possible combinations of the fundamental interactions coming from the Feynman rules (intertwiners). Thus one is led to consider generalized quantum field theories QFT′ living inside arbitrary Hermitian ribbon categories R, where the braiding and twist need not be trivial.

Now Tannaka-Krein duality tells us that one can recover the group G from the category Rep(G). In a certain sense, every ribbon category is a category of representations – in the general case not of a group, but of a quantum group. When we do quantum field theory in ribbon categories, we are replacing the symmetry group by a quantum group.

We encounter this phenomenon in Chern-Simons theory, where the Lie group G is replaced by its quantum deformation, Uq(g). The words “Chern–Simons theory” can mean various things to various people, but it generally refers to the three-dimensional topological quantum field theory whose configuration space is the space of principal bundles with connection on a bundle and whose Lagrangian is given by the Chern-Simons form of such a connection (for simply connected  G, or rather, more generally, whose action functional is given by the higher holonomy of the Chern-Simons circle 3-bundle. The reason for this deformation of the underlying symmetry group, as one passes from the classical to the quantum theory, has not been altogether elucidated, and remains an interesting problem. In Witten’s approach, three dimensional Chern-Simons theory defines a two dimensional conformal field theory on the boundary, the Wess-Zumino-Witten (WZW) model. The corresponding affine lie algebra g of the WZW model defines, for each k ∈ Z+, a category Ck(g) of integrable modules of level k, and these categories are modular.

On the other hand, in Turaev’s approach, one deforms the lie algebra g into a quantum group Uq(g), where q = eπi/k, which for k ∈ Z+ is a root of unity. The representation categories Rep(Uq(g)) of these quantum groups are also modular, and are the starting point in Turaev’s approach.

Despite this theorem, the relationship between the Witten and Turaev approaches is still not completely understood. Ordinary Lie groups are the symmetry groups of manifolds. Quantum groups are the symmetry groups of noncommutative spaces – deformed, noncommutative versions of the commutative algebra of functions on a manifold. Thus the process of passing from QFT to QFT′ is associated with the philosophy of noncommutative geometry, a relatively recent trend in physics.

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