Geometry and Localization: An Unholy Alliance? Thought of the Day 95.0

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There are many misleading metaphors obtained from naively identifying geometry with localization. One which is very close to that of String Theory is the idea that one can embed a lower dimensional Quantum Field Theory (QFT) into a higher dimensional one. This is not possible, but what one can do is restrict a QFT on a spacetime manifold to a submanifold. However if the submanifold contains the time axis (a ”brane”), the restricted theory has too many degrees of freedom in order to merit the name ”physical”, namely it contains as many as the unrestricted; the naive idea that by using a subspace one only gets a fraction of phase space degrees of freedom is a delusion, this can only happen if the subspace does not contain a timelike line as for a null-surface (holographic projection onto a horizon).

The geometric picture of a string in terms of a multi-component conformal field theory is that of an embedding of an n-component chiral theory into its n-dimensional component space (referred to as a target space), which is certainly a string. But this is not what modular localization reveals, rather those oscillatory degrees of freedom of the multicomponent chiral current go into an infinite dimensional Hilbert space over one localization point and do not arrange themselves according according to the geometric source-target idea. A theory of this kind is of course consistent but String Theory is certainly a very misleading terminology for this state of affairs. Any attempt to imitate Feynman rules by replacing word lines by word sheets (of strings) may produce prescriptions for cooking up some mathematically interesting functions, but those results can not be brought into the only form which counts in a quantum theory, namely a perturbative approach in terms of operators and states.

String Theory is by no means the only area in particle theory where geometry and modular localization are at loggerheads. Closely related is the interpretation of the Riemann surfaces, which result from the analytic continuation of chiral theories on the lightray/circle, as the ”living space” in the sense of localization. The mathematical theory of Riemann surfaces does not specify how it should be realized; if its refers to surfaces in an ambient space, a distinguished subgroup of Fuchsian group or any other of the many possible realizations is of no concern for a mathematician. But in the context of chiral models it is important not to confuse the living space of a QFT with its analytic continuation.

Whereas geometry as a mathematical discipline does not care about how it is concretely realized the geometrical aspects of modular localization in spacetime has a very specific geometric content namely that which can be encoded in subspaces (Reeh-Schlieder spaces) generated by operator subalgebras acting onto the vacuum reference state. In other words the physically relevant spacetime geometry and the symmetry group of the vacuum is contained in the abstract positioning of certain subalgebras in a common Hilbert space and not that which comes with classical theories.

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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.