Topological quantum field theory
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A topological quantum field theory (or topological field theory or TQFT) is a quantum field theory which computes topological invariants.
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Basics
In a topological field theory, the correlation functions do not depend on the metric on spacetime. This means that the theory is not sensitive to changes in the shape of spacetime; if the spacetime warps or contracts, the correlation functions do not change. Consequently, they are topological invariants. (Strictly speaking, the argument above only shows that they are diffeomorphism invariants. Showing that they are homotopy invariants takes more effort.)
Topological field theories are not very interesting on the flat Minkowski spacetime used in particle physics. Minkowski space can be contracted to a point, so a TQFT on Minkowski space computes only trivial topological invariants. Consequently, TQFTs are usually studied on curved spacetimes, such as, for example, Riemann surfaces. Most of the known topological field theories are defined on spacetimes of dimension less than five. It seems that a few higher dimensional theories exist, but they are not very well understood.
Although TQFTs were invented by physicists (notably Witten), they are primarily of mathematical interest, being related to, among other things, knot theory and the theory of four-manifolds in algebraic topology, and to the theory of moduli spaces in algebraic geometry. (Donaldson, Jones, Witten, and Kontsevich have all won Fields Medals for work related to topological field theory.)
Physically speaking topological field theories are not especially interesting. They are toy models, far simpler in structure than the quantum field theories which describe real-world physics. They can be thought of as warm-up exercises for the much harder task of quantizing gravity. Quantum gravity is believed to be background-independent (in some suitable sense), and TQFTs provide examples of background independent quantum field theories. Unfortunately, they are only background independent in a fairly trivial sense: they are independent of the spacetime metric (thought of as a background gravitational field), but they do not admit any local degrees of freedom. There is no radiation in a TQFT: no propagating waves, no gluons, no gravitons.
(Caveat: It is often said that TQFTs have only finitely many degrees of freedom. This is not a fundamental property. It happens to be true in most of the examples that physicists and mathematicians study, but it is not necessary. A topological sigma model with target infinite-dimensional projective space, if such a thing could be defined, would have countably infinitely many degrees of freedom.)
Specific Models
The known topological field theories fall into two general classes: Schwarz-type TQFTs and Witten-type TQFTs. Witten TQFTs are also sometimes referred to as cohomological field theories.
Schwarz-type TQFTs
In Schwarz-type TQFTs, the correlation functions computed by the path integral are topological invariants because the path integral measure and the quantum field observables are explicitly independent of the metric. For instance, in the BF model, the spacetime is a two-dimensional manifold M, the observables are constructed from a two-form F, an auxiliary scalar B, and their derivatives. The action (which determines the path integral) is
- <math>S=\int_M B F</math>
The spacetime metric does not appear anywhere in this theory, so the theory is explicitly topologically invariant.
Another, more famous example is Chern-Simons theory, which can be used to compute knot invariants.
Witten-type TQFTs
In Witten-type topological field theories, the topological invariance is more subtle.
NEEDS FINISHING
Mathematical Formulations
Atiyah-Segal Axioms
There is a formulation of an n-dimensional topological quantum field theory using cobordisms and category theory. Take the category of n-1 dimensional compact boundaryless manifolds with smooth maps. Cobordisms form a category with the n-1 dimensional compact boundariless manifolds as objects. There is a functor to the category of complex vector spaces. Both the cobordism category and the category of complex vector spaces are monoidal categories. The functor respects that.
