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Topological quantum field theory

A topological quantum field theory (or topological field theory or TQFT ) is a [quantum field theory] which computes [topological invariant] s.

Although TQFTs were invented by physicists, they are also of mathematical interest, being related to, among other things, [knot theory] and the theory of [four-manifold] s in [algebraic topology] , and to the theory of [moduli spaces] in [algebraic geometry] . [Donaldson] , [Jones] , [Witten] , and [Kontsevich] have all won [Fields Medal] s for work related to topological field theory.

In [condensed matter physics] , topological quantum field theories are the low energy effective theories of [topologically ordered] states, such as [fractional quantum Hall] states, [string-net] condensed states, and
other strongly correlated quantum liquid states.


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.

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.

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