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Topological quantum field theory

The correlation functions and physical predictions of a QFT depend on the spacetime metric g. For a special class of QFTs called topological quantum field theories (TQFTs), all correlation functions are independent of continuous changes in the spacetime metric. QFTs in curved spacetime generally change according to the geometry (local structure) of the spacetime background, while TQFTs are invariant under spacetime diffeomorphisms but are sensitive to the topology (global structure) of spacetime. This means that all calculational results of TQFTs are topological invariants of the underlying spacetime. Chern–Simons theory is an example of TQFT and has been used to construct models of quantum gravity. Applications of TQFT include the fractional quantum Hall effect and topological quantum computers. The world line trajectory of fractionalized particles (known as anyons) can form a link configuration in the spacetime, which relates the braiding statistics of anyons in physics to thelink invariants in mathematics. Topological quantum field theories (TQFTs) applicable to the frontier research of topological quantum matters include Chern-Simons-Witten gauge theories in 2+1 spacetime dimensions, other new exotic TQFTs in 3+1 spacetime dimensions and beyond.