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Topological quantum field theories. (English) Zbl 0692.53053

The starting point to topological quantum field theory was given by E. Witten [J. Differ. Geom. 17, 661-692 (1982; Zbl 0499.53056)] where he explained the geometric meaning of super-symmetry, pointing out that for super-symmetric quantum mechanics the Hamiltonian is just the Hodge- Laplacian. He then outlined the corresponding ideas for super-symmetric quantum field theories viewed as differential geometry of certain infinite dimensional manifolds, including the associated analysis and topology. This shows that there may be interesting topological aspects of quantum field theory and that these should be important for physics. On the other hand one can use these quantum field theories as a conceptual tool to suggest new mathematical results. Indeed this reversed process led to spectacular progress in the understanding of geometry in low dimensions.

The author starts with presenting a set of axioms for topological quantum field theories following G. B. Segal [The definition of conformal field theory, Differential geometrical methods in theoretical physics, Proc. 16th Int. Conf., NATO Adv. Res. Workshop, Como/Italy 1987, NATO ASI Ser., Ser. C 250, 165-171 (1988; Zbl 0657.53060)], then lists examples of theories (known to exist) satisfying such axioms. These include in $d=1$ the Floer/Gromov theory and holomorphic conformal field theories (Segal); in $d=2$ the Jones/Witten theory, Casson theory, Johnson theory and Thursten theory and in $d=3$ the Floer/Donaldson theory.

Reviewer: R.Schmid

##### MSC:
 53C80 Applications of global differential geometry to physics 57N99 Topological manifolds 81T99 Quantum field theory
##### References:
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