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Quantum field theory and low-dimensional geometry. (English) Zbl 0804.53098

Eguchi, T. (ed.) et al., Common trends in mathematics and quantum field theories. 1990 Yukawa international seminar school: Kansai Seminar House, Kyoto, Japan, May 10-16, 1990. Workshop: RIMS, Kyoto University, Japan, May 17-19, 1990. Tokyo: Yukawa Institute for Theoretical Physics, Prog. Theor. Phys., Suppl. 102, 1-13 (1990).
The meeting points between Quantum Field Theory and Low-Dimensional Geometry are (1) topological field theories, Lagrangian field theories which do not depend on the metric of background space, but only on topological invariants, (2) the solution manifold of self-duality equation in gauge theories. Whether this connection is purely mathematical, or really physical is unclear. These two points are related to Donaldson and Jones theories on the geometry of 3 and 4 manifolds. This is a very clear review on the connection between Quantum field theory and low-dimensional geometry; discussing also the still many open problems.
For the entire collection see [Zbl 0777.00029].

MSC:

53Z05 Applications of differential geometry to physics
81T40 Two-dimensional field theories, conformal field theories, etc. in quantum mechanics
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