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The Mathai-Quillen formalism and topological field theory. (English) Zbl 0780.57024

Summary: These lecture notes give an introductory account of an approach to cohomological field theory due to Atiyah and Jeffrey which is based on the construction of Gaussian shaped Thom forms by Mathai and Quillen. Topics covered are: an explanation of the Mathai-Quillen formalism for finite dimensional vector bundles; the definition of regularized Euler numbers of infinite dimensional vector bundles; interpretation of supersymmetric quantum mechanics as the regularized Euler number of loop space; the Atiyah-Jeffrey interpretation of Donaldson theory; the construction of topological gauge theories from infinite dimensional vector bundles over spaces of connections.

MSC:

57R57 Applications of global analysis to structures on manifolds
55N99 Homology and cohomology theories in algebraic topology
81T25 Quantum field theory on lattices
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