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Localization and Diagonalization: A Review of Functional Integral Techniques for Low-Dimensional Gauge Theories and Topological Field Theories

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Functional Integration

Part of the book series: NATO ASI Series ((NSSB,volume 361))

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Abstract

We review localization techniques for functional integrals which have recently been used to perform calculations in and gain insight into the structure of certain topological field theories and low-dimensional gauge theories. These are the functional integral counterparts of the Mathai—Quillen formalism, the Duistermaat—Heckman theorem, and the Weyl integral formula respectively. In each case, we first introduce the necessary mathematical background (Euler classes of vector bundles, equivariant cohomology, topology of Lie groups), and describe the finite dimensional integration formulae. We then discuss some applications to path integrals and give an overview of the relevant literature. The applications we deal with include supersymmetric quantum mechanics, cohomological field theories, phase space path integrals, and two-dimensional Yang—Mills theory.

(Extended version of lectures given by M.B. This article originally appeared in the Special Issue on Functional Integration of the Journal of Mathematical Physics, Jour. Math. Physics 36 No. 5 (1995) 2192-2236. We thank the American Institute of Physics (AIP) for permission to reproduce this article here. It has been slightly updated for inclusion in these Proceedings.)

supported by EC Human Capital and Mobility Grant ERB-CHB-GCT-93-0252

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Authors and Affiliations

  1. ICTP, P.O. Box 586, 34014, Trieste, Italy

    G. Thompson

  2. LPTHE-Enslapp, ENS-Lyon, 46 Allée d’Italie, F-69364, Lyon Cedex 07, France

    M. Blau

  3. URA 14-36 du CNRS, associée à l’E.N.S. de Lyon, l’Université de Savoie, France

    G. Thompson

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  1. M. Blau
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Editors and Affiliations

  1. The University of Texas at Austin, Austin, Texas, USA

    Cecile DeWitt-Morette

  2. Ecole Normale Supérieure, Paris, France

    Pierre Cartier

  3. Université de Corse, Corte, France

    Antoine Folacci

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Blau, M., Thompson, G. (1997). Localization and Diagonalization: A Review of Functional Integral Techniques for Low-Dimensional Gauge Theories and Topological Field Theories. In: DeWitt-Morette, C., Cartier, P., Folacci, A. (eds) Functional Integration. NATO ASI Series, vol 361. Springer, Boston, MA. https://doi.org/10.1007/978-1-4899-0319-8_12

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