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

  • M. Blau
  • G. Thompson
Part of the NATO ASI Series book series (NSSB, volume 361)


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.)


Partition Function Modulus Space Vector Bundle Loop Space Euler Number 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


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Copyright information

© Springer Science+Business Media New York 1997

Authors and Affiliations

  • M. Blau
    • 2
  • G. Thompson
    • 1
    • 3
  1. 1.ICTPTriesteItaly
  2. 2.LPTHE-EnslappENS-LyonLyon Cedex 07France
  3. 3.URA 14-36 du CNRS, associée à l’E.N.S. de Lyonl’Université de SavoieFrance

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