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Fermionic Integration

In: Supersymmetry and Equivariant de Rham Theory

Author

Listed:
  • Victor W. Guillemin

    (Massachusetts Institute of Technology, Department of Mathematics)

  • Shlomo Sternberg

    (Harvard University, Department of Mathematics)

  • Jochen Brüning

    (Humboldt-Universität Berlin, Institut für Mathematik Mathematisch-Naturwissenschaftliche Fakultät II)

Abstract

Fermionic integration was introduced by Berezin [Be] and is part of the standard repertoire of elementary particle physicists. It is not all that familiar to mathematicians. However it was used by Mathai and Quillen [MQ] in their path breaking paper constructing a “universal Thom form”. In this chapter we will develop enough of Berezin’s formalism to reproduce the MathaiQuillen result. We will also discuss the Fermionic Fourier transform and combine Bosonic and Fermionic Fourier transforms into a single “super” Fourier transform. We will see that there is an equivariant analogue of compactly supported cohomology which can be obtained from the Koszul complex by using this super Fourier transform, and use this to explain the Mathai-Quillen formula. In Chapter 10 we will apply these results to obtain localization theorems in equivariant cohomology.

Suggested Citation

  • Victor W. Guillemin & Shlomo Sternberg & Jochen Brüning, 1999. "Fermionic Integration," Springer Books, in: Supersymmetry and Equivariant de Rham Theory, chapter 0, pages 77-93, Springer.
    • Handle: RePEc:spr:sprchp:978-3-662-03992-2_7
    DOI: 10.1007/978-3-662-03992-2_7
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