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In $\S$ 9.3 of the book "Mirror symmetry" (Vafa, Zaslow eds.) the authors formulate the following general localization principle for computation of integrals with respect to both even and odd variables: Assume there is some supersymmetry transformation of variables. Then the integral becomes localized on the field configurations for which the fermionic variables are invariant under the supersymmetry.

I would like to understand the precise meaning of this sentence. Though several non-trivial and beautiful applications are discussed in the book, I do not understand how they are deduced from this principle due to the fact that the principle itself seems to be too vague. The explanation on pp.158-159 how it works is too concise for me, and it is not clear how that example can be generalized to other situations.

Question: how to formulate a precise statement behind this principle (the situation of finite dimensional integrals would be enough for me for the moment)? Can I read a proof of it somewhere?

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  • $\begingroup$ I know nothing about Mirror symmetry and the book you mentioned. But what you are asking looks like the Atiyah Bott localization formula. en.wikipedia.org/wiki/… $\endgroup$ – shu Feb 14 '15 at 15:46
  • $\begingroup$ I think it is a different localization formula. $\endgroup$ – MKO Feb 14 '15 at 17:30
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    $\begingroup$ There is a thorough answer at physicsoverflow.org/28079 $\endgroup$ – Arnold Neumaier Mar 10 '15 at 14:54
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I think what you are looking for is Theorem 1 in "Supersymmetry and Localization" by Schwarz and Zaboronsky.

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