Abstract
This is the introductory chapter to this issue. We review the main idea of the localization technique and its brief history both in geometry and in QFT. We discuss localization in diverse dimensions and give an overview of the major applications of the localization calculations for supersymmetric theories. We explain the focus of the present issue.
Highlights
This is a contribution to the review volume “Localization techniques in quantum field theories”
Later independently in 1982 Berline and Vergne [4] and in 1984 Atiyah and Bott [5] generalized the Duistermaat-Heckman formula to the case of a general compact manifold M with a U(1) action and an integral α of an equivariantlyclosed form α, that is (d + ιV )α = 0, where V (x) is the vector field corresponding to the U(1) action
The more detailed overview of this formula and its relation to equivariant cohomology is given in Contribution [6]
Summary
This is a contribution to the review volume “Localization techniques in quantum field theories” Like the Υ-function defined by infinite products like (2.3) are infinite-dimensional versions of the equivariant Euler class of the tangent bundle to the space of all fields appearing after localization of the path integral by Atiyah-Bott fixed point formula
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