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**Supersymmetry and equivariant de Rham theory. With reprint of two seminal notes by Henri Cartan.**
*(English)*
Zbl 0934.55007

Mathematics: Past and Present. Berlin: Springer. xxiii, 228 p. (1999).

In 1950 two papers by H. Cartan appeared on characteristic classes and connections in the principal \(G\)-bundles and the transgression in the cohomologies of homogeneous spaces. They contained a construction of the De Rham complex for the equivariant cohomologies known as the Cartan model. The book under review gives a modern exposition of this approach using the Matthai-Quillen isomorphism from supergeometry. Another goal is to expose some more recent developements. The authors write in the introduction: “As for the papers themselves, our efforts to update them have left us with a renewed admiration for the simplicity and elegance of Cartan’s original exposition of this material. We predict they will be as timely in 2050 as they were fifty years ago and as they are today”. V. Guillemin and S. Sternberg are known as the authors of several very transparent and attractively written books in differential geometry and analysis on manifolds. The style of the present book is as lively as in their previous publications. This is a good introduction to the whole area connected with equivariant cohomologies and their applications. The only general remark is that the word “supersymmetry” in the title is rather misleading. The book does not contain any introduction into supersymmetry as it exists in mathematical physics. It contains some superconstructions borrowed from the physical supersymmetry. It would be nice to have a book on the latter issue written in the same clear and coordinate free style. In the first part (6 chapters) there is an exposition of equivariant De Rham theory. The main result here is of course to prove the coincidence of the topological equivariant cohomologies and the De Rham ones. To do that the authors develop the necessary formalism of differential graded algebras and their cohomologies. The next main result is to prove equivalence of two complexes, or better say models, which serve for a computation of the equivariant De Rham theory. These are the Weil model known from the Weil approach to the Chern classes and the Cartan model. The latter one appeared in Cartan’s note and is much more effective for computations. One of the next effective computational tools is a spectral sequence which leads to the equivariant cohomologies of some \(G\)-manifold \(M\). Its first term is simply \(H(M)\otimes Sym(g^{*})^G\), where \(g\) is the Lie algebra of the Lie group \(G\). In many cases the sequence will degenerate. As an application one gets a reduction of the equivariant De Rham \(G\)-cohomologies to the equivariant \(T\)-cohomologies for a maximal torus \(T\). The next part is an initiation into Fermionic integration, characteristic classes, moment map and symplectic reduction. One important result discussed here in detail is the Matthai-Quillen construction of the universal equivariant Thom class using the Berezin integral and Fermionic Fourier transform. Another interesting application is the Duistermaat-Heckman theorem. Except that, one can find the Kirillov-Kostant theorem and group valued moment maps. The last two chapters develop localization formalism, particularly a fixed point theorem for the torus actions. This part also contains some very recent results such as the Goresky-Kottwitz-McPherson theorem which gives a simple description of the equivariant cohomologies for the so-called equivariantly formal \(G\)-manifolds. At the end of the book there is an appendix with reprints of Cartan’s papers.

Reviewer: A.N.Parshin (Moskva)