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Hyperkähler metrics and supersymmetry. (English) Zbl 0612.53043
In this article the authors ”attempt to clarify the relation between supersymmetry and aspects of modern differential geometry, along the way reviewing many basic and well known ideas in the hope of making them accessible to a new audience”. - The paper is organized as follows: The first paragraph ”Introduction” presents some basic material on Kähler and hyperkähler manifolds. In the second paragraph ”Construction of new hyperkähler metrics” the authors deal with hyperkähler manifolds (i) induced by a Legendre transform, and (ii) induced by symplectic quotients. In the third paragraph ”Geometric interpretation” these construction procedures are discussed in some more detail including a chapter on twistor spaces. In the remaining part of the paper the concepts introduced up to now are related to supersymmetric nonlinear \(\sigma\)-models: The fourth paragraph presents an introduction to the theory of ”Nonlinear \(\sigma\)-models” and the fifth paragraph to ”Supersymmetry”. The sixth paragraph entitled ”The supersymmetric construction of hyperkähler metrics” presents the supersymmetric version of the construction methods introduced in the second paragraph.
Reviewer: V.Perlick

MSC:
53C55 Global differential geometry of Hermitian and Kählerian manifolds
53C80 Applications of global differential geometry to the sciences
53-02 Research exposition (monographs, survey articles) pertaining to differential geometry
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