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On elliptic genera and theta-functions. (English) Zbl 0858.57034
The author gives a unified new proof of the Witten rigidity theorem for some elliptic operators commuting with a circle action. He makes essential use of a new feature of loop spaces and loop groups, the modular invariance. Actually he shows that modular invariance implies rigidity. Some new properties of elliptic genera and their relationships with theta functions are also discussed.
Reviewer: F.Gomez (Malaga)

MSC:
57R20 Characteristic classes and numbers in differential topology
58J20 Index theory and related fixed-point theorems on manifolds
58J26 Elliptic genera
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