## On the rigidity theorems of Witten.(English)Zbl 0667.57009

This paper proves that the elliptic genus is rigid, as was predicted by Edward Witten in 1968. Specifically, the paper describes a sequence of genera having the property that for an $$S^ 1$$ action on a Spin manifold the corresponding equivariant extension is a trivial representation. The proof given here is based on the standard methods of index theory for elliptic complexes. Taubes also gave a proof, based upon Witten’s outline, using the idea of elliptic complexes on the loop space of a manifold.
Reviewer: R.E.Stong

### MSC:

 57R20 Characteristic classes and numbers in differential topology 58J10 Differential complexes 58J20 Index theory and related fixed-point theorems on manifolds 55N91 Equivariant homology and cohomology in algebraic topology 55N99 Homology and cohomology theories in algebraic topology 55M20 Fixed points and coincidences in algebraic topology
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### References:

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