On the rigidity theorems of Witten

@article{Bott1989OnTR,
  title={On the rigidity theorems of Witten},
  author={Raoul Bott and Clifford H. Taubes},
  journal={Journal of the American Mathematical Society},
  year={1989},
  volume={2},
  pages={137-186}
}
  • R. Bott, C. Taubes
  • Published 1989
  • Mathematics
  • Journal of the American Mathematical Society
In this paper we prove the rigidity theorems predicted by Witten in 1986, about the index of certain elliptic operators on manifolds with an S1 action [W]. Witten's insight was the culmination of an interesting interchange of ideas between him and Hopkins, Landweber, Ochanine, and Stong. For the detailed history, we refer the reader to [La]. The present account is essentially a reinterpretation of the second author's (Taubes' [T]) original proof of the theorem. The senior author's contribution… 

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References

SHOWING 1-10 OF 31 REFERENCES
On the Fixed Point Formula and the Rigidity Theorems of Witten Lectures at Cargése 1987
These lectures are meant as an introduction to the interesting ideas which had their origin on the one hand in some string theoretic considerations of Witten and on the other in the more topological
Elliptic Genera of Level N for Complex Manifolds
My lecture at the Como Conference was a survey on the theory of elliptic genera as developed by Ochanine, Landweber, Stong and Witten. A good global reference are the Proceedings of the 1986
Note on the Landweber-Stong elliptic genus
(ii) P(u), an even power series with leading term I, the Hirzebruch characteristic power series of ~. This means that if ~ denotes the stable H*(.;R)-valued exponential characteristic class on
Orientability of fixed point sets
Consider a smooth orientation-preserving action of the cyclic group of order 2, Z2, on a manifold M. As is well known, in contrast to the case of odd order group actions, the fixed point set F is a
Elliptic genera and quantum field theory
It is shown that in elliptic cohomology — as recently formulated in the mathematical literature — the supercharge of the supersymmetric nonlinear signa model plays a role similar to the role of the
Remark on Witten’s modular forms
On donne une demonstration simple de l'invariance modulaire d'une serie de puissances que Witten attache a une variete fermee de dimension paire dont la premiere classe de Pontryagin est la torsion
Spin-manifolds and group actions
Let X be a compact oriented differentiable n-dimensional manifold (all manifolds are without boundary except in § 4) on which a Riemannian metric is introduced. Let Q be the principal tangential SO
Complex Analysis
...
...