• DOI:10.1090/S0894-0347-1989-0954493-5
• Corpus ID: 37066326

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 was solely to notice that the rather densely written arguments of the original manuscript could be formulated in terms of the well known fixed point formulae of equivariant index theory and equivariant cohomology. In this context, the final proof then appears in direct lineage of the Atiyah-Hirzebruch theorem concerning the vanishing of the A genus of spin manifolds admitting a circle action [A-H] and of the even older idea of Lusztig concerning circle actions in the complex case. There remains, however, the beautiful, physics inspired novelty of connecting these techniques with elliptic function theory.