On the rigidity theorems of Witten
On the rigidity theorems of Witten
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DOI：
10.1090/s0894-0347-1989-0954493-5
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发表时间：
1989
影响因子：
3.9
通讯作者：
R. Bott;C. Taubes
中科院分区：
文献类型：
--
作者：
R. Bott;C. Taubes
本文证明了维滕在1986年预言的关于流形上具有S1作用的椭圆算子的指数的刚性定理[W]。维滕的洞察力是他与霍普金斯、兰德韦伯、奥恰宁和斯通之间有趣的思想交流的顶峰。关于详细的历史，我们请读者参阅[La]。本说明实质上是对第二作者（Taubes [T]）的定理原始证明的重新解释。资深作者的贡献仅仅是注意到，相当密集的书面论点的原始手稿可以制定在著名的不动点公式的等变指数理论和等变上同调。在这种情况下，最终的证明直接继承了Atiyah-Hirzebruch定理，该定理涉及允许圆作用的自旋流形的A亏格的消失[A-H]，以及Lusztig关于复情形中圆作用的更古老的想法。然而，仍然存在着将这些技术与椭圆函数理论联系起来的美丽的、物理学启发的新奇。
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.