The index of the dirac operator in loop space
The index of the dirac operator in loop space
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DOI：
10.1007/bfb0078045
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发表时间：
1988
影响因子：
--
通讯作者：
E. Witten
中科院分区：
文献类型：
--
作者：
E. Witten
设M是一个自旋流形，eM是映射8 1-+ M的自由循环空间。虽然分析无限维空间，如。eM是数学文献中的一个新课题，在过去的60年里，物理学家们在量子场论的框架下对它进行了大量的研究。特别是，某些量子场论可以被解释为在有限维情况下具有既定数学意义的结构的无限维类似物。例如，我在[1]中证明了1+ 1维（目标空间为M）超对称非线性sigma模型的超荷Q可以解释为. eM上的Dirac类算子。这些讲义将致力于正式讨论. eM上Dirac算子的指数。这个问题与舍勒肯斯和华纳最近关于异常的工作密切相关，而且正如我们将看到的，它对Landweber和Ochanine在这次会议上讨论过的某个拓扑问题有着有趣的应用。从本质上讲，我们将看到，所讨论的拓扑猜想是由超对称非线性σ模型的某些简单（固定）性质得出的。由于非线性sigma模型的截止版本就足够了，因此有理由希望sigma模型的必要属性可以在几年内得到证明。[3]中简要讨论了sigma模型在拓扑学中的应用，沿着还有一些关于物理学应用的推测。我们的考虑也密切相关的使用不动点定理的循环空间的一组获得Weyl-Kac字符公式仿射李代数;见[4]的阐述这一主题。
Let M be a spin manifold and. eM the free loop space of maps 8 1-+ M. Although analysis on infinite dimensional spaces such as. eM is a new subject in the mathematical literature, it has been in the last sixty years much studied by physicists in the framework of quantum field theory. In particular, certain quantum field theories can be interpreted as infinite dimensional analogues of structures which have an established mathematical significance in the finite dimensional case. For instance, I showed in [1] that the supercharge Q of the supersymmetric nonlinear sigma model in 1+ 1 dimensions (with target space M) can be interpreted as a Dirac-like operator on. eM.These lecture notes will be devoted to discussing formally the index of the Dirac operator on. eM. This subject is closely related to the recent work of Schellekens and Warner on anomalies [2], and it has interesting applications, as we will see, to a certain topological problem which has been discussed at this conference by Landweber and Ochanine. In essence we will see that the topological conjecture in question would follow from certain simple (conjectured) properties of the supersymmetic nonlinear sigma model. Since a cutoff version of the nonlinear sigma model would be adequate, there is a reasonable hope that the requisite properties of the sigma model can be proved within a few years. This application of the sigma model to topology was briefly discussed in [3], along with some speculations about applications to physics. Our considerations are also closely related to the use of fixed point theorems on the loop space of a group to obtain the Weyl-Kac character formula for affine Lie algebras; see [4] for an exposition of this subject.