论文标题
横向类型的本地索引定理，具有本地免费的$ \ mathbb {s}^1 $ -ACTION的横向索引定理
A Local Index Theorem of Transversal Type on Manifolds with Locally Free $\mathbb{S}^1$-action
论文作者
论文摘要
我们在一个奇数歧管上$ x $，在本地免费$ \ mathbb {s}^1 $ -Action上研究横向狄拉克操作员的索引。使用加热核法的一个困难在于将渐近扩展为$ t \至0^+$。通过通过Feynman-kac公式的概率方法，$ x $上的横向加热内核可以链接到普通的热核，用于Orbifold $ M = x/\ Mathbb {s}^1 $上的功能上的功能，这更易于tract。经过统一约束估计为$ t \ to 0^+$的技术结果后，我们从横向的，Orbifold的情况降低到经典状况，尤其是在主要阶层的位置。一项应用程序断言，对于一定类别的旋转Orbifolds $ m $，在Riemannian的经典指数问题中，川崎的经典索引问题是由低维地层产生的净贡献，而不是主要的一个相同消失。
We study an index of a transversal Dirac operator on an odd-dimensional manifold $X$ with locally free $\mathbb{S}^1$-action. One difficulty of using heat kernel method lies in the understanding of the asymptotic expansion as $t\to 0^+$. By a probabilistic approach via the Feynman-Kac formula, the transversal heat kernel on $X$ can be linked to the ordinary heat kernel for functions on the orbifold $M=X/\mathbb{S}^1$ which is more tractable. After some technical results for a uniform bound estimate as $t\to 0^+$, we are reduced from the transversal, orbifold situation to the classical situation particularly at points of the principal stratum. One application asserts that for a certain class of spin orbifolds $M$, to the classical index problem of Kawasaki in the Riemannian setting the net contributions arising from the lower-dimensional strata beyond the principal one vanish identically.
