题目： Loop spaces, localization and the Atiyah-Singer index theorem （I）&（II）&（III）
时间：6月22日 下午 2:00-5:30
摘要： Atiyah and Witten discovered the remarkable fact that the index of the Dirac operator on a spin manifold can be formally interpreted as an integral of an equivariantly differential form over loop space and a formal application of the localization formula of Duistermaat-Heckman leads to the Atiyah-Singer index theorem for Dirac operator. Bismut extends this approach to Dirac operator coupled by a vector bundle. In doing so, for a vector bundle with connection, he constructed an equivariantly closed form on the loop space, lifting the Chern character form of the vector to the loop space. Actually as pointed out by Bismut, there is a loop space functor, which translates many natural objects of the original manifold to geometric objects on its loop space. In these talks, I will briefely introduce the theories of Atiyah-Witten and Bismut.