Witten, Edward The index of the Dirac operator in loop space. (English) Zbl 0679.58045 Elliptic curves and modular forms in algebraic topology, Proc. Conf., Princeton/NJ 1986, Lect. Notes Math. 1326, 161-181 (1988). [For the entire collection see Zbl 0642.00007.] The article consist of five sections. Section 1 is introductory. Section 2 is devoted to the Dirac index on the free loop space of maps \(S^ 1\to M\), where M is a spin manifold. In Section 3 the author indicates a few aspects of the relation between anomalies and the Dirac operator in loop space and Section 4 sketches the application of Dirac operators in loop space - or in other words of supersymmetric nonlinear sigma models - to a topological problem involving the vanishing of certain equivariant characteristic numbers. Section 5 is devoted to the case of complex manifolds. Reviewer: S.Bajzaev Cited in 12 ReviewsCited in 60 Documents MSC: 58J20 Index theory and related fixed-point theorems on manifolds 83C99 General relativity Keywords:spin manifold; index; Dirac operator; loop space; supersymmetric nonlinear sigma models Citations:Zbl 0642.00007 PDFBibTeX XML