Noncommutative Geometry and An Index Theory of Infinite-Dimensional Manifolds

Abstract

The Atiyah–Singer index theorem is one of the monumental works in geometry and topology. My dream is to construct an infinite-dimensional version of it. Although this project is very hard, I constructed several core objects for an analytic index theory for infinite-dimensional manifolds. The problem is the following: For an infinite-dimensional “\(Spin^c\)”-manifold \(\mathcal {M}\) on which a loop group of a circle acts, construct a \(C^*\)-algebra A which carries some information of \(\mathcal {M}\), a Hilbert space which can be regarded as an “\(L^2\)-space consisting of sections of the Spinor bundle”, and an operator \(\mathcal {D}\) which can be regarded as a Dirac operator on \(\mathcal {M}\), and define a spectral triple coming from them for A. The core idea for the construction comes from representation theory of loop groups and Higson–Kasparov–Trout’s algebra [4].