"Dirac Operators with Potential"

Abstract:  A Dirac operator is a first-order elliptic differential operator on a manifold. If the manifold is compact, the Dirac operator is Fredholm, with an index theory that has been well studied since Atiyah and Singer first proved their famous index theorem in the 1960s. This theory has had a fundamental impact on geometry and topology, with a vast number of applications and offshoots. For example, it can be applied to the problem of determining existence of metrics of positive scalar curvature. On the other hand, if the manifold is non-compact, a Dirac operator is not in general Fredholm. In this talk, I will begin by describing a class of modified Dirac operators on non-compact manifolds - first studied by Constantine Callias - that do have a well-defined index theory, before considering an equivariant generalization of this theory that takes into account the action of a symmetry group on the manifold. This can then be applied to obtain a refinement of Lichnerowicz's vanishing theorem on obstructions to positive scalar curvature that can be applied to manifolds equipped with the isometric action of an arbitrary Lie group.

Host: Yanli Song