
Physics 2000
Index theorem for equivariant Dirac operators on noncompact manifoldsAbstract: Let $D$ be a (generalized) Dirac operator on a noncompact complete Riemannian manifold $M$ acted on by a compact Lie group $G$. Let $v:M > Lie(G)$ be an equivariant map, such that the corresponding vector field on $M$ does not vanish outside of a compact subset. These data define an element of $K$theory of the transversal cotangent bundle to $M$. Hence a topological index of the pair $(D,v)$ is defined as an element of the completed ring of characters of $G$. We define an analytic index of $(D,v)$ as an index space of certain deformation of $D$ and we prove that the analytic and topological indexes coincide. As a main step of the proof, we show that index is an invariant of a certain class of cobordisms, similar to the one considered by Ginzburg, Guillemin and Karshon. In particular, this means that the topological index of Atiyah is also invariant under this class of noncompact cobordisms. As an application we extend the AtiyahSegalSinger equivariant index theorem to our noncompact setting. In particular, we obtain a new proof of this theorem for compact manifolds.
