Index theorem for equivariant Dirac operators on non-compact manifolds

TL;DR

This paper extends the equivariant index theorem to non-compact manifolds with group actions, defining both topological and analytic indexes and proving their equality, thus broadening the theorem's applicability.

Contribution

It introduces a new framework for the equivariant index on non-compact manifolds, including a proof of index invariance under certain cobordisms and an extension of the Atiyah-Segal-Singer theorem.

Findings

Analytic and topological indexes coincide for non-compact manifolds.

Index is invariant under a specific class of non-compact cobordisms.

Provides a new proof of the Atiyah-Segal-Singer theorem in the non-compact setting.

Abstract

Let $D$ be a (generalized) Dirac operator on a non-compact 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 non-compact cobordisms. As an application we extend the Atiyah-Segal-Singer equivariant index theorem to our non-compact setting. In particular, we obtain a new proof of this theorem for compact manifolds.