A KK-theoretic perspective on deformed Dirac operators

We study the index theory of a class of perturbed Dirac operators on non-compact manifolds of the form $\mathsf{D}+\mathrm{i}\mathsf{c}(X)$, where $\mathsf{c}(X)$ is a Clifford multiplication operator by an orbital vector field with respect to the action of a compact Lie group. Our main result is that the index class of such an operator factors as a KK-product of certain KK-theory classes defined by $\mathsf{D}$ and $X$. As a corollary we obtain the excision and cobordism-invariance properties first established by Braverman. An index theorem of Braverman relates the index of $\mathsf{D}+\mathrm{i}\mathsf{c}(X)$ to the index of a transversally elliptic operator. We explain how to deduce this theorem using a recent index theorem for transversally elliptic operators due to Kasparov.