A KKtheoretic perspective on deformed Dirac operators
Abstract
We study the index theory of a class of perturbed Dirac operators on noncompact 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 KKproduct of certain KKtheory classes defined by $\mathsf{D}$ and $X$. As a corollary we obtain the excision and cobordisminvariance 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.
 Publication:

arXiv eprints
 Pub Date:
 July 2019
 arXiv:
 arXiv:1907.06150
 Bibcode:
 2019arXiv190706150L
 Keywords:

 Mathematics  KTheory and Homology;
 Mathematics  Differential Geometry
 EPrint:
 24 pages