Local index theory for geometric first-order differential operators
Abstract
Abstract We introduce the concept of chiral geometric operators and use Gilkey’s invariance theory to prove the local index theorem for these operators. In other words, we demonstrate that the supertrace of the heat kernel of a given geometric operator converges as time approaches zero and that this limit is the Chern–Weil form of the Atiyah–Singer integrand. In addition to classical Dirac-type operators that appear in geometry, chiral geometric operators include all higher Dirac operators. This includes in particular the Rarita–Schwinger operator. We also construct a new class of such operators on four-manifolds called higher signature operators.
Fields of Study & Subjects
Spectral Theory in Mathematical Physics
Differential Equations and Boundary Problems
Algebraic and Geometric Analysis
Publication details
- Journal / source
- Annals of Global Analysis and Geometry
- Publisher
- Springer Science and Business Media LLC
- Volume
- 69
- Issue
- 4
- Publication date
- 18 May 2026
- Language
- EN