Local index theory for geometric first-order differential operators

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.