The generalized Lichnerowicz formula and analysis of Dirac operators.

Abstract

We study Dirac operators acting on sections of a Clifford module e over a Riemannian manifold Μ. We prove the intrinsic decomposition formula for their square, which is the generalisation of the well-known formula due to Lichnerowicz [L]. This formula enables us to distinguish Dirac operators of simple type. For each Dirac operator of this natural class the local Atiyah-Singer index theorem holds. Furthermore, if Μ is compact and dim Μ = 2n ≥ 4, we derive an expression for the Wodzicki function We, which is defined via the noncommutative residue on the space of all Dirac operators D(e). We calculate this function for certain Dirac operators explicitly. From a physical point of view this provides a method to derive gravity, resp. combined gravity/Yang-Mills aetions from the Dirac operators in question.