1995/03/23 by T. Ackermann, Ackermann, T., J. Tolksdorf +1
Physics and Astronomy · #FOS: Physical sciences #High Energy Physics - Theory (hep-th) #hep-th
paper · pdf · doi:10.48550/arxiv.hep-th/9503153
25 pages, plain tex
arxiv created 1995/03/23 · arxiv updated 2009/11/30
We study Dirac operators acting on sections of a Clifford module \cal E over a Riemannian manifold M. 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 M is compact and \petit \rm dim M=2n≥ 4, we derive an expression for the Wodzicki function W\cal E, which is defined via the non-commutative residue on the space of all Dirac operators \cal D(\cal 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 actions from the Dirac operators in question.