The generalized Lichnerowicz formula and analysis of Dirac operators
Abstract
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 wellknown 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 AtiyahSinger index theorem holds. Furthermore, if $M$\ is compact and ${{\petit \rm dim}\;M=2n\ge 4}$, we derive an expression for the Wodzicki function $W_{\cal E}$, which is defined via the noncommutative 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/YangMills actions from the Dirac operators in question.
 Publication:

arXiv eprints
 Pub Date:
 March 1995
 arXiv:
 arXiv:hepth/9503153
 Bibcode:
 1995hep.th....3153A
 Keywords:

 High Energy Physics  Theory
 EPrint:
 25 pages, plain tex