A generalized Lichnerowicz formula, the Wodzicki Residue and Gravity

Abstract

We prove a generalized version of the well-known Lichnerowicz formula for the square of the most general Dirac operator $\tilde{D}$\ on an even-dimensional spin manifold associated to a metric connection $\tilde{\nabla }$. We use this formula to compute the subleading term ${\Phi }_1(x,x,{\tilde{D}}^2)$\ of the heat-kernel expansion of ${\tilde{D}}^2$. The trace of this term plays a key-r$\hat {\petit\rm o}$le in the definition of a (euclidian) gravity action in the context of non-commutative geometry. We show that this gravity action can be interpreted as defining a modified euclidian Einstein-Cartan theory.

10 pages, plain tex

References in corpus (2)

• Gravity, Non-Commutative Geometry and the Wodzicki Residue
• The generalized Lichnerowicz formula and analysis of Dirac operators

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