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 D on an even-dimensional spin manifold associated to a metric connection ∇. We use this formula to compute the subleading term Φ 1 (x, x, D 2 ) of the heat-kernel expansion of D 2 . The trace of this term plays a key-rôle 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. ⋆ Supported by the European Communities, contract no. ERB 401GT 930224 expansion, Wodzicki residue, gravity for the most general Dirac operator D associated to a metric connection ∇ on a compact spin manifold M with dim M = 2n ≥ 4. We proceed as follows: According to the main theorem of [KW], there is a relation between the Wodzicki residue Res(△ −n+1 ) of a generalized laplacian△ on a hermitian bundle E over M and

Parallel spinors and connections with skew-symmetric torsion in string theory

“…In case of a Riemannian manifold the well-known Schrodinger-Lichnerowicz-formula expresses the square of the Dirac operator with respect to the Levi-Civita connection by the spinorial Laplace operator and some curvature term (see [46,14]). In the articles [5,1], a generalization of this formula for connections with arbitrary torsion is indicated. For connections with totally skew-symmetric torsion we shall derive the curvature term and prove the following explicit formula :…”

Classical gravity on fuzzy space-time

“…Indeed we are not in a position to argue that there is even a valid action principle. A discussion of this point has been made by Connes and co-workers in a series of articles [15,16,17,18] but the definition which these authors propose is valid only on the noncommutative generalizations of compact spaces with euclidean-signature metrics.…”

Introduction to Non-Commutative Geometry

“…The expression Tr log ∆[g] has a natural generalization to the noncommutative case [70,1,19]. See also Example 7.3.5 of Madore [91].…”