A Note on the Square of Dirac Type Differential Operators

Jürgen Tolksdorf

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Submission date: 24. Aug. 2006
Pages: 12
published in: Journal of geometry and physics, 57 (2007) 10, p. 1999-2013 
DOI number (of the published article): 10.1016/j.geomphys.2007.04.004
Bibtex
with the following different title: On the square of first order differential operators of Dirac type and the Einstein-Hilbert action
MSC-Numbers: 53C05, 53C07
PACS-Numbers: 02.40.Hw, 02.40.Ma
Keywords and phrases: Clifford Module Bundles, Dirac Type Differential Operators, Linear Connections
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Abstract:
The aim of this note is to present a new global formula for the

Lichnerowicz decomposition of a general Dirac type first order

differential operator. This formula generalizes the well-known

Lichnerowicz formula for Dirac type operators which are determined

by Clifford connections on an arbitrary Clifford module bundle.

Moreover, we also show that the connection class of a general Dirac

type operator has a natural representative. In this sense, each

Dirac type operator defines a natural connection on a given Clifford

module bundle. This generalizes the fact that the connection class of

a Dirac type operator possesses at most one Clifford connection.