The generalized Lichnerowicz formula and analysis of Dirac operators

Ackermann, Thomas ; Tolksdorf, Jürgen

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URN: urn:nbn:de:bsz:180-madoc-16565
Document Type: Working paper
Year of publication: 1995
Publication language: English
Institution: School of Business Informatics and Mathematics > Sonstige - Fakultät für Mathematik und Informatik
MADOC publication series: Veröffentlichungen der Fakultät für Mathematik und Informatik > Institut für Mathematik > Mannheimer Manuskripte
Subject: 510 Mathematics
Classification: MSC: 58G15 81E13 53A50 58G10 ,
Subject headings (SWD): Differentialgeometrie , Dirac-Operator , Eichtheorie
Keywords (English): Lichnerowicz formula , Dirac operator , Wodzicki function , gauge theory
Abstract: We study Dirac operators acting on sections of a Clifford module ε over a Riemannian manifold Μ. 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 Μ is compact and dim Μ = 2n ≥ 4, we derive an expression for the Wodzicki function Wε, which is defined via the noncommutative residue on the space of all Dirac operators D(ε). 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 aetions from the Dirac operators in question.
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