Refined analytic torsion as an element of the determinant line

Braverman, Maxim; Kappeler, Thomas

doi:10.2140/gt.2007.11.139

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Refined analytic torsion as an element of the determinant line

Maxim Braverman and Thomas Kappeler

Geometry & Topology 11 (2007) 139–213

DOI: 10.2140/gt.2007.11.139

arXiv: math.GT/0510532

Abstract

We construct a canonical element, called the refined analytic torsion, of the determinant line of the cohomology of a closed oriented odd-dimensional manifold $M$ with coefficients in a flat complex vector bundle $E$ . We compute the Ray–Singer norm of the refined analytic torsion. In particular, if there exists a flat Hermitian metric on $E$ , we show that this norm is equal to 1. We prove a duality theorem, establishing a relationship between the refined analytic torsions corresponding to a flat connection and its dual.

Keywords

determinant line, Ray–Singer, eta-invariant, analytic torsion

Mathematical Subject Classification 2000

Primary: 58J52

Secondary: 58J28, 57R20

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Publication

Received: 29 March 2006

Accepted: 21 November 2006

Published: 16 March 2007

Proposed: Peter Teichner

Seconded: Wolfgang Lueck, Leonid Polterovich

Authors

Maxim Braverman
Department of Mathematics Northeastern University Boston MA 02115 USA

Thomas Kappeler
Institut fur Mathematik Universitat Zürich Winterthurerstrasse 190 CH-8057 Zürich Switzerland

Volume 11, issue 1 (2007)