Volume 11, issue 1 (2007)

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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
 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