Complex-valued Ray-Singer torsion
type: Item
instance of: scholarly article
title: Complex-valued Ray-Singer torsion
zbMATH Open document ID: 1131.58020
author: Dan Burghelea
author: Stefan Haller
published in: Journal of Functional Analysis
publication date: 2007-07-16T00:00:00Z
full work available at URL: https://arxiv.org/abs/math/0604484
review text: The analytic torsion of a closed Riemannian manifold \(M\) with a flat complex vector bundle \(E \to M\) was defined by \textit{D. B. Ray} and \textit{I. M. Singer} in [Adv. Math. 7, 145--210 (1971; Zbl 0239.58014)]. It can be regarded as a norm on the alternating determinant det \(H^{*}(M;E)\). The authors define a complex bilinear form \(\tau^{\roman{an}}_{E, e^{*},b}\) on det \(H^{*}(M;E)\), where \( e^{*}\) is a co Euler structure on \(M\), and \(b\) is a nondegenerate symmetric bilinear form on \(E\) -- such a bilinear form exists at least on the direct sum of \(N\) copies of \(E\) for some \(N\). The authors also give a combinatorial construction of a symmetric bilinear form \(\tau^{\roman{comb}}_{E, e}\) and state a conjecture that relates both torsions. In the meantime, two preprints exist that prove the conjecture -- one by Su and Zhang, the other by the authors of this paper. \textit{M. Braverman} and \textit{T. Kappeler} gave a different complex extension of the analytic torsion in [Geom. Topol. 11, 139--213 (2007; Zbl 1135.58014)]. In a recent preprint, they relate their construction to the one in the article under review.
reviewed by: Sebastian Goette
Mathematics Subject Classification ID: 58J52
Mathematics Subject Classification ID: 57Q10
zbMATH DE Number: 5171409
zbMATH Keywords: determinant line
zbMATH Keywords: Ray-Singer torsion
zbMATH Keywords: Franz-Reidemeister torsion
zbMATH Keywords: Milnor-Turaev torsion
zbMATH Keywords: Euler structures
MaRDI profile type: Publication
OpenAlex ID: W2035037633
arXiv ID: math/0604484
cites work: Q4002484
cites work: Q4023901
cites work: A refinement of the Ray-Singer torsion
cites work: Refined analytic torsion
cites work: Refined analytic torsion as an element of the determinant line
cites work: Ray-Singer type theorem for the refined analytic torsion
cites work: Comparison of the refined analytic and the Burghelea-Haller torsions
cites work: Removing metric anomalies from Ray-Singer torsion
cites work: Euler structures, the variety of representations and the Milnor-Turaev torsion
cites work: Q3426446
cites work: Dynamics, Laplace transform and spectral geometry
cites work: Q3509496
cites work: Complex valued Ray-Singer torsion II
cites work: On the determinant of elliptic differential and difference finite operators in vector bundles over \(S^ 1\)
cites work: Asymptotic expansion of the Witten deformation of the analytic torsion
cites work: Analytic torsion and Reidemeister torsion
cites work: Analytic torsion and the heat equation
cites work: Characteristic forms and geometric invariants
cites work: Poincar\'e--Reidemeister metric, Euler structures, and torsion
cites work: Q3756189
cites work: A short proof of the local Atiyah-Singer index theorem
cites work: Riemannian geometry and geometric analysis
cites work: The projectivity of the moduli space of stable curves. I: Preliminaries on "det" and "Div".
cites work: Superconnections, Thom classes, and equivariant differential forms
cites work: Whitehead torsion
cites work: Analytic torsion and R-torsion of Riemannian manifolds
cites work: R-torsion and the Laplacian on Riemannian manifolds
cites work: Morse homology
cites work: Q5542418
cites work: Q2736011
cites work: A Cheeger-Müller theorem for symmetric bilinear torsions
cites work: EULER STRUCTURES, NONSINGULAR VECTOR FIELDS, AND TORSIONS OF REIDEMEISTER TYPE
DOI: 10.1016/J.JFA.2007.03.027
description: scientific article