We propose a refinement of the Ray–Singer torsion, which can be viewed as an analytic counterpart of the refined combinatorial torsion introduced by Turaev. Given a closed, oriented manifold of odd dimension with fundamental group Γ, the refined torsion is a complex valued, holomorphic function defined for representations of Γ which are close to the space of unitary representations. When the representation is unitary the absolute value of the refined torsion is equal to the Ray–Singer torsion, while its phase is determined by the η-invariant. As an application we extend and improve a result of Farber about the relationship between the absolute torsion of Farber–Turaev and the η-invariant.
Nous proposons un raffinement de la torsion analytique de Ray–Singer, qui peut être consideré comme un équivalent analytique du raffinement de la torsion combinatoire introduit par Turaev. Soit M une variété fermée et orientée de dimension impaire et de groupe fondamental Γ. La torsion analytique raffinée est une fonction holomorphe à valeurs complexes, définie pour les représentations de Γ, qui sont proches de l'espace des représentations unitaires. Dans le cas où la représentation est unitaire, la valeur absolue de la torsion analytique raffinée est égale à la torsion de Ray–Singer dès que sa phase est déterminée par l'invariant η. Comme application, nous généralisons et améliorons un resultat de Farber concernant la relation entre la torsion absolue de Farber–Turaev et l'invariant η.
Accepted:
Published online:
Maxim Braverman 1; Thomas Kappeler 2
@article{CRMATH_2005__341_8_497_0,
author = {Maxim Braverman and Thomas Kappeler},
title = {A refinement of the {Ray{\textendash}Singer} torsion},
journal = {Comptes Rendus. Math\'ematique},
pages = {497--502},
publisher = {Elsevier},
volume = {341},
number = {8},
year = {2005},
doi = {10.1016/j.crma.2005.09.015},
language = {en},
}
Maxim Braverman; Thomas Kappeler. A refinement of the Ray–Singer torsion. Comptes Rendus. Mathématique, Volume 341 (2005) no. 8, pp. 497-502. doi : 10.1016/j.crma.2005.09.015. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.1016/j.crma.2005.09.015/
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