Comparison of the refined analytic and the Burghelea-Haller torsions
Annales de l'Institut Fourier, Volume 57 (2007) no. 7, pp. 2361-2387.

The refined analytic torsion associated to a flat vector bundle over a closed odd-dimensional manifold canonically defines a quadratic form τ on the determinant line of the cohomology. Both τ and the Burghelea-Haller torsion are refinements of the Ray-Singer torsion. We show that whenever the Burghelea-Haller torsion is defined it is equal to ±τ. As an application we obtain new results about the Burghelea-Haller torsion. In particular, we prove a weak version of the Burghelea-Haller conjecture relating their torsion with the square of the Farber-Turaev combinatorial torsion.

La torsion analytique raffinée, associée à un fibré vectoriel plat sur une variété fermée et orientée de dimension impaire, définit d’une manière canonique une forme quadratique τ sur le déterminant de la cohomologie. La torsion introduite par Burghelea et Haller et la forme quadratique τ sont des concepts raffinés de la torsion analytique de Ray-Singer. On démontre que dans le cas où la torsion de Burghelea-Haller est définie, elle est identique à ±τ. Comme application, on obtient des résultats nouveaux pour la torsion de Burghelea-Haller. En particulier, on démontre une version faible de la conjecture de Burghelea-Haller concernant leur torsion et le carré de la torsion combinatoire de Farber-Turaev.

DOI: 10.5802/aif.2336
Classification: 58J52,  58J28,  57R20
Keywords: Determinant line, analytic torsion, Ray-Singer torsion, eta-invariant, Turaev torsion and Farber-Turaev torsion
Braverman, Maxim 1; Kappeler, Thomas 2

1 Northeastern University Department of Mathematics Northeastern University Boston, MA 02115 (USA)
2 Universität Zürich Institut für Mathematik Winterthurerstrasse 190 8057 Zürich (Switzerland)
@article{AIF_2007__57_7_2361_0,
     author = {Braverman, Maxim and Kappeler, Thomas},
     title = {Comparison of the refined analytic and the {Burghelea-Haller} torsions},
     journal = {Annales de l'Institut Fourier},
     pages = {2361--2387},
     publisher = {Association des Annales de l{\textquoteright}institut Fourier},
     volume = {57},
     number = {7},
     year = {2007},
     doi = {10.5802/aif.2336},
     mrnumber = {2394545},
     zbl = {1147.58033},
     language = {en},
     url = {https://aif.centre-mersenne.org/articles/10.5802/aif.2336/}
}
TY  - JOUR
AU  - Braverman, Maxim
AU  - Kappeler, Thomas
TI  - Comparison of the refined analytic and the Burghelea-Haller torsions
JO  - Annales de l'Institut Fourier
PY  - 2007
DA  - 2007///
SP  - 2361
EP  - 2387
VL  - 57
IS  - 7
PB  - Association des Annales de l’institut Fourier
UR  - https://aif.centre-mersenne.org/articles/10.5802/aif.2336/
UR  - https://www.ams.org/mathscinet-getitem?mr=2394545
UR  - https://zbmath.org/?q=an%3A1147.58033
UR  - https://doi.org/10.5802/aif.2336
DO  - 10.5802/aif.2336
LA  - en
ID  - AIF_2007__57_7_2361_0
ER  - 
%0 Journal Article
%A Braverman, Maxim
%A Kappeler, Thomas
%T Comparison of the refined analytic and the Burghelea-Haller torsions
%J Annales de l'Institut Fourier
%D 2007
%P 2361-2387
%V 57
%N 7
%I Association des Annales de l’institut Fourier
%U https://doi.org/10.5802/aif.2336
%R 10.5802/aif.2336
%G en
%F AIF_2007__57_7_2361_0
Braverman, Maxim; Kappeler, Thomas. Comparison of the refined analytic and the Burghelea-Haller torsions. Annales de l'Institut Fourier, Volume 57 (2007) no. 7, pp. 2361-2387. doi : 10.5802/aif.2336. https://aif.centre-mersenne.org/articles/10.5802/aif.2336/

[1] Berline, Nicole; Getzler, Ezra; Vergne, Michèle Heat kernels and Dirac operators, Grundlehren Text Editions, Springer-Verlag, Berlin, 2004 (Corrected reprint of the 1992 original) | MR | Zbl

[2] Bismut, Jean-Michel; Zhang, Weiping An extension of a theorem by Cheeger and Müller, Astérisque (1992) no. 205, pp. 235 (With an appendix by François Laudenbach) | MR | Zbl

[3] Braverman, Maxim; Kappeler, Thomas A canonical quadratic form on the determinant line of a flat vector bundle (arXiv:math.DG/0710.1232)

[4] Braverman, Maxim; Kappeler, Thomas Refined Analytic Torsion (arXiv:math.DG/0505537, To appear in J. of Differential Geometry)

[5] Braverman, Maxim; Kappeler, Thomas A refinement of the Ray-Singer torsion, C. R. Math. Acad. Sci. Paris, Volume 341 (2005) no. 8, pp. 497-502 | MR | Zbl

[6] Braverman, Maxim; Kappeler, Thomas Ray-Singer type theorem for the refined analytic torsion, J. Funct. Anal., Volume 243 (2007) no. 1, pp. 232-256 | DOI | MR | Zbl

[7] Braverman, Maxim; Kappeler, Thomas Refined analytic torsion as an element of the determinant line, Geom. Topol., Volume 11 (2007), pp. 139-213 | DOI | MR | Zbl

[8] Burghelea, D. Removing metric anomalies from Ray-Singer torsion, Lett. Math. Phys., Volume 47 (1999) no. 2, pp. 149-158 | DOI | MR | Zbl

[9] Burghelea, D.; Friedlander, L.; Kappeler, T. Asymptotic expansion of the Witten deformation of the analytic torsion, J. Funct. Anal., Volume 137 (1996) no. 2, pp. 320-363 | DOI | MR | Zbl

[10] Burghelea, Dan; Haller, Stefan Torsion, as a function on the space of representations (arXiv:math.DG/0507587)

[11] Burghelea, Dan; Haller, Stefan Euler structures, the variety of representations and the Milnor-Turaev torsion, Geom. Topol., Volume 10 (2006), p. 1185-1238 (electronic) | DOI | MR | Zbl

[12] Burghelea, Dan; Haller, Stefan Complex-valued Ray-Singer torsion, J. Funct. Anal., Volume 248 (2007) no. 1, pp. 27-78 | DOI | MR | Zbl

[13] Cheeger, Jeff Analytic torsion and the heat equation, Ann. of Math. (2), Volume 109 (1979) no. 2, pp. 259-322 | DOI | MR | Zbl

[14] Farber, M. Absolute torsion and eta-invariant, Math. Z., Volume 234 (2000) no. 2, pp. 339-349 | DOI | MR | Zbl

[15] Farber, Michael; Turaev, Vladimir Absolute torsion, Tel Aviv Topology Conference: Rothenberg Festschrift (1998) (Contemp. Math.), Volume 231, Amer. Math. Soc., Providence, RI, 1999, pp. 73-85 | MR | Zbl

[16] Farber, Michael; Turaev, Vladimir Poincaré-Reidemeister metric, Euler structures, and torsion, J. Reine Angew. Math., Volume 520 (2000), pp. 195-225 | DOI | MR | Zbl

[17] Gilkey, Peter B. The eta invariant and secondary characteristic classes of locally flat bundles, Algebraic and differential topology-global differential geometry (Teubner-Texte Math.), Volume 70, Teubner, Leipzig, 1984, pp. 49-87 | MR | Zbl

[18] Huang, R.-T. Refined analytic torsion: comparison theorems and examples (arXiv:math.DG/0602231, To appear in Illinois J. Math.)

[19] Ma, X.; Zhang, W. η -invariant and flat vector bundles II (arXiv:math.DG/0604357) | MR

[20] Mathai, Varghese; Quillen, Daniel Superconnections, Thom classes, and equivariant differential forms, Topology, Volume 25 (1986) no. 1, pp. 85-110 | DOI | MR | Zbl

[21] Müller, Werner Analytic torsion and R-torsion of Riemannian manifolds, Adv. in Math., Volume 28 (1978) no. 3, pp. 233-305 | DOI | MR | Zbl

[22] Müller, Werner Analytic torsion and R-torsion for unimodular representations, J. Amer. Math. Soc., Volume 6 (1993) no. 3, pp. 721-753 | DOI | MR | Zbl

[23] Ray, D. B.; Singer, I. M. R-torsion and the Laplacian on Riemannian manifolds, Advances in Math., Volume 7 (1971), pp. 145-210 | DOI | MR | Zbl

[24] Turaev, V. G. Reidemeister torsion in knot theory, Russian Math. Survey, Volume 41 (1986), pp. 119-182 | DOI | MR | Zbl

[25] Turaev, V. G. Euler structures, nonsingular vector fields, and Reidemeister-type torsions, Math. USSR Izvestia, Volume 34 (1990), pp. 627-662 | DOI | MR | Zbl

[26] Turaev, Vladimir Introduction to combinatorial torsions, Lectures in Mathematics ETH Zürich, Birkhäuser Verlag, Basel, 2001 (Notes taken by Felix Schlenk) | MR | Zbl

Cited by Sources: