Comparison of the refined analytic and the Burghelea-Haller torsions
[Comparaison de la torsion raffinée et la torsion de Burghelea-Haller]
Braverman, Maxim ; Kappeler, Thomas
Annales de l'Institut Fourier, Tome 57 (2007), p. 2361-2387 / Harvested from Numdam

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.

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.

Publié le : 2007-01-01
DOI : https://doi.org/10.5802/aif.2336
Classification:  58J52,  58J28,  57R20
Mots clés: déterminant, torsion analytique, torsion de Ray-Singer, invariant eta, torsion de Turaev et de Farber-Turaev
@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},
     volume = {57},
     year = {2007},
     pages = {2361-2387},
     doi = {10.5802/aif.2336},
     zbl = {1147.58033},
     mrnumber = {2394545},
     language = {en},
     url = {http://dml.mathdoc.fr/item/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, Tome 57 (2007) pp. 2361-2387. doi : 10.5802/aif.2336. http://gdmltest.u-ga.fr/item/AIF_2007__57_7_2361_0/

[1] Berline, Nicole; Getzler, Ezra; Vergne, Michèle Heat kernels and Dirac operators, Springer-Verlag, Berlin, Grundlehren Text Editions (2004) (Corrected reprint of the 1992 original) | MR 2273508 | Zbl 1037.58015

[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 1185803 | Zbl 0781.58039

[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, Tome 341 (2005) no. 8, pp. 497-502 | MR 2180817 | Zbl 1086.58015

[6] Braverman, Maxim; Kappeler, Thomas Ray-Singer type theorem for the refined analytic torsion, J. Funct. Anal., Tome 243 (2007) no. 1, pp. 232-256 | Article | MR 2291437 | Zbl 05129774

[7] Braverman, Maxim; Kappeler, Thomas Refined analytic torsion as an element of the determinant line, Geom. Topol., Tome 11 (2007), pp. 139-213 | Article | MR 2302591 | Zbl 05136060

[8] Burghelea, D. Removing metric anomalies from Ray-Singer torsion, Lett. Math. Phys., Tome 47 (1999) no. 2, pp. 149-158 | Article | MR 1682302 | Zbl 0946.58026

[9] Burghelea, D.; Friedlander, L.; Kappeler, T. Asymptotic expansion of the Witten deformation of the analytic torsion, J. Funct. Anal., Tome 137 (1996) no. 2, pp. 320-363 | Article | MR 1387514 | Zbl 0858.57029

[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., Tome 10 (2006), p. 1185-1238 (electronic) | Article | MR 2255496 | Zbl 05117940

[12] Burghelea, Dan; Haller, Stefan Complex-valued Ray-Singer torsion, J. Funct. Anal., Tome 248 (2007) no. 1, pp. 27-78 | Article | MR 2329682 | Zbl 1131.58020

[13] Cheeger, Jeff Analytic torsion and the heat equation, Ann. of Math. (2), Tome 109 (1979) no. 2, pp. 259-322 | Article | MR 528965 | Zbl 0412.58026

[14] Farber, M. Absolute torsion and eta-invariant, Math. Z., Tome 234 (2000) no. 2, pp. 339-349 | Article | MR 1765885 | Zbl 0955.57022

[15] Farber, Michael; Turaev, Vladimir Absolute torsion, Tel Aviv Topology Conference: Rothenberg Festschrift (1998), Amer. Math. Soc., Providence, RI (Contemp. Math.) Tome 231 (1999), pp. 73-85 | MR 1705570 | Zbl 0934.57030

[16] Farber, Michael; Turaev, Vladimir Poincaré-Reidemeister metric, Euler structures, and torsion, J. Reine Angew. Math., Tome 520 (2000), pp. 195-225 | Article | MR 1748274 | Zbl 0938.57020

[17] Gilkey, Peter B. The eta invariant and secondary characteristic classes of locally flat bundles, Algebraic and differential topology-global differential geometry, Teubner, Leipzig (Teubner-Texte Math.) Tome 70 (1984), pp. 49-87 | MR 792686 | Zbl 0584.58040

[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 2313340

[20] Mathai, Varghese; Quillen, Daniel Superconnections, Thom classes, and equivariant differential forms, Topology, Tome 25 (1986) no. 1, pp. 85-110 | Article | MR 836726 | Zbl 0592.55015

[21] Müller, Werner Analytic torsion and R-torsion of Riemannian manifolds, Adv. in Math., Tome 28 (1978) no. 3, pp. 233-305 | Article | MR 498252 | Zbl 0395.57011

[22] Müller, Werner Analytic torsion and R-torsion for unimodular representations, J. Amer. Math. Soc., Tome 6 (1993) no. 3, pp. 721-753 | Article | MR 1189689 | Zbl 0789.58071

[23] Ray, D. B.; Singer, I. M. R-torsion and the Laplacian on Riemannian manifolds, Advances in Math., Tome 7 (1971), pp. 145-210 | Article | MR 295381 | Zbl 0239.58014

[24] Turaev, V. G. Reidemeister torsion in knot theory, Russian Math. Survey, Tome 41 (1986), pp. 119-182 | Article | MR 832411 | Zbl 0602.57005

[25] Turaev, V. G. Euler structures, nonsingular vector fields, and Reidemeister-type torsions, Math. USSR Izvestia, Tome 34 (1990), pp. 627-662 | Article | MR 1013714 | Zbl 0692.57015

[26] Turaev, Vladimir Introduction to combinatorial torsions, Birkhäuser Verlag, Basel, Lectures in Mathematics ETH Zürich (2001) (Notes taken by Felix Schlenk) | MR 1809561 | Zbl 0970.57001