Comptes Rendus
no. 10
Analytic geometry/Differential topology
Scattering matrix and analytic torsion

Presented by: Jean-Michel Bismut

Martin Puchol1; Yeping Zhang2; Jialin Zhu3
1 Institut Camille-Jordan, Université Lyon-1, Bâtiment Braconnier, 43, boulevard du 11-Novembre-1918, 69622 Villeurbanne cedex, France
2 Département de mathématiques, Bâtiment 425, Faculté des sciences d'Orsay, Université Paris-Sud, 91405 Orsay cedex, France
3 Shanghai Center for Mathematical Sciences, Fudan University, Shanghai 200433, PR China
Comptes Rendus. Mathématique, Volume 355 (2017) no. 10, pp. 1089-1093.

We consider a compact manifold with a piece isometric to a (finite-length) cylinder. By making the length of the cylinder tend to infinity, we obtain an asymptotic gluing formula for the zeta determinant of the Hodge Laplacian and an asymptotic expansion of the L2 torsion of the corresponding Mayer–Vietoris exact sequence. As an application, we give a purely analytic proof of the gluing formula for analytic torsion.

On considère une variété compacte ayant une partie isométrique à un cylindre fini. En faisant tendre la longueur du cylindre vers l'infini, on obtient une formule asymptotique pour le déterminant du laplacien de Hodge et un développement asymptotique de la torsion L2 associée à la suite exacte de Mayer–Vietoris. On obtient une preuve analytique de la formule de recollement pour la torsion analytique.

Received:
Accepted:
Published online:
DOI: 10.1016/j.crma.2017.09.018

Martin Puchol 1; Yeping Zhang 2; Jialin Zhu 3

1 Institut Camille-Jordan, Université Lyon-1, Bâtiment Braconnier, 43, boulevard du 11-Novembre-1918, 69622 Villeurbanne cedex, France
2 Département de mathématiques, Bâtiment 425, Faculté des sciences d'Orsay, Université Paris-Sud, 91405 Orsay cedex, France
3 Shanghai Center for Mathematical Sciences, Fudan University, Shanghai 200433, PR China 
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Martin Puchol; Yeping Zhang; Jialin Zhu. Scattering matrix and analytic torsion. Comptes Rendus. Mathématique, Volume 355 (2017) no. 10, pp. 1089-1093. doi : 10.1016/j.crma.2017.09.018. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.1016/j.crma.2017.09.018/

[1] J.-M. Bismut; H. Gillet; C. Soulé Analytic torsion and holomorphic determinant bundles. I. Bott–Chern forms and analytic torsion, Comment. Phys.-Math., Volume 115 (1988) no. 1, pp. 49-78

[2] J.-M. Bismut; J. Lott Flat vector bundles, direct images and higher real analytic torsion, J. Amer. Math. Soc., Volume 8 (1995) no. 2, pp. 291-363

[3] J.-M. Bismut; W. Zhang An extension of a theorem by Cheeger and Müller, Astérisque, Volume 205 (1992), p. 235 (With an appendix by François Laudenbach)

[4] J. Brüning; X. Ma An anomaly formula for Ray–Singer metrics on manifolds with boundary, Geom. Funct. Anal., Volume 16 (2006) no. 4, pp. 767-837

[5] J. Brüning; X. Ma On the gluing formula for the analytic torsion, Math. Z., Volume 273 (2013) no. 3–4, pp. 1085-1117

[6] J. Cheeger Analytic torsion and the heat equation, Ann. of Math. (2), Volume 109 (1979) no. 2, pp. 259-322

[7] R.G. Douglas; K.P. Wojciechowski Adiabatic limits of the η-invariants. The odd-dimensional Atiyah–Patodi–Singer problem, Comment. Phys.-Math., Volume 142 (1991) no. 1, pp. 139-168

[8] W. Müller Analytic torsion and R-torsion of Riemannian manifolds, Adv. Math., Volume 28 (1978) no. 3, pp. 233-305

[9] W. Müller Eta invariants and manifolds with boundary, J. Differ. Geom., Volume 40 (1994) no. 2, pp. 311-377

[10] J. Park; K.P. Wojciechowski Adiabatic decomposition of the ζ-determinant and scattering theory, Mich. Math. J., Volume 54 (2006) no. 1, pp. 207-238

[11] D.B. Ray; I.M. Singer R-torsion and the Laplacian on Riemannian manifolds, Adv. Math., Volume 7 (1971), pp. 145-210

[12] K. Reidemeister Homotopieringe und Linsenräume, Abh. Math. Semin. Univ. Hamb., Volume 11 (1935) no. 1, pp. 102-109

[13] J. Zhu Gluing formula of real analytic torsion forms and adiabatic limit, Isr. J. Math., Volume 215 (2017) no. 1, pp. 181-254

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