Burghelea-Haller analytic torsion for manifolds with boundary

Su, Guangxiang

doi:10.1090/S0002-9939-09-10003-5

Burghelea-Haller analytic torsion for manifolds with boundary
HTML articles powered by AMS MathViewer

by Guangxiang Su PDF

Proc. Amer. Math. Soc. 137 (2009), 4295-4306 Request permission

Abstract:

In this paper, we extend the complex-valued Ray-Singer torsion, introduced by Burghelea-Haller, to compact connected Riemannian manifolds with boundary. We also compare it with the refined analytic torsion.

References

J.-M. Bismut, H. Gillet, and C. Soulé, Analytic torsion and holomorphic determinant bundles. I. Bott-Chern forms and analytic torsion, Comm. Math. Phys. 115 (1988), no. 1, 49–78. MR 929146, DOI 10.1007/BF01238853
Jean-Michel Bismut and Weiping Zhang, An extension of a theorem by Cheeger and Müller, Astérisque 205 (1992), 235 (English, with French summary). With an appendix by François Laudenbach. MR 1185803
J.-M. Bismut and W. Zhang, Milnor and Ray-Singer metrics on the equivariant determinant of a flat vector bundle, Geom. Funct. Anal. 4 (1994), no. 2, 136–212 (English, with English and French summaries). MR 1262703, DOI 10.1007/BF01895837
Maxim Braverman and Thomas Kappeler, A refinement of the Ray-Singer torsion, C. R. Math. Acad. Sci. Paris 341 (2005), no. 8, 497–502 (English, with English and French summaries). MR 2180817, DOI 10.1016/j.crma.2005.09.015
Maxim Braverman and Thomas Kappeler, Refined analytic torsion as an element of the determinant line, Geom. Topol. 11 (2007), 139–213. MR 2302591, DOI 10.2140/gt.2007.11.139
Maxim Braverman and Thomas Kappeler, Comparison of the refined analytic and the Burghelea-Haller torsions, Ann. Inst. Fourier (Grenoble) 57 (2007), no. 7, 2361–2387 (English, with English and French summaries). Festival Yves Colin de Verdière. MR 2394545, DOI 10.5802/aif.2336
Dan Burghelea and Stefan Haller, Torsion, as a function on the space of representations, $C^\ast$-algebras and elliptic theory II, Trends Math., Birkhäuser, Basel, 2008, pp. 41–66. MR 2408135, DOI 10.1007/978-3-7643-8604-7_{2}
Dan Burghelea and Stefan Haller, Complex-valued Ray-Singer torsion, J. Funct. Anal. 248 (2007), no. 1, 27–78. MR 2329682, DOI 10.1016/j.jfa.2007.03.027
D. Burghelea and S. Haller, Complex-valued Ray-Singer torsion. II. Preprint, math.DG/0610875.
J. Brüning and M. Lesch, Hilbert complexes, J. Funct. Anal. 108 (1992), no. 1, 88–132. MR 1174159, DOI 10.1016/0022-1236(92)90147-B
Jeff Cheeger, Analytic torsion and the heat equation, Ann. of Math. (2) 109 (1979), no. 2, 259–322. MR 528965, DOI 10.2307/1971113
Michael Farber and Vladimir Turaev, Poincaré-Reidemeister metric, Euler structures, and torsion, J. Reine Angew. Math. 520 (2000), 195–225. MR 1748274, DOI 10.1515/crll.2000.021
Peter B. Gilkey, Invariance theory, the heat equation, and the Atiyah-Singer index theorem, 2nd ed., Studies in Advanced Mathematics, CRC Press, Boca Raton, FL, 1995. MR 1396308
Tosio Kato, Perturbation theory for linear operators, Die Grundlehren der mathematischen Wissenschaften, Band 132, Springer-Verlag New York, Inc., New York, 1966. MR 0203473
Finn Faye Knudsen and David Mumford, The projectivity of the moduli space of stable curves. I. Preliminaries on “det” and “Div”, Math. Scand. 39 (1976), no. 1, 19–55. MR 437541, DOI 10.7146/math.scand.a-11642
J. Milnor, Whitehead torsion, Bull. Amer. Math. Soc. 72 (1966), 358–426. MR 196736, DOI 10.1090/S0002-9904-1966-11484-2
Jeff Cheeger, Analytic torsion and the heat equation, Ann. of Math. (2) 109 (1979), no. 2, 259–322. MR 528965, DOI 10.2307/1971113
Werner Müller, Analytic torsion and $R$-torsion for unimodular representations, J. Amer. Math. Soc. 6 (1993), no. 3, 721–753. MR 1189689, DOI 10.1090/S0894-0347-1993-1189689-4
D. Kvillen, Determinants of Cauchy-Riemann operators on Riemann surfaces, Funktsional. Anal. i Prilozhen. 19 (1985), no. 1, 37–41, 96 (Russian). MR 783704
D. B. Ray and I. M. Singer, $R$-torsion and the Laplacian on Riemannian manifolds, Advances in Math. 7 (1971), 145–210. MR 295381, DOI 10.1016/0001-8708(71)90045-4
Guangxiang Su and Weiping Zhang, A Cheeger-Müller theorem for symmetric bilinear torsions, Chinese Ann. Math. Ser. B 29 (2008), no. 4, 385–424. MR 2429629, DOI 10.1007/s11401-007-0307-8
V. G. Turaev, Euler structures, nonsingular vector fields, and Reidemeister-type torsions, Izv. Akad. Nauk SSSR Ser. Mat. 53 (1989), no. 3, 607–643, 672 (Russian); English transl., Math. USSR-Izv. 34 (1990), no. 3, 627–662. MR 1013714, DOI 10.1070/IM1990v034n03ABEH000676
B. Vertman, Refined analytic torsion on manifolds with boundary. Geom. and Topol. 13 (2009), 1989–2027.
Edward Witten, Supersymmetry and Morse theory, J. Differential Geometry 17 (1982), no. 4, 661–692 (1983). MR 683171