Advertisement

Annals of Global Analysis and Geometry

, Volume 41, Issue 1, pp 61–90 | Cite as

The metric anomaly of analytic torsion at the boundary of an even dimensional cone

  • Boris VertmanEmail author
Article

Abstract

The formula for analytic torsion of a cone in even dimensions is composed of three terms. The first two terms are well understood and given by an algebraic combination of the Betti numbers and the analytic torsion of the cone base. The third “singular” contribution is an intricate spectral invariant of the cone base. We identify the third term as the metric anomaly of the analytic torsion coming from the non-product structure of the cone at its regular boundary. Hereby we filter out the actual contribution of the concial singularity and identify the analytic torsion of an even-dimensional cone purely in terms of the Betti numbers and the analytic torsion of the cone base.

Keywords

Analytic torsion Conical singularities Bounded generalized cone Metric anomaly Cheeger-Müller Theorem 

Mathematics Subject Classification (2000)

58J52 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    Abramowitz, M., Stegun, I.A. (eds.): Handbook of Mathematical functions with formulas, graphs, and mathematical tables. Reprint of the 1972 edition. Dover Publications, Inc., New York (1992)Google Scholar
  2. 2.
    Bordag M., Geyer B., Kirsten K., Elizalde E.: Zeta function determinant of the Laplace Operator on the D-dimensional ball. Comm. Math. Phys. 179(1), 215–234 (1996a)MathSciNetzbMATHCrossRefGoogle Scholar
  3. 3.
    Bordag M., Kirsten K., Dowker J.S.: Heat-kernels and functional determinants on the cone. Comm. Math. Phys. 182, 371–394 (1996b)MathSciNetzbMATHCrossRefGoogle Scholar
  4. 4.
    Brüning J., Ma X.: An anomaly-formula for Ray-Singer metrics on manifolds with boundary. Geom. Funct. Anal. 16(4), 767–837 (2006)MathSciNetzbMATHCrossRefGoogle Scholar
  5. 5.
    Brüning J., Seeley R.: An index theorem for first order regular singular operators. Amer. J. Math. 110, 659–714 (1988)MathSciNetzbMATHCrossRefGoogle Scholar
  6. 6.
    Burghelea D., Friedlander L., Kappeler T.: On the determinant of elliptic boundary value problems on a line segment. Proc. Amer. Math. Soc. 123, 3027–3038 (1995)MathSciNetzbMATHCrossRefGoogle Scholar
  7. 7.
    Cheeger J.: Analytic torsion and Reidemeister torsion. Proc. Natl Acad. Sci. USA 74, 2651–2654 (1977)MathSciNetzbMATHCrossRefGoogle Scholar
  8. 8.
    Cheeger J.: Spectral geometry of singular Riemannian spaces. J. Differential Geom. 18, 575–657 (1983)MathSciNetzbMATHGoogle Scholar
  9. 9.
    Dar A.: Intersection R-torsion and analytic torsion for pseudo-manifolds. Math. Z. 194, 193–216 (1987)MathSciNetzbMATHCrossRefGoogle Scholar
  10. 10.
    de Melo T., Hartmann L., Spreafico M.: Reidemeister torsion and analytic torsion of discs. Boll. Unione Mat. Ital. (9) 2(2), 529–533 (2009)MathSciNetzbMATHGoogle Scholar
  11. 11.
    de Rham G.: Complexes a automorphismes et homeomorphie differentiable. Ann. Inst. Fourier 2, 51–67 (1950) The MetricMathSciNetzbMATHCrossRefGoogle Scholar
  12. 12.
    Dowker J.S., Kisten K.: Spinors and forms on the ball and the cone. Comm. Anal. Geom. 7(3), 641–679 (1999)MathSciNetzbMATHGoogle Scholar
  13. 13.
    Franz W.: Über die Torsion einer Überdeckung. J. Reine Angew. Math. 173, 245–254 (1935)zbMATHCrossRefGoogle Scholar
  14. 14.
    Gradsteyn I.S., Ryzhik I.M.: Alan Jeffrey Table of Integrals, Series and Products, 5th edn. Academic Press Inc., Boston (1994)Google Scholar
  15. 15.
    Hartmann L., Spreafico, M.: The Analytic Torsion of the Cone Over an Odd Dimensional Manifold (2010). Preprint on arXiv:math.DG/1001.4755v1Google Scholar
  16. 16.
    Hartmann, L., Spreafico, L.: An extension of the Cheeger-Müller theorem for a cone. arXiv:1008.2987vl [math.DG] (2010)Google Scholar
  17. 17.
    Hartmann L., Spreafico M.: The Analytic Torsion of the Cone over an Odd Dimensional Manifold. J. Geom. Phys. 61, 624–657 (2011)MathSciNetzbMATHCrossRefGoogle Scholar
  18. 18.
    Lesch, M.: The Analytic Torsion of the Model Cone. Columbus University, unpublished notes (1994)Google Scholar
  19. 19.
    Lesch M.: Determinants of regular singular Sturm-Liouville operators. Math. Nachr. 194, 139–170 (1998)MathSciNetzbMATHCrossRefGoogle Scholar
  20. 20.
    Lück W.: Analytic and topological torsion for manifolds with boundary and symmetry. J. Diff. Geom. 37, 263–322 (1993)zbMATHGoogle Scholar
  21. 21.
    Müller W.: Analytic torsion and R-torsion of Riemannian manifolds. Adv. Math. 28(3), 233–305 (1978)zbMATHCrossRefGoogle Scholar
  22. 22.
    Müller, W., Vertman, B.: The Metric Anomaly of Analytic Torsion on Manifolds with Conical Singularities, announced in [33] (in preparation)Google Scholar
  23. 23.
    Olver F.W.: Asymptotics and Special Functions AKP Classics, p. xviii+572. A K Peters Ltd., Wellesley, MA (1977)Google Scholar
  24. 24.
    Paquet L.: Probl’emes mixtes pour le syst’eme de Maxwell. Ann. Fac. Sci. Toulouse IV, 103–141 (1982)MathSciNetCrossRefGoogle Scholar
  25. 25.
    Ray D.B., Singer I.M.: R-Torsion and the Laplacian on Riemannian manifolds. Adv. Math. 7, 145–210 (1971)MathSciNetzbMATHCrossRefGoogle Scholar
  26. 26.
    Reidemeister K.: Die Klassifikation der Linsenräume. Abh. Math. Sem. Hamburg 11, 102–109 (1935a)zbMATHCrossRefGoogle Scholar
  27. 27.
    Reidemeister K.: Überdeckungen von Komplexen. J. Reine Angew. Math. 173, 164–173 (1935b)zbMATHCrossRefGoogle Scholar
  28. 28.
    Spreafico M.: Zeta function and regularized determinant on a disc and on a cone. J. Geom. Phys. 54(3), 355–371 (2005)MathSciNetzbMATHCrossRefGoogle Scholar
  29. 29.
    Spreafico M.: Zeta invariants for Dirichlet series. Pacific J. Math. 224(1), 185–200 (2006)MathSciNetzbMATHCrossRefGoogle Scholar
  30. 30.
    Vertman B.: Analytic torsion of a bounded generalized cone. Comm. Math. Phys. 290(3), 813–860 (2009a)MathSciNetzbMATHCrossRefGoogle Scholar
  31. 31.
    Vertman B.: Zeta determinants for regular-singular Laplace-type operators. J. Math. Phys. 50(8), 23 (2009b)MathSciNetCrossRefGoogle Scholar
  32. 32.
    Vertman, B.: The metric anomaly of analytic torsion at the boundary of an even dimensional cone. arXiv:1004.2069 [math.SP] (2010)Google Scholar
  33. 33.
    Vertman, B.: The metric anomaly at the regular boundary of the analytic torsion of a bounded generalized cone, I. Odd-dimensional generalized cone, arXiv:1004.2067vl [math.SP] (2010)Google Scholar
  34. 34.
    Vishik S.: Generalized Ray-Singer conjecture. I. A manifold with smooth boundary. Comm. Math. Phys. 167, 1–102 (1995)MathSciNetzbMATHCrossRefGoogle Scholar
  35. 35.
    Weidmann J.: Linear Operators in Hilbert Spaces Graduate Texts in Mathematics, vol. 68, p. xiii+402. Springer-Verlag, New York-Berlin (1980)Google Scholar

Copyright information

© Springer Science+Business Media B.V. 2011

Authors and Affiliations

  1. 1.Mathematisches InstitutUniversität BonnBonnGermany

Personalised recommendations