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The analytic torsion of a disc

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Abstract

In this article, we study the Reidemeister torsion and the analytic torsion of the m dimensional disc, with the Ray and Singer homology basis (Adv Math 7:145–210, 1971). We prove that the Reidemeister torsion coincides with a power of the volume of the disc. We study the additional terms arising in the analytic torsion due to the boundary, using generalizations of the Cheeger–Müller theorem. We use a formula proved by Brüning and Ma (GAFA 16:767–873, 2006) that predicts a new anomaly boundary term beside the known term proportional to the Euler characteristic of the boundary (Lück, J Diff Geom 37:263–322, 1993). Some of our results extend to the case of the cone over a sphere, in particular we evaluate directly the analytic torsion for a cone over the circle and over the two sphere. We compare the results obtained in the low dimensional cases. We also consider a different formula for the boundary term given by Dai and Fang (Asian J Math 4:695–714, 2000), and we compare the results. The results of these work were announced in the study of Hartmann et al. (BUMI 2:529–533, 2009).

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Authors and Affiliations

  1. UNESP, Universidade Estadual Paulista, Rio Claro, Brazil

    T. de Melo

  2. UFSCar, Universidade Federal de São Carlos, São Carlos, Brazil

    L. Hartmann

  3. ICMC, Universidade de São Paulo, São Carlos, Brazil

    M. Spreafico

Authors
  1. T. de Melo
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  2. L. Hartmann
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  3. M. Spreafico
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Correspondence to M. Spreafico.

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de Melo, T., Hartmann, L. & Spreafico, M. The analytic torsion of a disc. Ann Glob Anal Geom 42, 29–59 (2012). https://doi.org/10.1007/s10455-011-9300-2

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  • DOI: https://doi.org/10.1007/s10455-011-9300-2

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