Large time limit and local L^2-index theorems for families
Abstract
We compute explicitly, and without any extra regularity assumptions, the large time limit of the fibrewise heat operator for Bismut-Lott type superconnections in the L^2-setting. This is motivated by index theory on certain non-compact spaces (families of manifolds with cocompact group action) where the convergence of the heat operator at large time implies refined L^2-index formulas. As applications, we prove a local L^2-index theorem for families of signature operators and an L^2-Bismut-Lott theorem, expressing the Becker-Gottlieb transfer of flat bundles in terms of Kamber-Tandeur classes. With slightly stronger regularity we obtain the respective refined versions: we construct L^2-eta forms and L^2-torsion forms as transgression forms.
- Publication:
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arXiv e-prints
- Pub Date:
- June 2013
- DOI:
- 10.48550/arXiv.1306.5659
- arXiv:
- arXiv:1306.5659
- Bibcode:
- 2013arXiv1306.5659A
- Keywords:
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- Mathematics - Differential Geometry;
- Mathematics - Geometric Topology
- E-Print:
- 34 pages, minor corrections