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Journal of Noncommutative Geometry

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Volume 9, Issue 2, 2015, pp. 621–664
DOI: 10.4171/JNCG/203

Published online: 2015-06-15

Large time limit and local $L^2$-index theorems for families

Sara Azzali[1], Sebastian Goette[2] and Thomas Schick[3]

(1) Universität Potsdam, Germany
(2) Universität Freiburg, Germany
(3) Georg-August-Universität Göttingen, Germany

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–Tondeur 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.

Keywords: Local index theory, eta forms, torsion forms, $L^2$-invariants

Azzali Sara, Goette Sebastian, Schick Thomas: Large time limit and local $L^2$-index theorems for families. J. Noncommut. Geom. 9 (2015), 621-664. doi: 10.4171/JNCG/203