JournalsjncgVol. 9, No. 2pp. 621–664

Large time limit and local L2L^2-index theorems for families

  • Sara Azzali

    Universität Potsdam, Germany
  • Sebastian Goette

    Universität Freiburg, Germany
  • Thomas Schick

    Georg-August-Universität Göttingen, Germany
Large time limit and local $L^2$-index theorems for families cover

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 L2L^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 L2L^2-index formulas.

As applications, we prove a local L2L^2-index theorem for families of signature operators and an L2L^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 L2L^2-eta forms and L2L^2-torsion forms as transgression forms.

Cite this article

Sara Azzali, Sebastian Goette, Thomas Schick, Large time limit and local L2L^2-index theorems for families. J. Noncommut. Geom. 9 (2015), no. 2, pp. 621–664

DOI 10.4171/JNCG/203