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 -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 -index formulas.
As applications, we prove a local -index theorem for families of signature operators and an -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 -eta forms and -torsion forms as transgression forms.
Cite this article
Sara Azzali, Sebastian Goette, Thomas Schick, Large time limit and local -index theorems for families. J. Noncommut. Geom. 9 (2015), no. 2, pp. 621–664DOI 10.4171/JNCG/203