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–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.