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.