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

TL;DR

This paper explicitly computes the large time limit of fibrewise heat operators for superconnections in L^2-sets, leading to refined L^2-index theorems for families of manifolds with group actions.

Contribution

It provides the first explicit large time limit computation for Bismut-Lott superconnections without extra regularity, enabling new L^2-index formulas and refined invariants.

Findings

Established a local L^2-index theorem for families of signature operators.

Proved an L^2-Bismut-Lott theorem relating transfer and classes.

Constructed L^2-eta and torsion forms as transgression forms.

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-Tandeur 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.