Nonlocal invariants in index theory

TL;DR

This paper reviews the interplay between local and nonlocal invariants in Atiyah-Singer index theory, highlighting their roles in refined index theorems and geometric analysis.

Contribution

It provides a comprehensive survey of local and nonlocal invariants, including eta invariants and higher torsion forms, and discusses their relations and applications in index theory.

Findings

Connections between local and nonlocal invariants are elucidated.

Nonlocal invariants like eta and torsion are crucial in refined index theorems.

Higher torsion forms relate to the topology of diffeomorphism groups.

Abstract

This article surveys the relations among local and nonlocal invariants in Atiyah-Singer index theory. We discuss the local invariants that arise from the heat equation approach to the index theorem for geometric operators, as well as the nonlocal invariants (the eta invariant, the determinant of the Laplacian/analytic torsion) that occur in more refined index theorems, such as the determinant line bundle setting and the index theorem for families of manifolds with boundary. We also discuss the higher torsion forms of Bismut and Lott and their conjectured relation to the rational homotopy of the diffeomorphism group of aspherical manifolds.