Eta forms and determinant lines

TL;DR

This paper constructs a canonical zeta connection on the determinant line bundle for families of elliptic boundary problems, linking its curvature to the eta form, advancing understanding of geometric analysis and index theory.

Contribution

It introduces a canonical construction of a zeta connection on determinant line bundles associated with APS elliptic boundary problems, relating curvature to eta forms.

Findings

Curvature of the zeta connection equals the 2-form part of the eta form

Provides a canonical method for constructing connections on determinant line bundles

Enhances understanding of the geometric structure of elliptic boundary problems

Abstract

We show that there is a canonical construction of a zeta (Bismut-Quillen) connection on the determinant line bundle of a family of APS elliptic boundary problems and that it has curvature equal to the 2-form part of a relative eta form.