Eta-Invariants and Determinant Lines

TL;DR

This paper investigates eta-invariants on odd-dimensional manifolds with boundary, establishing a gluing law and variation formula, leading to a simplified proof of the holonomy formula for determinant line bundles of Dirac operators.

Contribution

It introduces new gluing and variation formulas for eta-invariants, simplifying the proof of the global anomaly formula for Dirac operator determinant lines.

Findings

Proved a gluing law for eta-invariants.

Derived a variation formula for eta-invariants.

Provided a simpler proof of the holonomy formula.

Abstract

We study eta-invariants on odd dimensional manifolds with boundary. The dependence on boundary conditions is best summarized by viewing the (exponentiated) eta-invariant as an element of the (inverse) determinant line of the boundary. We prove a gluing law and a variation formula for this invariant. This yields a new, simpler proof of the holonomy formula for the determinant line bundle of a family of Dirac operators, also known as the ``global anomaly'' formula. This paper is written using AMSTeX 2.1, which can be obtained via ftp from the American Mathematical Society (instructions included). A postscript file with figures was submitted separately in uuencoded tar-compressed format.