Eta invariant and holonomy, the even dimensional case

TL;DR

This paper develops an intrinsic spectral interpretation of eta invariants for even dimensional manifolds, relating them to holonomy of determinant line bundles, thus extending concepts analogous to Witten's holonomy theorem.

Contribution

It introduces an intrinsic spectral interpretation of eta invariants in even dimensions using adiabatic limits, connecting them to holonomy of determinant line bundles.

Findings

Eta invariants for even-dimensional manifolds are related to holonomy of determinant line bundles.

The spectral interpretation is achieved via adiabatic limit techniques.

The work extends Witten's holonomy theorem to even dimensions.

Abstract

In previous work, we introduced eta invariants for even dimensional manifolds. It plays the same role as the eta invariant of Atiyah-Patodi-Singer, which is for odd dimensional manifolds. It is associated to $K^1$ representatives on even dimensional manifolds and is closely related to the so called WZW theory in physics. In fact, it is an intrinsic interpretation of the Wess-Zumino term without passing to the bounding 3-manifold. Spectrally the eta invariant is defined on a finite cylinder, rather than on the manifold itself. Thus it is an interesting question to find an intrinsic spectral interpretation of this new invariant. We address this issue here using adiabatic limit technique. The general formulation relates the (mod $\mathbb Z$ reduction of) eta invariant for even dimensional manifolds with the holonomy of the determinant line bundle of a natural family of Dirac type operators. In this sense our result might be thought of as an even dimensional analogue of Witten's holonomy theorem proved by Bismut-Freed and Cheeger independently.