Anomaly inflow for local boundary conditions

TL;DR

This paper investigates the eta-invariant of Dirac operators on manifolds with boundary under local boundary conditions, establishing relations to boundary invariants and emphasizing the importance of strong ellipticity.

Contribution

It provides a new analysis of eta-invariants with local boundary conditions, clarifies the role of strong ellipticity, and examines specific boundary conditions like Witten--Yonekura.

Findings

Relates eta-invariants to boundary Dirac operators in even dimensions

Expresses eta-invariants via boundary operator indices in odd dimensions

Shows Witten--Yonekura boundary conditions are nearly but not strongly elliptic

Abstract

We study the $\eta$-invariant of a Dirac operator on a manifold with boundary subject to local boundary conditions with the help of heat kernel methods. In even dimensions, we relate this invariant to $\eta$-invariants of a boundary Dirac operator, while in odd dimension, it is expressed through the index of boundary operators. We stress the necessity of the strong ellipticity condition for the applicability of our methods. We show that the Witten--Yonekura boundary conditions are not strongly elliptic, though they are very close to strongly elliptic ones.