Spectral asymmetry for bag boundary conditions

TL;DR

This paper derives a formula for spectral asymmetry of the Euclidean Dirac operator with local boundary conditions in two dimensions, and applies it to specific geometries like cylinders and disks.

Contribution

It provides a new boundary spectral function expression for spectral asymmetry and evaluates it explicitly for finite cylinders and disks.

Findings

Explicit spectral asymmetry formula for 2D Dirac operator with boundary conditions

Consistent results for finite-length cylinder case

Proposed interpretation for non-product boundary case in a disk

Abstract

We give an expression, in terms of boundary spectral functions, for the spectral asymmetry of the Euclidean Dirac operator in two dimensions, when its domain is determined by local boundary conditions, and the manifold is of product type. As an application, we explicitly evaluate the asymmetry in the case of a finite-length cylinder, and check that the outcome is consistent with our general result. Finally, we study the asymmetry in a disk, which is a non-product case, and propose an interpretation.