The spectral flow for Dirac operators on compact planar domains with local boundary conditions

TL;DR

This paper computes the spectral flow for a family of Dirac operators on a disk with holes, considering fixed symbol and boundary conditions, revealing a result dependent on the number of holes and explicitly calculating the case of an annulus.

Contribution

It provides an explicit computation of the spectral flow for Dirac operators with fixed symbol and boundary conditions on multiply-holed disks, including a specific calculation for the annulus.

Findings

Spectral flow depends only on the number of holes in the disk.

Explicit formula for spectral flow in the case of an annulus.

Spectral flow is determined up to an integer constant related to the topology.

Abstract

Let $D_t$, $t \in [0,1]$ be an arbitrary 1-parameter family of Dirac type operators on a two-dimensional disk with $m-1$ holes. Suppose that all operators $D_t$ have the same symbol, and that $D_1$ is conjugate to $D_0$ by a scalar gauge transformation. Suppose that all operators $D_t$ are considered with the same locally elliptic boundary condition, given by a vector bundle over the boundary. Our main result is a computation of the spectral flow for such a family of operators. The answer is obtained up to multiplication by an integer constant depending only on the number of the holes in the disk. This constant is calculated explicitly for the case of the annulus ($m=2$).