Aharonov-Casher theorems for Dirac operators on manifolds with boundary and APS boundary condition

TL;DR

This paper extends the Aharonov-Casher theorem to Dirac operators on two-dimensional manifolds with boundary, analyzing zero modes under APS boundary conditions with magnetic fields.

Contribution

It provides a new analysis of zero modes for Dirac operators on manifolds with boundary, incorporating APS boundary conditions and magnetic field configurations.

Findings

Derived zero mode counts for Dirac operators with boundary conditions.

Extended Aharonov-Casher theorem to manifolds with boundary.

Analyzed effects of magnetic fields on zero modes in bounded domains.

Abstract

The Aharonov-Casher theorem is a result on the number of the so-called zero modes of a system described by the magnetic Pauli operator in $\mathbb{R}^2$. In this paper we address the same question for the Dirac operator on a flat two-dimensional manifold with boundary and Atiyah-Patodi-Singer boundary condition. More concretely we are interested in the plane and a disc with a finite number of circular holes cut out. We consider a smooth compactly supported magnetic field on the manifold and an arbitrary magnetic field inside the holes.