Aharonov–Casher Theorems for Dirac Operators on Manifolds with Boundary and APS Boundary Condition

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 \(\mathbb {R}^2\) ${\mathbb{R}}^2$ . 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.