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 $\mathbb{R}^2$. 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.