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

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 $$\mathbb {R}^2$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msup> <mml:mrow> <mml:mi>R</mml:mi> </mml:mrow> <mml:mn>2</mml:mn> </mml:msup> </mml:math> . 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.