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

arXiv comment32 pages. Compared to previous version: Subsection 2.3, Section 5 and Appendix C were removed

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• Protected Fermionic Zero Modes in Periodic Gauge Fields