Boundary conditions, vacuum quantum numbers and the index theorem

Abstract

We present a version of the index theorem on a two-dimensional manifold with boundary, using local, rather than spectral, boundary conditions on the fermion fields. The index is defined through an appropriate regularization procedure and can take noninteger values. It is shown that local boundary conditions do not introduce spurious boundary contributions to vacuum quantities like the charge and angular momentum, in contrast with spectral conditions, and give results compatible with an open infinite space. We also examine the vacuum quantum numbers induced by an infinitely thin magnetic flux tube piercing a two-dimensional space, and in the case of a sphere containing a monopole, and clarify some issues concerning the observability of the Dirac string and the proper definition of the angular momentum.