The Riemann-Roch Theorem and Zero Energy Solutions of the Dirac Equation on the Riemann Sphere

Abstract

In this paper, we revisit the connection between the Riemann-Roch theorem and the zero energy solutions of the two-dimensional Dirac equation in the presence of a delta-function like magnetic field. Our main result is the resolution of a paradox - the fact that the Riemann-Roch theorem correctly predicts the number of zero energy solutions of the Dirac equation despite counting what seems to be the wrong type of functions.