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

TL;DR

This paper explores the link between the Riemann-Roch theorem and zero energy solutions of the 2D Dirac equation on the Riemann sphere, resolving a paradox about solution counting.

Contribution

It clarifies how the Riemann-Roch theorem accurately predicts zero energy solutions despite initial apparent discrepancies.

Findings

Confirmed the Riemann-Roch theorem's prediction of solution count

Resolved the paradox regarding function types in solution counting

Established a connection between geometric theorems and quantum solutions

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.