Topologically Nontrivial Sectors of the Maxwell Field Theory on Algebraic Curves

TL;DR

This paper explores the Maxwell field theory on algebraic curves with $Z_n$ symmetry, deriving metrics, analyzing differential equations, and explicitly constructing solutions related to nontrivial topological sectors.

Contribution

It introduces a large family of nondegenerate metrics on algebraic curves and explicitly constructs solutions of Maxwell equations with nontrivial Chern class on $Z_n$ symmetric curves.

Findings

Derived nondegenerate metrics for algebraic curves.

Connected differential equations on Riemann surfaces to those on the complex sphere.

Constructed explicit solutions for Maxwell equations with nontrivial topological sectors.

Abstract

In this paper the Maxwell field theory is considered on the $Z_n$ symmetric algebraic curves. As a first result, a large family of nondegenerate metrics is derived for general curves. This allows to treat many differential equations arising in quantum mechanics and field theory on Riemann surfaces as differential equations on the complex sphere. The examples of the scalar fields and of an electron immersed in a constant magnetic field will be briefly investigated. Finally, the case of the Maxwell equations on curves with $Z_n$ group of automorphisms is studied in details. These curves are particularly important because they cover the entire moduli space spanned by the Riemann surfaces of genus $g\le 2$. The solutions of these equations corresponding to nontrivial values of the first Chern class are explicitly constructed.