On the tetrahedrally symmetric monopole

TL;DR

This paper characterizes SU(2) BPS monopoles with specific spectral curves, showing that only tetrahedrally symmetric cases produce monopoles, using advanced theta function techniques and elliptic functions.

Contribution

It proves that only tetrahedrally symmetric spectral curves correspond to BPS monopoles within a certain family, employing new mathematical techniques involving Fay and Accola's factorization and Humbert varieties.

Findings

Only tetrahedral symmetric curves yield BPS monopoles.

Introduces new methods using theta function factorization and elliptic functions.

Provides a conjecture extending the main result.

Abstract

We study SU(2) BPS monopoles with spectral curves of the form $\eta^3+\chi(\zeta^6+b \zeta^3-1)=0$. Previous work has has established a countable family of solutions to Hitchin's constraint that $L^2$ was trivial on such a curve. Here we establish that the only curves of this family that yield BPS monopoles correspond to tetrahedrally symmetric monopoles. We introduce several new techniques making use of a factorization theorem of Fay and Accola for theta functions, together with properties of the Humbert variety. The geometry leads us to a formulation purely in terms of elliptic functions. A more general conjecture than needed for the stated result is given.