Finite Energy Monopoles in Non-Abelian Gauge Theories on Odd-dimensional Spaces

TL;DR

This paper explores finite energy monopoles in higher-dimensional non-Abelian gauge theories, introducing models with higher power field strength terms and analyzing Hedge-Hog solutions through complex geometric methods.

Contribution

It proposes new gauge theory models with higher power terms and scalar kinetic terms, and investigates conditions for finite energy monopole solutions using algebraic and geometric techniques.

Findings

Finite energy monopoles exist under specific conditions.

Hedge-Hog solutions satisfy Abel's differential equation.

Spaces of solutions relate to singular quartic surfaces.

Abstract

In higher dimensional gauge theory, we need energies with higher power terms of field strength in order to realize point-wise monopoles. We consider new models with higher power terms of field strength and extraordinary kinetic term of scalar field. Monopole charges are computed as integrals over spheres and they are related to mapping class degree. Hedge-Hog solutions are investigated in these models. Every differential equation for these solutions is Abel's differential equation. A condition for existence of finite energy solution is shown. Spaces of 1-jets of these equations are defined as sets of zeros of polynomials. Those spaces can be interpreted as singular quartic surfaces in three-dimensional complex projective space.