Yang-Mills solutions on Minkowski space via non-compact coset spaces

TL;DR

This paper constructs a family of Yang-Mills solutions on Minkowski space using non-compact coset space foliations, revealing new analytic solutions and properties of the field strength and energy-momentum tensor.

Contribution

It introduces a novel method of generating Yang-Mills solutions via non-compact coset space foliations on Minkowski space, including analysis of their properties.

Findings

Constructed a two-parameter family of solutions with singular field strength.

Reduced Yang-Mills equations to a mechanical system with an inverted double-well potential.

Identified vacuum solutions on the lightcone with simple energy-momentum tensor structure.

Abstract

We find a two-parameter family of solutions of the Yang-Mills equations for gauge group SO(1,3) on Minkowski space by foliating different parts of it with non-compact coset spaces with SO(1,3) isometry. The interior of the lightcone is foliated with hyperbolic space $H^3\cong \text{SO}(1,3)/\text{SO}(3)$, while the exterior of the lightcone employs de Sitter space dS$_3\cong \text{SO}(1,3)/\text{SO}(1,2)$. The lightcone itself is parametrized by SO(1,3)/ISO(2) in a nilpotent fashion. Equivariant reduction of the SO(1,3) Yang-Mills system on the first two coset spaces yields a mechanical system with inverted double-well potential and the foliation parameter serving as an evolution parameter. Its known analytic solutions are periodic or runaway except for the kink. On the lightcone, only the vacuum solution remains. The constructed Yang-Mills field strength is singular across the lightcone and of infinite action due to the noncompact cosets. Its energy-momentum tensor takes a very simple form, with energy density of opposite signs inside and outside the lightcone.