Solutions to Yang-Mills equations on four-dimensional de Sitter space

Abstract

We consider pure SU(2) Yang-Mills theory on four-dimensional de Sitter space dS${}_4$ and construct a smooth and spatially homogeneous magnetic solution to the Yang-Mills equations. Slicing dS${}_4$ as $\mathbb{R}\times S^3$, via an SU(2)-equivariant ansatz we reduce the Yang-Mills equations to ordinary matrix differential equations and further to Newtonian dynamics in a double-well potential. Its local maximum yields a Yang-Mills solution whose color-magnetic field at time $\tau \in \mathbb{R}$ is given by ${\tilde{B}}_a=-\frac{1}{2}{I}_a/(R^2{\cosh }^2\tau )$, where $I_a$ for $a=1,2,3$ are the SU(2) generators and $R$ is the de Sitter radius. At any moment, this spatially homogeneous configuration has finite energy, but its action is also finite and of the value $-\frac{1}{2}j(j+1)(2j+1){\pi }^3$ in a spin-$j$ representation. Similarly, the double-well bounce produces a family of homogeneous finite-action electric-magnetic solutions with the same energy. There is a continuum of other solutions whose energy and action extend down to zero.

1+7 pages; v2: introduction extended, gauge group representation dependence added, minor clarifications, 3 more references; v3: title change, published version

References in corpus (4)

• $\overline{D3}$ and dS
• Instantons and Yang-Mills Flows on Coset Spaces
• Rotating black holes with non-Abelian hair
• Yang-Mills Instantons and Dyons on Group Manifolds

Cited by in corpus (13)

• A new construction of rational electromagnetic knots
• Finite-action solutions of Yang-Mills equations on de Sitter dS${}_4$ and anti-de Sitter AdS${}_4$ spaces
• On rational electromagnetic fields
• Yang-Mills solutions on Minkowski space via non-compact coset spaces
• Yang-Mills solutions on de Sitter space of any dimension
• Instability of cosmic Yang-Mills fields
• Some Exact Solutions for Maximally Symmetric Topological Defects in Anti de Sitter Space
• A low-energy limit of Yang-Mills theory on de Sitter space
• A Topological Way of Finding Solutions to Yang-Mills Equation
• Classification of all constant solutions of $\mathrm{SU}(2)$ Yang-Mills equations with arbitrary current in pseudo-Euclidean space ${\mathbb{R}}^{p,q}$
• Generalization of Instanton-Induced Inflation and Dynamical Compactification
• Exact Multi-Instanton Solutions to Selfdual Yang-Mills Equation on Curved Spaces
• Characteristic Lightcone Sources in SO(1,3) Yang-Mills Theory