We consider pure SU(2) Yang-Mills theory on four-dimensional de Sitter space ${\mathrm{dS}}_4$ and construct a smooth and spatially homogeneous magnetic solution to the Yang-Mills equations. Slicing ${\mathrm{dS}}_4$ 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.

We consider pure SU(2) Yang-Mills theory on four-dimensional de Sitter dS₄ and anti-de Sitter AdS₄ spaces and construct various solutions to the Yang-Mills equations. On de Sitter space we reduce the Yang-Mills equations via an SU(2)-equivariant ansatz to Newtonian mechanics of a particle moving in $$ {\mathbb{R}}^3 $$ ℝ 3 under the influence of a quartic potential. Then we describe magnetic and electric-magnetic solutions, both Abelian and non-Abelian, all having finite energy and finite action. A similar reduction on anti-de Sitter space also yields Yang-Mills solutions with finite energy and action. We propose a lower bound for the action on both backgrounds. Employing another metric on AdS₄, the SU(2) Yang-Mills equations are reduced to an analytic continuation of the above particle mechanics from $$ {\mathbb{R}}^3 $$ ℝ 3 to $$ {\mathbb{R}}^{2,1} $$ ℝ 2 , 1 . We discuss analytical solutions to these equations, which produce infinite-action configurations. After a Euclidean continuation of dS₄ and AdS₄ we also present self-dual (instanton-type) Yang-Mills solutions on these backgrounds.

We investigate instantons on sine-cones over Sasaki-Einstein and 3-Sasakian manifolds. It is shown that these conical Einstein manifolds are Kähler with torsion (KT) manifolds admitting Hermitian connections with totally antisymmetric torsion. Furthermore, a deformation of the metric on the sine-cone over 3-Sasakian manifolds allows one to introduce a hyper-Kähler with torsion (HKT) structure. In the large-volume limit these KT and HKT spaces become Calabi-Yau and hyper-Kähler conifolds, respectively. We construct gauge connections on complex vector bundles over conical KT and HKT manifolds which solve the instanton equations for Yang-Mills fields in higher dimensions.

We consider SU (N) Yang–Mills theory on the space ℝ × S³ with Minkowski signature (-+++). The condition of SO(4)-invariance imposed on gauge fields yields a bosonic matrix model which is a consistent truncation of the plane wave matrix model. For matrices parametrized by a scalar ϕ, the Yang–Mills equations are reduced to the equation of a particle moving in the double-well potential. The classical solution is a bounce, i.e. a particle which begins at the saddle point ϕ = 0 of the potential, bounces off the potential wall and returns to ϕ = 0. The gauge field tensor components parametrized by ϕ are smooth and for finite time, both electric and magnetic fields are nonvanishing. The energy density of this non-Abelian dyon configuration does not depend on coordinates of ℝ × S³ and the total energy is proportional to the inverse radius of S³. We also describe similar bounce dyon solutions in SU (N) Yang–Mills theory on the space ℝ × S² with signature (-++). Their energy is proportional to the square of the inverse radius of S². From the viewpoint of Yang–Mills theory on ℝ^1,1 × S² these solutions describe non-Abelian (dyonic) flux tubes extended along the x³-axis.