Solutions of Yang-Mills theory in four-dimensional de Sitter space

This doctoral work deals with the analysis of some Yang-Mills solutions on 4-dimensional de Sitter space d$S_4$. The conformal equivalence of this space with a finite Lorentzian cylinder over the 3-sphere and also with parts of Minkowski space has recently led to the discovery of a family of rational knotted electromagnetic field configurations. These "basis-knot" solutions of the Maxwell equations, aka $U(1)$ Yang-Mills theory, are labelled with the hyperspherical harmonics of the 3-sphere and have nice properties such as finite-energy, finite-action and presence of a conserved topological quantity called helicity. We study their symmetry properties, compute their conserved Noether charges for the conformal group and study behaviour of charged particles in their presence. Moreover, in the non-Abelian case of the gauge group $SU(2)$ there exist time-dependent solutions of Yang-Mills equation on d$S_4$ in terms of Jacobi elliptic functions that are of cosmological significance. These could play a role in early-time cosmology where an $SO(4)$-symmetric Higgs vacuum is stabilized due to rapid fluctuations of these gauge fields. We analyze the linear stability of such $SU(2)$ "cosmic gauge fields" against arbitrary perturbations of the Yang-Mills equation, while keeping the FLRW metric frozen. The stablity analysis of the time-dependent normal modes that arise from the diagonalization of the Yang-Mills fluctuation operator is carried out using Floquet theory.