论文标题
四维设立空间中的杨利尔斯理论解决方案
Solutions of Yang-Mills theory in four-dimensional de Sitter space
论文作者
论文摘要
这项博士学位工作涉及对4维理性空间D $ S_4 $的一些Yang-Mills解决方案的分析。该空间与有限的洛伦兹圆柱体在三个球员上以及Minkowski空间的一部分的共形等效性最近导致发现了一个有理子打结的电磁场配置家族。麦克斯韦方程（aka $ u（1）$ u（1）$ yang-mills理论的这些“基础结”解决方案都标有三个球体的超球形谐波，并且具有良好的特性，例如有限能量，有限的作用和保守的拓扑数量，称为刺激性。我们研究了它们的对称特性，计算了它们在共形群体中的保守noe电荷，并在存在的情况下研究带电颗粒的行为。 此外，在量规组$ su（2）$的非阿布尔案中，Yang-Mills方程的时间依赖于D $ S_4 $的Yang-Mills方程，就宇宙学意义而言。这些可能在$（4）$ - 对称的希格斯真空的早期宇宙学中发挥作用，这是由于这些仪表场的快速波动而稳定的。我们分析了此类$ SU（2）$“宇宙量规场”的线性稳定性，以对Yang-Mills方程的任意扰动，同时保持FLRW度量冻结。 Yang-Mills波动算子的对角线化产生的时间依赖性正常模式的稳定性分析是使用Floquet理论进行的。
This doctoral work deals with the analysis of some Yang-Mills solutions on 4-dimensional de Sitter space d$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.
