标题： 阿贝尔规范理论中磁单极子的拓扑数据分析 显示英文标题

标题： Topological Data Analysis of Abelian Magnetic Monopoles in Gauge Theories

摘要： 受关于量子色动力学中可能存在第二个高温相变的最新文献的启发，我们使用拓扑数据分析的方法重新探讨了颜色禁闭与磁单极子动力学之间的关系，该方法提供了在格点上定义的量的拓扑性质的数学严格表征。 在介绍了拓扑数据分析的主要工具之一持久同调之后，我们将讨论这一概念如何用于定量分析磁单极子在去禁闭相变过程中的行为。 我们的方法首先在紧致的$U(1)$格点规范理论中进行了演示，该理论已知具有由磁荷守恒对称性恢复驱动的零温度去禁闭相变。 对于该系统，我们对捕捉磁电流环同调性的可观测量进行了有限尺寸标度分析，结果表明我们的分析再现了预期的去禁闭临界耦合常数的期望值。 然后我们将该方法扩展到$SU(3)$规范理论，在该理论中，通过在最大阿贝尔规范下的投影可以识别出阿贝尔磁单极子。 在时间尺寸为$N_t = 4$的阿贝尔磁电流环的同调可观测量的有限尺寸标度分析中，我们以比常规热力学方法在相似统计精度下获得的更高准确性得到了临界耦合常数的期望值，这暗示着磁单极子电流的拓扑性质可能与禁闭有关。 显示英文摘要

摘要： Motivated by recent literature on the possible existence of a second higher-temperature phase transition in Quantum Chromodynamics, we revisit the proposal that colour confinement is related to the dynamics of magnetic monopoles using methods of Topological Data Analysis, which provides a mathematically rigorous characterisation of topological properties of quantities defined on a lattice. After introducing persistent homology, one of the main tools in Topological Data Analysis, we shall discuss how this concept can be used to quantitatively analyse the behaviour of monopoles across the deconfinement phase transition. Our approach is first demonstrated for Compact $U(1)$ Lattice Gauge Theory, which is known to have a zero-temperature deconfinement phase transition driven by the restoration of the symmetry associated with the conservation of the magnetic charge. For this system, we perform a finite-size scaling analysis of observables capturing the homology of magnetic current loops, showing that the expected value of the deconfinement critical coupling is reproduced by our analysis. We then extend our method to $SU(3)$ gauge theory, in which Abelian magnetic monopoles are identified after projection in the Maximal Abelian Gauge. A finite-size scaling of our homological observables of Abelian magnetic current loops at temporal size $N_t = 4$ provides the expected value of the critical coupling with an accuracy that is generally higher than that obtained with conventional thermodynamic approaches at comparable statistics, hinting towards the relevance of topological properties of monopole currents for confinement.