Topological Data Analysis of Abelian Magnetic Monopoles in Gauge Theories

TL;DR

This paper applies Topological Data Analysis, specifically persistent homology, to study magnetic monopoles in lattice gauge theories, providing a new quantitative method to analyze phase transitions related to confinement in Quantum Chromodynamics.

Contribution

It introduces a novel application of persistent homology to analyze monopole configurations in gauge theories, improving the accuracy of critical coupling estimates over traditional methods.

Findings

Successfully reproduces known critical couplings in U(1) lattice gauge theory.

Extends the method to SU(3) gauge theory with improved accuracy.

Highlights the importance of topological properties of monopoles in confinement.

Abstract

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.