Novel insight into centre-vortex geometry in four dimensions

TL;DR

This paper investigates the geometry of centre-vortex surfaces in four-dimensional SU(3) lattice gauge theory, revealing how secondary loops and sheets evolve across the finite-temperature phase transition, providing new insights into vortex structure.

Contribution

It introduces a novel method to classify secondary loops as connected or disconnected, and analyzes their behavior in four dimensions during the phase transition.

Findings

Secondary loops are mainly in the same sheet at low temperatures.

Above the phase transition, vortex sheets align with the temporal dimension.

Connected secondary loops become rare above the phase transition.

Abstract

Centre-vortex surfaces are mapped out in four dimensions within the framework of SU(3) lattice gauge theory to understand the role of secondary loops that develop in three-dimensional visualisations of centre-vortex structure, appearing separate from the percolating cluster. Loops that initially appear disconnected in three-dimensional slices can originate from the same connected surface in four dimensions depending on the surface's curvature. For the first time, these secondary loops are identified as "connected" or "disconnected" with respect to the vortex sheet, allowing new insight into the evolution of centre-vortex geometry through the finite-temperature phase transition. At low temperatures, we find that secondary loops of any length primarily lie in the same sheet percolating the four-dimensional volume. Only a handful of small secondary sheets disconnected from the percolating sheet are identified. Above the phase transition, the vortex structure is still found to be dominated by a single large sheet but one that has aligned with the temporal dimension. With the near absence of any curvature orthogonal to the temporal dimension, connected secondary loops become vanishingly rare. Other novel quantities, such as the four-dimensional density of secondary sheets and the sheet sizes themselves, are analysed to build a complete picture of centre-vortex geometry in four dimensions.