Topology of Calorons Re-examined

TL;DR

The paper tackles the problem of topological characterization of calorons on $S^1\\times\\mathbb{R}^3$, showing that the standard second Chern number formula requires refinement in the presence of gauge singularities and gauge-dependence. It develops and applies an analytic framework to Harrington-Shepard calorons (HS1) across multiple gauges (LtH, magnetic, temporal) using the Nahm construction and 't Hooft ansatz, demonstrating that the topological charge $Q$ remains $-1$ but the intermediate contributions shift between the clutching-function (C), FA, and dAA terms. The key result is a modified expression for $Q$ that includes the dAA contribution explicitly, with explicit gauge-dependent demonstrations of how magnetic charge can appear or be redistributed via gauge transformations. The findings illuminate the gauge-structure of caloron topology on $S^1\\times\\mathbb{R}^3$, motivate extensions to HS^k and holonomy cases, and provide methods that incorporate singular gauges through a hyperfunction-based regularization, impacting semiclassical analyses of confinement on circle-compactified spaces.

Abstract

We reconsider the detailed structure of the topological character of the instantons in pure Yang-Mills theory on $S^1\times\mathbb{R}^3$, so-called calorons. The claim is that the standard formula for the topological character, the second Chern number, requires some modification through analytic consideration. For concreteness, we explicitly calculate the second Chern number of the gauge configuration of the Harrington-Shepard type with unit topological charge of the gauge group $\mathrm{SU}(2)$ in several gauges. The genuine formula is shown to be applicable even though the gauge connection is in singular gauge. The gauge dependence of the magnetic charge is also discussed.