Topology of Calorons Re-examined

TL;DR

This paper re-examines the topological structure of calorons in Yang-Mills theory, proposing modifications to the standard topological charge formula and analyzing gauge dependence of magnetic charge.

Contribution

It provides a detailed analytic reconsideration of the topological character of calorons, including explicit calculations of the second Chern number in various gauges.

Findings

Standard formula for topological charge requires modification.

Explicit calculation confirms the modified formula's applicability.

Gauge dependence of magnetic charge is analyzed.

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.