Topology of Calorons Re-examined

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.