Calorons in Weyl Gauge

TL;DR

This paper shows that caloron solutions in Euclidean gauge theory, when transformed to Weyl gauge, are periodic only up to a large gauge transformation, clarifying their topological and tunneling properties.

Contribution

It explicitly constructs caloron solutions in Weyl gauge, revealing their non-trivial periodicity related to topological charge and clarifying their tunneling interpretation.

Findings

Caloron gauge fields in Weyl gauge are periodic up to a large gauge transformation.

The winding number of the gauge transformation equals the caloron's topological charge.

The work clarifies the relation between calorons, Chern-Simons numbers, and tunneling processes.

Abstract

We demonstrate by explicit construction that while the untwisted Harrington-Shepard caloron $A_\mu$ is manifestly periodic in Euclidean time, with period $\beta=\frac{1}{T}$, when transformed to the Weyl ($A_0=0$) gauge, the caloron gauge field $A_i$ is periodic only up to a large gauge transformation, with winding number equal to the caloron's topological charge. This helps clarify the tunneling interpretation of these solutions, and their relation to Chern-Simons numbers and winding numbers.