Classical Yang-Mills Vacua on $T^{3}$ : Explicit Constructions

TL;DR

This paper explicitly constructs flat gauge connections on a 3-torus for various groups, including twisted and nontrivial cases, and verifies their Chern-Simons numbers through direct computation.

Contribution

It provides explicit constructions of flat connections with twisted boundary conditions and nontrivial holonomies for multiple gauge groups using Jacobi theta functions.

Findings

Explicit flat connections for unitary, orthogonal, and exceptional groups.

Verification of fractional Chern-Simons numbers.

Construction of nontrivial holonomies with nondiagonalizable triples.

Abstract

Flat connections for unitary gauge groups on a 3--torus with twisted boundary conditions as well as recently discovered periodic nontrivial flat connections with ``nondiagonalizable'' triples of holonomies for higher orthogonal and exceptional groups are constructed explicitly in terms of Jacobi theta functions with rational characteristics. The (fractional) Chern-Simons numbers of these vacuum gauge field configurations are verified by direct computation.