Constructing SU(N) fractional instantons

TL;DR

This paper constructs and analyzes self-dual SU(N) gauge configurations called fractional instantons on a 4-torus with twisted boundary conditions, extending known solutions and developing a power series expansion approach.

Contribution

It generalizes constant field strength solutions for fractional instantons to more complex geometries using a power series expansion method.

Findings

Explicit construction of vector potential and field strength in a deformation parameter expansion.

Next-to-leading order terms explicitly computed.

Solutions interpreted as self-dual configurations with a crystal structure in $\\mathbb{R}^4$.

Abstract

We study self-dual SU(N) gauge field configurations on the 4 torus with twisted boundary conditions, known as fractional instantons. Focusing on the minimum non-zero action case, we generalize the constant field strength solutions discovered by `t Hooft and valid for certain geometries. For the general case, we construct the vector potential and field strength in a power series expansion in a deformation parameter of the metric. The next to leading term is explicitly computed. The methodology is an extension of that used by the author for SU(2) fractional instantons and for vortices in two-dimensional Abelian Higgs models. Obviously, these solutions can also be seen as self-dual configurations in $\mathbb{R}^4$ having a crystal structure, where each node of the crystal carries a topological charge of $1/N$.