A loop of SU(2) gauge fields stable under the Yang-Mills flow

TL;DR

This paper constructs and analyzes a topologically nontrivial loop of SU(2) gauge fields on S4 that remains stable under the Yang-Mills flow, with implications for quantum field theory and instanton corrections.

Contribution

It explicitly constructs a stable, topologically nontrivial loop of SU(2) gauge fields under the Yang-Mills flow and provides evidence for stable manifolds, advancing understanding of gauge field stability.

Findings

Constructed a stable loop of SU(2) gauge fields on S4.

Demonstrated local stability via flow equations.

Presented evidence for stable 2-manifolds of gauge fields.

Abstract

The gradient flow of the Yang-Mills action acts pointwise on closed loops of gauge fields. We construct a topologically nontrivial loop of SU(2) gauge fields on S4 that is locally stable under the flow. The stable loop is written explicitly as a path between two gauge fields equivalent under a topologically nontrivial SU(2) gauge transformation. Local stability is demonstrated by calculating the flow equations to leading order in perturbations of the loop. The stable loop might play a role in physics as a classical winding mode of the lambda model, a 2-d quantum field theory that was proposed as a mechanism for generating spacetime quantum field theory. We also present evidence for 2-manifolds of SU(2) and SU(3) gauge fields that are stable under the Yang-Mills flow. These might provide 2-d instanton corrections in the lambda model. For Isidore M. Singer in celebration of his eighty-fifth birthday.