Classical Gauge Vacua as Knots

TL;DR

This paper explores the connection between classical gauge vacua and knot theory by rewriting a non-linear sigma model in terms of gauge fields, revealing that certain vacuum configurations correspond to knots characterized by the Hopf invariant.

Contribution

It introduces a novel interpretation of SU(2) pure gauge vacua as knots with the Hopf invariant, linking gauge theory and topological knot invariants.

Findings

Gauge vacua can be represented as knots with specific invariants.

Reformulation of the sigma model in terms of gauge fields.

Identification of the Hopf invariant with the winding number.

Abstract

The four dimensional O(3) non-linear sigma model introduced by Faddeev and Niemi, with a Skyrme-like higher order term to stabilise static knot solutions classified by the Hopf invariant, can be rewritten in terms of the complex two-component CP1 variables. A further rewriting of these variables in terms of SU(2) curvature free gauge fields is performed. This leads us to interpret SU(2) pure gauge vacuum configurations, in a particular maximal abelian gauge, in terms of knots with the Hopf invariant equal to the winding number of the gauge configuration.