Surface-invariants in 2D classical Yang-Mills theory

We study a method to obtain invariants under area-preserving diffeomorphisms associated to closed curves in the plane from classical Yang-Mills theory in two dimensions. Taking as starting point the Yang-Mills field coupled to nondynamical particles carrying chromo-electric charge, and by means of a perturbative scheme, we obtain the first two contributions to the on-shell action, which are area-invariants. A geometrical interpretation of these invariants is given.

Figure 1

Configuration with a nonvanishing $0\mathrm{\text{−}}\mathrm{th}$ order contribution

Figure 2

Example of a “linking-area” configuration detected by $Y({\gamma }_i,{\gamma }_j,{\gamma }_k)$