Surface-invariants in 2D classical Yang-Mills theory

Abstract

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 non dynamical 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.

Invariants from classical field theory

“…13. To our knowledge the results of this type have seldom been reported-unlike the relation between links and Chern-Simons theory-perhaps because the space of immersed curves in the plane, considered up to area preserving diffeomorphisms, has not been deeply studied in the mathematical literature.…”

“…13 we describe explicitly three planar curves-a planar version of the Borromean rings-for which S ͑1͒ is well defined and nonvanishing. A generic immersed curve in R 2 induces a partition of R 2 into a finite number of compact blocks and an unbounded block.…”

“…To our knowledge results of this type have seldom been reported -unlike the relation between links and Chern-Simons theory -perhaps because the space of immersed curves in the plane, considered up to area preserving diffeomorphisms, has not been deeply studied in the mathematical literature. The example consider in this section is studied in full details in [13], here we only highlight the results of that paper that are useful to illustrate yet another application of our method.…”

“…Again S (1) is only well-defined for an appropriated choice of curves. In [13] we describe explicitly three planar curves -a planar version of the Borromean rings -for which S (1) is well-defined and non-vanishing.…”

THE TOPOLOGICAL THEORY OF THE MILNOR INVARIANT $\bar\mu(1,2,3)$

“…This approach to obtain LIs from classical field theories can be rigorously proven and generalized to situations where the symmetry group is different from the group of diffeomorphisms of the base manifold. 10,11 Although we shall focus mainly in this classical approach, we shall also make some remarks about the "quantum method", which is the procedure usually employed to study the relation between LIs and topological theories.…”