Harald Dorn

The conformal nature of smooth curves in R 3 is characterised by conformal length, curvature and torsion. We present a derivation of these conformal parameters via a limiting process using inscribed polygons with circular edges. The procedure is based on elementary geometry in R 3 only and similar to the rectification of curves in the metrical case. It seems to be not available in the literature so far.

This short note is some obvious mathematical addendum to our papers on Wilson loops on polygon-like contours with circular edges \cite{Dorn:2020meb,Dorn:2020vzj}. Using the technique of osculating spheres and circles we identify the conformal invariants characterising the kinks (cusps) of generic piecewise smooth curves in 3-dimensional space.

We calculate Wilson loops in lowest order of perturbation theory for triangular contours whose edges are circular arcs. Based on a suitable disentanglement of the relations between metrical and conformal parameters of the contours, the result fits perfectly in the structure predicted by the anomalous conformal Ward identity. The conformal remainder function depends in the generic 4D case on three cusp and on three torsion angles. The restrictions on these angles imposed by the closing of the contour are discussed in detail and also for cases in 3D and 2D.

We calculate Wilson loops in lowest order of perturbation theory for triangular contours whose edges are circular arcs. Based on a suitable disentanglement of the relations between metrical and conformal parameters of the contours, the result fits perfectly in the structure predicted by the anomalous conformal Ward identity. The conformal remainder function depends in the generic 4D case on three cusp and on three torsion angles. The restrictions on these angles imposed by the closing of the contour are discussed in detail and also for cases in 3D and 2D.

We calculate both at leading weak and strong coupling the renormalised Maldacena-Wilson loop for contours formed by consecutive passage of two touching circles. At the touching point both circles should have the same normal direction but form cusps of non-zero opening angle α. Particular emphasis is put on the behaviour in the limit α → 0 and its comparison with the spiky situation studied in a previous paper, where α was set to zero before renormalisation.

We study the Wilson loops for contours formed by a consecutive passage of two touching circles with a common tangent, but opposite orientation. The calculations are performed in lowest nontrivial order for N = 4 SYM at weak and strong coupling and for QCD at weak coupling. After subtracting the standard linear divergence proportional to the length, as well the recently analysed spike divergence, we get for the renormalised Wilson loops log W ren = 0. The result holds for circles with different radii and arbitrary angle between the discs spanned by them.