Remarks on conformal invariants for piecewise smooth curves and Wilson loops

TL;DR

This paper discusses conformal invariants for piecewise smooth curves in 3D space, focusing on Wilson loops with polygon-like contours, and introduces techniques involving osculating spheres and circles to characterize kinks and cusps.

Contribution

It provides a mathematical framework for identifying conformal invariants at kinks of piecewise smooth curves, extending previous work on Wilson loops with polygonal contours.

Findings

Identification of conformal invariants for kinks and cusps

Use of osculating spheres and circles in analysis

Extension of Wilson loop analysis to more general curves

Abstract

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.