Quantization of the conformal arclength functional on space curves

TL;DR

This paper establishes a correspondence between conformal classes of closed critical curves in Euclidean space and rational points in a complex domain, revealing a quantization phenomenon in conformal geometry.

Contribution

It introduces a classification of conformal strings via invariants and links conformal geometry to a quantization framework through contact dynamical systems.

Findings

Conformal classes correspond to rational points in a specific complex domain.

Existence of a model conformal string called symmetrical configuration.

Quantization of closed trajectories of the associated contact dynamical system.

Abstract

By a conformal string in Euclidean space is meant a closed critical curve with non-constant conformal curvatures of the conformal arclength functional. We prove that (1) the set of conformal classes of conformal strings is in 1-1 correspondence with the rational points of the complex domain $\{q\in \mathbb{C} \,:\, 1/2 < \mathrm{Re}\, q < 1/\sqrt{2},\,\, \mathrm{Im}\, q > 0,\,\, |q| < 1/\sqrt{2}\}$ and (2) any conformal class has a model conformal string, called symmetrical configuration, which is determined by three phenomenological invariants: the order of its symmetry group and its linking numbers with the two conformal circles representing the rotational axes of the symmetry group. This amounts to the quantization of closed trajectories of the contact dynamical system associated to the conformal arclength functional via Griffiths' formalism of the calculus of variations.