Conformal symmetry breaking and self-similar spirals

Jemal Guven (ICN-UNAM)

RESUMEN: Self-similar curves are a recurring motif in nature. They also arise naturally as the tension-free equilibrium states of conformally invariant energies. Planar logarithmic spirals, for example, are associated with the simplest such energy, the conformal arc-length, and their remarkable properties follow as a consequence of this invariance and the manner of its breaking. I will show how to construct their three-dimensional analogues explicitly. The qualitative behavior of tension-free states is controlled by two parameters, the conserved scaling current S and the magnitude of the torque M. Planar logarithmic spirals occur when M and S are tuned so that 4MS=1. More generally, spirals exhibit internal structure, nutating between two cones aligned along the torque axis, expanding monotonically as the pattern precesses about this axis. If the spiral is supercritical (4MS>1) the exclusion cones are identical and oppositely oriented. The projection along the torque axis oscillates with increasing amplitude, turning when the torsion changes sign. I will argue that these elementary space curves provide useful templates for understanding a very broad range of self-similar spiral patterns appearing in nature.