Exotic Galilean Conformal Symmetry and its Dynamical Realisations

TL;DR

The paper investigates exotic Galilean conformal symmetry in $(2+1)$ dimensions, showing how the nonrelativistic conformal algebra with an exotic central charge $\theta$ emerges as a $c\to\infty$ contraction of the relativistic conformal group $O(3,2)$. It provides a concrete dynamical realisation via a higher-derivative Lagrangian and explicit phase-space generators, demonstrating the full exotic Galilean conformal algebra and its possible extension by $J_\pm$ in the planar setting. It further examines how Coulomb and magnetic vortex interactions and, more generally, constant electromagnetic fields modify the symmetry, leading to an enlarged 10-dimensional algebra with a second central charge $B$ and corresponding modified generators. The results highlight the role of $\theta$ in enabling dynamical realization of the symmetry in $(2+1)$D and point to future directions, including supersymmetric extensions and the status of $J_\pm$ in broader classes of invariant models.

Abstract

The six-dimensional exotic Galilean algebra in (2+1) dimensions with two central charges $m$ and $θ$, is extended when $m=0$, to a ten-dimensional Galilean conformal algebra with dilatation, expansion, two acceleration generators and the central charge $θ$. A realisation of such a symmetry is provided by a model with higher derivatives recently discussed in \cite{peterwojtek}. We consider also a realisation of the Galilean conformal symmetry for the motion with a Coulomb potential and a magnetic vortex interaction. Finally, we study the restriction, as well as the modification, of the Galilean conformal algebra obtained after the introduction of the minimally coupled constant electric and magnetic fields.