Invariant PDEs with Two-dimensional Exotic Centrally Extended Conformal Galilei Symmetry

Abstract

Conformal Galilei Algebras labeled by $d,\ell$ (where $d$ is the number of space dimensions and $\ell$ denotes a spin-${\ell}$ representation w.r.t. the $\mathfrak{sl}(2)$ subalgebra) admit two types of central extensions, the ordinary one (for any $d$ and half-integer $\ell$) and the exotic central extension which only exists for $d=2$ and ${\ell}\in\mathbb{N}$. For both types of central extensions invariant second-order PDEs with continuous spectrum were constructed in [1]. It was later proved in [2] that the ordinary central extensions also lead to oscillator-like PDEs with discrete spectrum. We close in this paper the existing gap, constructing \textcolor{black}{a new class of second-order invariant PDEs for the exotic centrally extended CGAs; they admit a discrete and bounded spectrum when applied to a lowest weight representation. These PDEs are markedly different with respect to their ordinary counterparts. The ${\ell}=1$ case (which is the prototype of this class of extensions, just like the $\ell=\frac{1}{2}$ Schr\"odinger algebra is the prototype of the ordinary centrally extended CGAs) is analyzed in detail.