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

TL;DR

This paper constructs a new class of second-order invariant PDEs with discrete spectra for exotic centrally extended Conformal Galilei Algebras in two dimensions, filling a gap in the understanding of their spectral properties.

Contribution

It introduces and analyzes a novel class of invariant PDEs for exotic central extensions, which exhibit discrete and bounded spectra, unlike previously known PDEs.

Findings

Constructed invariant PDEs with discrete spectra for exotic extensions

Analyzed the $ ext{l}=1$ case in detail

Demonstrated differences from ordinary central extension PDEs

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.