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 $\ell =\frac{1}{2}$ Schrödinger algebra is the prototype of the ordinary centrally extended CGAs) is analyzed in detail.

12 pages, no figure

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• Infinite-dimensional meta-conformal Lie algebras in one and two spatial dimensions
• Boundedness of meta-conformal two-point functions in one and two spatial dimensions
• Invariant PDEs of Conformal Galilei Algebra as deformations: cryptohermiticity and contractions
• Aspects of infinite dimensional $\ell$-super Galilean conformal algebra