Abstract
We present the minimal realization of the $\ell$-conformal Galilei group in 2+1 dimensions on a single complex field. The simplest Lagrangians yield the complex Pais-Uhlenbeck oscillator equations. We introduce a minimal deformation of the $\ell$=1/2 conformal Galilei (a.k.a. Schr\"odinger) algebra and construct the corresponding invariant actions. Based on a new realization of the d=1 conformal group, we find a massive extension of the near-horizon Kerr-dS/AdS metric.
Highlights
Dilatation D and the conformal boost K in the stability subgroup
We present the minimal realization of the -conformal Galilei group in 2+1 dimensions on a single complex field
Based on a new realization of the d = 1 conformal group, we find a massive extension of the near-horizon Kerr-dS/AdS metric
Summary
It has been shown [8, 9] that the Pais-Uhlenbeck oscillator enjoys an -conformal Newton-Hooke symmetry for half-integer or integer values of if the oscillation frequency is an odd or even integer multiple of the basis frequency ω, up to 2 ω, respectively. We are going to construct the minimal realization (on one complex bosonic field) of -conformal Galilei and Newton-Hooke symmetries for both integer and half-integer values of the parameter. From the general theory of nonlinear realizations [15,16,17,18] it follows that the forms ωα and ωα (2.12) are invariant with respect to the shift symmetries (2.8) and transform nontrivially under the conformal group (2.7), because ei(aL−1+bL0+cL1) g = g ei(b+ct)L0 eicL1 eihC (2.14). These equations follow from the conformally invariant Pais-Uhlenbeck oscillator actions. The nonlinear realization of the -conformal Galilei group in the coset (2.5) gives rise to the conformally invariant Pais-Uhlenbeck oscillators. We claim that the equations of motion of [10] may be decoupled from the dilaton by a nonlinear redefinition of the fields
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