Minimal realization of $\ell$-conformal Galilei algebra, Pais-Uhlenbeck oscillators and their deformation

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.


Introduction
Besides their application to the nonrelativistic AdS/CFT correspondence, nonrelativistic conformal algebras [1,2,3] attract attention due to their non-trivial structure. The conformal Galilei algebra is determined by a positive half-integer or integer number ℓ and the number d of spatial dimensions. Most of its features do not depend on the dimension, except in the special case of d=2, where an additional central-charge extension is admitted [4,5]. The (more relevant) parameter ℓ counts the number of vector generators G (n) i with i = 1, . . . , d and n = 0, . . . , 2ℓ (1.1) which span the ℓ-conformal Galilei algebra together with the so(d) generators M ij for 1 ≤ i < j ≤ d and the generators P , D and K of the one-dimensional conformal algebra [3,6]. Mechanical models invariant under ℓ-conformal Galilei transformations typically contain higher time derivatives for ℓ > 1 2 [6,7,9,10,11,12,13]. The method of nonlinear realizations [14,15] together with the inverse Higgs phenomenon [16] work quite well for the ℓ-conformal Galilei algebra, giving rise to interesting invariant Lagrangians [9,10]. However, recent results on the conformal invariance of Pais-Uhlenbeck oscillators [17] with specific frequencies [7,8] appear to be add odds with the nonlinear-realization approach. Indeed, in [7] the full ℓ-conformal Galilei group was realized on a single bosonic field, thus achieving conformal invariance for Pais-Uhlenbeck oscillators without a dilaton. To describe such a situation within the nonlinear-realization approach, one has to include the dilatation D and the conformal boost K in the stability subgroup. Usually, such an extension of the stability subgroup is undesirable, because the generators of the coset fail to form a representation of the stability subgroup, and the construction of invariant actions becomes problematic. In this paper we resolve this paradox by explicitly demonstrating (in Section 2) how actions can easily be constructed for such unusual cosets and transformation properties of the fields and Cartan forms. Moreover, we show that the minimal actions in d=2 describe Pais-Uhlenbeck oscillators with specific frequencies 1 ω, 3ω, 5ω, . . . , 2ℓω for ℓ ∈ Z+ 1 2 and 2ω, 4ω, 6ω, . . . , 2ℓω for ℓ ∈ Z , (1.2) where the basis frequency ω is a parameter entering the nonlinear realization.
For d=2 it turns out to be advantageous to relabel the vector generators as [11] G α and G α with α = −ℓ, . . . , ℓ . (1.3) In this notation (see (2.1) below) the ℓ= 1 2 -conformal Galilei algebra resembles the N =2 superconformal algebra in one dimension, except that the fermionic generators are commuting (to zero). This raises the question whether one can make these generators non-commuting, thereby introducing a deformation of the Schrödinger algebra. We analyze this possibility (in Section 3) and present the simplest invariant action describing this newly deformed variant of the harmonic oscillator. We also find the general solution for its equation of motion.
As a byproduct of our deformation, the one-dimensional conformal group is realized in an unusual way on a single complex bosonic field. We employ (in Section 4) our modified realization of the conformal group to generalize the recent investigation [18,19,20] of four-dimensional Ricci-flat metrics with SL(2,R) symmetry. Reproducing their near-horizon Kerr metric for ω=0, we find for ω =0 a very specific modification affecting only the radial and time variables. It is easy to include the effect of a cosmological constant, which yields new constant-curvature metrics. In this case we present analytic expressions only for a one-parameter family and for some isolated solutions.

Conformal Galilei group realization and Pais-Uhlenbeck oscillators
Recently it has been shown [7,8] that the Pais-Uhlenbeck oscillator enjoys an ℓ-conformal Newton-Hooke symmetry for half-integer or integer values of ℓ if the oscillation frequency is an odd or even integer multiple of the basis frequency ω, up to 2ℓω, respectively. In this Section we are going to construct the minimal realization (on one complex bosonic field) of ℓ-conformal Galilei and Newton-Hooke symmetries for both integer and half-integer values of the parameter ℓ. We will demonstrate that the simplest invariant actions describe the corresponding conformal Pais-Uhlenbeck oscillators.
This choice of stability subgroup H is quite unusual, because the dilatation L 0 and the conformal boost L 1 will then generate unbroken symmetries. Previously [9], nonlinear realizations of this group took L 0 and L 1 to be spontaneously broken and thus always featured a dilaton among the physical fields. In contrast, in our approach only G α and G α are spontaneously broken. Our parametrization of the coset space reads introducing a parameter ω. Although ω does not enter the ℓ-conformal Galilei algebra, it affects its dynamical realization, and in this context the term 'ℓ-conformal Newton-Hooke algebra' is often used [21]. 2 The ℓ-conformal Galilei group is realized by left multiplications on this coset, Thus, with respect to conformal transformations g 0 = e i(aL−1+bL0+cL1) our fields u α ,ū α and time t transform as 3 Up to a redefinition of the parameters a, b, c, these transformations exactly coincide with those in [7]. To find the transformation properties of the "lowest-weight" fields u −ℓ andū −ℓ under the shift symmetries generated by g α = e i(bαGα+bαGα) , (2.8) one has to commute g α past the factor e it(L−1+ω 2 L1) in the coset element (2.5). This is achieved by employing the relation [22] e it(L−1+ω 2 L1) = e i tan(ωt) ω L−1 e −2i log(cos(ωt))L0 e iω tan(ωt)L1 , (2.9) which easily yields Apparently, for any given value of ℓ the transformations (2.10) are just combinations of the shifts e −2i ℓωt , e −2i (ℓ−1)ωt , . . . , e 2i (ℓ−1)ωt , e 2i ℓωt (2.11) (including constants in the case of integer ℓ).
The next step is to calculate the Cartan forms for g in (2.5), (2.13) The above calculations are quite similar to those ones performed in [9]. From the general theory of nonlinear realizations [14,15] it follows that the forms ω α andω α (2.12) are invariant with respect to the shift symmetries (2.8) and transform nontrivially under the conformal group (2.7), because and, therefore, The factor e i(b+ct)L0 just rescales the forms ω α (2.12) and the factor e ihĈ is harmless for these forms, while the second factor e icL1 will seriously reshuffle them. This is the price we have to pay for the non-orthonormal coset (2.5). 4 Nevertheless, the conditions are invariant under all symmetries. Thus, the entire tower of fields (u α ,ū α |α = −ℓ+1, . . . , ℓ) may be covariantly expressed through time derivatives of the lowest-weight fields u −ℓ andū −ℓ as This is the inverse Higgs phenomenon [16]. In addition we impose the covariant constraints which are just the equations of motion. It is not hard to check that they coincide with the characteristic equation for the matrix A (2.13) written for the time derivative d dt , i.e. with the equations for half-integer and integer ℓ, correspondingly. Clearly, these equations follow from the conformally invariant Pais-Uhlenbeck oscillator actions respectively. Thus, the nonlinear realization of the ℓ-conformal Galilei group in the coset (2.5) gives rise to the conformally invariant Pais-Uhlenbeck oscillators.
We have checked in the lowest cases that the Lagrangian is just the Cartan form for the central chargeĈ. Unfortunately, we did not succeed to bring the intermediate calculations into readable form and, hence, a rigorous proof of this statement is lacking.
Let us complete this section with two comments.
• The main difference of our nonlinear realization with those considered in [9] is putting the generators L 0 and L −1 into the stability subgroup H (2.4). We may restore these generators via employing a coset parametrized by The additional factors will then seriously reshuffle the forms ω α ,ω α of (2.12). Nevertheless, the full set of constraints ω α =ω α = 0 for α = −ℓ, . . . , ℓ will produce the same set of equations of motion (2.19) and (2.20). The dilaton u will decouple and obey the standard equation of motion upon imposing the additional constraints ω L0 = ω L1 = 0 [22].
• However, if instead we use the coset as in [9], then the equations of motion will get modified by interactions between the dilaton u and the fields u −ℓ andū −ℓ . Passing from (2.23) to (2.22) requires a redefinition of all the fields u α ,ū α . In effect, we claim that the equations of motion of [9] may be decoupled from the dilaton by a nonlinear redefinition of the fields.

Deformed oscillator
The commutation relations of the ℓ-conformal Galilei algebra written in the form (2.1) are reminiscent of the relations of the wedge subalgebra in the Virasoro algebra extended by two commuting primary fields of the conformal weights ℓ+1. From this analogy it is natural to ask: Can one admit nontrivial relations between these primary fields, i.e. make the shift generators G α and G α non-commuting? A natural choice consists in the wedge subalgebra in some nonlinear, W-type algebra discovered by A.B. Zamolodchikov [23]. Let us consider the simplest case of a such deformation. The basic idea is to replace the ℓ = 1 2 conformal Galilei algebra by the factor algebra of the wedge subalgebra in W (2) 3 [24,25] over composite higher-spin generators, i.e. a linear su(1, 2) algebra with the following commutation relations, Here, γ is a deformation parameter: if γ = 0, we come back to the ℓ = 1 2 conformal Galilei algebra. The exact value of γ is inessential: if nonzero it can be put to unity by a rescaling of the generators G r and G r .
We choose the stability subalgebra H as and realize this deformed symmetry by left multiplications of g = e it(L−1+ω 2 L1) e i(uG −1/2 +ūG −1/2) e i(vG 1/2 +vG 1/2) . 3) The transformation properties of the time t and the lowest-weight fields u,ū get deformed for γ = 0: g 0 = e i aL−1 : δt = a sin 2 (ωt) + 4 cos(2ωt) 4−γ 2 ω 2 (uū) 2 , δu = − a 2 ωu sin(2ωt) + 4i γω cos(2ωt) 4−γ 2 ω 2 (uū) 2 uū , In what follows, we will need only the Cartan forms ω ±1/2 ,ω ±1/2 and ω U which read where The inverse Higgs constraints are the same as in the undeformed case, With these constraints taken into account, the form ω U simplifies to Observing that under all transformations (3.4) the form ω U only shifts by an exact differential, we can write down a simple invariant action, (3.9) The equations of motion following from this action coincide with those obtained from the constraints where v,v are defined in (3.7). We conclude that the deformation of the symmetry algebra, i.e. the passing from the ℓ = 1/2 Galilei algebra to the su(1, 2) algebra produces a non-polynomial velocity dependence in the action (3.9). The "free"(ω = 0) system shares this feature. The undeformed (γ = 0) case describes a harmonic oscillator (or, with ω = 0, a free particle). The intriguing question is whether our deformation preserves the integrability of the harmonic oscillator? In the next section we will prove this by explicit construction of the general solution for the system (3.9).
Thus, one may directly solve the equations of motion. However, the deformed oscillator (3.9) possesses some interesting properties which allow us to find the general solution in a purely algebraic way. Let us summarize these properties.
The energy of this solution is given by the Hamiltonian H (3.12) as We note that, for the special value C = − 1 γ , the system has only the trivial solution u =ū = 0.

The ω=0 case
It is worth commenting on the ω = 0 case. In this limit everything greatly simplifies. The action reads , (3.19) while its symmetry transformations acquire the form The expressions for the higher Goldstone fields v andv (3.7) become very simple, v =u 1 + i γ 2 (uu −ūu) ,v =u 1 + i γ 2 (uu −ūu) , (3.21) as do the equations of motion,v − iγv 2u = 0. and therefore obtain I = B,Ī =B, The angular momentum (3.11) is still conserved, and so one may algebraically find the solution of the equations of motion in the form Again, checking the relations (3.21) determines the coefficient µ in (3.27) to The energy of this solution is derived from the Hamiltonian H 0 (3.23), where, as before, the parameters a, b, c correspond to translations (L −1 ), dilatations (L 0 ) and conformal boosts (L 1 ), respectively. This form resembles the realization of the one-dimensional conformal symmetry found by Bardeen and Horowitz [26] in considering the near-horizon limit of the four-dimensional Kerr black hole. 6 In the same basis (4.1), the ω = 0 so(1, 2) transformations (3.4) read Does there exist some ω = 0 deformation of the near-horizon Kerr solution in which the conformal so(1, 2) symmetry is realized as in (4.3)? To answer this question we apply the procedure performed in [19]. The conformal invariants entering the near-horizon metric read where θ is the latitudinal angular variable, which is inert under the conformal transformations (4.3). With these invariants one may express the most general four-dimensional conformally invariant metric as (4.5) Note that shifting ρ → ωρ and t → ω −1 t corresponds to putting ω = 1, and redefining ρ → γρ amounts to setting γ = 1. The vacuum Einstein equations R µν = 0 (4.6) impose conditions merely on the coefficient functions F (θ) and G(θ): Somewhat surprisingly, the mass parameter ω does not enter these equations, and thus they are identical to the ω = 0 case studied in [19]. Referring to this paper for a detailed analysis, we reproduce here the general solution, with arbitrary integration constants C 1 and C 2 . The third integration constant hides in a trivial constant shift of θ. It is remarkable that the solution space is linear (i.e. it admits superpositions) although the equation is not. We conclude that the modified realization (3.4) of the conformal so(1, 2) symmetry introduces a "frequency" modification into the four-dimensional Ricci-flat metrics constructed (for ω = 0) in [19]. The physical interpretation of the metric (4.5) with the solution (4.8) requires passing back to Minkowski signature via γ → − i 2 γ and φ → i 2 φ and remains an open challenge.

Deforming the near-horizon Kerr-dS/AdS metric
It is easy to extend the construction to general Einstein metrics, i.e. constant-curvature metrics, by adding a cosmological constant Λ to (4.6). Demanding R µν + Λg µν = 0 (4.9) for the metric (4.5) changes the conditions (4.7) to which is no longer homogeneous under rescaling of F . However, the Λ dependence may be absorbed in a rescaling F → F/Λ. It follows that solutions blow up in the Λ → 0 limit, unless their overall scale is variable. The equation for F can be rewritten as (4.12) and so it reduces to solving 7 with some integration constant C 0 . In the limit of C 0 → ∞ we are led to F + F ′′ − ΛF 2 = 0, (4.14) so this possibility of solving (4.12) is already included in (4.13). The general solution to (4.13) contains one more integration constant besides C 0 and the trivial θ shift. In the limit Λ → 0, we can compare with the general solution (4.8) in the pervious subsection and confirm this count by the explicit relation The full explicit solution to (4.10) is given by elliptic functions and will not be displayed here. We would like to mention four special cases however. First, for C 0 = 3/4Λ we have the particular solution The other three cases occur at C 0 → ∞, i.e. are solutions to (4.14). Second, .
Third, there is the trivial constant solution to be discarded since it produces a singular metric. Fourth and most interesting, the C 1 = 0 family of (4.8) smoothly extends to a family of Λ = 0 solutions, since the naturally singular 1/Λ behavior could be absorbed into C 2 in this case.

Conclusions
We have constructed a minimal field realization (on a single complex boson) of the ℓ-conformal Galilei group in 2+1 dimensions. The simplest actions, given by the integral of the Cartan form for the central-charge generator, describe conformally invariant Pais-Uhlenbeck oscillators. The main difference between our approach and previous ones [9] is in the structure of the stability subgroup H, in which we put the generators of dilatation and conformal boost. Despite the non-orthonormality of the resulting coset space we could construct covariant equations of motion by imposing proper restrictions on the Cartan forms.
We have found it useful to employ a special basis for the ℓ-conformal Galilei algebra, in which the shift generators resemble primary spin-(ℓ+1) fields of a Virasoro algebra. This basis has also been used, for example, in [11]. This analogy may be prolonged further by deforming the conformal Galilei algebra to include a wedge subalgebra in some nonlinear W -type algebra. We did this for the simplest ℓ=1/2 Galilei algebra, i.e. for the Schrödinger algebra. We constructed the simplest action for this case and proved that the corresponding system and its ω = 0 extension are both integrable and solvable.
Concerning further developments, we make the following remarks.
• Our deformation is not limited to the case we presented. For example, one may contemplate an su(n) deformation of the Schrödinger algebra, based on the wedge subalgebra in a quasi-superconformal algebra (see e.g. [28] and references therein). One may also consider a deformation of the ℓ = 1 conformal Galilei algebra as well as the deformations for other values of ℓ.
• To shed light on the interpretation of the proposed deformation, it is important to look at the quantum deformed oscillator and its spectrum.
• One may investigate the supersymmetric extension of the deformed oscillator, which should be based on the wedge subalgebra in the N =2 super W (2) 3 algebra [29].
• Along the line we proposed, deformations of other conformal Galilei algebras will also yield modified transformation laws for the time parameter and the fields, thus providing novel realizations of the d = 1 conformal algebra. The latter may be employed for constructing new four-dimensional Einstein metrics along the line of [18,20].