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Conformal Galilei groups, Veronese curves, and Newton-Hooke spacetimes

Finite-dimensional nonrelativistic conformal Lie algebras spanned by polynomial vector fields of Galilei spacetime arise if the dynamical exponent is z=2/N with N=1,2,.... Their underlying group structure and matrix representation are constructed (up to a covering) by means of the Veronese map of degree N. Suitable quotients of the conformal Galilei groups provide us with Newton-Hooke nonrelativistic spacetimes with a quantized reduced negative cosmological constant \lambda=-N.


Introduction
Newton-Hooke spacetimes provide us with solutions of the nonrelativistic gravitational field equations with nonvanishing cosmological constant, and may play a role in cosmology [1,2,3,4,5,6]. They can be viewed as deformations of their Galilean counterparts to which they reduce when the cosmological constant is turned off, and can indeed be obtained as nonrelativistic limits of the de Sitter or anti-de Sitter solutions of Einstein's equations.
Another way of constructing these nonrelativistic spacetimes is to first contract the (anti-)de Sitter group to yield the "Newton-Hooke" group(s), and then factor out the homogeneous part of the latter [3].
On the other hand, various conformal extensions of the Galilei Lie algebra have attracted much recent attention [7,8,9,10,11,12], and one may wonder about their group structure and associated homogeneous spacetimes. This paper is devoted to studying this question. The most common, and historically first, of such extensions, referred to as the Schrödinger group [13,14,15], has been first discovered in classical mechanics [16], and then for the heat equation [17], before being forgotten for almost a century and then rediscovered as the maximal group of symmetries of the free Schrödinger equation [18].
The Schrödinger group admits, in addition to those of the Galilei group, two more generators given by their spacetime actions (x, t) → (x * , t * ), namely dilations with a ∈ R * , and expansions (also called inversions) where with c ∈ R. These transformations generate, along with Galilean time-translations: x * = x, t * = t + b, with b ∈ R, the unimodular group SL (2, R). Note that the dynamical exponent is z = 2; see (1.1). Schrödinger symmetry typically arises for massive systems, as it combines with the one-parameter central extension of the Galilei group.
The Conformal Galilei (CG) symmetry algebra 1 [9,11] was first found, and then discarded, by Barut in his attempt to derive the by then newly (re)discovered Schrödinger symmetry by contraction from the relativistic conformal Lie algebra [19]. At the group level, this new symmetry also features an SL(2, R) subgroup generated by time-translations augmented with modified dilations x * = a x, t * = a t, (1.4) and expansions x * = Ω 2 (t) x, t * = Ω(t) t, (1.5) with the same parameters and factor Ω(t) as above. This second type of nonrelativistic conformal symmetry has dynamical exponent is z = 1, and also contains accelerations where B 2 ∈ R d (our notation will be justified below, see (4.12) and (4.14)). Moreover, this second type of conformal extension only allows for a vanishing mass [9]. It is rather difficult therefore to find physical systems with this kind of symmetry [20].
Both types of nonrelativistic symmetries have been related to the geometric "Newton-Cartan" structure of nonrelativistic spacetime [18,21,22,23,12]. Now, as recognized by Negro et al. [23], and by Henkel [24,7], both infinitesimal Schrödinger and CG symmetry belong to a much larger, generally infinite dimensional, class of Lie algebras with arbitrary, possibly even fractional, dynamical exponent z; their "conformal nonrelativistic algebra" [23] is, however finite dimensional for the particular values The terminology is justified by that, for all z as in (1.7), the algebra has an sl(2, R) Lie subalgebra, highlighted by the dilation generator (1.8) Taking into account rotations, boosts, and translations, yields, for z = 2, the Schrödinger algebra; the CG algebra is obtained, for z = 1, after incorporating also accelerations. For both N = 1 (Schrödinger) and N = 2 (CG), the infinitesimal action integrates to a Lie group action, but for general z, the results known so far only concern Lie algebras. Our first new result is the derivation of the global group structure for all N as in (1.7).
A crucial observation for our purposes is the following: owing to the factor (1.3) in (1.2) and (1.5), neither Schrödinger, nor Conformal Galilei transformations are globally well-defined over ordinary Galilean spacetime. As explained in Sections 2.3 and 4, Galilei spacetime should be replaced by a "better one". Our investigations in Section 5 show indeed that the proper arena where our conformal Galilei symmetry groups act is in fact provided by Newton-Hooke spacetimes with quantized negative cosmological constant.

Nonrelativistic spacetimes
The standard Galilei spacetime is the affine space modeled on R d+1 , endowed with its canonical flat affine connection Γ, and a Galilei structure (γ, θ) defined by a pair of (covariantly) constant tensor fields: namely by a spatial "metric" and a "clock", expressed in an affine coordinate system (x 1 , . . . , x d , x d+1 ) as respectively, where t = x d+1 is an affine coordinate of the time axis, T ∼ = R [26,27,28,29]. Notice that θ spans ker(γ).

Newton-Cartan structures
Generalized Galilei structures consist therefore of triples (M, γ, θ) where M is a smooth (d + 1)-dimensional spacetime manifold, γ a twice-symmetric contravariant tensor field of M whose kernel is spanned by a nowhere vanishing closed 1-form θ.
Due to the lack of a canonical affine connection on a Galilei structure, one is compelled to introduce then Newton-Cartan (NC) structures as quadruples (M, γ, θ, Γ) where (M, γ, θ) is a Galilei structure, and Γ a symmetric affine connection compatible with (γ, θ) whose curvature tensor, R, satisfies non-trivial extra symmetries which read, locally, for all α, β, ρ, σ = 1, . . . , d + 1. Upon introducing field equations relating the Ricci tensor to the mass-density, ̺, of the sources and the cosmological constant, Λ, viz., the connection Γ is interpreted as the gravitational field in a purely geometric generalization of Newtonian gravitation theory [26,27,28]. See [14] for a formulation of Newton-Cartan theory in a Kaluza-Klein type ("Bargmann") framework.

The Galilei group and its Lie algebra
The Galilei group, Gal(d), consists of all diffeomorphisms g of space-time which preserve all three ingredients of the Galilei structure, i.e., such that g * γ = γ, g * θ = θ, g * Γ = Γ. (2.4) This is the group of symmetries that governs nonrelativistic physics in d spatial dimensions. It clearly consists of (d + 2) × (d + 2) matrices of the form [25] where A ∈ O(d), and B 0 , B 1 ∈ R d stand respectively for a space translation and a boost, and b ∈ R is a time translation. The (affine) action of Gal(d) on spacetime R d × R reads (see [27,21] for a generalization to (curved) NC structures); these vector field read where ω ∈ so(d), β 0 , β 1 ∈ R d , and ε ∈ R. Latin indices run in the range 1, . . . , d, and Einstein's summation convention is assumed throughout this article. The Lie algebra gal(d) admits the faithful (d+2)-dimensional anti-representation X → Z where, with the above notation.

The Schrödinger group and its Lie algebra
Let us first discuss the Schrödinger group, Sch(d), which includes, in addition to the standard Galilei generators, those of the projective group, PSL(2, R), of the time axis. Up to a quotient that we will make more precise later on, the Schrödinger group will be defined as the matrix group whose typical element reads [30,21] defined on the open subset of spacetime where ct + d = 0.
It is an easy matter to check that the action (2.11) is consistent with the one presented in the introduction; Schrödinger dilations (1.1) correspond to b = 0, c = 0, and expansions (1.2) to a = 1, b = 0, d = 1.
The group structure is Sch Now, in order to guarantee a well-behaved action of this group on spacetime, one must demand that time be compactified, T ∼ = RP 1 . In fact, the Schrödinger group does not act on "ordinary" Galilei spacetime, but rather on the Möbius manifold fibered above the projective line, RP 1 , as clear from (2.11). This point will be further developed in Section 5.1. See also [21]. Note that (2.12) can be recovered by factoring out the homogeneous subgroup generated by rotations, expansions, dilations, and boosts, where Aff(1, R) stands for the 2-dimensional group of lower-triangular matrices in SL(2, R), generated by dilations and expansions. Note that, unlike conformally compactified Minkowski spacetime (S d × S 1 )/Z 2 , only time, not space, is compactified here since (2.14) It will be shown in Section 5 that the Möbius manifold carries a nonrelativistic Newton-Cartan structure; it is, in fact a Newton-Hooke spacetime with cosmological constant Λ = −d, minus the dimension of space; see (5.47).
The Schrödinger group can, indeed, be defined in a geometric way, namely in the NC framework [21,14,12], as the group, Sch(d), of all (locally defined) diffeomorphisms g such that where Proj(R d+1 , Γ) denotes the set of all projective transformations of spacetime, namely of all (local) diffeomorphisms which permute the geodesics of spacetime w.r.t. the connection Γ. Let us stress that the conditions (2.15) imply, in particular, that the diffeomorphism g projects on the time axis as an element of PGL(2, R) which must also preserve time-orientation defined by θ, namely an element of PSL(2, R).
The general solution of (2.15) is therefore given by (2.10), up to a covering; see also (2.11).
The Schrödinger Lie algebra, sch(d), is then the Lie algebra of those vector fields X on spacetime such that In local terms, we thus require for some 1-form ϕ of R d+1 depending on X, and for all α, β, ρ = 1, . . . , d + 1.
We easily find that X ∈ sch(d) iff with the same notation as above.
A remarkable property of the Schrödinger group, arising as a symmetry group of the classical space of motions of free spinning particle [12], and also important for studying supersymmetric extensions [31], is that it can be faithfully imbedded into the affine-symplectic Lie algebra (2.20)

Conformal Newton-Cartan transformations & finite-dimensional conformal Galilei Lie algebras
In close relationship with the Lorentzian framework, we call conformal Galilei transformation of a general Galilei spacetime (M, γ, θ) any diffeomorphism of M that preserves the direction of γ. Owing to the fundamental constraint γ(θ) = 0, it follows that conformal Galilei transformations automatically preserve the direction of the time 1-form θ.
In terms of infinitesimal transformations, a conformal Galilei vector field of (M, γ, θ) is a vector field, X, of M that Lie-transports the direction of γ; we will thus define X ∈ cgal(M, γ, θ) iff for some smooth functions f, g of M, depending on X. Then, cgal(M, γ, θ) becomes a Lie algebra whose bracket is the Lie bracket of vector fields. The one-form θ being parallel-transported by the NC-connection, one has necessarily dθ = 0; this yields dg ∧ θ = 0, implying that g is (the pull-back of) a smooth function on T , i.e., that g(t) depends arbitrarily on time t = x d+1 , which locally parametrizes the time axis. We thus have dg = g ′ (t)θ.

Conformal Galilei transformations, cgal
One can, at this stage, try and seek nonrelativistic avatars of general relativistic infinitesimal conformal transformations. Given a Lorentzian (or, more generally, a pseudo-Riemannian) manifold (M, g), the latter Lie algebra is generated by the vector fields, X, of M such that where g −1 denotes the inverse of the metric g : T M → T * M. It has been shown [29] that one can expand a Lorentz metric in terms of the small parameter 1/c 2 , where c stands for the speed of light, as with the previous notation. Here U is an "observer", i.e., a smooth timelike vector field of spacetime M, such that g(U, U) = c 2 , around which the light-cone opens up in order to consistently define a procedure of nonrelativistic limit. The Galilei structure (γ, θ) is recovered via γ = − lim c→∞ g −1 , and θ = lim c→∞ (c −2 g(U)). In (3.3) the symmetric twice-covariant tensor field U γ will define the Riemannian metric of the spacelike slices in the limiting Galilei structure. We can thus infer that the nonrelativistic limit of Equation More generally, we consider for some m = 1, 2, 3, . . ., and n = 0, 1, 2, . . ., to be further imposed on the vector fields X ∈ cgal(M, γ, θ). Then the quantity matches the ordinary notion of dynamical exponent [15,7,12]. We will, hence, introduce the Galilean avatars, cgal 2/z (M, γ, θ), of the Lie algebra so(d + 1, 2) of conformal vector fields of a pseudo-Riemannian structure of signature (d, 1) as the Lie algebras spanned by the vector fields X of M satisfying (3.1), and (3.5). We will call cgal 2/z (M, γ, θ) the conformal Galilei Lie algebra with dynamical exponent z in (3.6). This somewhat strange notation will be justified in the sequel.
is the obvious generalization to Galilei spacetimes of the Schrödinger-Virasoro Lie algebra sv(d) = sv(R × R d , γ, θ) introduced in [15] (see also [7]) from a different viewpoint in the case of a flat NC-structure. The representations of the Schrödinger-Virasoro group and of its Lie algebra, sv(d), as well as the deformations of the latter have been thoroughly studied and investigated in [32,33]. Let us henceforth use the notation cgal 2/z (d) = cgal 2/z (R d+1 , γ, θ) with γ as in (2.1) and and θ = dt respectively. Then one shows [12] that X ∈ cgal 2/z (d) iff where ω(t) ∈ so(d), β(t), and ξ(t) depend arbitrarily on time, t. The Lie algebra cgal 0 (M, γ, θ) corresponding to the case z = ∞ is also interesting; it is a Lie algebra of symplectomorphisms of the models of massless and spinning Galilean particles [21,12].

The Lie algebra, cgal N (d), of finite-dimensional conformal Galilei transformations
Now we show that our formalism leads to a natural definition of a whole family of distinguished finite-dimensional Lie subalgebras of the conformal Galilei Lie algebra cgal 2/z (d) with prescribed dynamical exponent z, generated by the vector fields in (3.8), where ω(t) ∈ so(d), β(t), and ξ(t) depend smoothly on time, t.
Referring to [12] for details, let us restrict our attention to those vector fields X ∈ cgal Pol 2/z (d) that are polynomials of fixed degree N > 0 in the variables x 1 , . . . , x d , and t = x d+1 . We then have necessarily with κ, λ, ε ∈ R, and we find that a closed Lie algebra of polynomial vector fields of degree N > 0 is obtained provided with ω ∈ so(d), and ξ(t) quadratic as in (3.9), together with The finite-dimensional Lie algebras cgal Pol N (d) turn out to be isomorphic to the so-called alt 2/N (d) Lie algebras discovered by Henkel [15] in his study of scale invariance for strongly anisotropic critical systems (with d = 1), viz., From now on we drop the superscript "Pol" as no further confusion can occur. In the case N = 1, we recognize the Schrödinger Lie algebra cgal 1 (d) ∼ = sch(d), see (2.18), 2 while for N = 2 we recover the "Conformal Galilei Algebra" (CGA) cgal 2 (d), called cmil 1 (d) in [12].
With a slight abuse of notation, we will still denote by Ver N : R 2 → R N +1 the mapping defined by for all k = 1, . . . , N + 1. Put t = (t 1 , t 2 ) ∈ R 2 , and consider t * = Ct with The image u * of t * under the Veronese map is clearly a (N +1)-tuple of homogeneous polynomials of degree N in t; it thus depends linearly on u = (u 1 , . . . , u N +1 ) ∈ R N +1 , where the u k are as in (4.2). The general formula is as follows. If t * 1 = at 1 + bt 2 , t * 2 = ct 1 + dt 2 , with ad − bc = 1, then Ver N (Ct) = Ver N (C)Ver N (t), (4.4) where Ver N (C) a nonsingular (N + 1) × (N + 1) matrix with entries which constitutes (up to equivalence) the well-known (N +1)-dimensional irreducible representation of SL(2, R); see [35]. Let us introduce the sl(2, R) generators interpreted physically as the infinitesimal generators of time translations ξ −1 , dilations ξ 0 , and expansions ξ 1 . Their images under the tangent map of of Ver N at the identity read then We now prove that the Lie algebra of CGal N (d) is, indeed, cgal N (d) introduced in Section 3. In fact, putting t = t 1 /t 2 in (4.2), wherever t 2 = 0, we easily find that the projective action g R d+1 : (x, t) → (x * , t * ) of CGal N (d) reads, locally, as  which, with the help of (4.3) and (4.5), leaves us with 14)  Let us now write any vector in the Lie algebra of CGal N (d) as where we have used Equations (4.8)-(4.10) with Then the infinitesimal form of the transformation laws (4.14) and (4.15) writes as δx = δx * | δg=Z,g=Id , together with δt = δt * | δg=Z,g=Id , i.e., At last, the vector field where the vector field X is as in (3.11), proving our claim. Our terminology for the dynamical exponent is justified by verifying that the dilation generator is (1.8) with z = 2/N.

Conformal Galilei spacetimes & cosmological constant
As mentioned in the Introduction, the physical spacetime can be recovered by postulating some symmetry group -defining its geometry -and then by factoring out a suitable subgroup. The simplest example is to start with the neutral component of the Galilei group (2.5), namely and factor out rotations and boosts to yield (ordinary) Galilei spacetime, Similarly, one can start instead with a deformation of the Galilei group called Newton-Hooke group [1, 2, 3] where SO(d) × SO(2) is the direct product of spatial rotations and translations of (compactified) time acting on the Abelian subgroup R 2d of boosts and spacetranslations. Then, quotienting N + (d) by the direct product of rotations and boosts yields the Newton-Hooke spacetime [3]. The latter carries a non-flat nonrelativistic structure and satisfies the empty space Newton gravitational field equations with negative cosmological constant [3]. 3 Below, we extend the above-mentioned construction to our conformal Galilei groups CGal N (d), at any level N ≥ 1.

Conformal Galilei spacetimes
The conformal Galilei spacetimes M N , associated with z = 2/N where N = 1, 2, . . ., are introduced by starting with the conformal Galilei groups CGal 2/z (d), viz., Explicitly, the projection π N : CGal N (d) → M N in (5.5) is defined by the direction π N (g) = R * · g d+N +1 (5.6) of the last column-vector of the matrix, g, in (4.12). Therefore x = π N (g) gets locally identified with  Intuitively, this amounts to factoring out O(d), dilations and expansions and all higher-than-zeroth-order accelerations, and identifying space-time with "what is left over".
Then, the action g → g M N of CGal N (d) on spacetime M N is globally given by g M N (π N (h)) = π N (gh) and, in view of (5.7), retains the local form (4.14) and (4.15). Now, by the very definition (4.12) of the conformal Galilei group at level N, we get indeed the conformal Galilei spacetime fibered above the Veronese curve Ver N (RP 1 ) ⊂ RP N , interpreted as the time axis, T . Besides, it may be useful to view the projective line as RP 1 ∼ = S 1 /Z 2 , locally parametrized by an angle ϑ related to the above-chosen affine parameter t = tan ϑ, (5.9) highlighting that ϑ ∼ ϑ + π. This entails (see (5.8)) that where Ver N (S 1 ) stands for the image of S 1 by the mapping (4.2). To summarize, we have the following diagram where the horizontal arrow denotes the canonical fibration of spacetime M N onto the time axis T .
We will from now on focus our attention to the new CGal N (d)-homogeneous spacetimes M N .

Galilean conformal Cartan connections
Prior to introducing (flat) Cartan connections associated with our conformal Galilei groups, let us recall some basic facts about Galilei connections [28]. is the R d+1 -valued soldering 1-form (5.17) of P . 4 Then the structure equations provide us with the definition of the associated curvature 2-form ((Ω a b ), (Ω a )) on P , namely ) for all a, b = 1, . . . , d + 1.
Demanding now that the curvature 2-form be h-valued (the torsion (Ω a ) is set to zero), we end up with a symmetric connection (Γ ρ αβ ), entering the following local expression such that, if ∇ stands for the associated covariant derivative of spacetime tensor fields, ∇ ρ γ αβ = 0 and ∇ α θ β = 0, for all α, β, ρ = 1, . . . , d + 1. This finally entails that Γ (given by ω) is a Galilei connection on (M, γ, θ) in the sense of Section 2.1. At this stage, it is worthwhile mentioning that Galilei connections (5.18) are special instances of "Cartan connections" on which the next developments will rely.
Let us thus recall, for completeness, the definition of a Cartan connection on a principal fiber bundle P → M, with structural group a closed subgroup H of a Lie group G, where dim(M) = dim(G/H). Put g = Lie(G) and h = Lie(H).
Such a "connection" is given by a g-valued 1-form ω on the principal H-bundle P such that where the subscript P refers to the group or Lie algebra right-action on P . These connections provide a powerful means to encode the geometry of manifolds modeled on homogeneous spaces G/H, e.g., projective or conformal geometry.
We will show, in this section, that the homogeneous spaces M N (see (5.8)) indeed admit, for all N = 1, 2, . . ., a conformal Newton-Cartan structure together with a distinguished, flat, normal Cartan connection associated with CGal N (d). 5 The general construction of the normal Cartan connection associated with a Schrödingerconformal Newton-Cartan structure has been performed in [21].
• Likewise, if N = 1, the Schrödinger group Sch(d) may be thought of as a subbundle of the bundle of 2-frames of spacetime M 1 ∼ = Sch(d)/H 1 , see (5.5). 6 This time, Sch(d) is, indeed, interpreted as the bundle of conformal Galilei 2-frames associated with the conformal class γ ⊗ θ of a Galilei structure (γ, θ) over M 1 as given by Equation (3.5) with m = n = 1. The Maurer-Cartan 1-form Θ Sch(d) of this group actually gives rise to the canonical flat Cartan connection on the H 1 -bundle Sch(d) → M 1 . We refer to [21,22] for a comprehensive description of Schrödinger conformal Cartan connections.
Using the same notation as in (2.19), we find and, hence, the structure equations Furthermore, a Galilei structure can be introduced on spacetime M 1 viewed as the quotient of (5.3). Our clue is that embedding SO(2) into SL(2, R) through Introducing the pulled-back Maurer-Cartan 1-form, Θ New 1 (d) , we end up with the previous structure equations (5.31)-(5.36) specialized to the case λ = 0, and κ = ε. Comparison between the latter equations, and the (flat) Galilei structure equations shows that both sets coincide except for the equations characterizing dβ 1 . This entails that M 1 acquires curvature through the 2-form [21] Ω 1 = β 0 ∧ ε. (5.41) Let us present, for completeness, a local expression of the NC-Cartan connection on M 1 generated by Θ New 1 (d) . Putting x = B 0 defines, along with ϑ ∈ R/(2πZ) introduced in (5.37), a coordinate system on spacetime M 1 . We readily find the components of the soldering 1-form to be and ε = dϑ, (5.42) see (5.20), while those of the h-connection write where θ = dϑ; the "cosmological constant" is therefore given by the dimension of space.
The angular parameter ϑ being dimensionless, it might be worth introducing, at this stage, a (circular) time parameter where H 0 is a "Hubble constant" whose inverse would serve as a time unit. 8  In view of (5.49), our model therefore yields a reduced cosmological constant • For the case N > 1, the Maurer-Cartan 1-form of CGal N (d) reads with the same notation as before. Then, the preceding computations can be reproduced, mutatis mutandis, for the group CGal N (d) which serves as the bundle of conformal Galilei N-frames of M N . At that point, as a straightforward generalization of (5.38), we can define the Newton-Hooke group at level N as (5.53) Much in the same manner as in the case N = 1, the NC-connection obtained from the pull-back Θ New N (d) of the Maurer-Cartan 1-form (5.52) can be computed. It is easily shown that the curvature of the homogeneous spacetimes M N is now given by Ω N = NΩ 1 .

Conclusion and outlook
Our main results are two-fold: firstly, we have found that the infinite-dimensional Lie algebra of infinitesimal conformal Galilei transformations with rational dynamical exponent admits, in fact, finite dimensional Lie subalgebras provided the dynamical exponent is z = 2/N, where N = 1, 2, . . . . Then, we have proposed a natural construction devised to integrate, for each N, these Lie algebras into Lie groups, named Conformal Galilei groups at level N, by means of the classic Veronese embeddings [34]. The values N = 1 and N = 2 correspond to the Schrödinger Lie algebra, and to the CGA, respectively. Our results are somewhat unexpected in that, starting with the symmetry problem in the Galilean context, we end up with Newton-Hooke space-times and their symmetries. Let us shortly list some of the applications beyond the by-now standard N = 1, i.e., z = 2 Schrödinger symmetry.
The N = 2, i.e., z = 1 conformal extension of the [exotic] Galilean algebra was studied in [8], and later extended so as to include also accelerations [9].
The values N = 4, and N = 6, i.e., the dynamical exponents z = 1/2 and z = 1/3 arise in statistical mechanics, namely for the spin-spin correlation function in the axial next-nearest-neighbor spherical model at its Lifschitz points of first and second order [24,7].
Our Conformal Galilei groups, CGal N (d), which generalize the Schrödinger (N = 1) and the Conformal Galilei (N = 2) cases to any integer N, do not act regularly on ordinary flat Galilean spacetime, however; they act rather on manifolds constructed from them, which are analogous to the conformal compactification of Minkowski spacetime. Moreover, their group structure allows us to recover these spacetimes as homogeneous spaces of the Newton-Hooke groups New N (d) as defined in (5.53), and generalizing to level N the Newton-Hooke group (5.3). These associated spacetimes M N are endowed with a (conformal) Newton-Cartan structure by construction. Remarkably, they are identified as Newton-Hooke spacetimes with quantized negative reduced cosmological constant, λ N in (5.55). 9 An intuitive way of understanding our strategy is to consider, say, Schrödinger expansions in (1.2), or in (2.11), and observe that it is the denominator which makes the group action singular. Our way of removing this "hole" singularity is factorize the denominator in these expressions, as dictated by the projective action in (4.13).
Having constructed our finite-dimensional conformal extensions for each N, there remains the task to find physical realizations.
Firstly, in the Schrödinger case N = 1, one can verify directly that the geodesic equations, i.e., the free Newton equations, d 2 x dt 2 = 0, (6.1) are, indeed, preserved due to the following transformation of the acceleration where A ∈ O(d). Similarly, the equations of uniform acceleration, are preserved by the conformal Galilei group, CGal 1 (d), in view of the following transformation law, namely, These examples make it plausible that the N-conformal symmetry, CGal N (d), is realized by the higher-order geodesic equations, Such a statement is suggested by the quantum formulas in [24,7] and is consistent, for N = 2, with Ref. [10]. Another promising approach is [39]. We would finally like to mention that a complete, general, construction of Cartan connections for conformal Galilei structures, modeled on the CGal N (d)homogeneous manifolds M N , still remains to be undertaken in order to extend that of normal Schrödinger-Cartan connections carried out in [22]. This would lead to a satisfactory, brand-new, geometric definition of the group of N-conformal Galilei transformations of a NC-structure as the group of automorphisms of such an associated normal Cartan connection.
Let us now discuss the relationship of our procedures and technique to some other work on the same subject, namely to conformal Galilean symmetries.
No central terms are considered in this paper: our Lie algebras are represented by vector fields on (flat) Newton-Cartan spacetime. Central terms, and the mass in particular, are important, though, and are indeed necessary for physical applications. Henkel [24,7] does actually consider mass terms: he works with operators with such terms involving c −2 , where c is the speed of light. In his approach, those appear in the coefficients of powers of ∂/∂t. Considering higher-order terms in powers of ∂/∂t goes beyond our framework, though. For finite c, Henkel's boosts are Lorentzian, not Galilean, however; our center-free Lie algebras with genuine Galilei boosts are recovered as c → ∞, yielding also (N −1) accelerations in addition. At last, the mass terms disappear, as they should: mass and accelerations are indeed inconsistent [9].
Central extensions of the conformal Galilei algebra have been considered in [9,44] in the planar case, d = 2, and in [23] for d = 3.
Let us finally mention a natural further program, in the wake of the present study, namely the determination of the group-cohomologies of our Conformal Galilei groups. Also would it be worthwhile to classify all symplectic homogeneous spaces [25] of the latter, likely to unveil new, physically interesting, systems.