Bertrand Le Gal
email: bertrand.legal@ims-bordeaux.fr

Lilian Bossuet
email: lilian.bossuet@univ-st-etienne.fr

Automatic HLS based Low-Cost IP Watermarking

Currently, the Intellectual Properties (IP) and their reuse are common, however the use of IP is raising design security issues i.e. counterfeiter, reverse engineering. Watermarking is one of the efficient methods to detect an unauthorized IP use and a counterfeiter. In this context, many interesting works have been proposed. However, a few of them combine the watermarking process with the synthesis one. This article presents a new automatic and low cost watermarking solution. The design watermark is implanted in a high-level synthesis process. Some implementation results with Xilinx Virtex-5 FPGA assure the proposed solution low overhead compared to existing solution.

I.

INTRODUCTION

To handle the increased system complexity, companies reuse more and more IP cores. In consequence of the IP core business rise, the theft of IP is increasing [START_REF] Pecht | Bogus ! Electronic manufacturing and consumers confront a rising tide of counterfeit electronics[END_REF]. The trade group founded by Ciso, HP, Nortel and 3COM (Alliance for Gray Market and Counterfeit Abatement [START_REF]fosList ← SearchFreeOutputSlotsDuringExecution[END_REF]) estimates that the legitimate electronic companies loose almost $100 billion revenue per year due to counterfeiting. Counterfeiting prevention requires the technological solution in addition to the legal process i.e. patents.

In order to identify an IP theft, a novel watermarking solution (design methodology, which originally answers to the IP security required by actual design reuse market) is presented in this article. To reduce the watermarking overhead (area, delay, power consumption and design time), a mark is automatically inserted in the design. It is done during the behavioral synthesis process by using a High Level Synthesis (HLS) tool [3,[START_REF]High-Level Synthesis from Algorithm to Digital Circuit[END_REF].

The paper is structured as follows. Section II describes a state of the art of the watermarking solutions. Section IV details the new proposition of watermarking. Section V presents the main parts of the modified HLS automatic flow. Finally section VI gives experimental results with signal processing benchmarks on Xilinx Virtex-5 FPGA target.

II. RELATED WORKS

According to [START_REF] Chapman | IP Protection of DSP algorithms for system on chip implementation[END_REF] the goals of IP protection are: [START_REF] Pecht | Bogus ! Electronic manufacturing and consumers confront a rising tide of counterfeit electronics[END_REF] to enable IP providers to protect their IPs against unauthorized use, (2) to protect all types of design data used to produce and deliver IPs, (3) to detect an unauthorized use of IPs, (4) to trace an unauthorized use of IPs.

IP unauthorized use detection involves the ability to determine that an unauthorized use has occurred and then to trace the source of theft. To answer the detection issue, IP providers introduce shadow digital signature.

Digital watermarking is an indirect protection scheme which demonstrates the ownership of an IP. The concept of active watermarking consists of a digital signature insertion into an IP. Many approaches have been developed depending on the watermarking abstraction level:

• Application level digital signature hide examples can be found in [6][START_REF] Torunoglu | Watermarking-based copyright protection of sequential functions[END_REF][START_REF] Oliveira | Techniques for the creation of digital watermarks in sequential circuits designs[END_REF][9]. A Digital Signal Processing (DSP) watermarking is introduced in [6].

• Authors in [START_REF] Torunoglu | Watermarking-based copyright protection of sequential functions[END_REF] target an algorithmic level watermarking in the design flow. Both approaches [6] and [START_REF] Torunoglu | Watermarking-based copyright protection of sequential functions[END_REF] are based on the idea of slightly changing the gain of filters, without affecting the system behavior. Two different watermarking techniques at behavioral level are introduced in [START_REF] Oliveira | Techniques for the creation of digital watermarks in sequential circuits designs[END_REF] and [9]. Both algorithms are based on adding new input/output sequences in the design finite state machine (FSM) representation.

• Digital signature hiding at logical synthesis level can be found in [START_REF] Kirovski | Intellectual property protection by watermarking combinational logic synthesis solutions[END_REF]11,12,[START_REF] Chang | Synthesis-for-testability watermarking for field authenticatioil of VLSI intellectual property[END_REF]. In [START_REF] Kirovski | Intellectual property protection by watermarking combinational logic synthesis solutions[END_REF] Hong presents the watermarking combinational logic synthesis solutions. The watermarking behavioral synthesis techniques for IP protection are described in [11].

• Mostly the post-synthesis watermarking introduces constraints [START_REF] Jain | Zero overhead watermarking technique for FPGA designs[END_REF] and [START_REF] Sun | A watermarking system for ip protection by buffer insertion technique[END_REF]. An example is the addition of extra hardware, like buffer [START_REF] Fan | Watermarking for intellectual property protection[END_REF] or dedicated embedded tester [START_REF] Gal | Dynamic memory access management for high-performance DSP applications using high-level synthesis[END_REF].

The pre-synthesis watermarking techniques are application dependant, and their over-cost is not measurable. The postsynthesis techniques are time consuming and design/device dependant. The in-synthesis watermarking techniques introduce power/area/timing overhead. Generalizing the watermarking usage for IP identification is important. However, it requires enough generic watermarking techniques with low overhead cost. Such techniques must be implemented in automatic design flows due to time to market constraint. It is for a fast component tagging in a designer friendly process. For these reasons, a new in-synthesis watermarking scheme is devised in the following section.

III. ARCHITECTURAL SYNTHESIS CONCEPTS

Our methodology targets custom hardware architectures, dedicated to the computational intensive applications for signal and video processing. In most usual video and signal processing applications, full pipeline architecture is quite inefficient. In fact, such architectures are too area and power consuming. Usually, there is a trade-off between the highperformance and the low-cost (area, power consumption) architectures [START_REF] Arvind | High-Level Synthesis: an essential ingredient for designing complex ASICs[END_REF] and [START_REF] Coussy | GAUT -a high-level synthesis tool for DSP applications[END_REF]. An efficient architecture shares the hardware resources (operators and registers) during an application execution. This resource sharing makes some input and output slots free and ready to tag.

The circuit inputs and outputs permit it to receive and send data from/to the system. Figure 1 presents the IO behavior for a FIR filter application. This IP component receives data from the system (Xn), performs computations and then provides valid result to the system (Yn). The output time slots between two successive high levels of data valid signal are unused. These free output slots are grey-colored in Figure 1.

The proposed idea is to employ the empty output slots for the design watermarking. The circuit watermark is composed of a set of mathematical relations. These relations are based on circuit input values, initial values and internal results. Each mathematical relation is a sub-mark. The sub-marks are read like an output value during free output slots (when data valid is inactive). The watermark is invisible for an IPs integrator. It is because these sub-marks results look like the dynamic transient output values. Consequently, the watermark is invisible from static analysis.

The proposed method is well adapted for general-purpose applications like digital signal, image or video processing etc. Nevertheless, this method is not appropriate for data-security application such as data encryption, data integrity or data authentication. In such cases, the IP watermark can cause a dramatic data security failure

The next section details this novel watermarking technique, which uses output dynamic sub-marks to watermark the IP.

IV. NEW IPP PROPOSITION BY WATERMARKING

This novel technique is based on the free output slots behavior. These output slots can be modified by introducing custom design singularities (that are IP internal values). The proposed method is based on the assertion that the hardware IP has free output slots.

Depending on the required protection level and the watermark cost allowed, we propose two watermarking solutions with different area costs: random low-cost watermark and cost-less watermark.

Random low cost watermark is characterized by a set of randomly chosen architecture internal values. A special datapath generates each sub-mark by transferring the chosen internal value (during the choosen clock cycle) to a free output slot. The watermarking area cost is due to the data path modification (output multiplexer resizing and FSM controller modifications). The data path area overcost could be very low. It is particularly true for FPGA implementation. In fact, while using FPGA, the reconfigurable data paths do not cost area. It is because the data paths set is directly available in the device. However, increasing input multiplexer size could cost area.

Cost-less watermark is a low-cost one with an usable reduced set of internal values. It employs the existing dynamic transient outputs during the free slots (internal values which do not required new path creation in the datapath). To introduce this kind of watermark, the only change required consists in modifying IP control unit to drive selected internal data to output ports. HLS tools design the IP control unit with a Finite State Machine (FSM). The cost-less watermark affects only the FSM. Experimental results on the FPGA target presented in section IV will assure that the modifications are cost-less and even could decrease the IP area occupation. In fact, the area increase or decrease depends on the FSM modification and the logical synthesis process. Hence, it is hard to estimate. inaccessible data to outputs, increasing the design singularities. Figure 3 presents the case of second watermarking scheme (cost-less watermarking). Here, the data paths are already allocated to implement the behavior computations. They drive the internal data to outputs, only design controller is modified.

V. WATERMARKING AUTOMATION FLOW

The proposed technique is integrated as a part of a HLS design flow. It permits the designer to automatically include copyright information in generated circuits. The designer configures the watermark generation specifying: (1) the watermark length (number of sub-marks). ( 2) the number of distinct clock cycles to mark (3) the choice between cost-less or low-cost watermarking technique. After the automatic watermarking step, the designer has a watermarked IP, generated under the system constraints. The tool provides him a file containing the design watermark. This file contains the output time slots where the sub-marks are produced and the existing mathematical relations. This information may keep secret from a consumer point of view, as it may be used to proof the circuit ownership.

A. HLS design flow modifications

Figure 4 presents the design flow modifications compared to the usual HLS ones [START_REF] Gal | Dynamic memory access management for high-performance DSP applications using high-level synthesis[END_REF]. The watermarking stage composed of four steps: (1) Graph analysis, to find the usable internal data (2) Possible watermark enumeration (3) Internal values for random selection. (4) Architecture modifications: datapath change, multiplexer resizing and control unit modification.

B. Selecting the marks and modifying the design

By using the designer provided parameters, an automatic process computes the number of possible watermarks (depending on the number of registers, the number of internal values and their associated lifetime). Depending on this result, it computes the average number of marks to introduce per clock cycle. The watermark repartition is then randomly performed to tag the required number of clock cycles. Once these computations are performed, a mapping algorithm is applied to find the most singular data from the design. It is done [START_REF] Pecht | Bogus ! Electronic manufacturing and consumers confront a rising tide of counterfeit electronics[END_REF] to select them for sub-mark usage [START_REF]fosList ← SearchFreeOutputSlotsDuringExecution[END_REF] to drive them to one of the output port.

For each internal data mapping to output, the tool analyzes the required logical glue over-cost (due to multiplexer input ports increase) in order to find the best couple (internal data, output) for low area design cost. An overview of the watermarking algorithm is provided in Figure 5. This process is repeated for each sub-mark that the tool must insert in the design to respect the watermark length constraint given by designer.

C. Generating the watermark file

The last step of the watermarking procedure is the Watermark properties file creation. This stage aims to create a file, containing all the inserted IO marks related information (1) the couples {output port, sub-mark} (2) the clock cycle during which the sub-mark is produced (3) the mathematical equation that defines the sub-mark (see Equation 2).

These data correspond to the information required to identify a cloned IP.

VI. EXPERIMENTS

To evaluate the proposed watermark design methodology efficiency in terms of area and critical path delay, experiments with signal and image processing benchmarks are conducted. The watermarked IP implementations are done using a Xilinx Virtex-5 FPGA. Results obtained using cost-less watermark are presented in Table 1.

For each design (i.e. IP), the following circuit parameters are provided: the number of FSM states, the number of I/O ports, the maximum mark length (number of free output slots), the introduced watermark length (0% for reference design, 50% or 100% for watermarked ones). The right columns provide: the number of available different watermarks and the circuit area and critical path obtained after logical synthesis. Penalties for watermarked IP are obtained from comparison to unprotected one. Logical synthesis results were obtained using the Xilinx ISE 10.1 tool.

Area and timing overhead are functions of the watermark selection algorithm. As it is shown in section III, area and critical path length overhead come from the datapath changes (some multiplexers are allocated) and from the control unit changes (FSM instruction decoder is modified to drive data and control new multiplexer). the average number of marks to introduce per clock cycle. The watermark repartition is then randomly performed to tag the required number of clock cycles. Once these computations are performed, a mapping algorithm is applied to find the more singular data from the design. It is done (1) to selected them for sub-mark usage [START_REF]fosList ← SearchFreeOutputSlotsDuringExecution[END_REF] to drive them to one of the output port. These singular data may be data not erased by rewrite in share registers, inputs stored, temporally computation data, etc.

For each internal data mapping to output, the tool analyzes the required logical glue over-cost in order to find the best couple (internal data, output) for Table 1 shows, in the case of cost-less watermarking, that the controller changes have a very low impact on the global IP component characteristics. Moreover the number of different watermarks that can be used to protect the design is quite important in the overall examples. Design area varies from -0,36% up to +0,17% when critical path progress from -1, 31% to +1, 05%. However, Table 1 shows that the mark length is much smaller for a high-level security. For some designs, such as SSD 16×16, unusual area and critical path reductions come from FSM signal command modifications. It may involuntarily results into a better logical equation simplification during logical synthesis. While considering the low-cost watermarking experimentations (result table is not provided due to article page limit), the area and timing deterioration are higher like the possible number of watermarks available. Design area increase from 0.07% to 1.02% while circuit critical path evolve from -1.29% to 1.45%. However, watermarking penalties are still low for most of the experiments (area increase average = 0.55% and latency average = 0.08%) These experimental results confirm the interest and the low cost of the proposed watermarking techniques which required low runtimes (only a few seconds).

VII. CONCLUSION

In this paper, a new watermarking technique for behavioral IP components and its synthesis design flow have been presented. The proposed technique is used for automatic Intellectual Property protection by using the HLS tools. The essence of this new approach is the set of mathematical marks on the design output ports which encode the IP watermark (copyright information). The mathematical marks are selected and inserted during the synthesis process. It is done in such a way that they result into the minimal hardware overhead while embedding the signature. Finally, the watermark are difficult to detect and remove.

Figure 2 andFigure 1 .Figure 2 .

 212 Figure 2 and Figure 3 present the main distinction between the low-cost and the cost-less watermark. In these examples the architecture required modifications are represented with dotted lines. Figure 2 describes the case of low-cost watermarking mode. It allows the algorithm to allocate new data-path (multiplexers and wires). It is to drive the

Figure 3 .

 3 Figure 3. Cost-less watermarked architecture reusing existing paths controller with multiplexer control modifications (dotted line).

Figure 2 .

 2 Figure 2.

Figure 4 .

 4 Figure 4. High-Level Synthesis design flow including the proposed watermarking technique (grey-colored).

Fig. 12

 12 Fig. 12 High-Level Synthesis design flow including the proposed watermarking technique (grey-colored).

Figure 5 .

 5 Figure 5. Generic cost-less and low-cost watermarking procedure.


C Duval

P A Horv Áthy

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

Keywords: Conformal Galilei groups, Veronese curves, and Newton-Hooke spacetimes CPT-P006-2011 Schrödinger algebra, conformal Galilei algebras and Galilei groups, Newton-Cartan theory, Veronese maps, Newton-Hooke spacetimes, cosmological constant

published or not. The documents may come

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 [START_REF] Bacry | Possible kinematics[END_REF][START_REF] Aldrovandi | Nonrelativistic spacetimes with cosmological constant[END_REF][START_REF] Gibbons | g spacetimes, Hpp-waves and the cosmological constant[END_REF][START_REF] Tian | Mechanics and Newton-Cartan-Like Gravity on the Newton-Hooke Space-time[END_REF][START_REF] Lukierski | Acceleration-enlarged symmetries in nonrelativistic space-time with a cosmological constant[END_REF][START_REF] Arratia | 2+1) Newton-Hooke Classical and Quantum Systems[END_REF]. 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 [START_REF] Gibbons | g spacetimes, Hpp-waves and the cosmological constant[END_REF].

On the other hand, various conformal extensions of the Galilei Lie algebra have attracted much recent attention [START_REF] Henkel | Phenomenology of local scale invariance: from conformal invariance to dynamical scaling[END_REF][START_REF] Stichel | A new type of conformal dynamics[END_REF][START_REF] Lukierski | Acceleration-Extended Galilean Symmetries with Central Charges and their Dynamical Realizations[END_REF][START_REF] Fedoruk | Galilean Conformal Mechanics from Nonlinear Realizations[END_REF][START_REF] Bagchi | Galilean Conformal Algebras and AdS/CFT[END_REF][START_REF] Duval | Non-relativistic conformal symmetries and Newton-Cartan structures[END_REF], 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 [START_REF] Jackiw | Introducing scaling symmetry[END_REF][START_REF] Duval | Bargmann Structures And Newton-Cartan Theory[END_REF][START_REF] Henkel | Schrödinger invariance and strongly anisotropic critical systems[END_REF], has been first discovered in classical mechanics [START_REF] Jacobi | Vorlesungen über Dynamik[END_REF], and then for the heat equation [START_REF] Lie | Über die Integration durch bestimmte Integrale von einer Klasse linearer partieller Differentialgleichungen[END_REF], before being forgotten for almost a century and then rediscovered as the maximal group of symmetries of the free Schrödinger equation [START_REF] Havas | Conformal extensions of the Galilei group and their relation to the Schrödinger group[END_REF].

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

x * = a x, t * = a 2 t, (1.1)
with a ∈ R * , and expansions (also called inversions)

x * = Ω(t) x, t * = Ω(t) t, (1.2)
where

Ω(t) = 1 ct + 1 , (1.3)
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 algebra1 [START_REF] Lukierski | Acceleration-Extended Galilean Symmetries with Central Charges and their Dynamical Realizations[END_REF][START_REF] Bagchi | Galilean Conformal Algebras and AdS/CFT[END_REF] 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 [START_REF] Barut | Conformal Group → Schrödinger Group → Dynamical Group -The Maximal Kinematical Group of the Massive Schrödinger Particle[END_REF]. 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

x * = x + B 2 t 2 , t * = t, (1.6)
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 [START_REF] Lukierski | Acceleration-Extended Galilean Symmetries with Central Charges and their Dynamical Realizations[END_REF]. It is rather difficult therefore to find physical systems with this kind of symmetry [START_REF] Cherniha | The exotic conformal Galilei algebra and nonlinear partial differential equations[END_REF].

Both types of nonrelativistic symmetries have been related to the geometric "Newton-Cartan" structure of nonrelativistic spacetime [START_REF] Havas | Conformal extensions of the Galilei group and their relation to the Schrödinger group[END_REF][START_REF] Duval | Quelques procédures géométriques en dynamique des particules[END_REF][START_REF] Burdet | Cartan Structures on Galilean Manifolds: the Chronoprojective Geometry[END_REF][START_REF] Negro | Nonrelativistic conformal groups[END_REF][START_REF] Duval | Non-relativistic conformal symmetries and Newton-Cartan structures[END_REF]. Now, as recognized by Negro et al. [START_REF] Negro | Nonrelativistic conformal groups[END_REF], and by Henkel [START_REF] Henkel | Local Scale Invariance and Strongly Anisotropic Equilibrium Critical Systems[END_REF][START_REF] Henkel | Phenomenology of local scale invariance: from conformal invariance to dynamical scaling[END_REF], 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" [START_REF] Negro | Nonrelativistic conformal groups[END_REF] is, however finite dimensional for the particular values

z = 2 N , N = 1, 2, . . . (1.7)
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

X = 1 z x • ∂ ∂x + t ∂ ∂t . (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

γ = d i=1 ∂ ∂x i ⊗ ∂ ∂x i , θ = dt (2.1)
respectively, where t = x d+1 is an affine coordinate of the time axis, T ∼ = R [START_REF] Cartan | Sur les variétés à connexion affine et la théorie de la relativité généralisée[END_REF][START_REF] Trautman | Sur la théorie newtonienne de la gravitation[END_REF][START_REF] Künzle | Galilei and Lorentz structures on spacetime: Comparison of the corresponding geometry and physics[END_REF][START_REF] Künzle | Covariant Newtonian limit of Lorentz space-times[END_REF]. 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,

γ µβ R σ αµρ = γ µσ R β ρµα (2.2)
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.,

Ric = (4πG̺ -Λ)θ ⊗ θ, (2.3)
the connection Γ is interpreted as the gravitational field in a purely geometric generalization of Newtonian gravitation theory [START_REF] Cartan | Sur les variétés à connexion affine et la théorie de la relativité généralisée[END_REF][START_REF] Trautman | Sur la théorie newtonienne de la gravitation[END_REF][START_REF] Künzle | Galilei and Lorentz structures on spacetime: Comparison of the corresponding geometry and physics[END_REF]. See [START_REF] Duval | Bargmann Structures And Newton-Cartan Theory[END_REF] 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 [START_REF] Souriau | Structure des systèmes dynamiques, Dunod; Structure of Dynamical Systems[END_REF]

g =   A B 1 B 0 0 1 b 0 0 1   ∈ Gal(d), (2.5)
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 g R d+1 :   x t 1   →   Ax + B 1 t + B 0 t + b 1   . (2.6)
Infinitesimal Galilei transformations form hence a Lie algebra, gal(d), spanned by all vector fields X on space-time such that

L X γ = 0, L X θ = 0, L X Γ = 0 (2.7)
(see [START_REF] Trautman | Sur la théorie newtonienne de la gravitation[END_REF][START_REF] Duval | Quelques procédures géométriques en dynamique des particules[END_REF] for a generalization to (curved) NC structures); these vector field read

X = ω i j x j + β i 1 t + β i 0 ∂ ∂x i + ε ∂ ∂t , (2.8)
where ω ∈ so(d),

β 0 , β 1 ∈ R d ,
Z =   ω β 1 β 0 0 0 ε 0 0 0   ∈ gal(d) (2.9)
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 [START_REF] Perroud | Projective representations of the Schrödinger group[END_REF][START_REF] Duval | Quelques procédures géométriques en dynamique des particules[END_REF]

g =   A B 1 B 0 0 a b 0 c d   ∈ Sch(d), (2.10)
where

A ∈ O(d), B 0 , B 1 ∈ R d , and a, b, c, d ∈ R with ad -bc = 1. The projective "action" of g ∈ Sch(d) on spacetime R d × R takes the form g R d+1 :   x t 1   →       Ax + B 1 t + B 0 ct + d at + b ct + d 1       (2.11)
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

(d) = (O(d) × SL(2, R)) ⋉ (R d × R d ).
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

M = R d × (R 2 \{0}) /R * (2.12)
fibered above the projective line, RP 1 , as clear from (2.11). This point will be further developed in Section 5.1. See also [START_REF] Duval | Quelques procédures géométriques en dynamique des particules[END_REF]. Note that (2.12) can be recovered by factoring out the homogeneous subgroup generated by rotations, expansions, dilations, and boosts,

M = Sch(d)/H where H = O(d) × Aff(1, R) ⋉ R d , (2.13)
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

M ∼ = (R d × S 1 )/Z 2 .
(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 [START_REF] Duval | Quelques procédures géométriques en dynamique des particules[END_REF][START_REF] Duval | Bargmann Structures And Newton-Cartan Theory[END_REF][START_REF] Duval | Non-relativistic conformal symmetries and Newton-Cartan structures[END_REF], as the group, Sch(d), of all (locally defined) diffeomorphisms g such that

g * (γ ⊗ θ) = γ ⊗ θ & g ∈ Proj(R d+1 , Γ), (2.15)
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

L X (γ ⊗ θ) = 0 & X ∈ proj(R d+1 , Γ). (2.16)
In local terms, we thus require

L X γ αβ θ ρ + γ αβ L X θ ρ = 0 & L X Γ ρ αβ = δ ρ α ϕ β + δ ρ β ϕ α (2.17)
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

X = ω i j x j + κtx i + λx i + β i 1 t + β i 0 ∂ ∂x i + κt 2 + 2λt + ε ∂ ∂t , (2.18)
where ω ∈ so(d), β 0 , β 1 ∈ R d , and κ, λ, ε ∈ R. The Schrödinger dilation (or homothety) generator is, indeed, (1.8) with dynamical exponent z = 2. The Lie algebra sch(d) admits the faithful (d + 2)-dimensional anti-representation X → Z, where

Z =     ω β 1 β 0 0 λ ε 0 -κ -λ     ∈ sch(d) (2.19)
with the same notation as above.

Note that sch(d) is, in fact, the (centerless) Schrödinger Lie algebra. Physical applications also involve a central extension associated with the mass; see, e.g., [START_REF] Duval | Quelques procédures géométriques en dynamique des particules[END_REF][START_REF] Burdet | Cartan Structures on Galilean Manifolds: the Chronoprojective Geometry[END_REF][START_REF] Jackiw | Introducing scaling symmetry[END_REF][START_REF] Duval | Bargmann Structures And Newton-Cartan Theory[END_REF][START_REF] Henkel | Schrödinger invariance and strongly anisotropic critical systems[END_REF][START_REF] Henkel | Phenomenology of local scale invariance: from conformal invariance to dynamical scaling[END_REF][START_REF] Duval | Non-relativistic conformal symmetries and Newton-Cartan structures[END_REF].

A remarkable property of the Schrödinger group, arising as a symmetry group of the classical space of motions of free spinning particle [START_REF] Duval | Non-relativistic conformal symmetries and Newton-Cartan structures[END_REF], and also important for studying supersymmetric extensions [START_REF] Duval | On Schrödinger superalgebras[END_REF], is that it can be faithfully imbedded into the affine-symplectic Lie algebra

Sch(d) ⊂ Sp(d, R) ⋉ R 2d .
(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

L X γ = f γ hence L X θ = g θ (3.1)
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 2/z (d), with dynamical exponent z

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

L X (g -1 ⊗ g) = 0, (3.2)
where g -1 denotes the inverse of the metric g : T M → T * M.

It has been shown [START_REF] Künzle | Covariant Newtonian limit of Lorentz space-times[END_REF] 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

g = c 2 θ ⊗ θ -U γ + O(c -2 ), g -1 = -γ + c -2 U ⊗ U + O(c -4 ), (3.3)
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 (3.2) would be

L X lim c→∞ (c -2 g -1 ⊗ g) = 0, viz. L X (γ ⊗ θ ⊗ θ) = 0. (3.4)
More generally, we consider

L X (γ ⊗m ⊗ θ ⊗n ) = 0, (3.5)
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

z = 2 q where q = n m (3.6)
matches the ordinary notion of dynamical exponent [START_REF] Henkel | Schrödinger invariance and strongly anisotropic critical systems[END_REF][START_REF] Henkel | Phenomenology of local scale invariance: from conformal invariance to dynamical scaling[END_REF][START_REF] Duval | Non-relativistic conformal symmetries and Newton-Cartan structures[END_REF]. 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.

The Lie algebra

sv(M, γ, θ) = cgal 1 (M, γ, θ) (3.7)
is the obvious generalization to Galilei spacetimes of the Schrödinger-Virasoro Lie algebra sv(d) = sv(R × R d , γ, θ) introduced in [START_REF] Henkel | Schrödinger invariance and strongly anisotropic critical systems[END_REF] (see also [START_REF] Henkel | Phenomenology of local scale invariance: from conformal invariance to dynamical scaling[END_REF]) 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 [START_REF] Roger | The Schrödinger-Virasoro Lie group and algebra: from geometry to representation theory[END_REF][START_REF] Roger | A Hamiltonian action of the Schrödinger-Virasoro algebra on a space of periodic time-dependent Schrödinger operators in (1+1)-dimensions[END_REF].

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 [START_REF] Duval | Non-relativistic conformal symmetries and Newton-Cartan structures[END_REF] that X ∈ cgal 2/z (d) iff

X = ω i j (t)x j + 1 z ξ ′ (t)x i + β i (t) ∂ ∂x i + ξ(t) ∂ ∂t , (3.8)
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 [START_REF] Duval | Quelques procédures géométriques en dynamique des particules[END_REF][START_REF] Duval | Non-relativistic conformal symmetries and Newton-Cartan structures[END_REF].

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 [START_REF] Duval | Non-relativistic conformal symmetries and Newton-Cartan structures[END_REF] 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

ξ(t) = κt 2 + 2λt + ε, (3.9)
with κ, λ, ε ∈ R, and we find that a closed Lie algebra of polynomial vector fields of degree N > 0 is obtained provided

z = 2 N . (3.10)
At last, we find that X ∈ cgal Pol N (d) iff

X = ω i j x j + N 2 ξ ′ (t)x i + β i (t) ∂ ∂x i + ξ(t) ∂ ∂t , (3.11)
with ω ∈ so(d), and ξ(t) quadratic as in (3.9), together with

β(t) = β N t N + • • • + β 1 t + β 0 , (3.12)
where β 0 , . . . β N ∈ R d . 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 [START_REF] Henkel | Schrödinger invariance and strongly anisotropic critical systems[END_REF] in his study of scale invariance for strongly anisotropic critical systems (with d = 1), viz.,

cgal Pol N (d) ∼ = alt 2/N (d).
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 [START_REF] Duval | Non-relativistic conformal symmetries and Newton-Cartan structures[END_REF].

4 Conformal Galilei Groups with dynamical exponents z = 2/N 4.1 Veronese curves and finite-dimensional representations of SL(2, R)

A Veronese curve is an embedding Ver N : RP 1 → RP N defined, for N ≥ 1 by Ver N (t 1 : t 2 ) = (t N 1 : t N -1 1 t 2 : • • • : t N 2 ), (4.1)
where (u 1 :

u 2 : • • • : u N +1 ) stands for the direction of (u 1 , u 2 , . . . , u N +1 ) ∈ R N +1 \{0},
that is, a point in RP N . See, e.g., [START_REF] Ovsienko | Projective differential geometry old and new[END_REF].

With a slight abuse of notation, we will still denote by Ver N : R 2 → R N +1 the mapping defined by

Ver N (t 1 , t 2 ) = (u 1 , u 2 , . . . , u N +1 )
where

u k = t N -k+1 1 t k-1 2 (4.2)
for all k = 1, . . . , N + 1. Put t = (t 1 , t 2 ) ∈ R 2 , and consider t * = Ct with

C = a b c d ∈ SL(2, R). (4.3)
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

Ver N (C) m m ′ = min(N -m+1,m ′ -1) k=max(0,m ′ -m) N -m + 1 k m -1 m ′ -k -1 × ×a N -m-k+1 b k c m-m ′ +k d m ′ -k-1 (4.5)
for all m, m ′ = 1, . . . , N + 1. Our mapping provides us with a group homomorphism

Ver N : SL(2, R) → SL(N + 1, R) (4.6)
which constitutes (up to equivalence) the well-known (N +1)-dimensional irreducible representation of SL(2, R); see [START_REF] Knapp | Representation Theory of Semisimple Groups[END_REF]. Let us introduce the sl(2, R) generators

ξ -1 = 0 1 0 0 , ξ 0 = 1 0 0 -1 , ξ 1 = 0 0 1 0 , (4.7)
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

ver N (ξ -1 ) = N +1 n=1 (N -n + 1) u n+1 ∂ ∂u n , (4.8
)

ver N (ξ 0 ) = N +1 n=1 (N -2n + 2) u n ∂ ∂u n , (4.9
)

ver N (ξ 1 ) = N +1 n=1 (n -1) u n-1 ∂ ∂u n . ( 4
g = A B N • • • B 0 0 Ver N (C) (4.12)
where

A ∈ O(d), B 0 , B 1 , . . . , B N ∈ R d , and C ∈ SL(2, R).
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

       x * t N * . . . t * 1        = R * • g        x t N . . . t 1        , (4.13)
which, with the help of (4.3) and (4.5), leaves us with Let us now write any vector in the Lie algebra of CGal N (d) as

x * = Ax + B N t N + • • • + B 1 t + B 0 (ct + d) N , (4.14)
t * = at + b ct + d . ( 4
Z = ω β N • • • β 0 0 ver N (ξ) , (4.17)
where we have used Equations (4.8)-(4.10) with

ξ = λ ε -κ -λ ∈ sl(2, R). ( 4
Z R d+1 = X ∈ cgal N (d), (4.20)
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. • For N = 2, i.e., z = 1, in particular, we get

g =     A B 2 B 1 B 0 0 a 2 2ab b 2 0 ac ad + bc bd 0 c 2 2cd d 2     ∈ CGal 2 (d), (4.23) with A ∈ O(d), B 0 , B 1 , B 2 ∈ R d , a, b, c, d ∈ R with ad -bc = 1.
In addition to the usual space translations B 0 , and Galilei boosts B 1 , we also have extra generators, namely accelerations B 2 [START_REF] Lukierski | Acceleration-Extended Galilean Symmetries with Central Charges and their Dynamical Realizations[END_REF][START_REF] Bagchi | Galilean Conformal Algebras and AdS/CFT[END_REF]. It is an easy matter to check that the actions (1.4), (1.5), and (1.6) of dilations, expansions, and accelerations, respectively, are recovered by considering their projective action given by (4.13) with N = 2.

The Lie algebra cgal 2 (d) of the Conformal Galilei Group, CGal 2 (d), is plainly isomorphic to the (centerless) Conformal Galilei Algebra (CGA), cf. [START_REF] Duval | Non-relativistic conformal symmetries and Newton-Cartan structures[END_REF][START_REF] Lukierski | Acceleration-Extended Galilean Symmetries with Central Charges and their Dynamical Realizations[END_REF][START_REF] Bagchi | Galilean Conformal Algebras and AdS/CFT[END_REF]. It has been shown [START_REF] Lukierski | Acceleration-Extended Galilean Symmetries with Central Charges and their Dynamical Realizations[END_REF] that cgal 2 (d) admits a nontrivial 1-dimensional central extension in the planar case, d = 2, only.

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

Gal + (d) = (SO(d) × R) ⋉ R 2d , (5.1)
and factor out rotations and boosts to yield (ordinary) Galilei spacetime,

R d × R = Gal + (d)/(SO(d) ⋉ R d ). (5.2)
Similarly, one can start instead with a deformation of the Galilei group called Newton-Hooke group [1, 2, 3]

N + (d) = SO(d) × SO(2) ⋉ R 2d , (5.3)
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 [START_REF] Gibbons | g spacetimes, Hpp-waves and the cosmological constant[END_REF]. The latter carries a non-flat nonrelativistic structure and satisfies the empty space Newton gravitational field equations with negative cosmological constant [START_REF] Gibbons | g spacetimes, Hpp-waves and the cosmological constant[END_REF]. 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.,

M N = CGal N (d)/H N where H N = (O(d) × Aff(1, R)) ⋉ R N d .
(5.5)

3 Starting with the group

N -(d) = SO(d) × SO(1, 1) ↑ ⋉ R 2d (5.4)
we would, similarly, end up with a spacetime with positive cosmological constant. Here we will focus our attention to N + (d) in (5.3).

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

       x t N . . . t 1        = R * •        B 0 b N . . . bd N -1 d N        . (5.7)
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

M N = (R d × Ver N (R 2 \{0}))/R * , (5.8)
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

M N ∼ = (R d × Ver N (S 1 ))/Z 2 (5.10)
where Ver N (S 1 ) stands for the image of S 1 by the mapping (4.2). To summarize, we have the following diagram CGal N (d)

H N   M N ∼ = (R d × Ver N (S 1 ))/Z 2 R d ---→ T ∼ = Ver N (RP 1 ) (5.11)
where the horizontal arrow denotes the canonical fibration of spacetime M N onto the time axis T .

Note that for N = 1 the Schrödinger-homogeneous spacetime M 1 , i.e., the Möbius spacetime (2.12), is obtained.

We will from now on focus our attention to the new CGal N (d)-homogeneous spacetimes M N .

Let us lastly provide, for the record, explicit formulae for the projective action (x, ϑ) → (x * , ϑ * ) of CGal N (d) on our Newton-Hooke manifold M N , in terms of the angular coordinate introduced in (5.9). We will use the local polar decomposition of any element in SL(2, R), viz.,

C = cos α sin α -sin α cos α a 0 0 a -1 1 0 c 1 , (5.12)
where α ∈ R/(2πZ) is a "time-translation", a ∈ R * a dilation, and c ∈ R an expansion. Then, Equations (4.14) and (4.15) yield

A : x * = Ax, ϑ * = ϑ, B N : x * = x + B N tan N ϑ, ϑ * = ϑ, . . . . . . . . . B 1 : x * = x + B 1 tanϑ, ϑ * = ϑ, B 0 : x * = x + B 0 , ϑ * = ϑ, α : x * = x cos N ϑ cos N (ϑ + α) , ϑ * = ϑ + α, a : x * = a N x, ϑ * = arctan(a 2 tan ϑ), c : x * = x (c tan ϑ + 1) N , ϑ * = arctan tan ϑ c tan ϑ + 1 , (5.13)
with the same notation as before. These formulae extend those derived before, at the Lie algebraic level, for N = 1, and N = 2 [START_REF] Lukierski | Acceleration-enlarged symmetries in nonrelativistic space-time with a cosmological constant[END_REF][START_REF] Galajinsky | Conformal mechanics in Newton-Hooke spacetime[END_REF]. Those in [START_REF] Galajinsky | Conformal mechanics in Newton-Hooke spacetime[END_REF], for example, are obtained from (5.13) by putting N = 2l and introducing, in view of (5.7) and (5.9), the new coordinates

X = x cos N ϑ, t = tan ϑ.
(5.14)

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 [START_REF] Künzle | Galilei and Lorentz structures on spacetime: Comparison of the corresponding geometry and physics[END_REF].

Given a Galilei structure (M, γ, θ) as introduced in Section 2.1, we define the bundle of Galilei frames of M as the bundle P → M of those frames (x, e 1 , . . . , e d+1 ) such that

d K=1 e K ⊗ e K = γ and θ d+1 = θ, (5.15)
where (θ 1 , . . . , θ d+1 ) is the coframe at x ∈ M. The bundle of Galilei frames is a principal H-subbundle of the frame-bundle of M, with H ⊂ G(= Gal(d)) the homogeneous Galilei group consisting of those matrices

h = A B 1 0 1 , (5.16)
where A ∈ O(d), and B 1 ∈ R d . A local coordinate system (x α ) on M (where α = 1, . . . , d + 1) induces a local coordinate system ((x α ), (e α a )) on P , where e a = e α a ∂/∂x α for all a = 1, . . . , d + 1; with the definition (θ a α ) = (e α a ) -1 , the 1-forms

θ a = θ a α dx α (5.17)
constitute the component of soldering 1-form of P . Let us denote by g (resp. h) the Lie algebra of G (resp. H) so that g = h⋉R d+1 . A Galilei connection is a g-valued 1-form of P such that

ω = (ω a b ) (θ a ) 0 0 , (5.18)
where, with the notation of (2.9),

(ω a b ) = ω β 1 0 0 (5.19)
is an ordinary h-valued connection 1-form on the H-bundle P → M, and

(θ a ) = β 0 ε (5.20)
is the R d+1 -valued soldering 1-form (5.17) of P . 4Then the structure equations provide us with the definition of the associated curvature 2-form ((Ω a b ), (Ω a )) on P , namely

Ω a b = dω a b + ω a c ∧ ω c b , (5.21
)

Ω a = dθ a + ω a c ∧ θ c , (5.22)
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

ω a b = θ a ρ (de ρ b + Γ ρ αβ dx α e β b ), (5.23)
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 1. ω(Z P ) = Z for all Z ∈ h 2. (h P ) * ω = Ad(h -1 )ω for all h ∈ H

ker ω = {0}

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 [START_REF] Duval | Quelques procédures géométriques en dynamique des particules[END_REF].

• The Galilei group Gal(d) can be viewed as the bundle of Galilei frames over spacetime R d × R = Gal(d)/(O(d) ⋉ R d ), cf. (5.2). Using (2.5), we find that the (left-invariant) Maurer-Cartan 1-form Θ Gal(d) = g -1 dg reads

Θ Gal(d) =     ω β 1 β 0 0 0 ε 0 0 0     , (5.24)
where β 0 , and ε are interpreted as the components of the soldering 1-form (5.17 Clearly, the Maurer-Cartan 1-form (5.24) endows the bundle Gal(d) → R d × R with a Cartan connection in view of the above defining properties of the latter. This connection is canonical and flat. Indeed, Equations (5.25, 5.26), specializing (5.21), entail that the connection 2-form (ω, β 1 ) is flat, while Equations (5.27, 5.28), corresponding to (Ω a ) = 0 in (5.22), guarantee having zero torsion. See, e.g., [START_REF] Künzle | Galilei and Lorentz structures on spacetime: Comparison of the corresponding geometry and physics[END_REF][START_REF] Duval | Quelques procédures géométriques en dynamique des particules[END_REF]. We note that, with the standard notation used throughout our article, the Galilei structure is given here by γ = δ ij A i ⊗A j , where A ∈ O(d) represents an orthonormal frame, together with the clock 1-form θ = ε; see (2.5). At last, in view of the general form (5.23) of Galilei connections, there holds Γ ρ αβ = 0 (5.29) for all α, β, ρ = 1, . . . , d + 1, in the spacetime coordinate system (x i = B i 0 , x d+1 = b) provided by the matrix realization (2.5) of Gal(d).

• 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). 6This 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 [START_REF] Duval | Quelques procédures géométriques en dynamique des particules[END_REF][START_REF] Burdet | Cartan Structures on Galilean Manifolds: the Chronoprojective Geometry[END_REF] for a comprehensive description of Schrödinger conformal Cartan connections. Now, cosmologists usually introduce the reduced cosmological constant, λ, in terms of the cosmological constant, Λ, the Hubble constant, H 0 , and the dimension of space, d, via Λ = λH 2 0 d.

(5.50)

In view of (5.49), our model therefore yields a reduced cosmological constant

λ 1 = -1.
(5.51)

• For the case N > 1, the Maurer-Cartan 1-form of CGal N (d) reads

Θ CGal N (d) =                 ω β N β N -1 • • • β 1 β 0 0 Nλ Nε • • • 0 0 0 -κ (N -1)λ • • • 0 0 . . . . . . . . . . . . 0 0 0 0 • • • • • • 2ε 0 0 0 • • • • • • (1 -N)λ ε 0 0 0 0 • • • 0 -Nκ -Nλ                 (5.52)
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

New N (d) = (O(d) × SO(2)) ⋉ R d(N +1) ⊂ CGal N (d).
(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 .

(5.54)

This connection turns out to produce an exact solution of the vacuum NC-field equations (2.3), the reduced cosmological constant at level N being now given by λ N = -N (5.55) for all N = 1, 2, . . ..

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 [START_REF] Ovsienko | Projective differential geometry old and new[END_REF]. 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 [START_REF] Stichel | A new type of conformal dynamics[END_REF], and later extended so as to include also accelerations [START_REF] Lukierski | Acceleration-Extended Galilean Symmetries with Central Charges and their Dynamical Realizations[END_REF].

Newton-Hooke symmetry and spacetime have been considered in nonrelativistic cosmology [START_REF] Bacry | Possible kinematics[END_REF][START_REF] Aldrovandi | Nonrelativistic spacetimes with cosmological constant[END_REF][START_REF] Gibbons | g spacetimes, Hpp-waves and the cosmological constant[END_REF][START_REF] Tian | Mechanics and Newton-Cartan-Like Gravity on the Newton-Hooke Space-time[END_REF][START_REF] Lukierski | Acceleration-enlarged symmetries in nonrelativistic space-time with a cosmological constant[END_REF][START_REF] Arratia | 2+1) Newton-Hooke Classical and Quantum Systems[END_REF]. It has been proved [START_REF] Sakaguchi | Super Galilean conformal algebra in AdS/CFT[END_REF] that the N = 1 Galilean Conformal algebra is isomorphic to the Newton-Hooke string algebra studied in string theory [START_REF] Brugues | Newton-Hooke algebras, nonrelativistic branes and generalized pp-wave metrics[END_REF].

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 [START_REF] Henkel | Local Scale Invariance and Strongly Anisotropic Equilibrium Critical Systems[END_REF][START_REF] Henkel | Phenomenology of local scale invariance: from conformal invariance to dynamical scaling[END_REF].

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). 9An 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, These examples make it plausible that the N-conformal symmetry, CGal N (d), is realized by the higher-order geodesic equations,

d N +1
x dt N +1 = 0. (6.5) Such a statement is suggested by the quantum formulas in [START_REF] Henkel | Local Scale Invariance and Strongly Anisotropic Equilibrium Critical Systems[END_REF][START_REF] Henkel | Phenomenology of local scale invariance: from conformal invariance to dynamical scaling[END_REF] and is consistent, for N = 2, with Ref. [START_REF] Fedoruk | Galilean Conformal Mechanics from Nonlinear Realizations[END_REF]. Another promising approach is [START_REF] Galajinsky | Many-body conformal mechanics with rational dynamical exponent in d = 1[END_REF]. 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 [START_REF] Burdet | Cartan Structures on Galilean Manifolds: the Chronoprojective Geometry[END_REF]. 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.

Gibbons and Patricot [START_REF] Gibbons | g spacetimes, Hpp-waves and the cosmological constant[END_REF] derive the Newton-Cartan structure of Newton-Hooke spacetime in the "Bargmann" framework of Ref. [START_REF] Duval | Bargmann Structures And Newton-Cartan Theory[END_REF]. This null "Kaluza-Kleintype" approach provides us indeed with a preferred way of defining a nonrelativistic structure "downstairs". A similar explanation holds in a uniform magnetic field [START_REF] Burdet | Time-dependent quantum systems and chronoprojective geometry[END_REF][START_REF] Duval | Conformal properties of Chern-Simons vortices in external fields[END_REF][START_REF] Hassaïne | The symmetries of the Manton superconductivity model[END_REF][START_REF] Gibbons | Kohn's Theorem, Larmor's Equivalence Principle and the Newton-Hooke Group[END_REF][START_REF] Alvarez | Anisotropic harmonic oscillator, noncommutative Landau problem and exotic Newton-Hooke symmetry[END_REF][START_REF] Zhang | Kohn's Theorem and Galilean symmetry[END_REF].

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 [START_REF] Henkel | Local Scale Invariance and Strongly Anisotropic Equilibrium Critical Systems[END_REF][START_REF] Henkel | Phenomenology of local scale invariance: from conformal invariance to dynamical scaling[END_REF] 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 [START_REF] Lukierski | Acceleration-Extended Galilean Symmetries with Central Charges and their Dynamical Realizations[END_REF].

Central extensions of the conformal Galilei algebra have been considered in [START_REF] Lukierski | Acceleration-Extended Galilean Symmetries with Central Charges and their Dynamical Realizations[END_REF][START_REF] Alvarez | Anisotropic harmonic oscillator, noncommutative Landau problem and exotic Newton-Hooke symmetry[END_REF] in the planar case, d = 2, and in [START_REF] Negro | Nonrelativistic conformal groups[END_REF] 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 [START_REF] Souriau | Structure des systèmes dynamiques, Dunod; Structure of Dynamical Systems[END_REF] of the latter, likely to unveil new, physically interesting, systems.

. 10 )

 10 One checks that ver N (ξ a ) is, indeed, divergence-free, and[ver N (ξ a ), ver N (ξ b )] = -ver N ([ξ a , ξ b ]) (4.11)for a, b = -1, 0, 1, i.e., that ver N : sl(2, R) → sl(N + 1, R) is a Lie algebra antihomomorphism.

. 15 )

 15 These formulae allow for the following interpretation for the parameters in (4.12), A : orthogonal transformation, B 0 : translation, B 1 : boost , B 2 : acceleration, . . . . . .B N : higher-order "acceleration", C : projective transformation of time.

  8) with z = 2/N. The above definition of the conformal Galilei groups, see (4.12), yield their global structure CGal N (d) ∼ = (O(d) × SL(2, R)) ⋉ R (N +1)d , (4.21) and dim(CGal N (d)) = Nd + 1 2 d(d + 1) + 3. • For N = 1, i.e., z = 2, we recover the Schrödinger group (2.10), and therefore CGal 1 (d) ∼ = Sch(d). (4.22)

  ) of the principal H-bundle Gal(d) → R d × R. Then, the Maurer-Cartan structure equations dΘ + Θ ∧ Θ = 0 read 0 = dω + ω ∧ ω, (5.25)0 = dβ 1 + ω ∧ β 1 ,(5.26)0 = dβ 0 + ω ∧ β 0 + β 1 ∧ ε,(5.27) 0 = dε.(5.28)

2

 2 , preserved due to the following transformation of the accelerationd 2 x * dt * A ∈ O(d).Similarly, the equations of uniform acceleration,d 3 x dt 3 = 0, (6.3)are preserved by the conformal Galilei group, CGal 1 (d), in view of the following transformation law, namely,

  4.2 Matrix realizations of the Conformal Galilei Groups CGal N (d) Just as in the case of the Schrödinger group, see (2.10), we will strive integrating the conformal Galilei Lie algebras cgal N (d) within the matrix group GL(d + N + 1, R). Let us, hence, introduce the Conformal Galilei Group with dynamical exponent z = 2/N cf. (3.10), which we denote by CGal N (d); it consists of those matrices of the form

  .18) 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.,

δx = ωx + β N t N + • • • β 1 t + β 0 + N(κt + λ)x, δt = κt 2 + 2λt + ε. (4.19)

At last, the vector field Z R d+1 = δx i ∂/∂x i + δt ∂/∂t associated with Z in (4.17) is such that

Henkel [7] refers to it as to "Alt 1 ".

Strictly speaking, for the lowest level, N = 1, the vector fields generating cgal 1 (d) are polynomials of degree 2 in the spacetime coordinates, although z = 2 holds true. The higher levels N ≥ 2 duly correspond to the actual degree of the vector fields generating cgal N (d).

The translation components of ω are precisely chosen as those of the soldering 1-form because Galilei connection are assumed to be affine connections.

Since we are dealing here with homogeneous spaces G/H, it will naturally be given by the (left-invariant) Maurer-Cartan 1-form of the corresponding groups G.

The bundle of 2-frames is called upon since the vector fields (2.18) spanning sch(d) are polynomials of degree 2 in the spacetime coordinates.

Unlike in general relativity, no specific parameter such as the de Sitter spacetime radius is available in our nonrelativistic framework; whence this somewhat arbitrary choice of a time unit.

Spacetimes with positive cosmological constant would be obtained by replacing SO(2) by SO(1, 1) ↑ as in (5.4); see also[START_REF] Gibbons | g spacetimes, Hpp-waves and the cosmological constant[END_REF].

Acknowledgments: P.A.H is indebted to the Institute of Modern Physics of the Lanzhou branch of the Chinese Academy of Sciences for hospitality extended to him, and to P.-M. Zhang for discussions. Useful correspondence is acknowledged to A. Galajinsky, J. Gomis, M. Henkel, J. Lukierski, M. Plyushchay, as well as enlightening discussions to S. Lazzarini.

Using the same notation as in (2.19), we find

and, hence, the structure equations 0 = dω + ω ∧ ω, (5.31)

)

Furthermore, a Galilei structure can be introduced on spacetime M 1 viewed as the quotient of (5.3). Our clue is that embedding SO (5.39)

Then, New 1 (d) can readily be identified with the bundle of Galilei frames of

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 [START_REF] Duval | Quelques procédures géométriques en dynamique des particules[END_REF]]

(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

see (5.20), while those of the h-connection write ω = A -1 dA and

see (5.19). From the last expression we infer (exploiting the general form (5.23) of Galilei connections) that the only nonzero components of the Christoffel symbols of the connection are

for all i = 1, . . . , d. This entails that the nonzero components of curvature tensor R 1 , associated with Ω 1 , are (R 1 ) i jϑϑ = δ i j for all i, j = 1, . . . , d. This Galilei connection is clearly a NC-connection since (2.2) holds true.

The only nonvanishing component of the Ricci tensor is therefore

Hence, our NC-connection (ω, β 1 ) provides us with a solution of the NC-field equations (2.3), Ric 1 + Λ 1 θ ⊗ θ = 0, (5.46) 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 This entails that (Ric 1 ) τ τ = H 2 0 d (compare (5.45)), and, hence, provides us with the cosmological constant Λ 1 = -H 2 0 d.

(5.49)