Exploring Lovelock theory moduli space for Schroedinger solutions

We look for Schroedinger solutions in Lovelock gravity in $D>4$. We span the entire parameter space and determine parametric relations under which the Schroedinger solution exists. We find that in arbitrary dimensions pure Lovelock theories have Schroedinger solutions of arbitrary radius, on a co-dimension one locus in the Lovelock parameter space. This co-dimension one locus contains the subspace over which the Lovelock gravity can be written in the Chern-Simons form. Schroedinger solutions do not exist outside this locus and on this locus they exist for arbitrary dynamical exponent z. This freedom in z is due to the degeneracy in the configuration space. We show that this degeneracy survives certain deformation away from the Lovelock moduli space.


Introduction
The Schrödinger solutions belong to a class of solutions to the gravitational equations of motion which asymptotically do not preserve the Lorentz symmetry. They, however, do respect some non-relativistic symmetries. The deviation from the relativistic symmetry is parametrized by the Schrödinger scaling exponent z, or the dynamical exponent.
The Schrödinger solution was first obtained by Son [1] as well as Balasubramanian and Mcgreevy [2]. They assumed the stress tensor consisting of the cosmological constant term and the pressure-less dust. The Schrödinger solution possesses the Galilean boost invariance by assigning a specific transformation property to one of the light-like directions [1,2](nonrelativistic metrics in higher derivative gravity were discussed in [3]).
In this note we analyse this question in more detail by spanning the entire coupling parameter space of Lovelock theories in various dimensions. Up to four space-time dimensions, the Lovelock action is identical to the Einstein Hilbert action with the cosmological constant, but from five dimensions onwards the Lovelock action has additional Gauss-Bonnet term in the action. This term can be added in four dimensions as well but being total derivative term it does not contribute to the dynamics. In five and higher dimensions the Gauss-Bonnet term does contribute to the dynamics. Similarly the cubic order Lovelock term can be added from six dimensions onwards but it contributes to dynamics only from seven dimensions onwards.
We will show that the Schrödinger metric is generically not a solution to the Lovelock equations of motion, however, it exists as a solution on a co-dimension 1 locus in the Lovelock coupling space. We show that the Schrödinger solution exists precisely on the same locus on which the Lifshitz solution is known to exist 1 . In our computation we restrict ourselves to the Lovelock terms up to cubic order in the curvature tensor but we generalize our analysis to arbitrary dimensions. The co-dimension 1 locus on which we get the Schrödinger solution is interesting from another point of view. It is known that the Lovelock theories can be written in terms of the parity preserving Chern-Simons theory. However, this representation exists only for specific values of the Lovelock couplings. The Chern-Simons formulation exists at a point on this co-dimension 1 locus on which we find the Schrödinger solutions. We present these solutions in the Chern-Simons gauge field forms as well.
The Schrödinger solutions are relevant from the point of view of application to holographically dual condensed matter physics systems. It then naturally raises a question of relevance of these higher dimensional solutions to 2+1 and 3+1 dimensional condensed matter systems. In this regard it is worth pointing out that unlike the AdS and Lifshitz holography which relates D dimensional theory of gravity to D − 1 dimensional field theory, the Schrödinger holography relates D dimensional theory of gravity to D − 2 dimensional field theory. Therefore, 4 + 1 and 5 + 1 dimensional Lovelock theories are relevant to 2 + 1 and 3 + 1 dimensional boundary physics. Higher dimensional theories can be dimensionally reduced to lower dimensional theories. Such higher dimensional theories typically give rise to scalar-tensor theories 1 For closely related solutions of Kasner type in the Lovelock theory, see [4]. of gravity which are either referred to as Galileon or Horndeski theories [5][6][7][8]. For example, let us consider D = d + n + 1 dimensional theory of gravity with the cosmological constant, the Einstein-Hilbert, the Gauss-Bonnet term where, L 2 is the Gauss-Bonnet term. We will dimensionally reduce it down to d+1 dimensions by using an n-dimensional compact manifoldK n such that This is a simple but consistent diagonal toroidal compactification which give rise to one extra scalar degree of freedom, that is the size of the internal space. All terms with a tilde refers to internal n dimensional space, while terms with a bar refers to the d + 1 dimensional space-time. As we integrate out the internal space the effective action looks like [9] where,Ḡ is d + 1 dimensional Gauss-Bonnet term,G is n dimensional Gauss-Bonnet term andḠ µν is the Einstein tensor of d + 1 dinemsional space. The effective action written above can be related to the so called Galileon action, with the Galileon field is realized as the scalar parametrising the volume of the internal space. As we have reduced from the Einstein Gauss Bonnet action, in the reduced action all the terms are up to quartic order in derivatives of the metric or the scalar or of both, but the equations of motion following from it will still be of second order. The term of the form (∂φ∂φ)∇ 2 φ is often called the DGP term [10] appearing in the decoupling limit of the DGP model, and the term of the form (∂φ∂φ) 2 is the standard Galileon term [5]. As we will argue that the Schrödinger solution exists in a co-dimension 1 subspace of Lovelock moduli space. If we restrict to only the Gauss-Bonnet term then it exists on a subspace which relates a 2 to Λ. We will get back to the issue of dimension reduction in this context in the discussion section. As long as n ≤ 2, neither the DGP term nor the Galileon term appear in the dimensionally reduced theory. In addition for Ricci-flat compact spaces the lower dimensional action, up to the addition of higher derivative curvature terms, has a familiar form.
This note is organized in the following manner. We will first give basics of the Lovelock theory in arbitrary dimensions. Most of the information in this section is not new but is useful to fix the notation. In the next section we will look at various solutions to the Lovelock equations of motion. It is well known that the AdS solution generically exists for arbitrary values of Lovelock couplings in any dimension. This feature is not shared by the Schrödinger solution. We present the solution for general value of D and in explicit form for dimensions D = 5, 6, 7. We also comment on the solutions with anisotropic scaling in spatial direction and their relation to AdS×R type solutions. In section 4, we analyse branches of the AdS solution [11,12]. Our interest in presenting this result is to emphasize that the non-relativistic solutions exists only when the discriminant vanishes. Degeneracy in the configuration space [13,14] has been well studied for Lifshitz solutions [15,16]. We show that this degeneracy is responsible for unconstrained z for Schrödinger case as well. Unlike Lifshitz, in the case of Schrödinger solutions this degeneracy extends beyond the Lovelock moduli space. In this sense our results provide a template for a dynamical exponent z for which all values are equally likely. Any suitable value of z then can be obtained by either appropriately modifying the couplings in this theory or by adding new interactions.
It is known that the Lovelock theory in odd space-time dimensions can be written in the Chern-Simons form and in even space-time dimensions in the Born-Infeld form exactly when the discriminant vanishes [11]. We discuss the relation between vanishing discriminant and locus of non-relativistic solutions in section 5 and write down the Schrödinger solution in the Chern-Simons gauge field form in odd space-time dimensions and in the Born-Infeld gauge field form in even space-time dimensions. Finally, we point out the relation with the causality and stability constraints obtained in the Lovelock theories in higher dimensions [17][18][19].

The Lovelock Gravity Theory
Let us consider following action where G is the D dimensional Newton's constant, a p are coupling constants with a 0 = −2Λ = (D − 1)(D − 2)/ 2 , a 1 = 1, a 2 is the Gauss-Bonnet coupling etc. 2 , and L p are terms in the Lagrangian density of the Lovelock action, where δ ν 1 ν 2 ···ν 2p µ 1 µ 2 ···µ 2p is totally antisymmetric product of 2p Kronecker deltas normalized to take values 0 and ±1, and hence is completely antisymmetric in all its upper and lower indices separately. It can also be considered as the determinant of an (2p × 2p) matrix whose (ij)-th element is given by δ ν i µ j . We are using notation of [11] and the equation of motion can be written in the compact form as In this way of writing the Lovelock terms make is obvious that up to D = 4 only relevant couplings are a 0 and a 1 . The term with the coupling a 2 is topological in D = 4. In D = 5, 6 the Gauss-Bonnet term is dynamical. In these dimensions we will explore the parameter space spanned by a 0 and a 2 to find the range of values for which the Lifshitz solution is possible. In D = 6, the term with coupling a 3 can be written down but like the Gauss-Bonnet in D = 4, this terms is topological in D = 6 and hence does not affect the equation of motion. However, this term becomes relevant in D > 6. We will explore the three dimensional parameter space spanned by a 0 , a 2 and a 3 and find conditions for Lifshitz solutions.
We will start with the study of solution to the D = 5 equations of motion. We therefore write the action of pure gravity in D-dimensions as where Λ is the cosmological constant and L hd is the Lagrangian for the higher derivative terms of the Lovelock form.
As mentioned earlier in D = 5 space-time dimensions the higher derivative Lagrangian density contains the quadratic Lovelock term, also known as the Gauss-Bonnet term, which appears in the Lagrangian with the coupling constant, a 2 , L hd =a 2 L 2 , where, For D = 7 space-time dimensions we will have, in addition to the Gauss-Bonnet term, the cubic Lovelock term L 3 , where a 3 is the coupling constant of the cubic Lovelock term. The cubic term is explicitly written as The equation of motion that follows from here are as written below, where the explicit forms of G µν , G µν and G µν are given in appendix A.1. The equations of motion in D = 5 are obtained by setting a 3 = 0 in (2.8). We will now study specific solutions to the equations of motion.

Solutions to the Lovelock Gravity
In this section we will analyse solutions to the Lovelock gravity equations of motion. In Appendix A.2 we will summarise known results about the AdS and Lifshitz solutions. In particular the AdS solution is possible for generic values of the Lovelock couplings a 2 , a 3 , etc. On the other hand the Lifshitz solutions exist only on the co-dimension one subspace. As we will see below the Schrödinger solutions can be obtained only on the same co-dimension one locus in the parameter space. Furthermore, as we will see, on this locus the solution admits arbitrary dynamical scaling exponent z, which is also true for the Lifshitz solutions.

The Schrödinger Solution
We consider the Schrödinger space-times as solutions of the higher derivative Lovelock gravity theories. This is an example of non-relativistic space-time, known apart from the Lifshitz solution. We will look for the Schrödinger solution to the Lovelock gravity equations of motion in arbitrary dimensions but we will restrict to terms cubic in curvature tensor. Generalization to higher order Lovelock terms is tedious but straightforward.
The metric ansatz for the Schrödinger solution looks like Note that this metric for the Schrödinger solution also has two parameters, z which is the Schrödinger exponent and L sch which is the "Schrödinger radius". We will first state the results for arbitrary D but restricting upto cubic Lovelock terms and then write down explicit expressions for D = 5, 6, 7.
The Schrödinger space-time solution in general D dimensions exists subject to following two constraints, The dynamical exponents z is unconstrained. If we eliminate L sch , then it gives one relation between the parameters in the Lovelock action. Thus the Schrödinger solutions exist on co-dimension one subspace of the Lovelock moduli space.
In D = 5 space-time as we have the Gauss-Bonnet term in the Lovelock action besides the Einstein-Hilbert and the cosmological constant term. The constraint (3.2) corresponds to The non-zero components of the Ricci tensor and the Ricci scalar R for the metric of Schrödinger space-time are given by R tt = 2 ( and R = −20/L 2 Sch . In D = 6 space-time the Gauss-Bonnet term is important but the curvature cubed Lovelock term being a total derivative is not. The constraint eq.(3.2) becomes Λ = −5/L 2 Sch and a 2 = L 2 Sch /12. We again write down the components of the Ricci tensor R tt = (2z 2 + z + 2) r −2z , R tξ = R rr = R x i x i = −5/r 2 and the Ricci scalar R = −(30/L 2 Sch ). In D = 7 space-time apart from the the Gauss-Bonnet term, the cubic order Lovelock term will also be important and hence the action will contain three parameters, Λ, a 2 and a 3 . The constraint (3.2) in this case takes the form The Ricci tensor components are R tt = 2 (

Other Solutions
Lifshitz solutions 3 , where the time coordinate (t) scaled differently compared to the other coordinates, are known to be solutions of Lovelock gravity. As an alternative one can consider a different version of the Lifshitz solutions where instead of the time coordinate one of the spatial coordinates may scale differently compared to others. We call it "spatial Lifshitz" space-time. More specifically, we take in D = 5 dimensions the following metric It is obvious from the metric in eq.(3.5) that the x 3 coordinate scales differently compared to the other coordinates with a Lifshitz exponent parametrized by z.
Following the similar procedure as in the previous two subsections, we can obtain this "spatial Lifshitz" as a solution to the Lovelock action eq.(2.4). In D = 5, for the Gauss-Bonnet case eq.(2.5), we find the solution to be identical to both the Schrödinger case eq.(3.3), and the Lifshitz case eq.(A.9). Similarly, for D = 7 dimensions in the cubic Lovelock theory we find this "spatial Lifshitz" as a solution, again, identical to both the Schrödinger case eq.(3.4), and the Lifshitz case eq.(A.10). In all these cases, this solution exists at the same point in the coupling space at which the Schrödinger and Lifshitz solution exist. Thus we see that the special point continues to be relevant as long as one direction, whether spatial or temporal, has anisotropic scaling property. In case of multiple anisotropic directions also one can show that such solutions exist at special points in the coupling space but generically this point is different from the one under consideration.
It is also worth mentioning that the situation here is analogous to what happens in the case of Schrödinger solutions discussed earlier, namely the dynamical exponent z remains unconstrained for this solution as well. This, in particular, allows us to consider a special case when the dynamical exponent for the "spatial Lifshitz" solution vanishes, i.e., z = 0. A vanishing z in eq.(3.5) corresponds to the metric in the x 3 direction being invariant under scaling of the radial coordinate. Since the metric in the directions transverse to x 3 is simple AdS metric in the Poincaré coordinates, we obtain a AdS 4 × R solution of the form This kind of anisotropic solution was studied earlier in [20], where additional matter in the form of a linearly varying dilaton was coupled to two derivative Einstein gravity with negative cosmological constant to construct such anisotropic solutions.
Finally we would like to mention that Lovelock equations of motion do not support black brane solutions with either Lifshitz or Schrödinger asymptotic.
We illustrate this in five dimensions by considering a finite temperature metric ansatz for the Lifshitz solutions of the following form

The Phase-Space of Solutions
In this section we analyse the parameter space of the higher derivative theory and understand in some more detail the phase space of the solutions we obtained in the previous section. We are working with the action eq.(2.4) and the number of parameters of the theory depend on D. For D ≤ 4, the cosmological constant Λ is the only parameter. For D = 5, 6 we have Λ and the Gauss-Bonnet coupling constant a 2 and from D = 7 onwards we also have a 3 the cubic term coupling. While AdS solutions is parametrized only by its radius L AdS , the Schrödinger and Lifshitz solution has two parameters, radius L Sch , respectively L Lif and the dynamical exponent z. Note that the Schrödinger solutions obtained in the last section and the Lifshitz solutions exist on the locus which does not pass through the origin of the Lovelock moduli space. Thus these solutions would cease to exist if we turn off higher derivative couplings and they cannot be obtained perturbatively in Lovelock couplings.

The Phase-Space of Solutions in D = 5
The AdS solution in D = 5 is written in eq.(A.5). One can re-express it as L AdS being determined in terms of Λ.
which indicates that there are two branches for the AdS solution with different AdS-radius. This AdS solution exists when the term within the square root is non-negative There is one more constraint on the parameters for the existence of AdS solution coming from the demand that L 2 AdS > 0, which is a 2 > 0. The two branches of the AdS solutions meet at a point in the 2-dimensional phase-space spanned by the parameters {Λ, a 2 }, given by Λ = −3/(4a 2 ). The Schrödinger solution and the Lifshitz solution in D = 5 eq.(3.3) and eq.(A.9) exist at this point in the phase-space with arbitrary value of the dynamical exponent z. We will discuss the relation between unconstrained z and the degeneracy of the configuration space later in this section.

The Phase-Space of Solutions in D = 7
The AdS solution in D = 7 dimensions is given in eq.(A.6). The AdS radius squared L 2 AdS can be expressed in terms of the parameters Λ, a 2 and a 3 . From eq.(A.6) it is easy to see that one has to solve a cubic equation for L 2 AdS and the solutions are where (4.4) One can see that the last two roots are complex conjugate of each other. Now demanding that the term within the square root in eq.(4.4) vanishes, i.e., and also using Λ = 1 These relations are same as those encountered in our study of Schrödinger and Lifshitz solutions in D = 7. For these values, the imaginary parts of the second and third roots in eq.(4.3) vanish and they become equal, whereas the first root remains different, Interestingly, there is yet another choice which simplifies solutions to the cubic equation.
Consider the choice a 2 = −5/(12Λ), and set the discriminant equal to zero then the quantity s in (4.4), vanishes and all three roots of the cubic become equal to L 2 AdS = −5/Λ This point is on the co-dimension one locus and admits solutions of Schrödinger and Lifshitz kind.

Degeneracy of the Configuration Space
We will now take up the issue of unconstrained dynamical exponent in the Lifshitz and the Schrödinger metrics in eq.(A.7) and eq.(3.1)respectively. It is known in the literature, see [13,14], that in pure Lovelock theories, unconstrained dynamical exponent z of the Lifshitz solutions follows from the existence of degeneracy of the configuration space. The degeneracy of the configuration space corresponds to complete arbitrariness in specifying the metric component g tt (r) on this locus. Since this metric component is completely unconstrained, it naturally follows that the dynamical exponent is not constrained. While this result is known for the Lifshitz metrics, we believe our results for the Schrödinger metrics are new. More specifically, in D = 5 space-time dimensions, the statement of degeneracy in configuration space amounts to the fact that a metric ansatz of the following form for the Schrödinger geometry The degeneracy of the configuration space was studied mostly in the context of the Chern-Simons representation of the Lovelock theory. Since the Chern-Simons action does not have any free parameters, this representation exists only at a point in the Lovelock moduli space. However, in dimensions D > 6 the Lifshitz and Schrödinger solutions exist on a subspace which extends way beyond the Chern-Simons point. In order to understand the relation between the degeneracy of the configuration space and the special locus better, we deform the five dimensional Lovelock theory by adding R 2 and R µν R µν terms to it. Since neither Lifshitz nor Schrödinger solution exist in the Lovelock moduli space away from this locus, only way to study dependence of the degeneracy on the couplings is to expand the coupling constant space by adding new terms.
Let us deform the Lovelock action in D = 5 (2.4) by adding R 2 and R µν R µν terms. This deformation of the Lovelock theory is given in terms of two parameters b 1 and b 2 , We are now considering the most general action of gravity up to quadratic order in curvatures in D = 5. This action now contains 4 parameters, Λ, a 2 , b 1 and b 2 .
The Lifshitz solution, eq.(A.7), as is known in the literature [14], occurs at (4.10) When b 1 = b 2 = 0 we get back eq.(A.9), the Lifshitz solution exists for non-zero b 1 , b 2 but with fixed dynamical exponent z, determined by the parameters of the theory. This agrees with [14] that the degeneracy in the configuration space for the Lifshitz solution occurs only at Interestingly, the Schrödinger solution with the metric ansatz, eq.(3.1) in D = 5, for a general theory of higher derivative gravity beyond Gauss-Bonnet theory with action eq.(4.9), is obtained as (4.12) We recover eq.(3.3), as expected, when we put b 1 = b 2 = 0. But, it is interesting to notice that in eq.(4.12), we get a solution with unconstrained z when b 2 = 0 but b 1 = 0. Which in turn means, if we go beyond the Gauss-Bonnet theory and deform it with only R 2 term, but with no R µν R µν term, we still obtain a Schrödinger solution with arbitrary dynamical exponent. It is then natural to ask, if the degeneracy of the configuration space is still present at this locus in parameter space beyond Gauss-Bonnet point, and it indeed turns out to be true. More specifically, a Schrödinger metric with the ansatz is a solution to the equations of motion obtained from the action in eq.(4.9) with b 2 = 0, for with arbitrary f (r). We thus conclude that the degeneracy in configuration space and the solutions with arbitrary dynamical exponent belong to the same locus on the parameter space. The special locus in Gauss-Bonnet theory on which both Lifshitz and Schrödinger solutions co-exist also a Chern-Simons description but the degeneracy of the configuration space of Schrödinger solution is neither confined to Chern-Simons description nor to the Lovelock subspace. Although, we have carried out the study of degeneracy of the configuration space in D = 5 for Gauss-Bonnet theory and its deformation to more general quadratic curvature theories, similar analysis can be done in D > 5 dimensions.

Lovelock Gravity as AdS Chern-Simons Gravity and Born-Infeld Gravity
The Lovelock theory has the property that the action has general covariance and the field equations contain at most two derivatives of the metric. We parametrize the Lovelock theory using a set of real coefficients a p , p = 0, 1, · · · , [D/2] which are coupling constants of the higher derivative terms. It is convenient to adopt the first order approach, with the dynamical variables being the vielbein, e a = e a µ dx µ , and the spin connection, ω ab = ω ab µ dx µ , obeying first order equations of motion. It is straightforward to solve the vanishing of the torsion for the connection and eliminate them by writing them in terms of the vielbeins to obtain the standard second order form in terms of metric.
The action is constructed as a polynomial of degree [D/2] in R ab = (1/2)R ab µν dx µ ∧ dx ν and Imposing the condition that the theory possesses maximum possible degrees of freedom determines all Lovelock couplings in terms Λ and G N . The action in odd dimensions can then be written as a Chern-Simons action with AdS, dS or Poincaré symmetry [21][22][23], and in even dimensions as a Born-Infeld like action [24] 4 .

The Chern-Simons Theory
It is well known that gravity in (2 + 1) dimensions can equivalently be written as a Chern-Simons theory for the gauge groups ISO(2, 1) or SO(2, 2), but with no propagating bulk degrees of freedom. In higher dimensions, D = 2n − 1, n ≥ 2, the essential idea for constructing a Chern-Simons theory is to utilize the fact that there exists a 2n-form in D = 2n, This form is closed, i.e. dQ 2n = 0, where A is the Lie Algebra valued connection 1-form A = A a µ T a dx µ and F = dA + A ∧ A is the corresponding field strength or curvature 2-form, with T a being the generators of the Lie algebra g of the gauge group G [25]. The fact that Q 2n is closed leads to the existence of a (2n − 1)-form L 2n−1 CS such that which can always be solved as with α being some arbitrary closed (2n − 1)-form. This way one constructs a Chern-Simons Lagrangian L 2n−1 CS (A) in D = 2n − 1 dimensions with an action

Connection with the Lovelock Gravity
In odd dimensions, i.e., D = 2n − 1, it was argued that the requirement of having maximum possible degrees of freedom fixes the Lovelock coefficients as [25] leaving the action depending on only two parameters, gravitational constant κ and the cosmological constant Λ 5 . The precise connection of Lovelock theories with Chern-Simons gravity theories in odd dimensions (D = 2n − 1) is that the Lagrangian for the Lovelock theory can be cast as a Chern-Simons theory for the group AdS. This can be demonstrated through the packaging of the Lovelock vielbeins e a and connections ω ab in the following connection 1-form as where the indices a, b = 1, · · · , D and A, B = 1, · · · , D + 1. Note that the A, B-indices are raised or lowered with respect to the AdS metric This connection defines a curvature 2-form, also called the AdS curvature, as where R ab = dω ab + ω a c ∧ ω c b is the curvature 2-form for the 1-form ω ab , which is (2n − 1)dimensional and not to be confused with the 2n-dimensional AdS curvature 2-form F AB . T a is the torsion form and setting it to zero corresponds to imposing torsion-free constraint.
Next, using the invariant tensor for the AdS group A 1 ···A 2n along with the 2n-dimensional AdS curvature F AB one constructs the Euler form in 2n-dimension Note that L is the length parameter related to cosmological constant as Λ = ± (D−1)(D−2) Using the Bianchi identity for the AdS curvature F AB we can show that this Euler density is closed, dE 2n = 0, and eq.(5.9) one can write the (2n − 1)-form L 2n−1 CS in terms of the (2n − 1)-dimensional curvature 2-form R ab and the vielbeins e a such that a p a 1 ···a D R a 1 a 2 · · · R a 2p−1 a 2p e a 2p+1 · · · e a D , (5.11) and dL 2n−1 CS = E 2n . The coefficients a p are completely fixed here due to the relation between the Chern-Simons density and the Euler density and they turn out to be exactly same as those in eq.(5.6). The field equations obtained from the action in eq.(5.5) are a 1 a 2 a 3 ···α 2n−1 F a 2 a 3 · · · F a 2n−2 a 2n−1 = 0, a 1 a 2 a 3 ···α 2n−1 F a 3 a 4 · · · F a 2n−3 a 2n−2 T a 2n−1 = 0.

Even Dimensions: Lovelock Gravity as Born-Infeld Gravity
As we have seen in odd dimensions there are gravity actions which are invariant not just under Lorentz group but also under some its extensions, e.g. AdS group SO(D − 1, 2). On the contrary, this is not possible in even dimensions, D = 2n. However the requirement of having maximum possible number of degrees of freedom fixes the Lovelock coefficients as [25] The Lovelock action depends on two constants only, the gravitational constant and the cosmological constants, and the Lagrangian, given in eq.(5.1), becomes L = κ 2n a 1 ···a D F a 1 a 2 · · · F a D−1 a D (5.14) which is pfaffian of the two form F ab = R ab + e a e b L 2 and can be cast in Born-Infeld form [25] It is important to note that the two forms F ab are no longer a part of any AdS curvature. The field equations in even dimensions take the form (5.16)

Schrödinger Space-Time as a Solution to Chern-Simons Gravity in D = 5 Dimensions
We will now explicitly show that the Schrödinger solution obtained earlier from the Lovelock action can also be seen as a solution to the Chern-Simons gravity in odd dimensions. We will work in D = 5 dimensions. The metric in 5-dimension looks like We make the following choice for vielbeins corresponding to the metric in eq.(5.17), The spin connections ω ab and the AdS curvature F AB for the Schrödinger metric are listed in appendix A.3. It is easy to see that the AdS curvature F AB does indeed satisfy the field equations eq.(5.12).

Relation with Causality and Stability Constraints
Stability analysis of Lovelock theories in higher dimensions has been carried out in the past [17][18][19]. These studies derive constraints on the values of Gauss-Bonnet coupling (a 2 ) and the cubic Lovelock coupling (a 3 ) by demanding causality and stability condition on the solutions of the Lovelock theory in D = 7. These two conditions are satisfied in a region in the neighbourhood of the origin of the (a 2 , a 3 ) plane and at an isolated point, which in our choice of normalization corresponds to (a 2 = L 2 /36, a 3 = L 4 /648). The Lovelock parameters (a 2 , a 3 ) used in this paper are related to the parameters (β 2 , β 3 ) or (λ 1 , λ 2 ) used in [17] in the following way a 2 = β 2 L 2 = (λ 1 /12)L 2 and a 3 = β 3 L 4 = (λ 2 /72)L 4 . In terms of these parameters the isolated point, mentioned above, corresponds to (λ 1 = 1/3, λ 2 = 1/9) 6 . It is interesting to note that this isolated apex point in the phase diagram, see figure 1 in [19], is also the same point where we have the Chern-Simons representation for the Lovelock theory. The Schrödinger and Lifshitz solutions exist only at this point in the Lovelock coupling space, which presumably also implies that they also satisfy the causality and stability constraints. It would be interesting to check this explicitly for these solutions.

Discussion
We studied the coupling constant parameter space of Lovelock gravity theories in arbitrary dimensions, while restricting our analysis to the Lovelock terms up to cubic in curvatures. We demonstrated that Schrödinger solutions exist on co-dimension 1 subspace in the parameter space. Similar results for Lifshitz solutions already exist in the literature. Interestingly, both the solutions exist on the same locus. We found that on this locus, both Schrödinger and Lifshitz exponents were completely unconstrained. Even if we couple the Maxwell or Yang-Mills fields to these Lovelock theories, the Schrödinger and the Lifshitz moduli space would continue to be the same co-dimension 1 subspace.
As already mentioned earlier in the introduction, Schrödinger holography relates a theory of gravity to field theories living on a co-dimension 2 subspace. Therefore, D = 5 and D = 6 dimensional Schrödinger geometries are directly relevant for studying field theories with this symmetry in 2+1 and 3+1 dimensions. However, higher dimensional Schrödinger geometries in D ≥ 7 dimensions are also relevant for studying field theory systems in lower dimensions, since the higher dimensional Schrödinger space-times can be first dimensionally reduced to lower dimensions and then we can analyse those dimensionally reduced theories. As was pointed out in eq.(1.3), the dimensional reductions do generally lead to effective theories in the lower dimensions similar to Galileon type theories. In fact, in the case when n = 1 in eq.(1.3), that is starting from D = d + 2 dimensions we come down to D = d + 1 dimensions, the effective action takes even simpler form and for the case when n = 2, with simple toroidal reduction it becomes Since these effective actions are obtained are obtained by dimensionally reducing the higher dimensional theories, any solution of the higher dimensional theories continues to solve the equations of motion obtained from the reduced action.
We have seen in section 3.1 that for a specific value for the Gauss-Bonnet coupling constant a 2 , given in eq.(3.3), we obtain a Schrödinger solution in D = 5 dimensions with an unconstrained exponent z. Similarly, starting in D = 7 dimensions and for the sake of simplicity allowing only the Gauss-Bonnet term, that is assuming a 3 = 0, we can perform a toroidal compactification over a 2 dimensional internal manifold and the resulting effective theory in D = 5 dimension comes with action given in eq.(6.2). The Schrödinger solution in D = 7 dimension, given in eq.(3.4) with a 3 = 0, also becomes a solution to the dimensionally reduced effective theory eq.(6.2), with unconstrained dynamical exponent z for these particular values of the parameters Λ and a 2 . Therefore, from the view point of effective lower dimensional theories with the action motivated by the dimensional reductions from higher dimensional theories, we can take a phenomenological bottom-up approach to study various aspects of field theoretical systems with Schrödinger symmetries. In this regard, our study in this paper provides a template for studying non-relativistic field theories with arbitrary dynamical exponent via holography, in theories of gravity coupled to matter systems through non-trivial but specific values of couplings of the Galileon terms, as dictated by eq.(3.3) and eq.(3.4). For n ≤ 2 there is further simplification because in that case neither the DGP term nor the Galileon type terms appear in the d + 1 dimensional theory as is evident from eq.(6.1) and eq.(6.2). Starting from these dimensionally reduced theories, one can then deform them appropriately by adding new terms to the action and engineer non-relativistic solutions with particular fixed values of the dynamical exponent z. Analysis of these deformations leading to specific values of z relevant for application to, say, the condensed matter systems is beyond the scope of this investigation.
We also pointed that the co-dimension 1 subspace of the Lovelock theories on which Schrödinger and Lifshitz solutions exist also supports Chern-Simons formulation in odd spacetime dimensions and Born-Infeld formulation in even space-time dimensions. We cast our non-relativistic metric in the gauge connection form suitable for these formulations. At this point it is interesting to note that [22,24] these gauge connection formulations have natural super-symmetric extension. It would be interesting to explore super symmetric nonrelativistic solutions in the Lovelock theories.
It is known that the Chern-Simons point in the coupling space of the Lovelock theories is maximally symmetric. However, most of the studies at this point are concentrated either on the AdS type solutions or on the black brane solutions. Neither of these solutions can shed direct light on the possible values of the dynamical exponent z that appears in the Schrödinger or Lifshitz solutions. Unconstrained dynamical exponent is related to the degeneracy of the configuration space [13,14], which was studied in the context of Chern-Simons formulation. We studied modification of the Gauss-Bonnet theory by doing general deformation using terms quadratic in curvature and found that the degeneracy of the configuration space in case of the Schrödinger solution is not confined to the Chern-Simons point but extends in the direction orthogonal to the Lovelock moduli space corresponding to deformation by the Ricci scalar squared term. Thus the unconstrained dynamical exponent is a result of the degeneracy and it has weak dependence on the special locus in the Lovelock moduli space in the case of the Schrödinger solutions.
Another point worth mentioning is that we do not find hyper-scaling violating solutions anywhere in the Lovelock moduli space, nor do we find black brane solutions with either Schrödinger or Lifshitz type scaling. This in turn means it is harder to turn on temperature in these geometries, however, these shortcomings can be remedied by deforming away from the Lovelock moduli space.
is the standard Einstein tensor coming from L 0 . The term G (2) µν , coming from L 1 is given by and, the third term, G µν , comes from the cubic term L 2 of the Lovelock action,

A.2 AdS and Lifshitz Solutions in Lovelock Gravity
Here we will list AdS and Lifshitz solutions to the Lovelock theories in various dimensions. These result are presented here so that we can compare them with the Schrödinger solutions in the main text. The metric ansatz for the AdS solution is where L AdS is the AdS radius. This condition can be inverted to write L AdS in term of the couplings in the Lagrangian.
We will now consider Lifshitz solutions to the Lovelock equations of motion. We consider the metric ansatz for the Lifshitz space-time as follows 7 The parameter z in eq.(A.7), which in principle can take any real value, is called the Lifshitz exponent. Asymptotic symmetries are non-relativistic whenever z = 1. The parameter L Lif is Lifshitz radius. The Lifshitz solutions are parametrized by the set of parameters {z, L Lif }.
The Lifshitz solution in D dimensional cubic Lovelock theory exists if [14,29](see also [30]) As in five dimensions, we do not get any condition on z. These values of Λ and a 2 are also related to the locus in the parameter space at which three AdS branches merge.

A.3 Spin Connections, and F AB for Schrödinger Solutions
In this appendix we will follow the discussion of section 5.1 and 5.2 to compute the spin connection 1-form ω ab by solving the torsion-free condition T a = de a + ω a b ∧ e b = 0. Using the connection we will then compute the corresponding curvature 2-form R ab and finally the AdS curvature 2-form F AB defined in eq.(5.9) for the Schrödinger solutions discussed earlier. 7 The general Lifshitz metric has g tt = −L 2 Lif /r 2(D−2)(z−1) D−2−θ +2 and g rr = L 2 Lif r 2θ D−2−θ −2 where, θ is the hyper-scaling violating exponent [26][27][28]. These solutions for non-zero θ do not exist in Lovelock theories.
For Schrödinger solutions in D = 5 we consider the vielbeins as given in eq.(5.18) and