Supersymmetric non-relativistic geometries in M-theory

We construct M-theory supergravity solutions with the non-relativistic Schrodinger symmetry starting from the warped AdS_5 metric with N=1 supersymmetry. We impose the condition that the lightlike direction is compact by making it a non-trivial U(1) bundle over the compact space. Sufficient conditions for such solutions are analyzed. The solutions have two supercharges for generic values of parameters, but the number of supercharges increases to six in some special cases. A Schrodinger geometry with SU(2)xSU(2)xU(1) isometry is considered as a specific example. We consider the Kaluza-Klein modes and show that the non-relativistic particle number is bounded above by the quantum numbers of the compact space.


Introduction
AdS/CFT correspondence has made a remarkable development in the past decade [1][2][3]. The correspondence between N = 4 super Yang-Mills theory and multiple D3-branes is the first and most widely studied example. On the other hand, the correspondence for multiple M2branes has been quite mysterious until recently. The situation changed when Bagger and Lambert [4][5][6] discovered N = 8 Chern-Simons-matter theory(see also [7]), by developing the idea of [8]. However, it was difficult to increase the rank of the gauge group. This is in some sense related to the fact that the maximally supersymmetric M2-brane solution does not have an adjustable parameter. Later, Aharony et al. [9] constructed N = 6 U(N) × U(N) Chern-Simons-matter theories that describe multiple M2-branes on the orbifold C 4 /Z k , where k becomes the level of the Chern-Simons action in the field theory. This orbifold provides us with an adjustable parameter, which enables us to treat weakly coupled field theories in some limit.
For multiple M2-branes in flat space, we can turn on an anti-self-dual four-form flux, which corresponds to adding a fermionic mass term to the field theory. The four-form flux polarizes M2-branes into M5-branes [10,11] and the discrete set of vacua of the theory has a one-to-one correspondence with the partition of N, the number of M2-branes [12]. For multiple M2-branes on the orbifold C 4 /Z k , we can consider a similar story. A mass-deformed version of ABJM theory was considered in [13] and its vacuum structure was identified in [14]. Especially, in the most symmetric vacuum, the system has SU(2) × SU(2) × U(1) × Z 2 symmetry. The mass term breaks the relativistic scaling symmetry. However, there is a non-relativistic limit of this theory that has the Schrödinger symmetry [15,16]. Note that the Chern-Simons-matter theory is a good model to study the non-relativistic limit since gauge fields are not propagating.
Therefore, it is natural to seek for a supergravity solution that corresponds to the nonrelativistic limit of the mass-deformed ABJM theory. Assuming the classical analysis of the vacuum structure of the field theory is still applicable to the supergravity limit, the solutions will have SU(2) × SU(2) × U(1) × Z 2 global symmetry and several additional U(1) symmetries corresponding to the non-relativistic particle number symmetry, depending on which fields to retain in the non-relativistic limit [15]. In the most supersymmetric case, it has 14 supercharges. Although we were not able to find a supergravity solution with the same number of supersymmetries, we will present a class of supersymmetric solutions with the Schrödinger symmetry in two space dimensions in M-theory, and then consider a specific case with the same global bosonic symmetry of the non-relativistic limit of the mass-deformed ABJM theory.
A geometry with the Schrödinger symmetry was found in [17,18] 1 . In this case, the AdS symmetry is explicitly broken to the Schrödinger symmetry due to the term − dx +2 r 4 in the metric, where x + is one of the two lightlike coordinates. Soon after, the geometry was embedded in string theory [21][22][23]. The supergravity solutions with the Schrödinger symmetry does not have supersymmetry mainly due to the term − dx +2 r 4 in the metric and the lightlike three form flux H 3 that supports it. Supersymmetry can be recovered if the coefficient of dx +2 r 4 depends on the compact space [24]. However, in their case, the coefficient is necessarily negative in some region of the compact space and the stability of the spacetime is not guaranteed. Recently, this problem was remedied and supersymmetric solutions were obtained with negative coefficient of dx +2 r 4 by turning on some lightlike fluxes, which can be related either to a Killing vector that leaves some Killing spinors invariant [25], or to the properties of the Calabi-Yau structure [26]. Also, it is possible to explicitly break the AdS symmetry by adding a term dx + C to the metric where C is a one-form that does not depend on the worldvolume coordinates [26,27]. There are also proposals where the breaking occurs due to the fact that the lightlike direction is compact without explicitly adding a term to the AdS metric [28,29].
In this paper, we will explore supergravity solutions having the Schrödinger symmetry in M-theory. Since a non-relativistic field theory has a discrete particle number, we expect the U(1) direction corresponding to the particle number is compact. Instead of imposing this as an additional assumption, we make the compact lightlike direction a non-trivial U(1) bundle over the compact space. Then the compactness is required without further assumption. We begin with the N = 1 warped AdS 5 solutions in M-theory given in [30], and modify the geometry to obtain the Schrödinger symmetry. Initially the AdS 5 solution has eight supercharges and they reduce to two after the modification in general. However, there is a special case when there remain six supercharges, which is the same number as in the DLCQ of AdS. After general remarks, we specialize to a specific example with SU(2) × SU(2) × U(1) isometry. We consider the Kaluza-Klein spectrum of the theory, and show that the non-trivial U(1) bundle structure of the lightlike compact direction sets an upper bound for the non-relativistic particle number for given quantum numbers of the compact space. The initial motivation to consider a Schrödinger invariant geometry with SU(2) × SU(2) × U(1) was to find a candidate theory for the dual of the non-relativistic mass-deformed ABJM theory. In line with this, we also provide a non-supersymmetric solution with the same global symmetry briefly at the end.

General consideration
In this section, we will deform the supergravity solutions given in [30] in such a way that the resulting solutions have the Schrödinger symmetry.

Warped AdS 5 solutions in M-theory
Before dealing with non-relativistic solutions, let us describe the general N = 1 supersymmetric solutions of the supergravity limit of M-theory consisting of a warped product of AdS 5 and a six-dimensional space considered in [30]. The metric is of the form and the four-form flux lies along the compact six dimensions. The overall coefficient e λ is a warping factor that depends on M 6 . The authors of [30] obtained the most general condition for N = 1 supersymmetry, and then specialized to a special case where the six-dimensional manifold M 6 is a complex manifold with a Hermitian metric. In this case, the supersymmetry condition becomes significantly simplified and they can obtain many explicit solutions. Let us describe the manifold M 6 first. The metric of M 6 is given by There is a four dimensional Kähler manifold M 4 , whose metric isĝ ij dx i dx j . The complex structure of the metric is independent of y and ψ. ∂ ∂ψ is a Killing vector of M 6 and the y dependence of the metric warps the spacetime.P is the canonical Ricci-form connection defined by the Kähler metricĝ. That is, the Ricci form R = dP .P is independent of y and ψ. ζ is a function of y which is implicitly defined by 2y = e 3λ sin ζ . (2.3) We fix the AdS 5 radius to be 1. The four-form field strength is given by whereV 4 is the volume form andĴ is the Kähler form of M 4 . In addition to these, we have two more constraints: ∂ y log ĝ = −3y −1 tan 2 ζ − 2∂ y log cos ζ .
(2.5) Given these conditions, the Bianchi identity and the equations of motion for F (0) 4 , and the Einstein equations are all satisfied.

Deformation to solutions with Schrödinger symmetry
Let us first write the AdS 5 metric in a form that will be suitable for later analysis: The DLCQ of AdS 5 makes the x − direction compact. The modification we do here is to make x − a coordinate for a U(1) bundle over the compact space. In the case when the U(1) bundle is non-trivial, the lightlike direction is necessarily compact and breaks AdS 5 symmetry down to the Schrödinger symmetry 2 . Let us call the geometry Sch 5 . Note that making the lightlike direction compact makes it subtle to deal with the system in the supergravity approximation. The situation gets better if we add large momenta along the compact lightlike direction [22]. This will involve making a black hole solution that asymptotes to the geometry that we give below. We will not consider such a finite temperature/finite density solution here, but we note that the compact lightlike direction changes the causal structure of the spacetime drastically. In particular, any two points in the geometry can be joined by a timelike or lightlike curve: Suppose we want to connect some point P = (x + , x − , x i , r) to Q = (0, 0, 0, 0) using a timelike curve when x + < 0. Due to the periodic identification, we can equally start at P = (x + , x − − N∆x − , x i , r) for some large N where ∆x − is the period of the x − direction. For large enough N, there is indeed a timelike curve connecting the points P and Q. This is a property that is expected for the dual theory of a non-relativistic system. Note that we can also add a term proportional to dx +2 r 4 , which does not break the Schrödinger symmetry. The coefficient depends on the compact space. Such a possibility was explored previously in [24]. Specifically, we consider the following metric: (2.7) 2 There was a paper [31] that also considers modification of the warped AdS 5 solutions of [30]. They added dx + C component to the metric, where C is a globally defined one-form on the compact space, which means the U (1) bundle corresponding to the the x − direction is trivial.
A is a gauge field on M 4 and f (y) is some function that depends only on y. We need to determine these two quantities. To support this geometry, we turn on the four-form field strength along the lightlike direction: (2.8) We demand that A depends only on x i , and not on y: otherwise, the second term includes a part proportional to dx + ∧ dr ∧ dy ∧ ∂ y (dA), and then it is impossible to satisfy the equations is the original four-form field strength of the warped AdS 5 solution, and s(y) is some function to be determined. By construction, dF 4 = 0. Just as in the original warped AdS 5 solution, we also require F 4 ∧ F 4 = 0. This requireŝ Let us consider the equations of motion for F 4 first. The dual seven-form F 7 is given by is the seven-form field strength of the corresponding warped AdS 5 solution. Since we only consider the case when F 4 ∧ F 4 = 0, the equation of motion of F 4 is satisfied when dF 7 = 0. This is satisfied provided The last equation is satisfied when due to the relation (2.3). In the cases we are interested, y takes values between two roots of cos ζ = 0. Since (2.12) is a second order differential equation and the coefficient of s ′′ (y) vanishes when cos ζ = 0, the other solution necessarily blows up when cos ζ = 0. Therefore, s(y) = 2y is the regular solution we want. The third equation implies dA ∧Ĵ = 0. We will see presently that the Einstein equations are also satisfied by choosing the coefficient f (y) of dx +2 r 4 appropriately. However, it is possible that the coefficient can take both positive and negative values over the compact space and, in the example that we consider in the next section, indeed this is the case. This is analogous to the situation considered in [24], where the coefficient of 1 r 4 dx +2 is a harmonic function, which implies that it is necessarily negative in some region of the compact space. They show that there is an instability of a field with sufficiently large particle number due to the unboundedness of the Hamiltonian H(the conjugate momentum to x + ). Supersymmetry cannot guarantee H is positive since there is no dynamical supercharge.
We expect a similar instability in our geometry unless f (y) vanishes. However, as we will see in section 2.3, when f (y) = 0, there are two dynamical supercharges and the Hamiltonian H is bounded by the condition {Q, Q † } = H for dynamical supercharges Q and Q † .
To sum up, if there is a harmonic (anti)self-dual two-form dA that satisfies then we can construct a supergravity solution with the Schrödinger symmetry as described above 3 . Note that A is a one-form on M 4 and does not depend on y. Since ∂ yĴ = − 2 3 ydP andP is independent of y, if (2.14) is satisfied at one y, it is automatically satisfied for all y.
One case where a solution is easily found is when the manifold M 4 is Kähler-Einstein and y and ψ give a CP 1 bundle over M 4 . The isometry of CP 1 is broken to U(1) by the warping factor that depends on y. In this case, dP , the Ricci form, is proportional toĴ. Since dP is y-independent,Ĵ factorizes into a y-dependent function and a y-independent form. Hence, given a harmonic (anti)self-dual two-form dA withĴ ∧dA = 0, we can construct a Schrödinger solution. To do that, the dimension of the second cohomology class has to be greater than 1, which means we cannot construct our solution on CP 3 . However, there are cases when the dimension of the second cohomology class is greater than 1, and we will consider such an Given the above requirement, the Einstein equations are satisfied by choosing a suitable f (y). Let us first introduce the following vielbeins: There are two linearly independent solutions and one obvious solution is f (y) = βy for an arbitrary constant β. In the case when y and ψ combine to give topologically a two-sphere S 2 , cos ζ = 0 at the two poles of the sphere, and we take the solution f (y) = βy as the smooth solution. The other solution diverges when cos ζ = 0.

Supersymmetry
The Killing spinor equation is given by where ǫ is a Killing spinor and (2.18) We use A, B, · · · for vielbein indices and M, N, · · · for coordinate indices of eleven dimensions.
Our strategy is to divide the operator D A into two: one is independent of β and A, while the other is not. Then, given a Killing spinor ǫ of the corresponding AdS solution, we impose the condition that ǫ is annihilated by β, A-dependent part. Let us denote by ∆∂ A the change of the derivative ∂ A due to the presence of β and A, and similarly denote by ∆ω A the change of the connection Then it is easy to see that the only components that depend on β are (Λ T ) −1 − 0 and (Λ T ) −1 − 3 , and those that depend on A are (Λ T ) −1 − i . Therefore, we keep Killing spinors of the AdS solution when it is independent of x − . We will see later that the Killing spinors consistent with the compactification of x − are all independent of x − . Hence it does not give a new condition.
Next, let us consider the change of the connection ∆ω A . They are given by The condition that the differential operators D 0 and D 3 still annihilate a Killing spinor ǫ of the AdS solution imposes β Γ + ǫ = 0; , The second equation is satisfied if, for example, the manifold M 4 is Kähler-Einstein and the two-form field strength F is a (1, 1)-form on M 4 . To see this, let us decompose gamma matrices and spinors into AdS 5 and M 6 parts(note that we are looking for a Killing spinor of the original AdS 5 geometry which survives after we change the metric to the Sch 5 geometry).
The Killing spinor equation D a ǫ = 0 implies that ξ has to satisfy is a gamma matrix expression using τ m constructed from the four-form field strength (2.4). Let us multiply the above equation by F (2) , where now F (2) is made up of τ m matrices: From (2.11), we obtainL ∧ F =L ∧ * 4 F = 0. This implies {F (2) ,L} = 0 since {τ mn , τ pq } = 2γ mnpq − 4δ pq mn . Also, since we assume M 4 is Kähler-Einstein, [F (2) ,L] is proportional to F ijĴ j k Γ ik , which vanishes if F is a (1,1)-form. Now, we can simplify the expression (2.27) in the form QF (2) ξ = 0 where Q is some linear combination of gamma matrices. By examining the explicit expression, we see that Q has determinant (1 − 4y 2 (λ ′ ) 2 ) 4 , which does not vanish.
In conclusion, a Killing spinor of the AdS solution survive if it satisfies Γ + ǫ = 0. Therefore, at each point, a Killing spinor has to lie in some four dimensional space. This does not necessarily mean that there are four Killing spinors, since higher order integrability condition may not be satisfied. In fact, a superconformal supercharge cannot satisfy Γ + ǫ = 0. To see this, note that a superconformal supercharge is represented in the Poincaré coordinates as in the second expression in (2.25). Γ + ǫ = 0 translates into γ + ψ − = 0, which is written as At x = 0, this implies γ + ψ − 0 = 0. Now, we move γ + to the right. Then, since {γ + , γ − } = 2, we end up getting ψ − 0 = 0, which means the only solution to this equation is the trivial one. Hence no superconformal supersymmetries survive, which means there remain only two Poincaré supercharges that are annihilated by γ + .
That implies that the corresponding Killing spinor in the AdS 5 part is a Poincaré supercharge. Therefore, when β = 0, we have four Poincaré supercharges.
However, there should be additional supercharges that we might have overlooked when we analyze (2.28). Indeed, if we keep all four Poincaré supercharges of the AdS solution, there are two kinematical supercharges and two dynamical ones 4 . In this case, the commutator of the special conformal generator C and a dynamical supercharge Q produces a superconformal supercharge S: [C, Q] ∼ S. Therefore, there has to be a way to obtain a superconformal supersymmetry. To see how it comes about, let us look at the expression for a superconformal supersymmetry in AdS 5 space which we have already verified. Therefore, two superconformal supercharges that are constructed from ψ − 0 with γ + ψ − 0 = 0 survive. In this section, we have shown that, if M 4 is Kähler-Einstein and F = dA is a harmonic anti-self-dual two-form of type (1,1) on M 4 , it preserves two Poincaré supercharges when β = 0. This corresponds to the kinematical supercharges. If β = 0, we additionally have two dynamical supercharges and two superconformal supercharges, adding up to six in total. The number of surviving supercharges are the same as those of DLCQ of the AdS solution.
Note that the presence of the dynamical supercharges guarantees that the Hamiltonian H(the conjugate momentum to the x + coordinate) is positive definite: {Q, Q † } = H for dynamical supercharges Q and Q † .

Specific example
Here we present a specific example of the above analysis. We consider the case when the four dimensional manifold M 4 is S 2 × S 2 and y and ψ describes a CP 1 bundle, but warped by the y coordinate. The symmetry of the six-dimensional compact space is SU(2)×SU(2)×U(1)×Z 2 where the U(1) is related to ∂ ∂ψ and Z 2 exchanges the two spheres. Such a solution may be interesting since this is the symmetry of the non-relativistic limit of ABJM theory [15,16].
Let us first consider the warped AdS 5 solution.
(3.4) θ 1 and φ 1 parametrize one S 2 , and θ 2 and φ 2 the other S 2 . The period of ψ is 2π to have a smooth geometry. y and ψ combine to give a S 2 fibration over S 2 × S 2 . However, due to the y dependence here and there, only U(1) symmetry survives. Also c is constant, 0 ≤ c < 4 and y runs between the two roots of the equation cos 2 ζ = 0. Since cos 2 ζ > 0 for y = 0, one root is positive and the other negative. It preserves 8 supercharges.

Transformation to Schrödinger solution
Now we modify the geometry (3.1) according to section 2.2. We make x − a non-trivial U(1) bundle over S 2 × S 2 with gauge field A = n(A 1 − A 2 ), where n is some integer. The metric is given by (1 − y 2 )(dθ 2 1 + sin θ 2 1 dφ 2 1 + dθ 2 2 + sin θ 2 2 dφ 2 2 ) + e −6λ sec 2 ζdy 2 + 1 9 cos 2 ζ(dψ +P ) 2 , Note that dA is anti-self-dual, dA ∧ dP = 0 and A does not depend on y. The four-form flux is modified as follows: Note that the solution exists for each c ∈ [0, 4) and each integer n. Given the general analysis in the previous section, the equations of motion for the four-form field and the metric are guaranteed to be satisfied. Note that −βy, the coefficient of 1 r 4 dx +2 , takes both positive and negative values over the compact space. As mentioned in section 2.2, this signals an instability due to the unboundedness of the Hamiltonian [24] unless we set β = 0.
Note that dA is an anti-self-dual two-form of type (1,1) in M 4 . Hence, according to the argument in section 2.3, there are two kinematical supercharges when β = 0, and six supercharges when β = 0. The six supercharges consist of two kinematical, two dynamical and two superconformal supercharges. Especially, when β = 0, the Hamiltonian will be bounded below due to the presence of the dynamical supercharges.

Solution with plane wave boundary
In the previous sections, we use the Poincaré coordinate system for (deformed)AdS 5 . In general, the AdS n+2 metric in Poincaré coordinates is given by where x = (x 1 , · · · , x n−1 ). The boundary is R 1,n . There is another coordinate system in which the boundary approaches the plane wave metric [28,29,37]. It is given by The relation between the two coordinate systems is (3.9) Note that ∂ ∂x − and ∂ ∂x ′− generate the same flow in different coordinates: both are related to the number operator of the Schrödinger algebra. This also suggests that not much will change even if the x ′ − direction is a line bundle over the compact space. That is, instead of (3.5), we may consider the metric ds 2 11 = e 2λ(y) ds 2 Sch 5 + ds 2 may look troublesome at first, but actually − dx +2 r 4 itself is invariant under (3.9). This form of the metric may be useful since the time direction in this coordinate system is associated to the harmonic oscillator potential of the Schrödinger algebra. Here H generates the time translation in the Poincaré coordinates and C is the special conformal generator.

Kaluza-Klein mass spectrum
The fact that the lightlike compact direction is a non-trivial bundle over the compact space has an interesting consequence on the spectrum of the Kaluza-Klein states. We will show below that the non-relativistic particle number is bounded above by the quantum numbers of the compact space. It seems at first a bit strange that there is such a bound. However, we can view the system from the compact space point of view and consider the Kaluza-Klein particles charged under the momentum conjugate to the x − coordinate. Due to the non-trivial gauge field A, we can think that the Kaluza-Klein particles are in a magnetic monopole background field. Then it is well-known [38] that the quantum numbers of the compact space of a wave function describing a Kaluza-Klein particle is bounded below by the 'electric' charge of the particle, which in this case means the U(1) charge along the x − direction. The eigenstates are expressed as monopole harmonics. Below, we will follow the classical analysis, but in a way that can be more easily applicable to our situation.
Let us first consider the three sphere S 3 as a preparation. The metric is given by where 0 ≤ ψ ≤ 4π, 0 ≤ θ ≤ π and 0 ≤ φ ≤ 2π. The manifest symmetry is SU(2) × U(1) of SO(4). The Killing vectors are (4.2) They satisfy [L i , L j ] = k ǫ ijk L k and [L i , L ψ ] = 0, which comprise SU(2) × U(1) Lie algebra. We will construct a wave function Φ(ψ, θ, φ) carrying definite quantum numbers of SU(2) and U(1). First, let us demand Since ψ has period 4π, m ψ ∈ Z 2 . For SU(2) part, the analysis is very similar to the standard angular momentum analysis in quantum mechanics. For the l representation of SU(2), let us consider the highest state (l, m) = (l, l). It will be annihilated by L + = L 1 + iL 2 . It is easy to see that The wave function is then given by Φ(ψ, θ, φ) = e −im ψ ψ e −imφ f (θ). Since we want a wave function not to diverge at θ = 0 or 2π, |m ψ | ≤ l. By applying the lowering operator L − = L 1 − iL 2 repeatedly, we obtain a wave function with definite quantum numbers (m ψ , l, m): where u = cos θ. Since Φ m ψ ,l,−l−k has to vanish for any k = 1, 2, · · · , l ± m ψ has to be integral and positive. In particular, l can be half-integral since m ψ can. The Laplacian of S 3 is written which means the eigenvalues of the Laplacian ∆ is l(1 + 1) with l ∈ Z 2 . That is, 1 4 L(L + 2) with L ∈ Z.
In sum, for a given quantum number (l, m) of SU(2), the possible m ψ range from −l to l with spacing 1. Of course, the fact that the possible values of m ψ are finite for a given pair of (l, m) is obvious since S 3 has actually SO(4) symmetry and for a given value of the quadratic Casimir, there are finite number of states. However, the analysis we have done shows that the finiteness can be derived by using SU(2) × U(1) symmetry alone as well as the existence of a well-defined wave function. For example, we would arrive at the same conclusion even though the coefficient of (dψ − cos θdφ) 2 in (4.1) were different from 1.
Let us turn to the case we are interested in. The metric is given in (3.5). There are two sets of SU(2) Killing vectors L (4.7) For each S 2 , the only change from the analysis of S 3 is that ∂ ∂ψ is replaced by ∂ ∂ψ ± n ∂ ∂x − . Denoting the quantum numbers for U(1) ψ and U(1) x − by m ψ and N, respectively, then we have the following constraints for given quantum numbers (l 1 , m 1 ; l 2 , m 2 ) of SU(2) × SU(2): (4.8) In particular, N has to satisfy |nN| ≤ l 1 + l 2 .
To see some implication of this result, let us consider the massive Klein-Gordon equation in eleven dimensions: 1 Due to the warping factor, the Laplacian becomes a little complicated. The result can be written as We put all y dependence except the overall factor into a function M 2 , which is given by (1 − y 2 ) 2 cos 2 ζ ∂Φ ∂y (4.11) ∆ 1 and ∆ 2 are the Casimir operators of the two SU(2) isometry groups, which are given by 3 ) 2 using (4.7). For a wave function with definite quantum numbers of SU(2) × SU(2) × U(1) ψ and definite particle number, this equation becomes an ordinary second order differential equation in y. Note that the last term in (4.11) looks problematic since, by increasing the momenta along the ψ and x − directions, this part can be negative and large in absolute value. However, this cannot happen since the quantum numbers m ψ and N are bounded. That is, from (4.8), we have It implies that the operator cannot have positive eigenvalues. Therefore, the last three terms in (4.11)(multiplied by e −2λ ) gives positive contribution to the mass parameter M 2 . That is, when β vanishes, the Kaluza-Klein mode does not suffer an instability due to the violation of the Breitenlohner-Freedman bound.
If we solve (4.11) and get the spectrum of the mass parameter M, the scaling dimensions and the correlation functions can be computed [17,18]. Let ν = √ M 2 + 4. The scaling dimension ∆ of the corresponding operator in the field theory is given by ∆ = 2 + ν and the two point correlation function of two such operators is given by (4.14) where ∆ i are the scaling dimensions of O i . ∆ = 2 − ν is possible if 0 < ν < 1 [17,39].

Solution with no supersymmetry
In the absence of supersymmetry, there may be many solutions with the symmetries we want. The solution given here can be thought of as a deformation of the non-supersymmetric AdS 5 × CP 3 solution in [40]. As such, the solution here does not preserve any supersymmetry.

Discussion
In this section, we present one possible way of achieving Galilean symmetry from Poincaré symmetry, which gives a reason why we consider the case where the lightlike direction is a U(1) bundle over the compact space.
Let us consider a non-relativistic limit of some geometry in general, which has (1 + d)dimensional Poincaré symmetry. It need not have scale invariance. Then, it has translational symmetries, whose associated Killing vectors are ∂ ∂t , ∂ ∂x i . Additionally, there are rotational symmetries. Killing vectors for spatial rotations are x i ∂ ∂x j −x j ∂ ∂x i and those for Lorentz boosts Suppose also that there is a U(1) isometry and ∂ ∂φ is the corresponding Killing vector. Now, one way to send this geometry to a non-relativistic limit is to change the coordinates so that, instead of {t, ǫ is some constant. Correspondingly, the Killing vectors are expressed in this new coordinate system as Translational symmetries along t, x i , φ directions and space rotational symmetries are expressed in the same way as in the original coordinate system. But the Lorentz boost symmetries in the new coordinate system are Since ǫ is just some constant, it can be freely multiplied to the generators as above. Given this form, take the limit ǫ → 0. Then the resulting generators are which are the Galilean boost symmetry generators.
This seems that we should be able to obtain the Galilean boost symmetry generators when we take a certain limit of a geometry with Poincaré symmetry. But, of course, we have glossed over an important requirement that the geometry should behave well when we take the coordinate transformation (6.1) and take the limit ǫ → 0.
Before examining an example, let us comment on the coordinate transformation (6.1). First, φ in (6.1) has been shifted by t, but in general it can be replaced by kt with some constant k. But we do not take a special limit for k. Note that t and x i are rescaled, but φ is not. The reason is that we have in mind the case where φ is compact, in which case rescaling is not possible. Also, shifting φ coordinate by t does not spoil the periodicity of φ.
The example considered in the following is the near M5-brane geometry of the LLM geometry with M2-branes polarized into a small number of M5-branes [11,12]. The metric takes the form ds 2 11 = (πN 5 ) − 1 3 r(−dt 2 + dx 2 1 + dx 2 2 + ds 2 S 3 ) + (πN 5 ) 2 3 1 r 2 dr 2 + r 2 (dθ 2 + sin 2 θds 2 S 3 ) . (6.5) N 5 is the number of polarized N 5 branes. M5-branes are wrapping t, x 1 , x 2 and S 3 . As r becomes large, the geometry becomes AdS 7 × S 4 . Although we do not have an exact field theory dual for multiple M2-branes in flat space with mass deformation, we expect that this geometry can be thought of as a vacuum in which matter fields have non-zero expectation values along the first four coordinates in R 8 [10][11][12]. As such, the geometry will not have a relativistic or non-relativistic scaling symmetry and we do not expect it to have a well-defined limit of the previous discussion. However, we can just see where the limit sends this geometry to.
In the case of the mass deformed ABJM theory, the analogue of the LLM geometry is not known. Especially, we lack a geometry dual to the most symmetric vacuum of the field theory, in which the matter fields have vanishing expectation values. If we have such a solution, we will be able to check whether we can apply the procedure mentioned above to that solution.
discussion. This work is supported in part by DOE grant DE-FG03-92-ER40701. The work of H.O. is also supported in part by a Grant-in-Aid for Scientific Research (C) 20540256 from the Japan Society for the Promotion of Science, by the World Premier International Research Center Initiative of MEXT of Japan, and by the Kavli Foundation. C.P. is supported in part by Samsung Scholarship.