May 21, 2022

[3mm] with non-relativistic conformal symmetry

Juan Maldacena, Dario Martelli, and Yuji Tachikawa

1 Einstein Drive, Princeton, NJ 08540, U.S.A.

We consider non-relativistic conformal quantum mechanical theories that arise by doing discrete light cone quantization of field theories. If the field theory has a gravity dual, then the conformal quantum mechanical theory can have a gravity dual description in a suitable finite temperature and finite density regime. Using this we compute the thermodynamic properties of the system. We give an explicit example where we display both the conformal quantum mechanical theory as well as the gravity dual. We also discuss the string theory embedding of certain backgrounds with non-relativistic conformal symmetry that were recently discussed. Using this, we construct finite temperature and finite density solutions, with asymptotic non-relativistic conformal symmetry. In addition, we derive consistent Kaluza-Klein truncations of type IIB supergravity to a five dimensional theory with massive vector fields.

On leave from: Blackett Laboratory, Imperial College, London SW7 2AZ, U.K.

## 1 Introduction

Recently some gravity backgrounds with non-relativistic conformal symmetry were discussed [1, 2, 3, 4]111For earlier work on non-relativistic conformal structures see e.g. [5, 6] and references therein.. The idea is that such backgrounds would be gravity duals of conformal quantum mechanical systems, which are useful for describing certain condensed matter systems (see [1, 2, 7] for references).

Here we make some comments on these constructions. These constructions involve performing the discrete light cone quantization (DLCQ) of certain field theories and their gravity duals. As it is well known, the DLCQ of a field theory gives a non-relativistic system. Thus the DLCQ of a conformal field theory is expected to give a conformal quantum mechanical system. On the other hand, DLCQ quantization is subtle. In particular, when we perform DLCQ of gravity backgrounds we cannot naively apply the gravity approximation, since there is a circle becoming very small. Fortunately, if one considers a sector with large non-zero light-cone momentum one can find regions in the geometry where the circle has a non-zero size so that computations can be trusted. In this paper we will discuss some backgrounds that arise in string theory that display a non-relativistic conformal symmetry. One particular example where we know the conformal quantum mechanical theory and the corresponding gravity background is the theory that arises when we do the DLCQ of the M5 brane theory. In this case the conformal quantum mechanics was discussed in  and we will review it below. We will discuss some gravity backgrounds that can be used to perform computations which can be trusted and are predictions for results in the conformal quantum mechanical theory. Conformal quantum mechanical systems were studied in the context of black hole physics and AdS, see  for a review and further references.

The backgrounds considered in [1, 2] enjoy non-relativistic conformal symmetry even before taking the DLCQ limit. In this paper we embed these backgrounds in string theory and discuss their deformation at finite temperature and finite density. In one particular case we can relate these backgrounds to certain non-commutative dipole theories studied in . There is a simple procedure that allows us to introduce such non-commutativity both in the field theory and in gravity . A non-relativistic quantum mechanical system is expected to arise when we perform a DLCQ quantization of such a theory. Due to the non-commutative interpretation of the background we find that certain quantities are independent of the non-commutativity in the planar approximation, since non-commutativity does not change certain planar diagrams . Thus for many observables computations in the backgrounds [1, 2] are the same as in asymptotically AdS backgrounds with compactified.

We also discuss consistent type IIB Kaluza-Klein reductions of AdS backgrounds, where is a Sasaki-Einstein manifold, to five-dimensional systems involving massive vector fields. Our motivation is that these truncations admit solutions with (asymptotic) non-relativistic conformal symmetries, of the type discussed in [1, 2]. Such Kaluza-Klein reductions might be useful also for other purposes.

The contents of the paper are organised as follows: we start in Sec. 2 by discussing how we obtain a quantum mechanical system with non-relativistic conformal symmetry via the discrete light cone quantisation of relativistic field theories. We also discuss subtleties inherent in the construction. In Sec. 3 we construct the gravity dual of these systems, which can be trusted when we put sufficient amount of momentum along the compactified dimension so that it becomes a spacelike circle with large radius. In Sec. 4 we study the consistent Kaluza-Klein reductions of type IIB supergravity on or any other Sasaki-Einstein space which retain massive gauge fields, and we show that these reductions can be used to construct some of the supergravity solutions discussed in Sec. 3. We conclude the paper with a short discussion in Sec. 5. We have a few Appendices where the reader can find the details of the calculation.

Note added: the authors learned just before completion that there will appear two papers, one by A. Adams, K. Balasubramanian, and J. McGreevy  and another by C. P. Herzog, M. Rangamani and S. F. Ross  which have some overlap with our present paper.

Note added in v5: Appendix 12 was added on Nov. 2008 to explain the simple scaling behavior of the thermodynamic quantities, which is a generic consequence of the gravity dual being a DLCQ of a higher dimensional theory.

## 2 Non-relativistic theories from DLCQ

### 2.1 DLCQ of relativistic theories in Minkowski space

It is well known that the light cone quantization of a relativistic theory looks like a non-relativistic theory. Choosing light cone coordinates we see that the mass shell condition for a massive particle looks like

 −p+=→p2(−4p−)+m2(−4p−) (2.1)

which looks like the energy of a non-relativistic particle of mass in a constant potential. We find it useful to write things in terms of with the lower index since that is the momentum that is canonically conjugate to translations. One minor disadvantage is that they are negative definite. Thus our if we start with the ordinary Minkowski metric, .

In a relativistic theory is a continuous variable. We can make it discrete by compactifying the light cone direction . We then find that is quantized as

 −p−=Nr− (2.2)

where . This is called “discrete light cone quantization”. Note that the parameter can be changed by doing a boost in the directions, which is a symmetry of the relativistic theory. This boost is broken by the compactification. However, the fact that it is a symmetry in the original theory implies that theories with different values of are related by a simple rescaling of the generators. Though in the formulas below we will keep explicitly, one could set without loss of generality.

If, in addition, the relativistic theory is conformal invariant, then this procedure would formally lead to a conformal invariant quantum mechanical theory, with a symmetry group which is called the ‘‘Schrödinger group’’222The name comes from the fact that it is the dynamical symmetry group of the ordinary Schrödinger equation for a free particle.. This is simply the subset of the conformal generators which commute with . See Appendix A for some details. This fact was already noted in  where the DLCQ of the theory on M5 brane was studied.

As explained in , DLCQ is very subtle. One has to be careful about the zero modes. In general these zero modes (modes with ) are described by an interacting theory which is obtained by taking the original theory and placing it on a very small spatial circle of vanishing size. Thus, one has to solve the problem of a field theory in one less dimension. For example, if we start with 3+1 dimensional super Yang Mills, the zero mode dynamics is described by a 2+1 dimensional conformal field theory which is the IR limit of 2+1 dimensional super Yang Mills. This is also the theory that lives on M2 branes. Thus, the proper analysis of the dynamics of a DLCQ theory is fairly non-trivial, but it can be done in principle. This point should be kept in mind when we discuss various theories in this paper. Proposals for the DLCQ of super Yang Mills were made in [17, 18]. We will not give a totally explicit description of the field theory side in this case, leaving a complete analysis of this issue for the future. Note that in this case we get a family of conformal quantum mechanical systems that arise by taking different expectation values of (and the dual photon) [17, 18].

The DLCQ procedure we outlined above is a way of generating examples of conformal quantum mechanical systems. Writing out explicitly the quantum mechanical system requires a proper analysis of the zero modes. This is an important issue for understanding precisely the nature of the corresponding non-relativistic quantum mechanical system. In particular, one would like to write down the Schrödinger equation for the quantum mechanical system. We discuss one specific example below.

Finally, note that the discussion in  that links AdS to AdS and a possible conformal quantum mechanical dual can be interpreted as a DLCQ quantization of AdS in a sector with nonzero .

If the parent relativistic theory has a gravity dual which is weakly coupled one can hope to have a gravity description of the corresponding conformal quantum mechanical system. For example, if the parent theory is super Yang Mills, then it also has a dual description as a gravity (or string) theory on AdS when the ’t Hooft coupling is large. We denote by the rank of the gauge group in order not to confuse it with , which is the number of quanta of the light cone momentum. Thus, one would hope that by performing the DLCQ procedure on both sides one would get a strongly coupled quantum mechanical system that is dual to a weakly coupled gravity solution. The DLCQ limit of the gravity dual is simply given by identifying the direction in the bulk [4, 3]. One should remember, though, that this identification is not as innocent as it looks. When we periodically identify the direction in the bulk we are performing a drastic change in the theory. For example, the correct graviton scattering amplitude in the DLCQ theory and the naive one (obtained by truncating tree level graviton scattering amplitudes in the theory before the DLCQ) are not the same .

The situation changes for the better when we introduce a large amount of momentum in the DLCQ direction. In fact, one is interested in the sector of the theory with non-zero values of . In the bulk, this has the effect of making the size of the circle spacelike in some interesting regions of the geometry. Thus, for large enough , or large enough density, one can indeed use the gravity description for computing certain properties of the system. We will describe this in more detail when we talk about the gravity solutions. Let us first describe a specific example and some simple variations of this construction.

### 2.2 DLCQ description of the M5 brane theory

There is one case where a fairly explicit description of the DLCQ is available in the literature. It is the case of the M5 brane theory, which is a 5+1 dimensional conformal field theory. In DLCQ with units of momentum, this becomes a certain conformal quantum mechanics theory constructed as follows . We start with fivebranes and units of momentum along a compactified spatial direction. The small radius limit, which leads to the DLCQ description, forces us to perform a duality to end up with D0 branes and D4 branes, and to take the low energy limit of this system. The result is a quantum mechanics theory which is a sigma model on the Higgs branch of a certain theory with 8 supercharges. It is the quantum mechanics on the moduli space of -instantons.

This system has four-dimensional Galilean invariance and conformal invariance, so it has the Schrödinger symmetry, but it is quite unlike the non-relativistic conformal system which is discussed recently in the literature, e.g.  fermions interacting via contact interaction. Namely, each of the instantons is not pointlike but has a size parameter which is affected by the dilatation; there is no obvious second-quantized framework, and so on.

The resulting quantum mechanics is the following. We start with a gauge theory with an adjoint hypermultiplet and fundamental hypermultiplets. The bosonic variables involve two complex adjoint matrices and complex scalars in the of and scalars in the of . These fields are constrained by

 [X,X†]−[~X,~X†]+qiq†i−(~qi)†(~qi)=0 ,            [X,~X]+qi~qi=0. (2.3)

and we quotient by gauge transformations. This gives a space of real dimensions which is a hyperkähler manifold. The metric is the induced metric in the ambient space. It can be found as follows: one constructs the Kähler potential of the ambient space , which also gives the Kähler potential of the metric in the moduli space, after choosing complex coordinates for the moduli. is also the special conformal generator in the quantum mechanical theory. This defines a conformal quantum mechanics, which becomes superconformal once we include also the fermionic degrees of freedom . For further details on the definition of the quantum mechanical theory and its symmetries see . See also  for a nice review on conformal and superconformal quantum mechanics.

### 2.3 DLCQ of a conformal field theory on a plane wave background

A small variation of the preceding theme is to start with a relativistic theory on a plane wave background

 ds2=−dx+dx−−→x2(dx+)2+d→x2 (2.4)

where stands for the transverse spatial directions. For a general field theory this also gives us a quantum mechanical system now with particles in a harmonic oscillator potential. The Hamiltonian is then

 −p+=→p2(−4p−)+(−p−)→x2 . (2.5)

The isometries of the plane wave are symmetries acting on the quantum mechanical theory. If the field theory is conformal we have further symmetries and the resulting theory is precisely the same as the one we obtained when we started from flat space but with a different choice of Hamiltonian . This can be understood considering the subgroup of the Schrödringer group which includes the Hamiltonian the dilatation and the special conformal transformation . Then the Hamiltonian on the plane wave background is

 Hosc=L0=12(H+C) . (2.6)

Another variation of the idea is to take the following form of the plane wave metric when the transverse direction is two-dimensional (or more generally even dimensional):

 ds2=−dx+(dx−−2ρ2d^ψ)+dρ2+ρ2d^ψ2 , (2.7)

where we took and chose . A particle with fixed moving on this metric reduces to a non-relativistic particle moving in the transverse space in the presence of a constant magnetic field with no potential. The new Hamiltonian is related to the one above by

 Hmag=Hosc−J , (2.8)

where is the angular momentum associated to the rotation in the plane.

### 2.4 DLCQ of dipole theories

Another variation is to consider a certain non-commutative theory, called a “dipole theory”  (see also [11, 22, 23]). This is a theory were the field multiplication is defined via a star product. In order to define the star product we use the conserved charge and also another global symmetry charge, of the system. The star product is then defined as follows

 f∗g=ei2πσ(Pf−Qg−Pg−Qf)fg , (2.9)

where is the ordinary product and is an arbitrary parameter. and are the values of and for and respectively. We are imagining that and have well defined values for both charges and , and we can get the product for more general functions of and by the ordinary distributive property of the product.

If the original theory has a symmetry that commutes with the two charges that appear in the definition of the star product (2.9), then it will also be a symmetry of the theory after the star product deformation. The generators of the conformal group that commute with are the Schrödinger subgroup. Thus we have the Schrödinger symmetry even before compactifying the coordinate . If we also compactify and perform a DLCQ quantization we expect to get a non-relativistic conformal system. Our reason for introducing these exotic theories is that, in some cases, their gravity duals, derived in [10, 11], are given by the metrics introduced in [1, 2]. For more details on these theories see [10, 11, 22, 23] .

## 3 DLCQ of string or M theory backgrounds

As we saw in the previous section, one can construct a system with Schrödinger symmetry by taking the DLCQ of a relativistic conformal theory. If the relativistic CFT is dual to AdS we should then consider the DLCQ of AdS space. We can write the AdS metric in the Poincaré patch as

 ds2=dr2r2+r2(−dx+dx−+d→x2) (3.1)

and take . This fact has been noticed also in the recent papers [4, 3].

Since we are taking a circle to have zero size, we cannot trust this geometry to do computations. A similar situation arises when we consider the DLCQ of eleven dimensional supergravity. In that case the conjectured correct description is in terms of a quantum mechanical theory given by D0 branes  (see also ). This description is not the same as the one we get by taking the naive DLCQ of the gravity theory. For example, the scattering amplitude of three gravitons to three gravitons in the naive supergravity approximation gives a different answer than in the matrix model . In fact, when one defines the DLCQ limit carefully, as a limit of the theory on a very small spatial circle, one finds that the correct answer is given by the D0 matrix quantum mechanics and not by supergravity .333Of course, the BFSS conjecture  is the idea that a suitable large limit allows one to recover the results with no compactification of the light like circle.

Fortunately, not all is lost. Of course, what we really want to do is to put units of momentum on this space. When is small, the dynamics cannot be computed in terms of particles moving in the metric (3.1), for reasons we have explained. Notice that this is true even when the radius of AdS is large. In fact, the correct description of units of momentum in the DLCQ of type IIB string theory in flat space is in terms of the field theory that lives on M2 branes on .

However, if we put a large amount of momentum, then the backreaction implies that the size of will be non-zero in some regions and we will be able to trust the metric. More concretely, we can consider the black three brane metric describing a finite temperature system

 ds2=11−r40r4dr2r2+r2[−dx+dx−+r404r4(λ−1dx++λdx−)2+d→x2] . (3.2)

This is simply a boosted version of the ordinary black brane metric, in the near-horizon region. We can compute the density of , which we interpret as the “particle density”, and the energy density. We obtain (see Appendix C.2)

where is the volume of the two spatial dimensions.

We can see that by tuning and we get different values of particle density as well as energy density. We can also write this in terms of the temperature and chemical potential of the non-relativistic system

 1T=πλr0 ,μNT=πr0r−λ . (3.4)

The entropy is given by

and these thermodynamic quantities satisfy the first law

 δH=TδS−μNδN . (3.6)

The physical value of the radius of the circle is

therefore we can trust the gravity description as long as . When it becomes smaller than one we might be able to do a T-duality and use an alternative description also. Notice that this size becomes small as we approach the boundary of AdS, . Thus, we cannot trust the metric near the boundary. It might be possible that after performing suitable U-dualities we might find a metric we can trust.

Of course this solution implies that the thermodynamic properties can be simply extracted from the thermodynamic properties of the ordinary black D3 brane. We are only looking at the same system in light cone gauge, so we just need to translate all quantities to light cone gauge.

### 3.2 DLCQ of AdS5 with a plane wave boundary: harmonic potential

In this subsection we consider the gravity dual of the field theory with plane wave boundary conditions, so that it is the gravity dual to the field theory on the plane wave. Since the plane wave metric is conformal to flat space the metric is simply AdS, which can be sliced in a way that make the plane wave at the boundary more manifest. One way to find the slicing is to start with and take the Penrose limit for a particle moving with large angular momentum in one of the angles of . Namely, we write the AdS metric in global coordinates,

 ds2=−(1+r2)dt2+dr21+r2+r2ds2(S3) , (3.8)

which has as a boundary. Here . We then define

 x+=t ,        x−2R2=t−φ ,     θ=ρR ,        r=Ry (3.9)

and we take the limit, keeping and , fixed. We find the metric

 ds2=dy2y2+y2(−dx+dx−−ρ2(dx+)2+dρ2+ρ2dψ2)−(dx+)2 . (3.10)

Notice that it is not necessary to take this limit to obtain (3.10), and we can also obtain it directly by writing AdS space in the appropriate coordinates444In fact, (3.10) can be transformed into the metric recently discussed in  (cf. equation (35) in that reference) by an obvious change of coordinates. (see also ). Explicitly, after the further change of coordinates

 sinhr=ρy ,z=ρ2+1y2 , (3.11)

the metric (3.10) becomes

 ds2=sinh2rdψ2+dr2+cosh2r[−(dx++dx−2z)2+(dx−)2+dz24z2] . (3.12)

This is AdS sliced by AdS, which is the metric in square brackets. However, obtaining the metric (3.10) as a limit will be useful later. The metric in (3.10) cannot be trusted if is compact, since is a null direction.

We would now like to add some momentum and also raise the temperature of the system so that we have a black hole in a space which is asymptotic to (3.10). We obtain this black hole by starting with the five-dimensional Kerr-AdS black hole  (see also ) and performing a limit similar to the limit we performed above (3.9). We describe this in detail in appendix B. The final metric can be written as

 ds2=(r2+sin2θ)dr2(1+r2)2−2m−(1+r2sin2θ)(dx+)2−λ(1+r2)cos2θdx+dx− +(r2+sin2θ)dθ2cos2θ+r2sin2θdψ2+m(−2dx++(dx+−λdx−)cos2θ)22(r2+sin2θ) . (3.13)

This metric depends only on one non-trivial parameter . The parameter can be absorbed into the redefinition of but it is convenient to keep it because it represents the amount of boost one performs when we take the limit. We can bring this metric to a form which asymptotes to (3.10) via the coordinate change:

 λ−1y2 = (r2+1)cos2θ , ρ2y2 = r2sin2θ . (3.14)

This is analogous to the coordinate change employed in  to display the AdS asymptotics of the Kerr-AdS black hole. The result does not have a simple analytic form. As an expansion in the new radial variable , it is given by

 ds2 =(1−2m(1+λρ2)2λ2y4)−1dy2y2+y2(−dx+dx−−ρ2(dx+)2+dρ2+ρ2dψ2) −(dx+)2+λmy2((1+2λρ2)dx++λdx−)22(1+λρ2)3+O(y−4) (3.15)

where we kept only terms of order lower than with respect to the solution.

The and can be calculated by starting from the expressions for the energy and the angular momentum in  (also see ) and taking the appropriate limit, see appendix B. We find

Recall that the Hamiltonian here is the oscillator Hamiltonian .

The horizon radius is given by the solution of . As the metric becomes singular. For (or ) the energy and particle densities from (3.16), (3.17) are non-zero. For smaller densities or energies this black hole would not be a good description. However, for the metric (3.2) does go to the plane wave metric (3.10)555 This is reminiscent to what happens for the BTZ black hole. .

The Killing vector degenerating at the horizon is

 ∂++1λr2H−1r2H+1∂− . (3.18)

Then the temperature and the chemical potential are

 T=rHπ ,μN=1r−λr2H−1r2H+1 . (3.19)

The entropy of the system is

and one can check again the first law

 δH=TδS−μNδN (3.21)

is satisfied, as it should be. We can also compute the size of the circle at the horizon which is

This becomes zero when . In fact, the circle shrinks at at any value of as is evident from (3.15). In the coordinates introduced in (3.14) these points correspond to the limit , . Since we do not have a large contribution to the thermodynamic quantities from this region we assume that we can ignore this part of the metric.

### 3.3 DLCQ of AdS5 with a plane wave boundary: magnetic field

Here we briefly discuss the gravity dual of the field theory with plane wave boundary conditions in the coordinates (2.7) which lead to a non relativistic particle in a magnetic field when we fix . The Hamiltonian of this system is obtained shifting the oscillator Hamiltonian with the angular momentum on the plane, as in 2.8. The metric is obtained by performing a limit similar to the one discussed in the previous subsection, starting with the two-parameter Kerr-AdS black hole . This is discussed in Appendix B.2 and here we present the final result

 (3.23)

The temperature and the chemical potential are given by

 T=12π2y2H−3λ2λ√y2H−λ2 ,μN=y2H−2λ2r−y2Hλ2 , (3.24)

and the entropy is

where the horizon is at with

 1−2mλ4y4H+2mλ4y6H=0 . (3.26)

The densities are constant (see Appendix B.2), thus the charges are infinite on the plane, and proportional to the volume , as is in the case of the translation invariant black three brane metric. In fact, after Kaluza-Klein reducing along we see that we get a translation invariant metric with a constant Kaluza-Klein magnetic field. The particle number and energy per unit volume are given by

respectively. Recall that the Hamiltonian here is given by . This implies that we can view this system as the system on with the Harmonic oscillator hamiltonian but at a critical value of the chemical potential for the spin in the transverse plane. One could find a metric interpolating between (3.2) and (3.23) by adding a general chemical potential for the spin. Such metrics should correspond to more general limits of the rotating Kerr black hole .

### 3.4 DLCQ of AdS7 with a plane wave boundary

In this subsection we repeat the previous discussion for the AdS case. We write the as . We now start with AdS in coordinates where the boundary is . After performing the rescaling (3.9) we get a metric similar to (3.10) except that . Starting from the seven dimensional Kerr-AdS black hole and performing the same limit (3.9) we get a metric very similar to (3.2) except for and ,

 ds2=(r2+sin2θ)dr2(1+r2)2−2mr2−(1+r2sin2θ)(dx+)2−λ(1+r2)cos2θdx+dx− +(r2+sin2θ)dθ2cos2θ+r2sin2θdΩ23+mr2(−2dx++(dx+−λdx−)cos2θ)22(r2+sin2θ) . (3.28)

We can also compute the entropy as a function of the temperature and the chemical potential. The expressions for the temperature and chemical potential are

 T=12π(3rH+1rH),μN=1r−λr2H−1r2H+1 , (3.29)

where now the horizon radius obeys the equation . The values of the energy and the momentum can be uniquely fixed by first noting that and from the asymptotic form of the metric, and second by demanding that the first law is satisfied. We obtain

The entropy is

In this case we see that as the energy and the particle number both go to zero. On the other hand, we see that the temperature in (3.29) has a minimum. This suggests that the thermal ensemble will display a phase transition similar to the Hawking–Page we see when we treat AdS in global coordinates [31, 32]. In fact, for a given temperature and chemical potential there is another solution which is simply thermal AdS space with a gas of particles (we can only trust this last solution if we take to be non-compact). Comparing their free energies we find a phase transition at . Of course, the black hole is the stable solution for . We can evaluate this quantities in the particular case of AdS background of M theory. In that case we find that

where and is the number of M5 branes. As in the five-dimensional case, the circle shrinks when . For moderate temperatures, with of order one, we can trust the M-theory background near the horizon in the regime and . By reducing to type IIA we can extend the region of validity of the above formulas to , . The existence of the IIA string theory dual suggests that this conformal quantum mechanics theory has an interesting ’t Hooft limit.

These quantities should describe the thermodynamic properties of the conformal matrix quantum mechanics described in  and reviewed in Sec. 2.2. That quantum mechanics theory is characterized by only two discrete parameters and . Here is the number of fivebranes and is the amount of momentum. When we consider the Hamiltonian we expect that the temperature would also be a non-trivial parameter since there is an energy gap of order one that is given by the confining potential. In the above metric the temperature translates directly into . Finally, we see that the combination , at fixed temperature, fixes the chemical potential for the light cone momentum . Thus the metrics we have depend on the right number of parameters. Here is a trivial parameter and can be set to one without loss of generality.

### 3.5 DLCQ of the dipole theory and its gravity dual

In this section we describe in more detail a particular example of the dipole theories introduced in [10, 22, 23]. As we explained above these theories are based on the non-commutative product in (2.9). As a particular example we can consider starting with super Yang Mills in 3+1 dimensions. This theory has an R symmetry that rotates six scalars, , and their fermionic partners. We can consider the particular symmetry that rotates all pairs of scalars. Defining , , the symmetry acts by . Planar diagrams in non-commutative theories are particularly simple. The only difference between the planar diagrams in the ordinary theory and the theory with the star product is an overall phase depending only on the external particles  (see also ). Thus, if we consider a quantity such as the free energy, which does not involve any external particles, then the final result is the same as in the ordinary theory.

The gravity duals of such theories are easy to construct. We simply need to do a particular transformation on the gravity solution. This transformation has the following origin. In the original solution we have two commuting isometry directions, and an angle that is conjugate to the charge we defined above. If we naively imagine doing a dimensional reduction of the ten dimensional gravity theory on these two dimensions we get a theory in eight dimensions that has an symmetry. This symmetry is a symmetry in the eight dimensional gravity theory but it is not a symmetry of the ten dimensional configuration. In fact, it maps ordinary solutions into the solutions we want. This transformation is the following simple set of steps. We first do a T-duality in the direction to a T-dual direction . We then do a shift of coordinates . Finally, we do a T-duality again on into . The step where we did a shift of is not a symmetry of the theory since is periodic but is not. Nevertheless this operation generates a new solution which is another solution of the gravity equations which is not equivalent via legal U-dualities to the original one. This procedure was applied in  to obtain the corresponding gravity solutions. We perform these steps more explicitly in appendix C. We start with AdS, where the AdS metric is written as in (3.1). The final type IIB background that we obtain through this procedure is

 ds2 = −σ2r4(dx+)2+r2[−dx+dx−+d→x2]+dr2r2+ds2(CP2)+η2, BNS = σr2dx+∧η , (3.34)

and the rest of the fields have the same form as they had before we applied the transformation. Here is the one-form present in any Sasaki–Einstein manifold, which is dual to the Reeb vector field . In the case of , this generates simultaneous rotation of the scalars as discussed above. Indeed, in (3.5) we can replace the space with any local Kähler-Einstein space , so that

 ds2(Y)=ds2(BKE)+η2 (3.35)

is a Sasaki-Einstein metric on , where is the Kähler two-form on . These backgrounds do not preserve any supersymmetry. See Appendix E.

The five-dimensional part of this metric has the same form as the ones considered in [1, 2]. These metrics have the feature that they break the symmetries to the Schrödinger symmetry even before compactifying the direction . However, if we do not compactify the direction we have a theory with continuous eigenvalues for so that there is no sense in which we have a non-relativistic system. On the other hand, if we do not compactify we can definitely trust the metric (3.5), at least for values of that are not too large.

However, we wanted to perform the DLCQ of this theory. When we compactify we find that we can no longer trust computations in the metric (3.5). As before, we could consider a configurations with finite . A simple case arises when we consider a finite temperature configuration. We can obtain the corresponding metric if we start with the black brane solution (3.2) and perform the TsT transformation that lead to (3.5). The final metric in the Einstein frame is (see appendix C for details)

 −σ2r2(1−r40r4)(dx+)2]+e−Φ2r2[1r4−r40dr2+d→x2] +e−Φ2ds2(BKE)+e32Φη2 (3.36)

with dilaton and -field given by

 e−2Φ = 1+σ2λ2r40r2 , BNS = σr22e2Φ[(2−r40r4)dx+−r40λ2dx−]∧η , (3.37)

where is the parameter used in the shift, . By construction, the solution has the desired asymptotic form at infinity as . However, there is now a horizon at some finite value of the coordinate , inherited from the asymptotically-AdS black three-brane metric (3.2).

We can now compute the energy and momentum of this solution, as well as its entropy. All these results are independent of , which is related to the non-commutativity parameter. In fact we simply get the results in (3.3), (3.4), (3.5). The reason is that the TsT transformation is a symmetry of the eight dimensional gravity theory and this eight dimensional theory is all that we are using for computing these quantities666We are also using the fact that the TsT transformation leaves invariant the eight dimensional Kaluza-Klein gauge fields associated to the two charges.. See Appendix C.2 for a more detailed explanation. As explained in the beginning of this subsection, this is due in the field theory language to the fact that the planar diagrams are the same in the two theories. Of course, it is not a symmetry of the full theory, so we expect that at order the two theories would yield different answers, where is the rank of the gauge group.

## 4 Consistent truncations

In this section we construct two different consistent truncations of type IIB supergravity with massive vector fields. As we will see, one of them admits the black hole background (3.36), (3.37), among its solutions. Usually consistent truncations involve massless vector fields. These truncations can be used to generate other solutions which have the same symmetries as certain conformal quantum mechanical theories. We think that these truncations are interesting in their own right and could perhaps be useful for the construction of other solutions. We have two truncations which we will discuss in turn. Both are consistent truncations of the type IIB equations for the bosonic fields. It might be possible that they can be supersymmetrized. We will state here the final results and leave details to appendix D.

Before proceeding, let us recall how the backgrounds with non-relativistic conformal symmetry

 ds2(Mz)=−σ2r2z(dx+)2+dr2r2+r2(−dx+dx−+d→x2) (4.1)

arise from AdS gravity with a massive gauge field. This metric is a deformation of the AdS metric and clearly is only invariant under a subgroup of , which for generic values of is the non-relativistic dilatation group, enhanced to the Schrödinger group for . See Appendix A. These metrics are solutions to the equations of motion derived from the action [1, 2]

 S=∫dd+2xdr√−g(R−2Λ−14FμνFμν−m22AμAμ) . (4.2)

In particular, the ansatz solves the equations of motion, provided

 Λ=−12(d+1)(d+2) ,m2=z(z+d) . (4.3)

In the following subsections we will describe consistent truncations of type IIB supergravity, containing the action (4.2), where , and respectively.

### 4.1 Non-linear Kaluza-Klein reduction with m2=8

The ansatz we consider here is suggested by the observation that in any background of the type AdS, where is a Sasaki-Einstein manifold, there exist certain Kaluza-Klein vector fields with mass squared precisely . These arise from standard Kaluza-Klein reduction of the NS field and the RR field along the one-form that exists on .

We then consider type IIB supergravity where only the metric, the dilaton , the five-form field and the NSNS three-form are non-trivial. The equations of motion in the Einstein frame are777We use capital Roman letters for the ten-dimensional indices.

 RMN =12∂MΦ∂NΦ+196FMABCDFNABCD +14e−Φ(HMABHNAB−112gMNHABCHABC) , (4.4) □10Φ =−112e−ΦHABCHABC , d(e−Φ⋆H) =0 , F5 =⋆F5,dF5=0 ,H∧F5=0 , (4.5)

where . The final equation arises from the equation of motion of the RR 3-form, which has been set to zero. Our ansatz is

 ds210 =e−23(4U+V)ds2(M)+e2Uds2(BKE)+e2Vη2 , B =A∧η+θω , F5 =(1+⋆)G5whereG5=4e−4U−Vvol(M) , (4.6)

where is a Sasaki-Einstein metric. Here, is the Lorentzian metric of the five-dimensional part and denotes its volume form. We use indices () to denote the directions along and () for those along . , , and are scalar functions, and is a one-form on , respectively. The warp factor in front of is inserted to obtain the reduced equations in the five-dimensional Einstein frame. This ansatz is closely related to the one used in , where they had scalars , , and , but not the gauge field . They also had non-trivial RR 2-form, which we do not have.

Our choices for and satisfy the equations (4.5) automatically. Then defining , the field strength of the two-form reads

 H=F∧η−(2A−dθ)∧ω . (4.7)

The gauge transformation of the -field with a function on induces the five dimensional gauge transformation

 A→A+dχ ,θ→θ−2χ , (4.8)

which shows that is the Stückelberg field giving mass to the gauge field . We choose the gauge for simplicity.

The equations of motion (4.5) follow from the Type IIB action

 S=12∫d10x√−g[R−12∂AΦ∂AΦ−12⋅3!e−ΦHABCHABC−12⋅5!G(5)ABCDEG(5)ABCDE] (4.9)

by varying , and . Inserting the ansatz (4.6) in (4.9) and integrating over the internal directions, we obtain the five-dimensional action

 (4.10)

where we defined and to diagonalize the kinetic terms for the scalars. The five-dimensional equations of motion which follow from the action above are

 Rab =−13(24e−u−4v−4e−6u−4v−8e−10v)gab +12∂aΦ∂bΦ+5∂au∂bu+152∂av∂bv +12e−Φ+4u+v(FacFcb−16gabFcdFcd)+4gabe−Φ−2u−3vAcAc, d(e−Φ+4u+v⋆5F) =−8e−Φ−2u−3v⋆5A, □5Φ =−14e−Φ+4u+vFabFab−4e−Φ−2u−3vAaAa, 10□5u =24(e−u−4v−e−6u−4v) +e−Φ+4u+vFabFab−8e−Φ−2u−3vAaAa, 15□5v =16(6e−u−4v−e−6u−4v−5e−10v) +14e−Φ+4u+vFabFab−12e−Φ−2u−3vAaAa (4.11)

where is the five-dimensional d’Alembertian.

Interestingly, this non-linear Kaluza-Klein reduction is consistent: any solution to the five-dimensional equations of motion (4.11) can be lifted to a solution of the ten-dimensional equations of motion of type IIB supergravity, using the ansatz (4.6). The details can be found in Appendix D.1. The masses of the excitations around the AdS background are

 m2A=8 ,m2Φ=0 ,m2u=12 ,andm2
{"ft_lang_label":"__label__en","ft_lang_prob":0.91317654,"math_prob":0.9863308,"size":37796,"snap":"2022-40-2023-06","text_gpt3_token_len":8343,"char_repetition_ratio":0.15924005,"word_repetition_ratio":0.02140384,"special_character_ratio":0.21436131,"punctuation_ratio":0.096482694,"nsfw_num_words":0,"has_unicode_error":false,"math_prob_llama3":0.99252295,"pos_list":[0],"im_url_duplicate_count":[null],"WARC_HEADER":"{\"WARC-Type\":\"response\",\"WARC-Date\":\"2023-01-28T10:30:50Z\",\"WARC-Record-ID\":\"<urn:uuid:cea23b88-c32c-489f-933b-923ca07c14d8>\",\"Content-Length\":\"1049408\",\"Content-Type\":\"application/http; msgtype=response\",\"WARC-Warcinfo-ID\":\"<urn:uuid:ea7c5787-4c28-438f-b2eb-1c6ba0cd777d>\",\"WARC-Concurrent-To\":\"<urn:uuid:ead30d63-b2a1-4712-afe1-4707c112945b>\",\"WARC-IP-Address\":\"104.21.14.110\",\"WARC-Target-URI\":\"https://www.arxiv-vanity.com/papers/0807.1100/\",\"WARC-Payload-Digest\":\"sha1:22CWZ2MXJZQRQYPAGY2EI4CSGHCJHLYT\",\"WARC-Block-Digest\":\"sha1:Q5XVHEB6A2ATMFJUT4V6ZUOSRNYUZNEM\",\"WARC-Truncated\":\"length\",\"WARC-Identified-Payload-Type\":\"text/html\",\"warc_filename\":\"/cc_download/warc_2023/CC-MAIN-2023-06/CC-MAIN-2023-06_segments_1674764499541.63_warc_CC-MAIN-20230128090359-20230128120359-00199.warc.gz\"}"}