Asymptotically Schroedinger Space-Times: TsT Transformations and Thermodynamics
aa r X i v : . [ h e p - t h ] N ov Asymptotically Schr¨odinger Space-Times:TsT Transformations and Thermodynamics
Jelle Hartong and Blaise Rollier
Albert Einstein Center for Fundamental Physics,Institute for Theoretical Physics,University of Bern,Sidlerstrasse 5, CH-3012 Bern, Switzerland email: [email protected] and [email protected]
Abstract
We study the complete class of 5-dimensional asymptotically Schr¨odingerspace-times that can be obtained as the TsT transform of an asymptot-ically AdS space-time. Based on this we identify a conformal class ofSchr¨odinger boundaries. We use a Fefferman–Graham type expansionto study the on-shell action for this class of asymptotically Schr¨odingerspace-times and we show that its value is TsT invariant. In the secondpart we focus on black hole space-times and prove that black hole ther-modynamics is also TsT invariant. We use this knowledge to argue thatthermal global Schr¨odinger space-time at finite chemical potential under-goes a Hawking–Page type phase transition. ontents black holes with an asymptotic null Killing vector . . . . 134.2 Geometric properties of the MMT black hole . . . . . . . . . . . 154.3 Thermodynamics . . . . . . . . . . . . . . . . . . . . . . . . . . . 174.4 Phase transitions . . . . . . . . . . . . . . . . . . . . . . . . . . . 19 Asymptotically Schr¨odinger space-times form a natural starting point to try toextend holographic techniques to the realm of nonrelativistic physics. Thereexists a nonrelativistic counterpart of a CFT that is based on the Schr¨odingergroup [1] and from a phenomenological point of view they could be of relevanceto the description of cold atoms at unitarity [2, 3]. The Schr¨odinger space-timecan be thought of as a deformation of an AdS space-time. This deformation issuch that certain properties of the AdS space-time carry over to the Schr¨odingercase. From the point of view of constructing holographic techniques the inter-pretation that a Schr¨odinger space-time is a deformation of an AdS space-timeis both reassuring that this may be achievable as well as a useful guideline. Onthe other hand it is from the point of view of describing interesting nonrela-tivistic phenomena a bit of a nuisance because one needs to make sure that thephenomenon crucially depends on the deformation and not on the AdS part ofthe Schr¨odinger space-time. For example the properties of the propagators fora free scalar field theory can be inferred from AdS propagators [4, 5, 6]. As an-other example of physics that is insensitive to the deformation we will show thata class of asymptotically Schr¨odinger black holes has the same thermodynamicsas the undeformed asymptotically AdS black hole.What underlies the relation between asymptotically Schr¨odinger space-times(ASch) and asymptotically AdS (AAdS) space-times is a transformation knowninterchangeably as a TsT (T-duality, shift, T-duality) transformation [7] or aMelvin twist [8]. We will refer to it as a TsT transformation. Such a transforma-tion can be thought of as a solution generating transformation relating differentsolutions of type II supergravities. In order to perform a TsT transformationone needs two compact commuting Killing vectors that form the isometries ofa 2-torus. Suppose that the two torus directions are parametrized by ξ and V . The transformation then involves a T-duality along ξ , followed by a shift2 → V + γ ˜ ξ where ˜ ξ parametrizes the T-dual cirlce and a second T-dualityalong ˜ ξ . If the initial field configuration was a solution of type IIA/B super-gravity then the final field configuration will be another solution of type IIA/Bsupergravity. When this solution generating technique is applied to space-timesof the form AAdS × S with ξ parametrizing a circle in the 5-sphere and V parametrizing a compact Killing vector of the AAdS space-time that asymptot-ically becomes null then the resulting space-time is asymptotically Schr¨odinger(with a compact circle that becomes asymptotically null) and the TsT trans-formation is referred to as a null dipole deformation [9]. From now on when wetalk about a TsT transformation we will mean a null dipole deformation. Onthe level of writing down asymptotically Schr¨odinger solutions the fact that the V coordinate was assumed to be compact plays no role, the TsT transformedAAdS space-times also exist when we take V to be non-compact.Up to date the only asymptotically Schr¨odinger space-time that was explic-itly studied is the TsT transformed black brane solution of [9, 10, 11, 12]. Wewill employ the TsT transformation to study a large class of ASch space-timesand characterize properties that are inherited from the AAdS space-time. Adominant role in our analysis is played by the Killing vector N of the AAdS space-time that becomes asymptotically null and that is used in the TsT trans-formation. We will focus on those AAdS space-times that admit N as boundarynull Killing vector and study the associated conformal class of boundaries andinterpret their images under TsT as providing a conformal class of Schr¨odingerboundaries. This allows us to study counterterms for this class of Schr¨odingerboundaries. The main result of this analysis will be that the on-shell action isTsT invariant. We then continue to study thermodynamic properties of TsTtransformed AAdS black holes and show that also thermodynamic quantitiessuch as entropy, temperature and chemical potentials are all TsT invariant. Weend with a discussion of the Schr¨odinger analogue of the Hawking–Page typephase transition. The details of the TsT transformation are summarized in appendix A. Theresult that will be most important to us is the form of the 5-dimensional fields A µ , φ and ¯ g µν that result from applying a TsT transformation to a pure AAdS space-time, i.e. without matter sources, which for the convenience of the readerare copied from appendix A and repeated here A = γe g V µ dx µ , (1) e − = 1 + γ g V V , (2)¯ g µν = e − / (cid:0) g µν − e − A µ A ν (cid:1) , (3)where g µν is an AAdS metric that has ∂ V as a Killing vector. As shown inappendix A the isometries of the TsT transformed metric ¯ g µν are precisely allthe AAdS Killing vectors that commute with the AAdS Killing vector ∂ V .The class of ASch space-times that we will focus on are all space-times thatresult from applying TsT to those pure AAdS space-times that admit a Killingvector N = ∂ V that asymptotically commutes with the Schr¨odinger algebra, so3hat N is asymptotically null. This follows from the fact that the embeddingof the Schr¨odinger algebra into the AdS isometry algebra is essentially unique[13] and that this embedding is such that the Schr¨odinger algebra consists of allthose AdS isometries that commute with a null Killing vector.We emphasize that not every TsT transformation of a pure AAdS space-time gives rise to an ASch space-time. For example the stationary black holesolution (with spherical horizon topology) of [14] can be obtained as a TsT trans-formation of Schwarzschild-AdS using a Killing vector that does not asymptot-ically commute with the Schr¨odinger algebra. Therefore the black hole solutionof [14] is not asymptotically Schr¨odinger. This can also be inferred from thefact that the curvature invariants of this black hole solution do not asymptoteto their Schr¨odinger values.We will next introduce Fefferman–Graham (FG) coordinates [15] and discussthe properties of the class of pure AAdS metrics that give rise to ASch space-times. In FG coordinates the pure AAdS metric g µν is given by ds = g µν dx µ dx ν = dz z + h ab ( z, x ) dx a dx b , (4)where h ab ( z, x ) = 1 z (cid:0) g (0) ab + z g (2) ab + z g (4) ab + O ( z ) (cid:1) , (5)in which g (2) ab and g (4) ab are given by [16] g (2) ab = − (cid:18) R (0) ab − R (0) g (0) ab (cid:19) , (6) g (4) ab = 14 a (4) g (0) ab + 12 g (2) ac g c (2) b + 124 R (0) g (2) ab + t ab . (7)In here the metric g (0) ab is a representative of the conformal class of boundarymetrics and t ab is proportional to the boundary stress energy tensor. The traceand divergence of t ab are given by t aa = − a (4) , and ∇ (0) a t ab = 0 , (8)where a (4) is the anomaly density a (4) = − R (0) ab R ab (0) + 124 R . (9)Indices on g (2) ab , t ab etc. are raised with g ab (0) . The coefficients of all the higherorder terms O ( z ) and beyond are determined in terms of both g (0) ab and g (4) ab .For AAdS space-times g (0) ab is conformally flat.In order to identify the class of FG coordinate systems that are suitable forthe study of ASch space-times let us first consider the case of pure AdS . Inthis case the FG expansion for h ab terminates at the order z and its coefficientin this case is given by [17] g (4) ab = 14 g (2) ac g c (2) b . (10)Imagine that we are in a Poincar´e coordinate system for which ds = dz z + 1 z (cid:0) − dT dV + ( dx i ) (cid:1) , (11)4o that g (2) ab = g (4) ab = 0. In this case we know that performing a TsT transfor-mation with the shift in the V-direction leads to a pure Schr¨odinger space-timein Poincar´e coordinates. We will use this fact to construct a family of FG coordi-nate systems that can be used to study the boundary of a pure Sch space-time.For this purpose we remind the reader about Penrose–Brown–Henneaux (PBH)coordinate transformations [18, 19, 20]. A PBH transformation is a diffeomor-phism ( z, T, V, x i ) → ( z ′ , T ′ , V ′ , x ′ i ) , (12)where the new coordinates ( z ′ , T ′ , V ′ , x ′ i ) are such that in the primed coordinatesystem the metric is again of the FG form ds = dz ′ z ′ + 1 z ′ (cid:16) g ′ (0) ab + z ′ g ′ (2) ab + z ′ g ′ (4) ab (cid:17) dx ′ a dx ′ b , (13)where g ′ (0) ab = e σ g (0) ab and where g ′ (2) ab and g ′ (4) ab depend on σ and g (0) ab aswell as their derivatives. For us the precise form of these transformations is notimportant (see [20, 21] for explicit formulas). The full class of FG coordinatesystems can be obtained by starting with the Poincar´e coordinate system andact on it with an arbitrary PBH transformation. Below we will formulate re-strictions that are suitable to the study of Sch (and later of ASch ) space-times.In the TsT transformation we identify the scalar Φ with some function ofthe metric component g V V , see (2). If we perform an arbitrary coordinatetransformation then in general in the new coordinate system Φ ′ ( x ′ ) = Φ( x ) willnot depend only on g ′ V ′ V ′ . The identification of Φ ′ with g ′ V ′ V ′ is preserved underthe restricted set of coordinate transformations that satisfy g V V = g ′ V ′ V ′ , i.e. V = V ′ + f ( z ′ , T ′ , x ′ i ) , x A = f A ( z ′ , T ′ , x ′ i ) , (14)where x A is any coordinate other than V . These coordinate transformationshave the property that N = ∂ V = ∂ V ′ . Therefore the TsT formulas (1), (2) and(3) apply to all coordinate systems that we can get to by starting with Poincar´eAdS and applying the coordinate transformation (14) to it. This classifies allpossible coordinate systems in which we can use adapted coordinates to performthe TsT transformation. If we additionally choose a time slicing and considerdiffeomorphisms of the type (14) that preserve the time slicing then the resultingclass of coordinate transformations is contained in the class of double foliationpreserving diffeomorphisms of [22].Even though we do not have a coordinate independent way of performingthe TsT operations we expect that the result can be written in a coordinateindependent manner as follows ¯ g µν = e − / (cid:0) g µν − e − A µ A ν (cid:1) , (15) A µ = γe g µρ N ρ , (16) e − = 1 + γ N µ N ν g µν , (17) For example the TsT of global AdS (taken to mean any coordinate system that does notdepend on global AdS time) can be done as follows. First go to an adapted coordinate systemsuch as Poincar´e AdS in light cone coordinates, perform the TsT transformation and applyto the TsT transformed metric (Poincar´e Schr¨odinger) the inverse of the coordinate trans-formation from global to Poincar´e AdS. This will give rise to a time-dependent Schr¨odingercoordinate system that can be thought of as the TsT of global AdS. The time dependence isa consequence of the fact that N and the global AdS Hamiltonian do not commute. N µ is an AdS Killing vector which asymptotically commutes with theSchr¨odinger algebra.We will restrict to FG coordinate systems that have the property that N z = 0so that N is a boundary Killing vector. When N z = 0 we cannot make N µ manifest in a FG coordinate system and therefore cannot directly perform theTsT transformation. We could circumvent this problem by first performing aPBH transformation to a coordinate system in which N z ′ = 0, perform the TsTtransformation and subsequently the inverse of the PBH transformation that wedid in the beginning. On the other hand it is physically plausible that N shouldbe a boundary Killing vector after TsT (and therefore also before), but withoutan independent definition of the Schr¨odinger boundary we cannot claim that itmust be so. In any case for the applications considered here it will suffice totake N z = 0.The family of FG coordinate systems for pure AdS that has N z = 0 for an N whose commutant is the Schr¨odinger algebra can be obtained by starting with(11) and applying to it any PBH transformation with a function σ that doesnot depend on V . We will next discuss the properties of these FG coordinatesystems. We have g (0) V V = 0 , ∂ V g (0) ab = 0 . (18)These properties are preserved under conformal rescalings that do not dependon V . The family of conformally flat boundary metrics g (0) ab each element ofwhich possesses a null Killing vector, ∂ V , can be parametrized by g (0) ab dx a dx b = exp[2 σ ( T, x i )] (cid:0) − dT dV + δ ij dx i dx j (cid:1) . (19)Next we show that all metrics within this conformal class have the propertythat g (2) V V = 0 . (20)Because g (2) V V is conformally invariant under the above restricted class of con-formal rescalings that do not depend on V to prove (20) one simply notes thatit vanishes for say the Poincar´e boundary metric. Further, it also follows that g (2) V c g c (2) V = 0 . (21)If we next consider AAdS space-times then what changes is the form of g (4) ab plus the fact that there will appear terms of order z and higher in h ab .Using the general expression for g (4) ab given in (7) as well as (18), (20) and (21)we see that g (4) V V is given by g (4) V V = t V V , (22)so that g (4) V V is fully given in terms of the
V V component of the AdS boundarystress tensor. This implies that in the FG coordinates we have A µ = γδ µV − γ z t V V δ µV + O ( z ) , (23)Φ = − γ z t V V + O ( z ) . (24)6 TsT invariance
In the previous section we have identified a conformal class of Schr¨odingerboundaries which can be thought of as the TsT image of that subclass of AdSboundaries admitting a null Killing vector. In this section we will study all possi-ble counterterms that are relevant for those ASch solutions that can be writtenas the TsT transformation of an AAdS solution. We restrict the analysis todemanding finiteness of the on-shell action.In the previous section we denoted the 5-dimensional TsT transformed met-ric by ¯ g µν . Here we will continue to use this notation. Barred expressions suchas ¯ R are used to indicate that the metric ¯ g is involved. However we warn thereader that we do not always carefully distinguish when an index is raised with¯ g µν or with g µν . It will be clear from the context which inverse metric has beenused.Consider the action (116) which we repeat here I bulk + I GH = 116 πG N Z M d x √− ¯ g (cid:18) ¯ R − ∂ µ Φ ∂ µ Φ − V (Φ) − e −
83 Φ F µν F µν − A µ A µ (cid:19) + 18 πG N Z ∂ M d ξ p − ¯ h ¯ K , (27)where the potential V is given by V (Φ) = 4 e
23 Φ (cid:0) e − (cid:1) . (28)In this section we will consider adding counterterms so that the on-shell valueof (27) evaluated on (1) to (3) is finite. We will show that the form of thecounterterms is constrained by the fact that the on-shell value of (27) (using acut-off boundary to make the result finite) is TsT invariant as will be provenbelow. We then use Fefferman–Graham coordinates with N a boundary Killingvector for the AAdS metric g µν and the form of the solutions (1) to (3) toconstruct the counterterms.The action (27) evaluated for the class of solutions (1) to (3) is TsT invariant.This means that on-shell we have116 πG N Z M d x √− ¯ g (cid:18) ¯ R − ∂ µ Φ ∂ µ Φ − V (Φ) − e −
83 Φ F µν F µν − A µ A µ (cid:19) + 18 πG N Z ∂ M d ξ p − ¯ h ¯ K = 116 πG N Z M d x √− g ( R + 12) + 18 πG N Z ∂ M d ξ √− hK . (29) We use the following two conventions for the Riemann tensor and the extrinsic curvature R µνρσ = ∂ ρ Γ µνσ + . . . , (25) K µν = h µρ ∇ ρ n ν , (26)where h µρ = δ ρµ − n µ n ρ and where n µ is the outward pointing unit vector at the timelikeboundary. solutions is, upon sub-stituting the solutions (1) to (3), the counterterms must cancel the divergencescoming from the right hand side of (29). This means that counterterms thatcontribute to the on-shell action divergently must equal, upon use of (1) to (3),the counterterms of the usual AAdS action [23] I AAdS = 116 πG N Z M d x √− g ( R + 12)+ 18 πG N Z ∂ M d ξ √− h (cid:18) K − − R ( h ) (cid:19) . (30)We will write down the most general expression consisting of terms that cancontribute to the on-shell action if we evaluate them using (1) to (3) togetherwith a FG coordinate system for g µν in which N is a boundary Killing vector.This will suffice for the purposes of this paper, but for completeness we willindicate at the end of this subsection what kind of terms could be added when N is not a boundary Killing vector. We denote by I the action I = I bulk + I GH + I (0)ct + I (2)ct + I ext , (31)where I bulk + I GH is given in (27), I (0)ct + I (2)ct contains only intrinsic countertermswithout derivatives ( I (0)ct ) or terms that are second order in derivatives ( I (2)ct )and where I ext consists of extrinsic counterterms for the massive vector field A µ and scalar Φ. We find for I (0)ct , I (2)ct and I ext I (0)ct = 18 πG N Z ∂ M d ξ p − ¯ h (cid:0) a + a Φ + a Φ + a A a A a + a Φ A a A a + a ( A a A a ) (cid:1) , (32) I (2)ct = 18 πG N Z ∂ M d ξ p − ¯ h (cid:16) ( b + b Φ + b A a A a ) ¯ R (¯ h ) + b F ab F ab (cid:1) , (33) I ext = 18 πG N Z ∂ M d ξ p − ¯ h (cid:0)(cid:0) c + c Φ + c A b A b (cid:1) ¯ n µ A a F µa + ( c + c Φ + c A a A a ) ¯ n µ ∂ µ Φ) . (34)The counterterms in I (0)ct have also been considered in [10, 11, 24]. Some of theextrinsic counterterms have also been considered in [11].Since p − ¯ h is O ( z − ) we only need to include counterterms that are at mostof order z . In order to see that I (0) ct contains all possible terms note that forour class of solutions both Φ and A a A a are O ( z ) near z = 0. To show that I (2)ct consists of all possible terms that are nonvanishing for our class of solutionswe argue as follows. In general R ( h ) is an order z term. The extra conditions(18) do not change that. This means that also ¯ R (¯ h ) evaluated for the TsTtransformed solution, as given in (134), will be O ( z ). The term F evaluatedfor an arbitrary TsT transformed solution is of order z , but with the use of(18), i.e. for the case of a TsT transformation with a null Killing vector that is8lso a boundary null Killing vector, it can be seen to be of order z . A term suchas (cid:16) ¯ ∇ (¯ h ) a A b + ¯ ∇ (¯ h ) b A a (cid:17) is O ( z ) and so it does not contribute to the on-shellaction. Terms such as ¯ R (¯ h ) ab A a A b are not included because they are by partialintegration equivalent to a combination of F and (cid:16) ¯ ∇ (¯ h ) a A b + ¯ ∇ (¯ h ) b A a (cid:17) plus aterm of the form ( ¯ ∇ (¯ h ) a A a ) which does not contribute. Further a term such as¯ h ab ∂ a Φ ∂ b Φ is O ( z ) and so will never contribute to the on-shell action.The term I ext has been added for the following reason. The TsT identity (29)suggests that the only relevant extrinsic counterterm for our class of solutionsis the GH boundary term. However, due to TsT relations such as (126) and(129) it is possible to write down extrinsic counterterms for both the vectorfield and the scalar field which are such that they cancel divergent terms amongthemselves (we will confirm later that this must be so) rather then with the bulkaction. When this is the case they can still contribute to the on-shell action by afinite amount and secondly they may be very relevant for considerations relatedto the variation of the action since the extrinsic counterterms for the massivevector and the scalar will not cancel each other after variation.Evaluating I (0)ct , I (2)ct and I ext for the TsT transformed solutions we obtain I (0)ct = 18 πG N Z ∂ M d ξ √− h (cid:16) a + (cid:16) − a a − a (cid:17) Φ+ 118 ( a − a + 18 a − a − a + 72 a ) Φ (cid:19) , (35) I (2)ct = 18 πG N Z ∂ M d ξ √− h (cid:20) b R ( h ) + (cid:18) b b − b (cid:19) Φ R ( h ) + (cid:18) b + 14 b (cid:19) F ab F ab (cid:21) , (36) I ext = 18 πG N Z ∂ M d ξ √− h (( c − c ) n µ ∂ µ Φ+2 (cid:18) c − c + 163 c − c + 2 c (cid:19) Φ (cid:19) , (37)where one should think of the integrands as being expanded up to O ( z ). Thiswill involve the FG expansion coefficients g (5) V V and g (6) V V appearing in theexpansion of Φ at the orders z and z , respectively.In order to have equality with the divergent terms in the boundary actionof (30) we need a = − , (38) b = − , (39) c − c = 0 , (40) − a a − a = 0 . (41)The γ independent divergent terms must be equal to the corresponding countert-erms of (30). This fixes the a and b coefficients. The γ dependent divergentterms must cancel among themselves. There are two such terms Φ and n µ ∂ µ Φ.Assuming that in general g (5) V V = 0 these cannot cancel each other so their9oefficients must vanish independently. This leaves us with finite terms propor-tional to γ . Demanding that the counterterms reduce to the counterterms of(30) we set these terms to zero as well. This gives a − a + 18 a − a − a + 72 a )+2 (cid:18) c − c + 163 c − c + 2 c (cid:19) , (42) b − b = 112 , (43) b = 116 . (44)Even though it remains to be seen if by some appropriate definition of ASch space-times we should consider the case where N is not a boundary Killing vectorwe briefly discuss the effect of our assumption that in the FG expansion for g µν we take N to be a boundary null Killing vector. When we drop this assumptionwe have N z = 0 in which case both A and F are of order z (Φ is of coursestill of order z ). This means that we can now take arbitrary powers of A and F and add those as counterterms .In the AAdS case the counterterms of (30) respect the symmetries of theboundary theory. From a bulk space-time perspective this means that theyare invariant under both boundary diffeomorphisms as well as PBH transfor-mations (up to anomalous transformations [32]). When performing the TsTtransformation with a null Killing vector N we can in principle distinguish twocases: 1). N is a boundary vector and 2). N is not a boundary vector. Thefirst case is naturally selected by the TsT transformation and in this case thelocal symmetries are boundary diffeomorphisms and (possibly anomalous) PBHtransformations that do not depend on V . Our point of view has been to restrictto case one without claiming that this is the only possibility and to constructfor this case the local counterterms. In [25] it is suggested that one should allowfor counterterms that are nonlocal in the V direction. It would be interestingto explore these directions further.It should be clear that demanding that there is a well-posed variationalproblem for I lies outside the scope of this work. Rather, we will focus onthermodynamics for which the above counterterm analysis suffices. So far we have considered a rather general class of ASch space-times. In theremainder we will focus on asymptotically Schr¨odinger black holes that can beobtained as the TsT transform of a pure AAdS black hole and study theirthermodynamic properties.The type of AAdS black holes that we will consider have the followingproperties. The black hole space-time possesses three commuting Killing vectors Later on when we discuss black hole solutions we will introduce the grand potential. Thisquantity involves the difference of the (Euclidean) on-shell action for a black hole space-timeand some background space-time such as pure Schr¨odinger. This could therefore be computedusing a background subtraction method. In order that the background subtraction methodgives the same answer as the difference of two renormalized on-shell actions we need to impose(42) to (44). The formulas (131) to (134) change when N is no longer a boundary Killing vector. N and it possesses a Killing horizon that is generated by theKilling vector X that is a linear combination of these three Killing vectors. TheKilling vector X satisfies the usual equations X µ ∇ µ X ν = κX ν and X µ X µ = 0on the horizon, where κ is the surface gravity. It follows that on the horizon X is also hypersurface orthogonal. Below it will be assumed that κ = 0.The horizon properties of the TsT transformed black hole are inherited fromthe metric g µν . To see this note that, as follows from (122), after TsT all threeKilling vectors, and thus in particular X , remain to be Killing vectors. Thenorm of X with respect to ¯ g µν is given by || X || = e − / || X || − e − / ( A µ X µ ) . (45)The horizon of the metric g µν is the hypersurface at which || X || vanishes. Sincealso A µ X µ ∝ N µ X µ vanishes there we find that || X || vanishes at the samelocus as does || X || . To prove that N µ X µ vanishes on the horizon one can usehypersurface orthogonality of X , the fact that N and X commute and that X is null on the horizon. We conclude that the location of the horizon is preservedby TsT. Finally from the fact that on the horizon X µ ¯ ∇ µ X ν = X µ ∇ µ X ν (46)it follows that after TsT the horizon is still a Killing horizon that is generatedby X and that the surface gravity is TsT invariant.From equation (136) it follows that the determinant of the induced metricon the spatial part of the horizon (which is a co-dimension 2 surface having V as one of its tangential directions) is preserved. Hence the horizon area ispreserved under TsT (see also [8]).We assumed that the pure AAdS space-time before TsT possesses threeKilling vectors. Let us denote these as ∂ T , N = ∂ V and ∂ ψ , where T is sometime coordinate. Assume that we Wick rotate the metric before TsT and imposeperiodicities on iT , iV and iψ such that the Wick rotated metric does notcontain any conical singularities. These conical singularities can arise on a 2-dimensional surface that can be obtained as follows. Introduce coordinates thatare diagonal in a radial coordinate, r say, such that the horizon is at g rr = 0.Consider the equation X µ ∂ µ f = 0 where f is a function of the remaining fourcoordinates. This equation defines three independent hyperplanes. Considerthe induced metric on the common intersection of these three hyperplanes andexpand to leading order in r . This defines the induced metric on a 2-dimensionalsurface which upon Wick rotation (and removal of the conical singularity) isknown as the bolt [26], i.e. the fixed point set of the vector X . Now letus consider repeating this calculation for the TsT transformed metric ¯ g µν . Itcan be shown that N µ dx µ vanishes on the bolt (as follows essentially from theproperty that N µ X µ = 0 on the horizon) and hence the induced metric on thebolt conformally rescales under TsT (the rescaling is regular on the horizon aslong as the scalar field is regular on the horizon) and therefore the periodicitiesof iT , iV and iψ needed to remove the conical singularity are TsT invariant.This means that the temperature and chemical potentials associated with aparticular choice of Hamiltonian etc. are TsT invariant. This can be thoughtof as a generalization of the results of [8, 27].In order to do thermodynamics we will use the thermodynamical action for-malism of [28]. The thermodynamical action is the action evaluated for the11omplexified Wick rotated geometry obtained by setting T = − iτ in which theintegration is from the horizon to the boundary. We compute the thermodynam-ical action by integrating I bulk + I GH from the horizon to some cut-off boundaryand use a minimal subtraction method to regularize it. The difference betweenthe thermodynamical action for a TsT transformed black hole space-time anda TsT transformed space-time at the same temperature and chemical poten-tials but without the black hole, assuming that our system is well describedby a grand canonical ensemble, is the grand potential. From the results of theprevious subsection it follows that the grand potential is TsT invariant. One of the original questions that motivated this research was to find out ifthe global Schr¨odinger space-time of [29] in analogy with global AdS wouldshow a Hawking–Page type phase transition [30]. An important difference withglobal AdS is that the spatial sections of the global Schr¨odinger boundary arenon-compact. On the other hand in this case we have a harmonic trappingpotential so it could still be that there is some critical temperature at whicha phase transition occurs from pure thermal Schr¨odinger to an asymptoticallySchr¨odinger black hole. Now that we know that the thermodynamics is TsTinvariant we can answer this question by looking at the same problem beforeTsT, i.e. does there exist a phase transition between thermal plane wave AdSand some AAdS black hole space-time? We will show in this section that thisis indeed the case.In [9] a black hole is constructed that is time independent with respect toa Killing vector that becomes asymptotically the AdS plane wave Hamilto-nian. This solution is obtained by taking an appropriate scaling limit of a5-dimensional Kerr black hole in which the angular velocity is sent to the speedof light. The solution is asymptotic to thermal plane wave AdS at finite chem-ical potential. We will discuss the geometric and thermodynamic properties ofthis black hole and show that there is a phase transition between this black holeand thermal plane wave AdS at finite chemical potential. Then upon perform-ing a TsT transformation to both the black hole solution and pure plane waveAdS and using the TsT invariance of thermodynamics we find the sought forphase transition of the global Schr¨odinger space-time.One may object that we have actually only proven that local thermodynamicquantities are TsT invariant, but that this need not apply to the full saddlepoint approximation of the partition function. However, the conditions that weimpose on the saddle points are such that they always have a Killing vectorthat becomes asymptotically null and so the saddle points exist both before andafter TsT.In the next subsection we will review and slightly generalize the scaling argu-ment of [9]. After this we will study some of its geometric and thermodynamicproperties. At the end of this section we will discuss the phase transition.12 .1 AAdS black holes with an asymptotic null Killingvector Consider the 5-dimensional Kerr-AdS metric [31] in coordinates ( τ, r, θ, φ, ψ )that are stationary at infinity ds = − ∆ ( a,b ) θ (1 + r )Ξ a Ξ b dτ + 2 m Σ a,b ) ∆ ( a,b ) θ dτ Ξ a Ξ b − ω ( a,b ) ! + Σ a,b ) ∆ ( a,b ) r dr + Σ a,b ) ∆ ( a,b ) θ dθ + r + a Ξ a cos θdφ + r + b Ξ b sin θdψ , (47)where ω ( a,b ) = a cos θdφ Ξ a + b sin θdψ Ξ b , (48)Σ a,b ) = r + a sin θ + b cos θ , (49)∆ ( a,b ) r = 1 r ( r + a )( r + b )( r + 1) − m , (50)∆ ( a,b ) θ = 1 − a sin θ − b cos θ , (51)Ξ a = 1 − a , (52)Ξ b = 1 − b . (53)In the expression given in [31] we put the AdS radius l = 1 and replaced θ by π − θ following [9]. The ranges of the angular coordinates are 0 ≤ θ ≤ π/ − π ≤ φ < π and − π ≤ ψ < π . Contrary to the case of asymptotically flat Kerrblack holes, the parameters a and b are restricted to a < b < T and V via φ = T − − a ) V , (54) τ = T , (55)and substitute this into (47). The result is ds = − (cid:18) r + b Ξ b sin θ (cid:19) dT −
41 + a ( r + a ) cos θdT dV + 2 m Σ a,b ) (cid:18) a sin θ − b cos θ (1 + a )Ξ b dT + 2 a cos θ a dV − b sin θ Ξ b dψ (cid:19) + 4(1 − a )1 + a ( r + a ) cos θdV + Σ a,b ) ∆ ( a,b ) r dr + Σ a,b ) ∆ ( a,b ) θ dθ + r + b Ξ b sin θdψ . (56)Since φ is periodic with period 2 π we have that V is periodic with period π − a .13etting a = 1 we obtain ds = − (cid:18) r + b Ξ b sin θ (cid:19) dT − r + 1) cos θdT dV + m ,b ) (cid:18) θ − b cos θ Ξ b dT + 2 cos θdV − b Ξ b sin θdψ (cid:19) + Σ ,b ) ∆ (1 ,b ) r dr + Σ ,b ) Ξ b cos θ dθ + r + b Ξ b sin θdψ . (57)In this metric V runs from −∞ to + ∞ . We can however compactify V byidentifying V ∼ V + 2 πL . After this identification the resulting metric is nolonger obtainable from (47) via some scaling limit. Later we will see that, bychoosing different coordinates, (57) can be thought of as asymptotically planewave AdS with a compact null coordinate. We will refer to the solution (57)with b < a = 1 as the Maldacena–Martelli–Tachikawa (MMT) blackhole [9]. The limit a = 1 can be thought of as a limit in which one rotationapproaches the speed of light. We also refer to [31] for a discussion of the limitin which the rotation of an AAdS black hole is sent to the speed of light.We can define a second scaling limit in which we also send the parameter b to one. This can be done by defining the following limit b = 1 − λ , (58) θ = ρλ , (59)and sending λ → ∞ . The result is (after defining R − = 1 + r ) [9] ds = 1 R (cid:0) − ( R + ρ ) dT − dT dV + dρ + ρ dψ (cid:1) (60)+ 11 − m ( R − R ) dR R + m R (cid:0) dT + 2 dV − ρ ( dψ − dT ) (cid:1) . Both the metric (57) and (60) are asymptotically plane wave AdS and willafter TsT be asymptotically global Schr¨odinger.The difference between the scaling limits to a = 1 and to b = 1 is that inthe former case V can be compactified while in the latter case ρ cannot. Thisshows itself in the fact that for (60) the mass density is finite but the total massis infinite while for (57) the total mass is finite.Finally we can apply even a third scaling limit by starting with (60) anddefining R = ¯ rλ , T = tλ , V = ξλ , ρ = ρλ , m = λ ¯ m . (61)Then after sending λ → ∞ we obtain the AdS black hole (brane) solution thatis asymptotically Poincar´e AdS and whose metric reads ds = 1¯ r (cid:0) − dtdξ + dρ + ρ dψ (cid:1) + ¯ m r ( dt + 2 dξ ) + d ¯ r ¯ r (1 − m ¯ r ) . (62) An alternative to this scaling limit is to perform the coordinate transformation θ = (1 − b ) / ρ and to set b = 1 afterwards. and after TsT (shifting along ξ ) it gives rise to an asymptotically Poincar´e Schr¨odinger black brane. TheTsT-transform of this black brane solution has been studied in [9, 10, 11, 12]. In this subsection we will collect some geometric properties of the MMT blackhole (57) that are relevant for its thermodynamic properties.The horizon at r = r H is given by the largest positive root of the equation g rr = 0, i.e. ( r + b )( r + 1) − mr = 0 . (63)A necessary and sufficient condition for the existence of a horizon is providedby m ≥ (cid:16) b + 8 − b + | b | ( b + 8) / (cid:17) . (64)When (64) is a strict inequality there will be an inner and an outer horizon. Inthe extremal case, i.e. when (64) is an equality, the outer horizon is infinitelyfar away from any point in the space-time. It then looks as if both horizonshave coalesced at r H = − b | b | b + 8) / . (65)This is the lowest possible value r H can attain. When b = 0 there is only onepositive real root given by r = r H ≡ p − √ m assuming that 2 m > g rr = 0 the metric also has a stationary limitsurface at g T T = 0. When g T T > ∂∂T is no longer timelike.To see that the MMT black hole is asymptotically plane wave AdS performthe following coordinate transformation1 + r = 12 R (cid:0) b ( ρ + R ) + F ( R, ρ ) (cid:1) , (66)cos θ = 12Ξ b R (cid:0) b ( ρ + R ) − F ( R, ρ ) (cid:1) , (67)where F ( R, ρ ) = p (1 + Ξ b ( ρ − R )) + (2Ξ b ρR ) , (68)whose inverse is 1 R = (1 + r ) cos θ , (69) ρ R = r + b Ξ b sin θ . (70)In this coordinate system the metric (57) is asymptotically ds = 1 R (cid:0) − ( ρ + R ) dT − dT dV + dρ + ρ dψ + dR (cid:1) + 2 mR dR (1 + Ξ b ρ ) + mR b ρ ) (cid:16) (1 + 2 ρ ) dT + 2 dV − bρ dψ (cid:17) + O ( R ) . (71)15n order to study the metric near the horizon we define the Eddington–Finkelstein (EF) coordinates u ∓ , v ∓ and w ∓ via du ∓ = dT ± ( r + b )( r + 1) r ∆ (1 ,b ) r dr , (72) dv ∓ = dV ± ( r + b )( r − r ∆ (1 ,b ) r dr , (73) dw ∓ = dψ ± b ( r + 1) r ∆ (1 ,b ) r dr , (74)where each constant set of ( u + , v + , w + ) values describes an outgoing null geodesicwhile each constant set of ( u − , v − , w − ) values describes an ingoing null geodesic.In these EF coordinates the metric (57) takes the form ds = g T T du ∓ + 2 g T V du ∓ dv ∓ + 2 g T ψ du ∓ dw ∓ + g V V dv ∓ +2 g V ψ dv ∓ dw ∓ + g ψψ dw ∓ ± θdrdv ∓ ± (cid:18) b Ξ b sin θ (cid:19) drdu ∓ ∓ b Ξ b sin θdrdw ∓ + g θθ dθ . (75)Slices of constant r , including the horizon at r = r H , contain a non-compactdirection parametrized by θ . This is not so in the case of the 5-dimensionalKerr-AdS black hole (47) with a < a = 1 in(56) two things change: i). The m -independent term proportional to dV in(56) vanishes, so that for large r the coordinate V becomes a null coordinate.Secondly, because of the behavior of ∆ ( a,b ) θ as a goes to one the physical distancefrom a point P = ( T , V , r , θ , ψ ) to a point Q = ( T , V , r , π/ , ψ ) diverges.Therefore we can find two points on the horizon r = r H whose physical distanceis infinite. This is not possible for a <
1. After setting a = 1 and compactifyingthe V coordinate we can think of the spatial sections of the horizon as a surfacethat is homeomorphic to a 3-sphere that has been infinitely stretched in the θ direction. This can be confirmed by studying the metric in EF coordinates onslices of constant u − and r = r H . Despite the fact that the horizon contains anon-compact direction the area of the horizon is finite and given by A = 2 π L Ξ b ( r H + 1)( r H + b ) r H . (76)The normal to the horizon, the gradient g µν ∂ ν r , is proportional to the Killingvector X given by X = ∂∂T − µ N L ∂∂V − Ω ∂∂ψ . (77)When r H ≥ X is timelike outside the horizon. When 0 < r H < X is spacelike.Working in the EF coordinate system we can compute the surface gravity κ via X µ ∇ µ X ρ = κX ρ , (78)16valuated on the horizon. The result is that κ = 1 r H r H + b ( r H − r H + b . (79)In order to avoid conical singularities upon Wick rotating T = − iτ we mustmake the identifications τ ∼ τ + β , (80) iV ∼ iV + βLµ N , (81) iψ ∼ iψ + β Ω , (82)where the inverse temperature β and chemical potentials µ N and Ω are β = 2 πr H ( r H + b )2 r H + b ( r H − , (83) µ N = − L r H − r H + 1) , (84)Ω = − b ( r H + 1) r H + b . (85)The temperature has no local extrema in particular ∂β/∂r H < The Euclidean action or thermodynamical action defined by iI = − I E afterWick rotating T = − iτ is given by I E = − πG N Z M d x √ g ( R + 12) − πG N Z ∂ M d ξ √ h (cid:18) K − − R ( h ) (cid:19) . (86)The Wick rotation T = − iτ leads to a complex geometry for which the aboveEuclidean action is real-valued. The action is integrated from the horizon tothe boundary. The Wick rotated (complex) geometry is a saddle point of thisaction.We define ∆ I E as the difference between the renormalized on-shell actionevaluated for the MMT black hole in a given coordinate system and the renor-malized on-shell action evaluated for the pure AdS metric obtained by setting m = 0 in that coordinate system. Even though the on-shell action transformsanomalously under PBH transformations [32] the difference ∆ I E is the same inall coordinate systems [33]. Assuming that we can treat the thermodynamicsas the infinite volume limit of a grand canonical ensemble the grand potentialΦ G is given by Φ G = β − ∆ I E . We find for ∆ I E the following expression∆ I E = − πLβ b G N ( r H − r H + b ) r H . (87)17he difference ∆ I E satisfies the usual quantum statistical relations (cid:18) ∂ ∆ I E ∂β (cid:19) µ N , Ω = E − µ N N − Ω J , (88) (cid:18) ∂ ∆ I E ∂ Ω (cid:19) β,µ N = − βJ , (89) (cid:18) ∂ ∆ I E ∂µ N (cid:19) β, Ω = − βN , (90)where E is the energy, N the momentum conjugate to µ N and J the angularmomentum conjugate to Ω. The on-shell action ∆ I E ( β, µ N , Ω) has no stationarypoints.The quantities E , N and J can also be expressed in terms of the Brown–York(BY) charges Q ∂ T , Q ∂ V and Q ∂ ψ as E = Q ∂ T − Q ∂ T ( m = 0 solution) , (91) N = LQ ∂ V , (92) J = Q ∂ ψ , (93)where Q ∂ T ( m = 0 solution) is the Casimir energy computed in a coordinatesystem that has the same conformal boundary as the coordinate system in which Q ∂ T is computed. The conserved BY charges Q χ associated with a Killing vector χ that is also a boundary Killing vector are defined as [34, 23] Q χ = Z Σ d Σ √ σu a χ b T ab , (94)where the BY stress tensor is given by T ab = 18 πG N (cid:20) Kh ab − K ab + l (cid:18) G ab − l h ab (cid:19)(cid:21) . (95)In the expression for Q χ by Σ we denote an equal time surface at the boundary,i.e. Σ = Σ T ∩ ∂ M in which Σ T is an equal time T surface of the space-time M .The induced cut-off boundary metric on Σ is denoted by σ ij . The vector u µ denotes a future-directed timelike unit normal to the surface Σ T . Specificallylet T µ be some timelike vector which provides us with a time orientation andwhich satisfies T µ ∂ µ T = 1 then we have u µ = − g µν ∂ ν T p − g T T . (96)It has been shown in [35, 33] that Q ∂ T satisfies the quantum statisticalrelation Q ∂ T = β − S + µ N N + Ω J + β − I E , (97)From this it follows that in terms of E and Φ G we have the thermodynamicalrelation E = β − S + µ N N + Ω J + Φ G . (98)The energy E satisfies the first law of thermodynamics dE = β − dS + µ N dN + Ω dJ . (99)18he importance of writing the quantum statistical relation in terms of E andΦ G comes from the fact that E and Φ G (and not Q ∂ T and I E ), which are theonly two quantities that we need to compute on the boundary ( N and J can becomputed as surface integrals over the spatial sections of the horizon), are eachindependent of the choice of representative of the conformal boundary metric.Hence (98) is form invariant under PBH transformations that preserve the threeboundary Killing vectors ∂ T , ∂ V and ∂ ψ [33].The thermodynamics of (60) and (62) can be obtained by applying theappropriate scaling limits to the thermodynamics of the MMT black hole.We have tacitly assumed that in the computation of the conserved charges wehave been using a frame that does not rotate at infinity. Suppose we were usinga frame that does rotate at infinity. Such a frame can be obtained by startingwith (57) and performing the following coordinate transformation T ′ = T and ψ ′ = ψ − bT where the primed coordinates refer to the rotating frame. TheKilling vector ∂ T with respect to which we defined the mass in the frame thatis not rotating at the boundary transforms as ∂ T = ∂ T ′ − b∂ ψ ′ and further ∂ ψ = ∂ ψ ′ . Rewriting the Killing vector X that beomes null on the horizon interms of the primed Killing vectors we find that (98) becomes E ′ = β − S + µ N N + Ω ′ J + Φ G , (100)where E ′ is Q ∂ T ′ − Q ∂ T ′ ( m = 0 solution) and where Ω ′ = Ω + b . Because therotating and non-rotating frames have boundary metrics that are diffeomorphicwe have that Q ∂ T ′ ( m = 0 solution) is equal to Q ∂ T ( m = 0 solution). The energy E ′ is related to E via E ′ = E + bJ . The value of the on-shell action is the samein both coordinates systems (also this follows from the fact that the boundarymetrics in both frames are diffeomorphic). It can be checked that E ′ does notsatisfy a first law of thermodynamics like (99) with Ω replaced by Ω ′ . Insteadwe have dE ′ = β − dS + µ N dN + Ω ′ dJ + Jdb . (101)Since the grand potential Φ G is the same in the rotating and the non-rotatingframes we also have d Φ G = S dββ − N dµ N − Jd Ω ′ + Jdb = S dββ − N dµ N − Jd Ω . (102)We conclude that the grand potential in the rotating frame depends on Ω andnot on Ω ′ . See also [36, 35] for a discussion of the choice of Ω in the rotatingframe. The entropy S is given by S = β (cid:18) ∂ Φ G ∂β (cid:19) µ N , Ω = A G , (103)where A is the horizon area given in (76) and the specific heat is given by − β (cid:18) ∂S∂β (cid:19) µ N , Ω = π L ( r H + 1)(2 r H + b ( r H − r H − r H − b ( r H + 1))2 G N Ξ b r H ( r H − r H − b ( r H + 1)) . (104)19 - PSfrag replacements r H b Figure 1: Specific heat and grand potential as a function of the black holeparameters b and r H . The green and blue shaded areas are regions where thespecific heat is positive and where the grand potential is negative and positive,respectively. The red curve denotes the extremal black holes so that below thiscurve there are no black hole solutions. The white areas are places where thespecific heat is negative and the grand potential is positive.In figure 1 we have indicated by green the region where the specific heat ispositive and where the grand potential Φ G is negative, by blue the region wherethe specific heat is positive and the grand potential Φ G is positive and finallyby white the regions where the specific heat is negative and Φ G is positive.Further the red curve is the line of extremality at which the specific heat and thetemperature vanish. Below this line there are no black hole solutions. Regionswhere the specific heat is negative are thermodynamically unstable while regionswhere the specific heat is positive are locally thermodynamically stable.We will assume that besides the MMT black hole the only other saddle pointof the thermodynamical action (86) that has the same values for the temperatureand chemical potentials is plane wave AdS with iT ∼ iT + β , iV ∼ iV + βLµ N and iψ ∼ iψ + β Ω and with the V coordinate compactified V ∼ V + 2 πL .The grand potential Φ G is the difference between the on-shell action for theWick rotated MMT black hole and the on-shell action for an equal temperaturethermal plane wave AdS space-time at finite chemical potentials equal to thoseof the MMT black hole, which we simply refer to as thermal plane wave AdS .When r H > β − ( r H = 1) and chemicalpotentials µ N < r H <
1, thermal plane wave AdS withchemical potentials µ N > G > X µ that becomes null at thehorizon is everywhere timelike outside the horizon when r H >
1. In [37] it has20een proven that in this case there is no superradiant scattering of waves inci-dent on the black hole. When r H < X µ becomes spacelikein a large region of the space-time that lies strictly outside the horizon and thatextends to the boundary. It follows that there does not exist any Killing vectorfield which remains timelike everywhere outside the horizon. Subsequently, inthis case superradiant scattering does occur and since the space-time is asymp-totically AdS, superradiant modes are reflected back into the bulk either due tothe gravitational potential well at infinity or due to boundary conditions. Thisleads to a classical instability of black holes with r H < and the MMT black hole have a TsT trans-formed version and conversely since any saddle point after TsT which has thesame Killing vectors has a description before TsT as an AAdS metric, weshould observe the same behavior with thermal plane wave AdS replaced byglobal Sch and the MMT black hole replaced by its TsT transformed version. The way we employed the TsT transformation is sort of in reverse. We useit to parametrize the metric of an ASch space-time in terms of a metric thatis defined on a space-time (in this case pure AAdS ) that, unlike the ASch space-time, has a conformal boundary. A central role in the TsT transforma-tion is played by the vector N = ∂ V . We have identified a conformal class ofSchr¨odinger boundary metrics that are the TsT transform of those AdS bound-aries that possess a null Killing vector. The TsT transformation then guaranteesthat all AAdS properties that ‘commute’ with N carry over to the ASch case.For Schr¨odinger boundary metrics that can be viewed as the TsT transform of anAdS boundary metric possessing a null Killing vector we constructed local coun-terterms. These counterterms are invariant under boundary diffeomorphisms aswell as PBH transformations that respect the property that N is a boundary nullKilling vector (possibly up to some nonrelativistic conformal anomaly). Thesetransformations together form the local symmetries of the boundary theory.Another attractive feature of these Schr¨odinger boundaries is that in the pureSch case the V coordinate can be thought of as parametrizing the light likelines [13] which in turn form the boundaries of the light cones of the space-time.The Galilean-like causal structure is then naturally inherited by boundaries thathave V as a tangential coordinate.It is tempting to state that the near boundary region of any ASch space-time can be thought of as the TsT transformation of the near boundary regionof an AAdS space-time and it would be interesting to see if the TsT relation21etween ASch and AAdS solutions can be relaxed in the interior of the space-time. When relaxing some of the properties of the TsT transformation it mayno longer be a solution generating technique, but it could provide the startingpoint for a suitable Ansatz for solving the equations of motion (118) to (120).One possibility is to relax the condition that N is a global Killing vector ofthe space-time to the condition that it is only an asymptotic Killing vector. Suchsolutions break particle number in the bulk of the space-time. These space-timescan be physically interesting for the following reason. The Schr¨odinger invari-ant system of fermions at unitarity possesses a ground state that spontaneouslybreaks particle number (see [38] for a nice review of cold atom systems and see[39] for a discussion in the context of holography) and so it would be interest-ing to study geometries that break particle number away from the boundary.Another (perhaps related to the presence of a Killing vector) special featureof the TsT transformed AAdS metrics is the relation between the mass term A and the potential V (Φ) which in the bulk action cancel each other so asto leave a pure cosmological constant. We could try to relax this condition bydemanding that this happens only near the boundary but not in the interior ofthe space-time.We further showed that the on-shell action as well as the thermodynamicproperties of black hole space-times that relate to the horizon such as tem-perature, chemical potentials and entropy are all TsT invariant. We used thisto show that there is a Hawking–Page type phase transition between globalSchr¨odinger space-time at finite temperature and chemical potentials and theTsT transformed MMT black hole. It would be interesting to find a dual fieldtheory interpretation (much like as was done for the usual Hawking–Page phasetransition in [40]) for this phase transition and in particular to explain what therole of the harmonic trapping potential is.Hand in hand with constructing more general ASch solutions another inter-esting next step would be the formulation of a well-posed variational problemand the related question as to what the right notion of a boundary stress en-ergy tensor should be (see [11, 24, 25] for some proposals) and to see what theassociated algebra of conserved charges is. We hope to report on some of thesequestions in the future. Acknowledgments
We wish to thank Emiliano Imeroni for collaboration in initial stages of this re-search and for many useful discussions. Further we wish to thank Bom Soo Kim,Balt van Rees, Kostas Skenderis, Marika Taylor and Daiske Yamada for usefuldiscussions. Finally we express our gratitude to Matthias Blau for many usefuldiscussions and careful reading of this manuscipt. This work was supported inpart by the Swiss National Science Foundation and the “Innovations- und Koop-erationsprojekt C-13” of the Schweizerische Universit¨atskonferenz SUK/CRUS.
A TsT transformations
We review the details of a TsT transformation which has the direction for theT-duality in the 5-sphere and the direction for the shift in the pure AAdS ds E = g µν dx µ dx ν + ds S , (105)where ds S is the metric on a unit radius 5-sphere and where g µν dx µ dx ν is anysolution to the 5-dimenional Einstein equations G µν − g µν = 0 . (106)The above 10-dimensional metric is a solution of type IIB supergravity (Freund–Rubin compactification) if we also switch on a 5-form flux given by F = (1 + ⋆ ) G , (107)in which G = 4 ǫ µ ...µ dx µ ∧ · · · ∧ dx µ with ǫ µ ...µ = √− ge µ ...µ denotingthe volume form on g µν dx µ dx ν , where e µ ...µ is the Levi-Civit`a symbol. We usea normalization of the 5-form such that the 10-dimensional Einstein equationsread G MN = 196 F MP QRS F N P QRS + . . . . (108)All other type IIB fields are zero.The 5-sphere metric can be written, using a Hopf fibration, as ds S = dη + ds CP , (109)in which η is such that dη/ CP .Writing η = dξ + P the coordinate ξ parameterizes a Killing direction, i.e. ∂ ξ is a Killing vector (the Reeb vector).Let ∂ V denote a Killing vector of the metric g µν dx µ dx ν . Performing a T-duality along ξ with the T-dual circle being parametrized by ˜ ξ , subsequentlyshifting V → V + γ ˜ ξ and performing a second T-duality along ˜ ξ leads to thefollowing solution of type IIB supergravity which in Einstein frame reads [9, 12] ds E = e − Φ / (cid:0) g µν − e − A µ A ν (cid:1) dx µ dx ν + e − Φ / ds CP + e / η , (110) B = A ∧ η , (111) F = 4 √− g e µ ...µ dx µ ∧ · · · ∧ dx µ + 4 η ∧ Vol( CP ) , (112)where B denotes the NS-NS 2-form and where the 5-dimensional vector A andscalar Φ (dilaton) are given by A = γe g V µ dx µ , (113) e − = 1 + γ g V V . (114)The remaining IIB fields (RR potentials) are zero.The idea is now to reduce the Einstein frame type IIB action over thesquashed 5-sphere down to five dimensions. In doing so we obtain the 5-dimensional action and the 5-dimensional Einstein frame metric which we areinterested in. Using the results of [9] this procedure gives rise to the following5-dimensional Einstein frame metric ds = ¯ g µν dx µ dx ν = e − / (cid:0) g µν − e − A µ A ν (cid:1) dx µ dx ν . (115)23his metric together with the vector (113) and scalar field (114) solve the equa-tions of motion coming from the following action I bulk + I GH = 116 πG N Z M d x √− ¯ g (cid:18) ¯ R − ∂ µ Φ ∂ µ Φ − V (Φ) − e −
83 Φ F µν F µν − A µ A µ (cid:19) + 18 πG N Z ∂ M d ξ p − ¯ h ¯ K , (116)where the potential V is given by V (Φ) = 4 e
23 Φ (cid:0) e − (cid:1) . (117)This potential has an absolute minimum at Φ = 0 where it assumes the value −
12. For large negative values of Φ it asymptotes to zero and for large positivevalues it blows up.The bulk equations of motion that follow from this action are¯ G µν = 43 (cid:18) ∂ µ Φ ∂ ν Φ − ∂ ρ Φ ∂ ρ Φ¯ g µν (cid:19) − V (Φ)¯ g µν (118)+ 12 e −
83 Φ (cid:18) F ρµ F ρν − F ρσ F ρσ ¯ g µν (cid:19) + 4 A µ A ν − A ρ A ρ ¯ g µν , √− ¯ g ∂ µ (cid:18) √− ¯ ge −
83 Φ F µν (cid:19) = 8 A ν , (119)1 √− ¯ g ∂ µ (cid:0) √− ¯ g∂ µ Φ (cid:1) = 38 V ′ (Φ) − e −
83 Φ F µν F µν . (120)From the equation for A µ one derives that ∂ µ (cid:0) √− ¯ gA µ (cid:1) = 0 . (121)The reduction over the squashed 5-sphere is consistent and forms a specialcase of dimensional reductions over squashed Sasaki–Einstein manifolds [41, 42,43, 44, 45].We discuss the isometries that are preserved by the TsT transformation.Suppose that K is a Killing vector of the metric g µν . The Lie derivative of ¯ g µν along K is given by L K ¯ g µν = γe − / (cid:18) − e − A µ A ν A ρ + (cid:18) g µρ A ν + g νρ A µ − g µν A ρ (cid:19)(cid:19) [ ∂ V , K ] ρ . (122)Hence, only those Killing vectors K that commute with ∂ V are also Killingvectors of ¯ g µν . Conversely, given a Killing vector of the metric ¯ g µν then it mustalso be a Killing vector of g µν . To see this note that0 = ¯ g ρν ¯ ∇ µ K ρ + ¯ g ρµ ¯ ∇ ν K ρ = g ρν ∇ µ K ρ + g ρµ ∇ ν K ρ +terms proportional to γ . (123)24ince the first term on the right hand side only involves the AdS metric, whichcannot depend on γ , this term has to vanish by itself. We conclude that thecommutant of the Killing vector ∂ V used in the TsT transformation forms thecomplete isometry algebra of the metric ¯ g µν .We collect some TsT transformation formulas. In the bulk we have¯ g µν = e / g µν + e − / g µρ g νσ A ρ A σ , (124) √− ¯ g = e − / √− g , (125)¯ g ρσ A σ F ρµ = 2 e / ∂ µ Φ , (126)¯ g µσ ¯ g ρτ F µρ F στ = e / F µρ F στ g µσ g ρτ + 8 e / g ρτ ∂ ρ Φ ∂ τ Φ , (127)¯ g µν ∂ µ Φ ∂ ν Φ = e / g µν ∂ µ Φ ∂ ν Φ , (128)¯ g µν A µ A ν = e / (1 − e ) , (129)¯ R = e / (cid:18) R + (cid:18)
43 + 2 e (cid:19) g µν ∂ µ Φ ∂ ν Φ+ 14 e − F µν F ρσ g µρ g νσ + 23 (cid:3) Φ (cid:19) . (130)and on the boundary we have¯ h ab = e / h ab + e − / h ac h bd A c A d , (131) p − ¯ h = e − Φ / √− h , (132)¯ K = e Φ / (cid:18) K − n µ ∂ µ Φ (cid:19) , (133)¯ R (¯ h ) = e / (cid:20) R ( h ) + 14 e − F ab F cd h ac h bd + (cid:18)
43 + 2 e (cid:19) h ab ∂ a Φ ∂ b Φ (cid:21) . (134)In deriving equations (124) to (130) we used (113) to (115) as well as the factthat ∂ V is a bulk Killing vector. In deriving (131) to (134) we used that theboundary metric is ¯ h ab = e − / h ab − e − / A a A b with A a = γe h V a and e − = 1 + γ h V V and assumed that ∂ V is a Killing vector of h ab . Theseformulas are true for any TsT transformation and not just the ones that giverise to asymptotically Schr¨odinger space-times.In proving (125) we used that det(1 − C ) = 1 − Tr C for a matrix C thatis the product of two vectors. In establishing (132) it is important that V istangential to the boundary. In that case we can say the following. Suppose thatwe would compute the induced metric on a co-dimension n surface that has V as one of its tangential directions. The induced metric ¯ h ( n ) ij is¯ h ( n ) ij = δ µi δ νj ¯ g µν , (135)where i = ( V, I ) with I = 1 , . . . , − n . Then we find for the determinantdet(¯ h ( n ) ij ) of the induced metricdet(¯ h ( n ) ij ) = e n − / det( h ( n ) ij ) , (136)where h ( n ) ij = δ µi δ νj g µν . 25 eferences [1] Y. Nishida and D. T. Son, Nonrelativistic conformal field theories , Phys.Rev. D (2007) 086004, arXiv:0706.3746 [hep-th].[2] D. T. Son, Toward an AdS/cold atoms correspondence: a geometric re-alization of the Schroedinger symmetry , Phys. Rev. D (2008) 046003,arXiv:0804.3972 [hep-th].[3] K. Balasubramanian and J. McGreevy, Gravity duals for non-relativisticCFTs , Phys. Rev. Lett. (2008) 061601, arXiv:0804.4053 [hep-th].[4] A. Volovich and C. Wen,
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