Non-Relativistic Strings and Limits of the AdS/CFT Correspondence
aa r X i v : . [ h e p - t h ] M a y Non-Relativistic Strings and Limits of the AdS/CFT Correspondence
Troels Harmark , Jelle Hartong , Niels A. Obers ∗ The Niels Bohr Institute, University of CopenhagenBlegdamsvej 17, 2100 Copenhagen Ø, Denmark and Institute for Theoretical Physics and Delta Institute for Theoretical Physics, University of AmsterdamScience Park 904, 1098 XH Amsterdam, The Netherlands
Using target space null reduction of the Polyakov action we find a novel covariant action for stringsmoving in a torsional Newton–Cartan geometry. Sending the string tension to zero while rescalingthe Newton–Cartan clock 1-form, so as to keep the string action finite, we obtain a non-relativisticstring moving in a new type of non-Lorentzian geometry that we call U (1)-Galilean geometry. Weapply this to strings on AdS × S for which we show that the zero tension limit is realized by theSpin Matrix theory limits of the AdS/CFT correspondence. This is closely related to limits of spinchains studied in connection to integrability in AdS/CFT. The simplest example gives a covariantversion of the Landau-Lifshitz sigma-model. Introduction
Non-Lorentzian geometry has appeared in recent yearsin a wide variety of settings such as non-AdS holography[1, 2], effective actions of non-relativistic field theoriesincluding those relevant for the fractional quantum Halleffect [3–6] and gravity theories with non-relativistic localsymmetries such as Hoˇrava–Lifshitz gravity and Chern–Simons gravity theories on non-relativistic algebras [7–9].By non-Lorentzian geometry we mean a manifold thatis locally flat in the sense of a kinematical principle of rel-ativity that is different from Einstein’s equivalence prin-ciple. Examples are Newton–Cartan and Carrollian ge-ometries whose tangent space structure is dictated by theBargmann (centrally extended Galilei) and Carroll (zerospeed of light contraction of Poincar´e) algebras.There is considerable literature on non-relativisticstrings, see e.g. [10–13]. Of particular relevance forus will be the non-relativistic string spectra and associ-ated sigma-models, such as the Landau–Lifshitz model,observed before in the AdS/CFT context [11]. Besidesthe stringy Newton-Cartan geometry found in [12], theseworks are non-covariant with regards to the world-sheetand target space geometry. A natural question is thusto what extent non-Lorentzian geometries are importantfor sigma-models of non-relativistic strings.In this letter we show that target space null-reductionof the Polyakov action leads to a novel covariant ac-tion for the propagation of non-relativistic strings in a(torsional) Newton-Cartan target space. Furthermore,we uncover that taking a second non-relativistic limit,that affects both the target space and the world-sheet,leads to a new class of sigma-models that describes non-relativistic strings moving in a novel non-Lorentzian ge-ometry that we refer to as U (1)-Galilean geometry.Remarkably, we show that for a string on AdS × S ,the second non-relativistic limit corresponds to the SpinMatrix theory limits of the AdS/CFT correspondence.Spin Matrix theories are quantum mechanical theoriesthat arise as limits of N = 4 SYM on R × S [14]. Givena unitarity bound E ≥ J of N = 4 SYM, where J is a linear combination of commuting angular momenta and R -charges such that states with E = J are supersymmet-ric, one sends E − J and the ’t Hooft coupling λ = 4 πg s N to zero, keeping the ratio ( E − J ) /λ and N fixed. It isclear from the relativistic magnon dispersion relation [15] E − J = q λπ sin p − AdS × S considered in connection with integrabilityof the AdS/CFT correspondence, starting with Kruczen-ski [11]. The difference is that the Kruczenski limit doesnot decouple higher order terms in the string tension.However, the leading part of the sigma-model is the sameas for SMT.From the SMT or Kruczenski limit one gets the well-known Landau-Lifshitz sigma-model in the simplest case.Other limits give similar sigma-models that also are clas-sical limits of nearest-neighbor spin chains [17–20]. Usingthe results of this letter, these sigma-models can be madecovariant, thus providing a new interpretation in termsof non-relativistic string theory. Strings on torsional Newton–Cartan geometry
The action of a non-relativistic particle moving in atorsional Newton–Cartan (TNC) geometry can be ob-tained by null reduction of the action of a relativisticmassless particle [21–23]. Here we will do something sim-ilar for the target space null reduction of the Polyakovaction for a relativistic string.Consider the Polyakov action, S = Z d σ L = − T Z d σ √− γ γ αβ g αβ , (1)with g αβ = ∂ α X M ∂ β X N G MN where G MN is the d + 2dimensional target space metric. Here ∂ α is the deriva-tive with respect to the world-sheet coordinates σ α with α = 0 ,
1, and T is the string tension. We consider closedstrings hence σ ∼ σ + 2 π . The Virasoro constraints are g αβ − γ αβ γ γδ g γδ = 0 . (2)Assume that the target space has a null Killing vector ∂ u . The most general metric with this property is G MN dx M dx N = 2 τ ( du − m ) + h µν dx µ dx ν , (3)where µ, ν = 0 , , ..., d , M = ( u, µ ) and τ = τ µ dx µ , m = m µ dx µ , det h µν = 0. The tensors τ µ , m µ and h µν areindependent of u . This decomposition of the line elementadmits the following local symmetries δτ µ = L ξ τ µ , δm µ = L ξ m µ + ∂ µ σ + λ a e aµ ,δh µν = L ξ h µν + τ µ λ a e aν + τ ν λ a e aµ , (4)where we defined e aµ via h µν = δ ab e aµ e bν with a = 1 , . . . , d .The transformation with parameter σ is a U (1) gaugetransformation that acts on u as δu = σ . The transfor-mation with parameter λ a is known as a local Galileanor Milne boost. The Lie derivatives along ξ µ correspondto the infinitesimal d + 1 dimensional diffeomorphisms.The fields and transformations (4) are those of torsionalNewton–Cartan (TNC) geometry [5–7, 24, 25] in agree-ment with the known fact that null reductions give riseto TNC geometry [21, 22, 26, 27].So far we are still describing a relativistic string in abackground with a null isometry. To turn this into anon-relativistic string moving in a TNC background weneed to remove the field X u from the description. Thisis achieved by putting the momentum P αu along u , P αu = ∂ L ∂ ( ∂ α X u ) = − T √− γ γ αβ τ β , (5)on-shell, i.e. imposing ∂ α P αu = 0, here defining τ β = ∂ β X µ τ µ as the pullback of τ µ . This requires considering P αu (as opposed to ∂ α X u ) as an independent variable.We thus perform the Legendre transformationˆ L = L − P αu ∂ α X u , (6)where ˆ L is the Lagrangian for the remaining embeddingcoordinates X µ whose dependence on P αu is such that ∂ ˆ L ∂P αu = − ∂ α X u . (7)We will use (5) to solve for γ αβ in terms of P αu and τ α .The solution to (5) can be written as √− γ γ αβ = e (cid:0) − v α v β + e α e β (cid:1) , (8)where we defined e = det( τ α , e α ) = T P αu τ α and e α = e αβ P βu T , v α = − P αu P γu τ γ , e α = − T e αβ τ β P γu τ γ . (9) Here e αβ and e αβ denote Levi-Civita symbols with e = − e = 1. Together with τ α the vectors (9) form anorthonormal system: v α τ α = − v α e α = 0, e α τ α = 0and e α e α = 1. We assume that P αu τ α = 0.The action associated with ˆ L can be written asˆ S = Z d σ ˆ L = − T Z d σe (cid:0) − v α v β + e α e β (cid:1) ¯ h αβ (10)where ¯ h αβ = ∂ α X µ ∂ β X ν ¯ h µν with ¯ h µν = h µν − τ µ m ν − τ ν m µ . Further m α and h αβ are the pullbacks of m µ and h µν . From (7) we obtain m α − τ α (cid:0) e γ e δ + v δ v γ (cid:1) h γδ + e α v γ e δ h γδ = ∂ α X u , (11)which is equivalent to the Virasoro constraints (2) for astring in the background with a null isometry (3). Thisfollows from contracting (2) with all combinations of e α and v α . Furthermore from (7) it follows that ∂ α ∂ ˆ L ∂P βu − ∂ β ∂ ˆ L ∂P αu = 0 , (12)which is independent of X u .We are now going to put P αu on-shell, i.e. impose ∂ α P αu = 0 which is equivalent to setting ∂ α e β − ∂ β e α = 0.We will write P αu = T e αβ e β where locally e β = ∂ β η andsubstitute this into the action ˆ S . This leads to the fol-lowing Lagrangian for X µ and η ,ˆ L = T − e αβ m α ∂ β η + e αα ′ e ββ ′ ( ∂ α ′ η∂ β ′ η − τ α ′ τ β ′ )2 e γγ ′ τ γ ∂ γ ′ η h αβ ! . (13)The equation of motion of η gives the constraint (12).The action (13) is invariant under world-sheet diffeomor-phisms δX µ = ξ α ∂ α X µ and δη = ξ α ∂ α η generated by ξ α ,as well as under all local symmetries of the target spaceTNC geometry that are generated by σ and λ a in (4).There can also be global symmetries generated by K µ forthose ξ µ = K µ in (4) for which 0 = δτ µ = δm µ = δh µν .Assume that the target space clock 1-form τ is closed.Write this as τ µ = ∂ µ X . We can then choose the gauge σ = πTP X and η = P πT σ with P = R π P u dσ theconserved total momentum. In this gauge the action (13)on a flat TNC background with m µ = 0, τ µ = δ µ and h µν = δ ab δ aµ δ bν reproduces the standard non-relativisticstring action which has 1+1 dimensional world-sheetPoincar´e symmetry [28]. This latter action was also stud-ied in [12]. However the coupling to the target space ge-ometry in [12] involves a doubling of the fields τ µ and m µ which we do not see here. It would be interesting tounderstand this difference. Non-relativistic sigma models from scaling limit
We will take a limit of ˆ S in which the tension T goesto zero. In order to keep the action finite we compensate T → τ µ . We can alwayswrite τ µ = N ∂ µ F + β µ with v µ β µ = v µ h µν = 0 and v µ τ µ = −
1. If we rescale F = c ˜ F , T = ˜ T /c , η = c ˜ η andsend c to infinity we obtain˜ S = − ˜ T Z d σ e αβ m α ∂ β ˜ η + e αα ′ e ββ ′ ˜ τ α ′ ˜ τ β ′ e γγ ′ ˜ τ γ ∂ γ ′ ˜ η h αβ ! , (14)where ˜ τ α = ∂ α X µ ˜ τ µ with ˜ τ µ = N ∂ µ ˜ F .The resulting action ˜ S has world-sheet diffeomorphisminvariance δX µ = ξ α ∂ α X µ and δ ˜ η = ξ α ∂ α ˜ η . Assuming˜ τ µ = ∂ µ X = δ µ , we can choose the gauge σ = π ˜ T P X and ˜ η = P π ˜ T σ , obtaining˜ S = − P π Z d σ (cid:18) m µ ∂ X µ + 12 h µν ∂ X µ ∂ X ν (cid:19) . (15)This is a non-relativistic world-sheet theory containingonly first order time derivatives. The equation of motionof ˜ η gives the constraint ∂ m − ∂ m + 12 ∂ h = 0 . (16)The action (14) is invariant under local transforma-tions that act on ˜ τ µ , m µ and h µν = δ ab e aµ e bν as δ ˜ τ µ = 0 , δm µ = ∂ µ σ , δh µν = 2˜ τ ( µ e aν ) ˜ λ a . (17)These transformations plus target space diffeomorphsimsfollow from (4) if we set λ a = ˜ λ a /c , τ µ = c ˜ τ µ + β µ andsend c to infinity. The action ˜ S has a global symmetrygenerated by K µ if the Lie derivatives along K µ of ˜ τ µ , m µ , h µν vanish up to the transformations (17).TNC geometry can be obtained by gauging theBargmann algebra [7, 12]. The transformations (4) followfrom the Bargmann algebra { H, P a , J ab , G a , N } with a =1 , . . . , d whose nonzero commutators are [ H, G a ] = P a and [ P a , G b ] = δ ab N where we left out the nonzero com-mutators with J ab . The TNC fields can be assembled inthe connection A µ = Hτ µ + P a e aµ + N m µ + . . . , where weleft out the connections associated with Galilean boosts G a and rotations J ab . If we consider the transforma-tion δ A µ = L ξ A µ + ∂ µ Σ + [ A µ , Σ], where ξ µ generatesdiffeomorphisms and where Σ = N σ + G a λ a + J ab λ ab we obtain all transformations of the TNC fields τ µ , m µ and h µν = δ ab e aµ e bν in (4). If we rescale H = c ˜ H and G a = c − ˜ G a and send c to infinity we find the Galileialgebra Gal direct sum with a U (1) generated by N ,where Gal is the Bargmann algebra with N removed.In a similar way the local transformations of ˜ τ µ , m µ and h µν = δ ab e aµ e bν can be obtained by gauging Gal ⊕ U (1)where ˜ τ µ is the connection associated with ˜ H , e aµ theconnection associated with P a and m µ the connectionassociated with N . The resulting geometry is what wecall U (1)-Galilean geometry.Interestingly, applying the same limit to the case ofa massless relativistic particle leads to an action pro-portional to R dλm µ dX µ dλ , so that a particle on a U (1)-Galilean geometry has no dynamics. We have thus found a geometry that is more naturally probed by strings thanby particles. Limits of strings on AdS × S We apply now the above scaling limit c → ∞ tothe case of strings on AdS × S . As we shall see,the Spin Matrix theory (SMT) limits introduced in [14]are realizations of the scaling limit. Consider type IIBstrings on AdS × S in the global patch with radius R = (4 πg s N ) / l s and five-form flux N where g s is thestring coupling and l s the string length. Introduce nowthe following six commuting charges, namely the energy E , the angular momenta S and S on the S in AdS and the angular momenta J , J and J on S . The uni-tarity bounds of N = 4 are dual to BPS bounds E ≥ J where J is a linear combination of the five angular mo-menta. Specifically, one has the five BPS bounds E ≥ J with J = J + J , J = J + J + J , J = S + J + J , J = S + S + J or J = S + S + J + J + J . For agiven BPS bound E ≥ J the SMT limits of N = 4 SYMare dual to limits of type IIB strings on AdS × S with E − J and g s going to zero with ( E − J ) /g s and N keptfixed. The effective string tension in AdS × S is T = 12 π p πg s N , (18)which goes to zero in the SMT limits.Four of the bounds do not involve all of the five angularmomenta. Let n denote the number of angular momentanot included in the bound. In the SMT limit the 2 n di-rections - here called external directions - that realize therotation planes for these n angular momenta have a con-fining potential with effective mass proportional to 1 /g s and hence these directions are forced to sit at the mini-mum of the potential. This gives an effective reductionof the number of spatial dimensions after the limit.One can show that AdS × S admits a coordinatesystem u , x µ , y I where µ = 0 , , , ..., d , d = 8 − n , and I = 1 , , ..., n , with the properties that i). y I are the 2 n external directions that are confined to be at y I = 0 inthe limit, ii). ∂ u and ∂ x are Killing vector fields with i∂ x = E − J and iii). the metric of AdS × S can beput in the form (3) when setting y I = 0, with τ µ , m µ and h µν such that τ = 1 and m = h = h i = 0 for i = 1 , , .., d .The scaling limit introduced above corresponds to theSMT limit if one identifies c − = 4 πg s N . Following this,one rescales x = c ˜ x such that the rescaled energy i∂ ˜ x = ( E − J ) / (4 πg s N ) is kept fixed in the limit. Therescaled tension is ˜ T = cT = π . After the scaling limitwe get the action (14). With the gauge choice σ = P X (with X = ˜ x on the world-sheet) and ˜ η = P σ this be-comes (15).We conclude that the SMT limit applied on type IIBstrings on AdS × S realizes the scaling limit c → ∞ introduced above, and therefore corresponds to a non-relativistic limit both on the target space, as well as onthe world-sheet. After the limit, the target space is a d + 1 dimensional U (1)-Galilean geometry and the world-sheet theory is a non-relativistic two-dimensional theory.Note that the action (15) is large if P is large, and onecan thus take a classical limit of the action, even if theSMT/scaling limit involves sending the effective tension T to zero [16]. See [16] for a discussion of quantum effectsin such limits. Examples
As the simplest example, consider the SMT/scalinglimit towards the BPS bound E ≥ J = J + J . Writethe metric of AdS × S as g MN dx M dx N = cos ψ [2 τ ( du − m ) + h µν dx µ dx ν ] − (sinh ρ + sin ψ )( dx − du ) + dρ + sinh ρ d Ω + dψ + sin ψ dα , (19)with d = 2 since n = 3, τ = dx − m and m = − cos θ dφ , h µν dx µ dx ν = 14 ( dθ +sin θdφ ) . (20)Note that the radius is set to one and instead includedin the tension (18). The six external directions have apotential proportional to (sinh ρ + sin ψ ) /g s that con-fines them to the point ρ = ψ = 0 [16]. The SMT limitleads then to the 2+1 dimensional U (1)-Galilean geome-try given by ˜ τ = d ˜ x and Eq. (20). The non-relativisticsigma-model (15) is the Landau-Lifshitz model with P = J . Thus, we get a new interpretation of the Landau-Lifshitz model as a non-relativistic string theory of theform (14) with a U (1)-Galilean target space geometry.SMT becomes a nearest-neighbor spin chain for N = ∞ , which is the ferromagnetic Heisenberg spin chainwith SU (2) symmetry for J = J + J . In a long-wavelength approximation with large J this is described by theLandau-Lifshitz model hence matching the SMT/scalinglimit on the string theory side.The connection between the emerging sigma-modelsfrom spin chains and limits of strings on AdS × S wasfirst pointed out in [11] by Kruczenski and later studiedfor other sectors in [17–20]. These cases can all be inter-preted in the framework of this paper as well. However,the Kruczenski limit does not correspond to our scalinglimit since it does not take the tension (18) to zero. In-stead, it takes J = J + J to infinity keeping T /J fixed[17], hence it includes terms of higher orders in T /J incontrast with the SMT limit. Moreover, one is in differ-ent regimes on the gauge theory and string theory sides.Another example is the limit towards the BPS bound E ≥ J = S + S + J + J + J . Write the metric ofAdS × S as g MN dx M dx N = − cosh ρ dt + dρ + sinh ρ d Ω + d Ω ,d Ω k +1 = ( d Σ k ) + ( dχ k + A k ) , (21)where E = i∂ t , S + S = − i∂ χ , J + J + J = − i∂ χ ,( d Σ k ) is the Fubini-Study metric on C P k and A k is a one-form on C P k , k = 1 ,
2. Using t = v − u , χ = v − u + w and χ = v + u the metric is of the form (3)for d = 8 with m = − sinh ρ ( dw + A ) − A ,h µν dx µ dx ν = dρ + sinh (2 ρ )( dw + A ) + sinh ρ d Σ + d Σ , (22)and τ = dx + m + A . Taking the scaling limit givesnow the 8+1 dimensional U (1)-Galilean geometry definedby ˜ τ = d ˜ x and (22) with sigma-model given by (14) and(15). This limit is of particular interest since it corre-sponds to the highest possible dimension of the targetspace, and the largest global symmetry group SU (1 , | Discussion
The results of this letter open up for a wide scope ofdirections. It would be worthwhile to understand betterthe nature of the U (1)-Galilean target space geometry.Another important problem is to consider the quantumtheory of the non-relativistic string actions (13) and (14)that we have found, including beta-functions and the dy-namical role played by the target space dimension (forwhich we naturally get d + 1 = 3 , , AdS × S ). In particular, since dynam-ical NC geometry is related to Hoˇrava–Lifshitz gravity[7, 9], it would be interesting to see if the couplings tothe target space objects τ µ , m µ , h µν in (13) and ˜ τ µ , m µ , h µν in (14) have to obey certain consistency conditionsthat can be interpreted as the equations of motion ofa non-relativistic gravity. For SMT, this could in turnbe interesting since one should then be able to see theemergence of U (1)-Galilean geometry and its associatedgravitational dynamics from a quantum theory.Important generalizations and extensions of our re-sults are: i). the effect of adding the NSNS B -field tothe limits, which could be useful to understand if thereis a notion of T-duality and if there is a relation withthe Gomis–Ooguri formulation of non-relativistic closedstrings [10], ii). the inclusion of fermions, and corre-sponding supersymmetric versions of the non-relativisticsigma-models and iii). a systematic study of higherderivative corrections to the sigma-models. Moreover,by applying similar limits to the DBI D-brane action(see also [29]) it seems very likely that higher-dimensionalnon-relativistic world-volume theories should exist. Acknowledgements.
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