A Non-Relativistic Limit of NS-NS Gravity
Eric Bergshoeff, Johannes Lahnsteiner, Luca Romano, Jan Rosseel, Ceyda Simsek
aa r X i v : . [ h e p - t h ] F e b February 13 th , 2021 A Non-Relativistic Limit of NS-NS Gravity
E. A. Bergshoeff a , J. Lahnsteiner b , L. Romano c , J. Rosseel d and C. S¸im¸sek e1 Van Swinderen Institute, University of GroningenNijenborgh 4, 9747 AG Groningen, The Netherlands Faculty of Physics, University of Vienna,Boltzmanngasse 5, A-1090, Vienna, Austria
Abstract
We discuss a particular non-relativistic limit of NS-NS gravity that can be taken at thelevel of the action and equations of motion, without imposing any geometric constraints byhand. This relies on the fact that terms that diverge in the limit and that come from theVielbein in the Einstein-Hilbert term and from the kinetic term of the Kalb-Ramond two-formfield cancel against each other. This cancelling of divergences is the target space analogueof a similar cancellation that takes place at the level of the string sigma model between theVielbein in the kinetic term and the Kalb-Ramond field in the Wess-Zumino term. The limitof the equations of motion leads to one equation more than the limit of the action, due to theemergence of a local target space scale invariance in the limit. Some of the equations of motioncan be solved by scale invariant geometric constraints. These constraints define a so-calledDilatation invariant String Newton-Cartan geometry. a Email: e.a.bergshoeff[at]rug.nl b Email: j.m.lahnsteiner[at]outlook.com c Email: lucaromano2607[at]gmail.com d Email: jan.rosseel[at]univie.ac.at e Email: c.simsek[at]rug.nl ontents A.1 Index Conventions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18A.2 Lorentzian Geometry Conventions . . . . . . . . . . . . . . . . . . . . . . . . . . . 19A.3 Conversion of Curved to Flat Indices . . . . . . . . . . . . . . . . . . . . . . . . . . 20
B Torsional String Newton Cartan Geometry 20
B.1 String Galilei and Dilatation Connections . . . . . . . . . . . . . . . . . . . . . . . 20B.2 Affine Connection . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22B.3 Curvatures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22
C Details on the NR Limit of NS-NS Gravity 23
Non-relativistic (NR) string theory in flat space-time has been proposed a long time ago [1, 2].The generalization from a flat to special curved backgrounds was considered a few years later[3]. Closed bosonic NR string theory in general curved backgrounds, on the other hand, hasbeen constructed only recently. This has been done either by taking a NR limit [4, 5] or by nullreduction [6–8] of the worldsheet action for a relativistic string in a generic background. Thiswork showed that the natural target space geometry of the NR string theory of [1, 2] in arbitrarybackgrounds, is given by a NR Newton-Cartan-like geometry with co-dimension two foliation thatis referred to as String Newton-Cartan (SNC) geometry [13]. The NR string then couples tothe background fields of SNC geometry, as well as to a Kalb-Ramond (KR) and dilaton field. Allthese background fields must satisfy equations of motion that ensure (one loop) quantum Weylinvariance of the NR string worldsheet action [16, 17]. The case of a NR open string in a curvedbackground has been discussed recently in [19, 20].SNC geometry is a particular case of what can be called ‘ p -brane Newton-Cartan geometry’.The latter term refers to D -dimensional Newton-Cartan-like geometries that can be written as aco-dimension p + 1 foliation. These geometries are then equipped with two degenerate metrics, oneof rank D − p − p + 1 on the foliation’s co-dimension p + 1 part. The directions spanned by the co-dimension p + 1 part can be viewed as lying along aNR p -brane worldvolume, while the leaves of the foliation represent directions that are transversalto this worldvolume.Such p -brane geometries can be obtained as a ‘ p -brane NR limit’ of the Lorentzian geometry thatunderlies General Relativity. This limit can be conveniently discussed in the Vielbein formulationof Lorentzian geometry. To do this, one considers the relativistic Vielbein E µ ˆ A and splits the For other recent work on NR strings in a curved background, see [9–12]. For earlier work on SNC geometry, see [3, 14, 15]. See also [18]. -dimensional flat SO(1 , D −
1) index ˆ A into a flat ‘worldvolume’ index A = 0 , · · · , p and a flat‘transversal’ index A ′ = p + 1 , · · · , D . One then redefines E µ ˆ A as follows: E µA = cτ µA + 1 c m µA , E µA ′ = e µA ′ , (1)where c is a contraction parameter. As it stands, this redefinition is not invertible. To make itinvertible one typically introduces (and redefines) a p + 1-form gauge field by hand. In the case ofstring theory, i.e. p = 1, the role of this 2-form gauge field is played by the KR 2-form field. TheNR limit of a quantity, constructed out of E µ ˆ A , is then obtained by plugging in this redefinition inthe object of interest, expanding the result in powers of c − and formally taking the limit c → ∞ ,by retaining only the leading order in this expansion. By doing this for the Lorentzian metric,one obtains the degenerate metric, with ‘longitudinal Vielbein’ τ µA , on the co-dimension p + 1 partof a p -brane Newton-Cartan geometry, while the inverse Lorentzian metric leads to the degeneratemetric, with ‘transversal Vielbein’ e µA ′ on the foliation leaves. The SO(1 , D −
1) local Lorentztransformations get contracted in the limit to ‘homogeneous p -brane Galilei symmetries’, consistingof worldvolume Lorentz transformations, transversal rotations and Galilean-type boosts betweentransversal and worldvolume directions. These homogeneous p -brane Galilei symmetries are partof a larger symmetry group that includes translations. The field m µA can then be identified as agauge field for a non-central (central in case p = 0) extension of this larger symmetry group.The NR limit can also be taken for the relativistic spin connection Ω µ ˆ A ˆ B , to obtain NR spinconnections for the above mentioned homogeneous p -brane Galilei symmetries. Plugging the re-definitions (1) in the spin connection components Ω µ ˆ A ˆ B in the second-order formulation andexpanding the result in powers of c − , one finds that the leading order terms of this expansion donot transform as NR spin connections for the homogeneous p -brane Galilei symmetries. Instead, itis the subleading order terms of this expansion that give rise to proper NR spin connections. Forthis reason, the leading order terms are considered to be divergent. In order to obtain correct NRspin connections in the NR limit, one then requires that these divergent terms vanish. This can bedone by imposing the following ‘zero torsion constraint’ on the longitudinal Vielbein τ µA : D [ µ ( ω ) τ ν ] A = 0 , (2)where the derivative D µ ( ω ) is covariantized with respect to longitudinal Lorentz transformations,using a spin connection ω µAB . Part of this constraint (2) is identically satisfied once the dependentexpression for ω µAB in terms of the τ µA and their projective inverses τ Aµ is plugged in. However,not all of (2) is identically satisfied in this way and (2) thus leads to a genuine constraint on τ µA and the geometry. In case p = 0 for instance, this geometric constraint entails that thetime-like Vielbein of Newton-Cartan geometry is closed, implying that the space-time admits anabsolute time direction. For the string case ( p = 1), the constraints (57) define the torsionlessSNC geometry that was considered as target space-time for the NR string in [4, 5]. Imposing thezero torsion constraint, it was shown in [21] that the 0-brane limit of the equations of motion ofGeneral Relativity leads to the equations of motion of Newton-Cartan gravity. Similarly, it wasshown that the target space equations of motion for the NR string theory of [4, 5] arise from the1-brane limit of the equations of motion of NS-NS gravity, upon imposition of (2). Note that inboth cases, the constraint (2) was imposed by hand and can not be considered as an equation ofmotion that follows from a NR action. For that reason, the limit was in both cases taken at thelevel of the equations of motion and not at the level of the action.When taking the NR limit of supersymmetric theories, imposing the constraint (2) (or similargeometric constraints) by hand can be problematic. Indeed, when imposing constraints by hand,one also needs to impose their supersymmetry variations as constraints in order to maintain super-symmetry. This typically leads to a tower of constraints on top of the equations of motion of the Strictly speaking, when taking the limit that c goes to infinity, we mean that one first redefines c → λc andthen takes the limit where the dimensionless contraction parameter λ goes to infinity. Here, we consider the second-order formulation of General Relativity, in anticipation of the extension of theresults of this paper to NS-NS supergravity, for which no first-order formulation is available in the literature. Theexpression for the relativistic dependent spin-connection Ω µ ˆ A ˆ B can be found in Appendix A.2. For an explicit expression of ω µAB in terms of τ µA and its inverse, see Appendix B.
3R theory. There is then a danger that the NR limit does not lead to the most general possibletheory or even leads to an overconstrained theory. When taking the NR limit of supergravitytheories, it would thus be better, if one were able to take the NR limit of the action or equationsof motion, without imposing the zero torsion constraint as an a priori constraint. Some of theNR equations of motion might then take the form of differential or algebraic constraints for thecomponents of D [ µ ( ω ) τ ν ] A . These, however, do not give rise to a tower of extra constraints ontop of the NR equations of motion, since they correspond to NR equations of motion themselves.If the NR limit is taken consistently, their supersymmetry variations should thus also give rise toequations of motion.In this paper, we will focus exclusively on the p = 1 case of SNC geometry. Motivated by super-symmetry, we then address the question whether it is possible to take the NR limit of relativisticgravity, without imposing the zero torsion constraint (2) by hand. In a similar spirit as in the workof [22], we will show that this is possible for the matter-coupled relativistic gravity theory thatcorresponds to NS-NS gravity. We will see that in order to achieve this, we not only have to adoptthe redefinitions (1), but we similarly have to expand the NS-NS two-form field B µν as B µν = − c τ µA τ ν B ǫ AB + b µν , (3)where b µν corresponds to the two-form field of the NR theory that results from taking the c → ∞ limit. The redefinitions (1) and (3) are then the same as the ones used to obtain the NR stringworldsheet action as the NR limit of the relativistic string action. In that case, a fine-tuningbetween the Vielbein and the NS-NS two-form field leads to a cancellation of divergences whentaking the NR limit of the relativistic string action, so that a non-trivial NR action is obtained. Wewill show here that a similar mechanism takes place when taking the NR limit of NS-NS gravity,so that a non-trivial NR theory is obtained without imposing any geometric constraints by hand.Since no constraints need to be imposed in the process, the NR limit can be taken both for theequations of motion and the action of NS-NS gravity. We then find that both the action andequations of motion of the resulting NR theory exhibit an emerging local scale invariance, underwhich the longitudinal Vielbein τ µA scales non-trivially.The appearance of this emerging local scale symmetry has two consequences. First, it impliesthat the NR action leads to one equation of motion less than its relativistic counterpart. It turnsout that the missing equation of motion is important, as it corresponds to the analog of the Poissonequation of NR gravity. We will see that this equation of motion is recovered by considering theNR limit of the equations of motion of NS-NS gravity, so that taking the NR limit of the action isnot equivalent to taking the NR limit of the equations of motion. Secondly, since the zero torsionconstraint (2) is not invariant under local scale symmetries, it can not arise as one of the NRequations of motion. We will indeed see that some of the NR equations of motion that we findamount to algebraic and differential equations for τ µA allowing as a solution the following set ofconstraints that is weaker than the SNC geometric constraints given in (2): e A ′ µ τ { A | ν ∂ [ µ τ ν ] | B } = 0 , and e A ′ µ e B ′ ν ∂ [ µ τ ν ] A = 0 . (4)Here, τ Aµ , e A ′ µ are (projective) inverses of τ µA , e µA ′ and { AB } indicates the symmetric tracelesspart of AB . This set of constraints is invariant under the local scale symmetry of the NR theory.Compared to the SNC constraints (2), we have that b A ′ ≡ e A ′ µ τ Aν ∂ [ µ τ ν ] A , (5)which acts like the (transverse components of the) gauge field of the local scale symmetry, isnon-zero. This paper is organized as follows. In section 2 we review the NR string worldsheet action andhow it can be obtained from the NR limit to motivate the way we define the NR limit of NS-NS The second constraint is sufficient to define a globally well-defined co-dimension two foliation, called integrabledistribution [23]. We call the geometry defined by (2) a String Newton-Cartan (SNC) geometry and (4) a Dilatation invariantString Newton-Cartan (DSNC) geometry. The geometry without constraints will be referred to as a Torsional StringNewton-Cartan (TSNC) geometry. Note that the geometry defined by the second constraint only in (4) is a stringversion of the Twistless Torsional Newton-Cartan (TTNC) geometry, found in Lifshitz holography [24].
The worldsheet action for the NR bosonic string in a generic background was derived in [4, 5],by taking a NR limit of the relativistic Polyakov string action, coupled to an arbitrary target spacebackground. This leads to the following NR string action in the Polyakov form: S P = − T Z d σ h √− hh αβ ∂ α x µ ∂ β x ν H µν + ǫ αβ (cid:0) λe α τ µ + ¯ λ ¯ e α ¯ τ µ (cid:1) ∂ β x µ i − T Z d σ ǫ αβ ∂ α x µ ∂ β x ν b µν + 14 π Z d σ √− h R (2) ( h ) (cid:18) φ −
14 ln G (cid:19) . (6)Here T is the string tension, σ α ( α = 0 ,
1) are the worldsheet coordinates and x µ ( σ ), µ = 0 , , · · · , h αβ , h and R (2) ( h ) respectively. We have furthermore introduced a Zweibein e αa ( a = 0 ,
1) for h αβ via h αβ = e αa e βb η ab (with η ab = diag( − , e α = e α + e α , ¯ e α = e α − e α . (7)The second term in (6) includes two extra worldsheet fields λ ( σ ), ¯ λ ( σ ) that appear as Lagrangemultipliers. We refer to [5] for details on how these fields appear in the NR limit.The NR string couples in the action (6) to background fields that we take to be ten-dimensionalones. They are given by { τ µA , e µA ′ , m µA , b µν , φ } , A = 0 , A ′ = 2 , · · · , , (8)representing the longitudinal Vielbein τ µA , the transverse Vielbein e µA ′ and the non-central chargegauge field m µA of SNC geometry, as well as the KR field b µν and the dilaton φ . The first term inthe action (6) is the kinetic term and contains the so-called ‘transverse metric’ H µν that is givenin terms of the SNC background fields by H µν = e µA ′ e νB ′ δ A ′ B ′ + ( τ µA m νB + τ νA m µB ) η AB . (9)The fields τ µ , ¯ τ µ in the second term correspond to τ µA in a light-cone basis: τ µ = τ µ + τ µ , ¯ τ µ = τ µ − τ µ . (10)The third term in the action (6) describes the Wess-Zumino coupling of the background KR field b µν to the string. Furthermore, the object G in the last term of (6) was defined in [5] as the limitof the metric determinant G = e ≡ − lim c →∞ ( c − det G µν ) , where e = det ( τ A , e A ′ ) ≡ ε µ ··· µ τ µ τ µ e µ · · · e µ . (11) Strictly speaking, the background fields for the critical NR bosonic string are 26-dimensional. Here however, weconsider ten-dimensional backgrounds, since we have the superstring in mind. We thus view (6) as the bosonic partof a NR superstring action. Note that this metric is strictly speaking only transverse in the absence of the second term. λ , ¯ λ [4]. The equations of motion of λ , ¯ λ correspond to the constraints ǫ αβ e α τ µ ∂ β x µ = 0 , ǫ αβ ¯ e α ¯ τ µ ∂ β x µ = 0 . (12)These constraints are solved by h αβ = α ( x ) τ αβ , (13)where α ( x ) is an arbitrary proportionality factor and τ αβ ≡ τ µA τ νB η AB ∂ α x µ ∂ β x ν . (14)Plugging the solution (13) in the NR Polyakov action (6), leads to the NR Nambu-Goto action,given by S NG = − T Z d σ (cid:20)q − det( τ γδ ) τ αβ ∂ α x µ ∂ β x ν H µν + ǫ αβ ∂ α x µ ∂ β x ν b µν (cid:21) + S dilaton , (15)where S dilaton is the last term of (6) (with h αβ replaced by the solution (13)) and τ αβ is the inverseof τ αβ . Ignoring S dilaton , this action can be obtained from a NR limit of the relativistic stringaction in Nambu-Goto form [8]: S rel − NG = − T Z d σ r − det (cid:16) E µ ˆ A E ν ˆ A ∂ α x µ ∂ β x ν (cid:17) − T Z d σ ǫ αβ ∂ α x µ ∂ β x ν B µν , (16)where E µ ˆ A is the relativistic ten-dimensional Vielbein and B µν the relativistic KR field. Splittingthe index ˆ A in A = 0 , A ′ = 2 , · · · ,
9, the first two terms of (15) are then obtained by pluggingthe following redefinitions (see [1, 2, 5, 8, 25] for early and recent references) E µA = cτ µA + 1 c m µA , E µA ′ = e µA ′ , B µν = − c τ µA τ νB ǫ AB + b µν , (17)in (16) and taking the limit c → ∞ . The first two terms of (15) constitute the terms at O ( c ) inan expansion of (16) in powers of c − (after the redefinitions (17) have been performed). Whenexpanding, both terms in (16) lead to a contribution at O ( c ) that diverges in the c → ∞ limit. Thedivergent contribution that comes from the second, Wess-Zumino term, of (16) however exactlycancels the contribution coming from the first, kinetic term, so that the c → ∞ limit is well-defined.The actions (6), (15) are invariant under an abelian two-form symmetry, with parameters θ µ ,of the KR field δb µν = 2 ∂ [ µ θ ν ] , (18)as well as under local transformations of the background fields that we will refer to as ‘StringGalilei symmetries’ in this paper. These consist of longitudinal SO(1 ,
1) Lorentz transformationswith parameter λ M , transversal SO(8) rotations with parameters λ A ′ B ′ and Galilean boosts withparameters λ AA ′ and their non-trivial transformation rules are given by δτ µA = λ M ǫ AB τ µB , δe µA ′ = λ A ′ B ′ e µB ′ − λ AA ′ τ µA ,δm µA = λ M ǫ AB m µB + λ AA ′ e µA ′ . (19)Note that H µν is invariant under these symmetries, so that the String Galilei invariance of theactions (6), (15) is manifestly realized.The actions (6), (15) are furthermore also invariant under the following Stueckelberg symmetrywith parameters c µA , given by δb µν = ( c µA τ Bν − c νA τ µB ) ǫ AB , δm µA = − c µA . (20) Invariance of (6) under String Galilei symmetries requires that one also assigns a non-trivial SO(1 , λ , ¯ λ . Similar remarks hold for the Stueckelberg invariance. Werefer to [5] for the details. b µν andthe transformation rule of H µν , as induced by δm µA = − c µA , are formally invariant under a gaugesymmetry, with singlet parameter c , given by δc µA = ǫ AB τ µB c . (21)The Stueckelberg symmetry is thus parametrized by only 19 independent parameters.One can rewrite the action (15) in a manifestly Stueckelberg invariant way, by moving the m µA terms, that are part of the definition of H µν , from the kinetic term of (15) to the Wess-Zuminoterm, where they form a Stueckelberg-invariant combination, b µν + ( m µA τ Bν − m νA τ µB ) ǫ AB . (22)Equivalently, one can also fix the Stueckelberg symmetry by imposing the gauge-fixing condition m µA = 0 , (23)after which the string action (15) reads as follows: S NGf = − T Z d σ (cid:20)q − det( τ γδ ) τ αβ e αA ′ e βB ′ δ A ′ B ′ + ǫ αβ ∂ α x µ ∂ β x ν b µν (cid:21) + S dilaton . (24)Note that in contrast to the actions (6) and (15), the Stueckelberg gauge-fixed action (24) exhibitsthe Galilean boost symmetry in a non-manifest way that involves the KR field in a non-trivialmanner. Indeed, the price one pays for fixing the Stueckelberg symmetry is that the KR fieldtransforms under compensating Galilean boosts. In the action (24), b µν thus transforms underGalilean boosts as follows: δb µν = − ǫ AB λ AA ′ τ [ µB e ν ] A ′ . (25)Checking boost invariance of (24) then requires cancelling a contribution from the boost variationof e µA ′ (given in (19)) in the first term of (24) against a contribution from the variation (25) of b µν in the second term.It is worth pointing out that ordinarily, the longitudinal components of m µA capture the in-formation of NR gravity that is contained in the Newton potential [13]. The effect of fixing theStueckelberg symmetry, as in (23), is that the Newton potential is contained in the longitudinalcomponent b AB of the KR field. This can be seen from the fact that the gauge-fixing (23) isequivalent to replacing b µν by the Stueckelberg invariant combination (22) that contains the field m µA . In the following sections of this paper, we will work with this Stueckelberg gauge-fixedformulation. We will therefore also refer to b AB as ‘the Newton potential’.A final non-trivial property of the string action (6) is that it has an emerging local dilatationsymmetry, with parameter λ D , given by δτ µA = λ D τ µA , δφ = λ D . (26)This symmetry was not present in the relativistic case. It arises due to the fact that the backgroundfields couple to a string and not, for instance, to a membrane. It will play an important role in theremainder of this paper. We refer to Appendix A.3 for details on how curved indices are turned into flat ones in the different sections ofthis paper. In order to show invariance one also needs the transformation rule for the Stueckelberg field δm µA = − λ D m µA ,and the Lagrange multipliers δλ = − λ D λ , δ ¯ λ = − λ D ¯ λ . All the other fields in (6) have zero charge under localdilatations. The NR Limit of the NS-NS Gravity Action
In the previous section, we reviewed how the NR limit, defined in eqs. (17), can be used toobtain the NR string worldsheet action (15). Here, we will show that this limit can also be appliedin a well-defined way to the NS-NS gravity action to yield an action for all target space backgroundfields of NR string theory. We will first discuss this NR limit in more detail and, in particular,show that it reproduces the correct transformation rules of the NR background fields under, e.g.,String Galilei symmetries. After having recalled the relativistic NS-NS gravity action, we will thentake its NR limit and discuss the resulting action.
The field content of ten-dimensional relativistic NS-NS gravity is given by the dilaton Φ, theKR two-form field B µν and the metric G µν that we will describe in terms of the Vielbein E µ ˆ A . Todefine the NR limit, we first redefine these fields, using a parameter c , as follows: E µA = c τ µA , E µA ′ = e µA ′ , B µν = − c ǫ AB τ µA τ ν B + b µν , Φ = φ + ln c . (27)This corresponds to the redefinitions (17), where we have however adopted the condition (23) thatfixes the Stueckelberg symmetries (20) of the NR string worldsheet action. As shown by the use ofthese redefinitions in deriving the NR string worldsheet action (15), the fields τ µA , e µA ′ , b µν and φ correspond to the background fields of NR string theory, once the limit c → ∞ has been taken.Let us first discuss how eqs. (27) can be used to derive the transformation rules of τ µA , e µA ′ , b µν and φ under String Galilei symmetries from the transformations of E µ ˆ A , B µν and Φ under SO(1 , τ µA = c − E µA , e µA ′ = E µA ′ , b µν = B µν + ǫ AB E µA E ν B , φ = Φ − ln c . (28)We can also introduce inverse Vielbeine τ Aµ = c E Aµ and e A ′ µ = E A ′ µ that satisfy τ Aµ τ µB = δ BA , e A ′ µ e µB ′ = δ B ′ A ′ ,τ Aµ e µA ′ = 0 , e A ′ µ τ µA = 0 , (29) τ Aµ τ ν A + e A ′ µ e νA ′ = δ µν . The transformation rules of E µ ˆ A , B µν and Φ under SO(1 ,
9) Lorentz transformations and theabelian two-form gauge symmetry of the KR field are given by δE µA = Λ M ǫ AB E µB + Λ AA ′ E µA ′ , δE µA ′ = − Λ AA ′ E µA + Λ A ′ B ′ E µB ′ ,δB µν = 2 ∂ [ µ Θ ν ] , δ Φ = 0 , (30)where Θ µ is the parameter of the two-form gauge symmetry and we have split SO(1 ,
9) into SO(1 , AB = Λ M ǫ AB ), SO(8) (with parameters Λ A ′ B ′ ) and the remaining boost trans-formations (with parameters Λ AA ′ ). Using the field redefinitions (27), their inverses (28) and thefollowing redefinitions of the symmetry parametersΛ M = λ M , Λ AA ′ = − Λ A ′ A = 1 c λ AA ′ , Λ A ′ B ′ = λ A ′ B ′ Θ µ = θ µ , (31)we derive the following non-relativistic transformation rules after taking the c → ∞ limit δτ µA = λ M ǫ AB τ µB , δe µA ′ = − λ AA ′ τ µA + λ A ′ B ′ e µB ′ ,δb µν = 2 ∂ [ µ θ ν ] − ǫ AB λ AA ′ τ [ µB e ν ] A ′ , δφ = 0 , (32)where λ M , λ A ′ B ′ , λ AA ′ and θ µ are now interpreted as parameters of the longitudinal SO(1 , b µν under Galilean boosts that was necessary to ensure boost invariance of the string worldsheetaction (24). In a similar way, one finds that the projective inverse Vielbeine transform as δτ Aµ = λ M ǫ AB τ Bµ + λ AA ′ e A ′ µ , δe A ′ µ = λ A ′ B ′ e B ′ µ . (33)The NR limit can similarly be performed on other quantities that are expressed in terms ofthe fields of relativistic NS-NS gravity. To do this, one plugs the redefinitions (27) in the quantityof interest, expands the result in powers of c − and retains only the terms that appear at leadingorder. In the next subsection, we will apply this procedure to the relativistic NS-NS gravity action.The NR limit of the equations of motion of NS-NS gravity will be considered in section 4. The dynamics of the fields of relativistic NS-NS gravity is governed by the following action (inthe string frame): S NS − NS = 12 κ Z d x E e − (cid:18) R + 4 ∂ µ Φ ∂ µ Φ − H (cid:19) . (34)Here, κ is the gravitational coupling constant, E = det( E µ ˆ A ), R is the Ricci scalar of G µν and H = 13! H µνρ H µνρ , with H µνρ = 3 ∂ [ µ B νρ ] . (35)We refer to Appendix A.2 for our conventions on Lorentzian geometry.To take the NR limit of the relativistic NS-NS gravity action (34), we plug the redefinitions (27)in (34) and expand the result in powers of c − . The leading order term of the resulting expansionthen appears a priori at order c : S NS − NS = c (2) S + c (0) S + c − ( − S + · · · . (36)Here, the explicit expression for (2) S is proportional to (2) S ∝ Z d x e e − φ (cid:18) (2) R − (2) H AA ′ B ′ (2) H AA ′ B ′ (cid:19) , (37)where e = det( τ µA , e µA ′ ), (2) R is the term at order c in the expansion of the Ricci scalar R and (2) H µνρ is the term at order c in the expansion of H µνρ . Both terms in (2) S are separately non-zero.Using the explicit expressions (see also Appendix C) (2) R = − η AB τ A ′ B ′ A τ A ′ B ′ B , (2) H AA ′ B ′ = 2 ǫ AB τ A ′ B ′ B (with τ A ′ B ′ A = e A ′ µ e B ′ ν ∂ [ µ τ ν ] A ) , (38)one however sees that the contributions of the two terms in (2) S exactly cancel. This is a non-trivial cancellation between the Ricci scalar and the kinetic term of the KR field, that mirrors thecancellation, mentioned under (17), in the string worldsheet action.The upshot of this cancellation is that the actual leading order term in the expansion (36) is (0) S ,appearing at order c . This term can be written in terms of geometric quantities, that characterizea non-Lorentzian geometry that we call ‘torsional string Newton Cartan geometry’ (TSNC). InTSNC geometry, we define spin connections ω µ , ω µA ′ B ′ , ω µAA ′ for SO(1 , b µ that we will call ‘the dilatation connection’. These connections aredefined in terms of τ µA , e µA ′ , b µν and φ as follows: b µ = e µA ′ e A ′ ν τ Aρ ∂ [ ν τ ρ ] A + τ µA τ Aν ∂ ν φ , (39a) Note that the expression (2) H AA ′ B ′ is not the term at order c in the expansion of H AA ′ B ′ . Rather, it denotesthe contraction with non-relativistic Vielbeine τ Aµ e A ′ ν e B ′ ρ (2) H µνρ . See appendices A.1 and C for more details. µ = (cid:18) τ Aν ∂ [ µ τ ν ] B − τ µC τ Aν τ Bρ ∂ [ ν τ ρ ] C (cid:19) ǫ AB − ǫ AB τ µA τ Bν ∂ ν φ , (39b) ω µAA ′ = − τ Aν ∂ [ µ e ν ] A ′ + e µB ′ τ Aν e A ′ ρ ∂ [ ν e ρ ] B ′ + 32 ǫ AB τ Bν e A ′ ρ ∂ [ µ b νρ ] + τ µB W ABA ′ , (39c) ω µA ′ B ′ = − e [ A ′ | ν | ∂ [ µ e ν ] B ′ ] + e µC ′ e A ′ ν e B ′ ρ ∂ [ ν e ρ ] C ′ − τ µA ǫ AB τ Bν e A ′ ρ e B ′ σ ∂ [ ν b ρσ ] , (39d)Here, W ABA ′ is a tensor that is symmetric traceless in the AB indices, but is otherwise arbitrary. We refer to Appendix B for more information on the definitions of these and related TSNC geo-metric quantities. Using these connections, the leading order term (0) S can then be written as (0) S = 12 κ Z d x e e − φ (cid:18) R( J ) + 4 ∂ A ′ φ ∂ A ′ φ − h A ′ B ′ C ′ h A ′ B ′ C ′ − D A ′ b A ′ − b A ′ b A ′ − τ A ′ { AB } τ A ′ { AB } (cid:19) . (40)Here, τ µν A = ∂ [ µ τ ν ] A , h µνρ = 3 ∂ [ µ b νρ ] and we have turned curved indices into flat ones using τ µA , e µA ′ , τ Aµ , e A ′ µ , as detailed in Appendix A.3. The curvature scalar R( J ) and derivative D µ b A ′ areexplicitly given byR( J ) = − e A ′ µ e B ′ ν (cid:16) ∂ [ µ ω ν ] A ′ B ′ + ω [ µA ′ C ′ ω ν ] B ′ C ′ (cid:17) − ω A ′ BB ′ τ A ′ B ′ B , (41a) D µ b A ′ = ∂ µ b A ′ − ω µA ′ B ′ b B ′ − ω µAB ′ τ A ′ B ′ A . (41b)Note that D µ is covariant with respect to SO(1 , × SO(8) and Galilean boosts. We refer toAppendix C for details on how (40) is obtained.Like the Stueckelberg gauge-fixed string worldsheet action (24), this action (40) is invariantunder String Galilei symmetries and dilatations that act as δτ µA = λ M ǫ AB τ µB + λ D τ µA , δe µA ′ = λ A ′ B ′ e µB ′ − λ AA ′ τ µA ,δb µν = − ǫ AB λ AA ′ τ [ µB e ν ] A ′ , δφ = λ D , (42)where λ M , λ A ′ B ′ , λ AA ′ , λ D are the parameters of SO(1 , ω µ , ω µA ′ B ′ and ω µAA ′ , defined in (39), then indeed transform as con-nections for SO(1 , b µ transforms as a gauge field underdilatations, as anticipated by calling it ‘dilatation connection’.The invariance under String Galilei symmetries is not surprising, since (40) appears as theleading order term in an expansion in powers of c − . As such, it is guaranteed to be invariantunder the NR limit of SO(1 ,
9) local Lorentz symmetries, i.e., under the String Galilei symmetries.The invariance under dilatations is more surprising. Like in the NR limit of the string worldsheetaction, it appears here as an emergent symmetry.One can however rewrite (40) in a way that is manifestly dilatation invariant. This rewritingis achieved by partially integrating the e e − φ D A ′ b A ′ term, using (94a). Doing this, one finds that(40) is equivalent to S NR [ τ µA , e µA ′ , b µν , φ ] = 12 κ Z d x e e − φ (cid:18) R( J ) + 4 ∇ A ′ φ ∇ A ′ φ − h A ′ B ′ C ′ h A ′ B ′ C ′ − τ A ′ { AB } τ A ′ { AB } +4 ω A ′ BB ′ τ A ′ B ′ B (cid:19) , (43)up to a boundary term − ∂ µ ( e e − φ e A ′ µ b A ′ ). Manifest dilatation invariance is then achieved byvirtue of the fact that the dependent field b µ corresponds to a gauge field for dilatations (see (89)) As explained in Appendix B, the connections (39) are found as solutions of conventional constraints. Thepresence of the arbitrary tensor W ABA ′ then indicates that the imposed conventional constraints do not suffice todetermine all boost connection components uniquely. It should be noted that the presence of this term is irrelevantfor what follows, as it drops out of the NR action and equations of motion. ∇ µ φ = ∂ µ φ − b µ is thus dilatation invariant. The occurrence of an explicit boost spinconnection in eq. (43) is indicative of the fact that the Lagrangian is boost invariant only up to atotal derivative. This is also clear from the form of the boundary term. To summarize, we presenttwo physically equivalent ways of writing the NR action—one, (40), in which boost symmetry ismanifest and dilatation symmetry is not, and one, (43), where dilatation symmetry is manifest butboost symmetry is not. In order to distinguish the two, we will continue to use (0) S for (40) and S NR for (43), even though they give rise to the same equations of motion.The non-relativistic action (43) contains the background fields of non-relativistic string theory.This should be contrasted to the situation that would occur, had the two contributions in (2) S notcancelled. In that case, one would have ended up with a non-relativistic action (namely (2) S ) thatonly contains τ µA .Let us now look at the equations of motion that are derived from (43). We denote the equationsof motion of τ µA , e µA ′ , b µν and φ by h τ i Aµ , h e i A ′ µ , h b i µν and h φ i and define them via the followingvariation: δ S NR = 12 κ Z d x e e − φ (cid:18) h τ i Aµ δτ µA + h e i A ′ µ δe µA ′ − h φ i δφ + 12 h b i µν δb µν (cid:19) . (44)Here, the pre-factors have been chosen for later convenience. In total, the equations of motion h τ i Aµ , h e i A ′ µ , h b i µν and h φ i consist of 20 + 80 + 45 + 1 = 146 components. Not all of thesecomponents are independent however. Indeed, the invariance of the action (43) under StringGalilei symmetries and dilatations implies the following algebraic relations (Noether identities): h τ i [ AB ] = 0 , h e i [ A ′ B ′ ] = 0 , h e i A ′ µ τ µA + ǫ AB h b i BA ′ = 0 , h τ i Aµ τ µA = 8 h φ i . (45)The first three relations are the Noether identities for the SO(1 , h τ i Aµ , h e i A ′ µ , h b i µν , h φ i , we are left with 100 equations.In order to give the equations of motion h τ i Aµ , h e i A ′ µ , h b i µν , h φ i explicitly, we turn the curvedindices into flat ( A or A ′ ) ones, and decompose the resulting tensors into irreducible representationsof SO(1 , × SO(8). Strictly speaking, this is not necessary at this point but turns out to beconvenient when comparing with the equations of motion in the next section, and the beta functionsin section 5. We rename the representations as follows: h S − i ǫ AB ≡ h b i AB , h G i { AB } ≡ − h τ i { AB } , h V − i AA ′ ≡ − h τ i AA ′ , h G i A ′ B ′ ≡ δ A ′ B ′ h φ i − h e i ( A ′ B ′ ) , h V + i AA ′ ≡ −h e i A ′ A = ǫ AB h b i BA ′ , h Φ i ≡ h φ i = 18 η AB h τ i AB , h B i A ′ B ′ ≡ h b i A ′ B ′ . (46)In terms of these irreducible representations, h τ i Aµ , h e i A ′ µ and h b i µν are decomposed as follows: h τ i Aµ = 4 τ Aµ h Φ i − τ µB h G i { AB } − e µA ′ h V − i AA ′ , (47a) h e i A ′ µ = 4 e A ′ µ h Φ i − e µB ′ h G i A ′ B ′ − τ µA h V + i AA ′ , (47b) h b i µν = 12 ǫ AB τ Aµ τ Bν h S − i + e A ′ µ e B ′ ν h B i A ′ B ′ + 2 τ A [ µ e ν ] A ′ ǫ AB h V + i BA ′ . (47c)Note that we have taken the redundancy due to the Noether identities (45) into account in (46)and (47). The 100 independent equations of motion, derived from (43) are thus given by h S − i = 0 , h V ± i AA ′ = 0 , h G i { AB } = 0 , h Φ i = 0 , h G i A ′ B ′ = 0 , h B i A ′ B ′ = 0 . (48)11hese are then explicitly found to be h Φ i = ∇ A ′ ∇ A ′ φ − ( ∇ A ′ φ ) + 14 R( J ) − h A ′ B ′ C ′ h A ′ B ′ C ′ − τ A ′ { AB } τ A ′ { AB } , (49a) h G i { AB } = − ∇ B ′ − ∇ B ′ φ )) τ B ′ { AB } , (49b) h V − i AA ′ = R C ′ A ( J ) A ′ C ′ + 2 ∇ A ∇ A ′ φ + 2 ∇ B τ A ′ { AB } , (49c) h G i A ′ B ′ = R C ′ ( A ′ ( J ) B ′ ) C ′ + 2 ∇ ( A ′ ∇ B ′ ) φ − h A ′ C ′ D ′ h B ′ C ′ D ′ − τ A ′ { AB } τ B ′ { AB } , (49d) h V + i AA ′ = − ∇ B ′ − ∇ B ′ φ )) τ B ′ A ′ A − τ B ′ { AB } τ B ′ A ′ B − ǫ AB h A ′ B ′ C ′ τ B ′ C ′ B , (49e) h S − i = 4 τ A ′ B ′ C τ A ′ B ′ C , (49f) h B i A ′ B ′ = ( ∇ C ′ − ∇ C ′ φ )) h C ′ A ′ B ′ + 2 ǫ AB ∇ A τ A ′ B ′ B , (49g)where the derivative ∇ µ is covariantized with respect to dilatations, SO(1 , × SO(8) and Galileanboosts. The curvature R µν ( J ) A ′ B ′ is defined in eqs. (95). One can show that none of theseequations depends on the undetermined symmetric traceless part of the boost spin connection(39c) W µAA ′ .Note that none of the above equations (49) contains a term of the form ∂ A ′ ∂ A ′ b at thelinearized level. Since b can be identified with the Newton potential (in the Stueckelberg gaugefixed formalism with m µA = 0 that we are working in), this means that none of the equations(49) can be interpreted as a covariant version of a Poisson-type equation of NR gravity. In thenext section, we will consider how to take the NR limit of the equations of motion of relativisticNS-NS gravity directly. As we will see, this limit can be defined such that it not only reproducesthe equations (49), but also leads to an extra Poisson-type equation. In this section, we will discuss the results of applying the NR limit (27) to the equations ofmotion of relativistic NS-NS gravity. We will first discuss how these equations of motion can bereorganized, such that their NR limit can be taken in an appropriate manner. Next, we will discussthe NR equations that result from the limit and compare them to the equations of motion (49)that are derived from the NR action (43).We will denote the equations of motion for the fields G µν , B µν and Φ of relativistic NS-NSgravity by [ G ] µν , [ B ] µν and [Φ] resp. They are derived from the action (34) and are given by [ G ] µν ≡ R µν + 2 ∇ µ ∂ ν Φ − H µρσ H νρσ = 0 , (50a)[ B ] µν ≡ ∇ ρ H ρµν − ∂ ρ Φ) H ρµν = 0 , (50b)[Φ] ≡ ∇ µ ∂ µ Φ + 14
R − ∂ µ Φ ∂ µ Φ − H µνρ H µνρ = 0 . (50c)Note that these constitute 55 + 45 + 1 = 101 relativistic equations of motion.In string theory, these equations (50) also ensure that scale invariance of the string worldsheetaction is maintained at the quantum mechanical level. Indeed, in the Polyakov action for therelativistic string, G µν , B µν and Φ can be viewed as coupling constants. Quantum scale invarianceof the string action, then requires that the beta functions β Gµν , β Bµν and β Φ of G µν , B µν and Φvanish.We can then take the NR limit of the equations of motion (50), by plugging (27) in (50),expanding the resulting equations in powers of c − and retaining only the terms at leading orderas NR equations of motion. If one does this for the 101 equations of motion, as they are given in(50), one finds that some of them give rise to the same NR equation. In particular, one finds that To be precise, the equations of motion, derived from the action (34), are equivalent to the equations, givenin (50). The action (34) leads to [ B ] µν = 0 and [Φ] = 0 as equations of motion for B µν and Φ, while it gives[ G ] µν − G µν [Φ] = 0 as equation of motion for G µν . G ] AA ′ and ǫ AB [ B ] BA ′ are proportional to each other. Remarkably, for η AB [ G ] AB and ǫ AB [ B ] AB , both the terms at leading and at subleading order areproportional to each other. To avoid this redundancy, one can instead consider the following 101equations[ S ± ] ≡ η AB [ G ] AB ± ǫ AB [ B ] AB = 0 , [ V ± ] AA ′ ≡ [ G ] AA ′ ± ǫ AB [ B ] BA ′ = 0 , [ G ] { AB } = 0 , [ G ] A ′ B ′ = 0 , [Φ] = 0 , [ B ] A ′ B ′ = 0 . (51)Plugging in the redefinitions (27) into (51) and expanding the resulting equations in powers of c − ,we then find that leading order terms for different equations occur at different powers of c . We willuse the following notation [ X ] = c n h X i + O ( c n − ) , (52)to denote the expansion of the relativistic equations [ X ] as in (51), and the terms at leading order n as h X i . For example, one can show that [ S − ] has n = +2, whereas [ S + ] has n = −
2. The factthat the singlet equations [ S ± ] have leading orders separated by a factor c is remarkable and hasimportant consequences for the structure of the non-relativistic theory. For future reference, we indicate here how all components of the above equations of motiontransform into each other under the boosts (with parameters Λ AA ′ ) of SO(1 , δ [Φ] = 0 , δ [ S ± ] = 2 Λ AA ′ [ V ∓ ] AA ′ , δ [ G ] { AB } = Λ { AA ′ [ V + ] B } A ′ + Λ { AA ′ [ V − ] B } A ′ ,δ [ V + ] AA ′ = Λ AB ′ [ G ] A ′ B ′ − ǫ AB Λ BB ′ [ B ] A ′ B ′ − Λ BA ′ [ G ] { AB } −
12 Λ AA ′ [ S − ] ,δ [ V − ] AA ′ = Λ AB ′ [ G ] A ′ B ′ + 12 ǫ AB Λ BB ′ [ B ] A ′ B ′ − Λ BA ′ [ G ] { AB } −
12 Λ AA ′ [ S + ] , (53) δ [ G ] A ′ B ′ = − Λ A ( A ′ ([ V + ] | A | B ′ ) + [ V − ] | A | B ′ ) ) , δ [ B ] A ′ B ′ = − A [ A ′ ǫ | AB ([ V + ] B | B ′ ] − [ V − ] B | B ′ ] ) . The NR limit is then obtained by retaining only the leading order terms h X i in eqs. (51). Inthis way, we obtain 101 NR equations as the NR limit of the relativistic NS-NS gravity equationsof motion. These 101 NR equations are given by the 100 equations given in eqs. (48), (49), as wellas the extra equation: h S + i ≡ − τ Aµ e A ′ ν R µν ( G ) AA ′ − ǫ AB τ Aµ τ Bν R µν ( M ) = 0 , (54)using the notation introduced in eq. (52). R µν ( G ) AA ′ and R µν ( M ) are defined in (95).We have seen that the non-relativistic action (43) is invariant under dilatations. Consequently,the equations of motion also transform covariantly under transformations (26). It turns out thatthe leading order n at which the equations occur in the expansion (52) is the dilatation weight ofthe corresponding non-relativistic equation of motion, i.e., δ D h X i = n λ D h X i . We summarize allthe dilatation weights in table 1. h X i h Φ i h S − i h S + i h G i { AB } h V − i AA ′ h V + i AA ′ h G i A ′ B ′ h B i A ′ B ′ n − − Table 1: Dilatation weights of the equations of motion δ D h X i = n λ D h X i Note that the NR action (43) gives rise to one equation of motion less than the relativistic NS-NS gravity action (34). This discrepancy is consistent with the fact that the NR action (43) enjoysthis extra dilatation symmetry, that emerges after taking the limit. The additional equation (54)that is not obtained from the NR action (43) corresponds to (a covariant version of) the Poissonequation of the NR gravity theory that is described by NR string theory. Indeed, the expression h S + i contains a term ∂ A ′ ∂ A ′ b , where b corresponds to the Newton potential, in the formulationwith m µA = 0 that we are currently using. Here, we have turned curved indices on components of (50) into flat indices using the relativistic (inverse)Vielbeine E Aµ , E A ′ µ . For example [ G ] AA ′ = E Aµ E A ′ ν [ G ] µν . For more details, see Appendix A.3. Similar observations have been made in the context of non-relativistic expansions of General Relativity [26]. , × SO(8) singlet—leavingeqs. (49f), (49a), and (54) as options. Let us now see why it is the Poisson equation h S + i that doesnot—and cannot—follow from (43). Recall that the action has dilatation weight zero. Hence everyequation of motion corresponds to a field component of opposite dilatation weight. The non-linearequation h S − i , for example, has n = +2 and follows as the field equation of ǫ AB b AB which hasweight n = −
2. Using this argument and the fact that the Poisson equation has n = −
2, we seethat it should correspond to an SO(1 , × SO(8) singlet field of dilatation weight n = +2. However,no such field component exists in our theory. Hence, with the field content at hand, it is impossibleto derive h S + i from a variational principle compatible with dilatations.Since the 100 NR equations of motion (48) can be obtained from varying a String Galileiinvariant action, one finds that they form a representation of the String Galilei symmetries. Theytransform under SO(1 , × SO(8) as indicated by their index structure and their transformationrules under Galilean boosts can be inferred from (53): δ h Φ i = 0 , δ h S − i = 0 ,δ h G i { AB } = λ { AA ′ h V + i B } A ′ , δ h V + i AA ′ = − λ AA ′ h S − i , (55) δ h V − i AA ′ = λ AB ′ h G i A ′ B ′ + 12 ǫ AB λ BB ′ h b i A ′ B ′ − λ BA ′ h G i { AB } ,δ h G i A ′ B ′ = − λ A ( A ′ h V + i | A | B ′ ) , δ h b i A ′ B ′ = − λ A [ A ′ h V + i | A | B ′ ] . Since, h S + i transforms under Galilean boosts and dilatations as δ h S + i = − λ D h S + i + 2 λ AA ′ h V − i AA ′ , (56)we see that adding (54) to the set of 100 equations of motion obtained from the action (43), givesa consistent set of 101 equations of motion that transform as a representation of String Galileiand dilatation symmetries, according to (55) and (56). The dilaton equation is a singlet underall the symmetries. The remaining set of 100 equations forms a reducible, but indecomposablerepresentation. Reducible means that the equations of motion contain smaller sets of equationsthat are closed under the symmetries of the theory. Indecomposable, on the other hand, meansthat the subrepresentations cannot be truncated consistently. For example, the nonlinear singletequation h S − i is inert under Galilean boosts—yet cannot be omitted since δ h V + i = − / λ AA ′ h S − i .This also demonstrates the special status of the Poisson equation (54) as it requires all the otherequations (49) to form a closed set under Galilei boosts. In other words, one could start fromthe Poisson equation h S + i , vary it under Galilean boost, and thereby generate the full set ofnon-relativistic equations. The general structure of the reducibility is in correspondence withthe dilatation weights. This allows for a schematic summary of the representation theory of theequations of motion, given in figure 1. Let us finally comment on the relation between the NR limit of NS-NS gravity, discussed above,and the beta functions of NR bosonic string theory. The latter were calculated in [16, 17] for theNR string moving in a SNC geometry with zero torsion, i.e., subject to the constraints: τ A ′ ( AB ) = 0 , and τ A ′ B ′ A = 0 . (57)In [5], it was then shown that these ‘zero torsion beta functions’ are reproduced by a NR limitof the beta functions of the relativistic bosonic string, i.e. of the equations of motion of NS-NS gravity. The limit of [5] is similar to the limit discussed in this paper. However, there isan important difference. The limit of [5] was taken in the first-order formalism, in which therelativistic spin connection Ω µ ˆ A ˆ B is treated as an independent field. To perform this first-orderlimit, the redefinitions (17) were supplemented with rescalings of the components of Ω µ ˆ A ˆ B with14 = − h S + i n = − h V − i AA ′ n = 0 h G i A ′ B ′ h B i A ′ B ′ h G i { AB } h Φ i n = +1 h V + i AA ′ n = +2 h S − i Figure 1: This diagram summarizes how the equations of motion transform under String Galilei anddilatation symmetries. The index structure indicates the SO(1 , × SO(8) representations and the layersdenote the dilatation weights δ D h X i = n λ D h X i . Under Galilean boosts, the equations of motion form areducible, but indecomposable representation. This is indicated by the red arrows. powers of c , not allowing them to be divergent in the limit c → ∞ . The zero torsion constraint(57) is then reproduced by the NR limit of the conventional constraints. One could also considerthe limit of [5] in the second-order formalism. This would lead to the same outcome if one requiresthe expansion of spin connections to be finite after taking the NR limit. However, in this paper,we took a NR limit of the NS-NS gravity action and the equations of motion in the second-orderformalism and arranged things, by canceling divergences when expanding the action or combiningequations when expanding the equations of motion, such that the leading order terms are alwaysof the order c before taking the NR limit and we could extract the maximum number of NRequations of motion. We showed that one can take the limit by adopting the redefinitions (27)and expanding the Kalb-Ramond field strength H µνρ and the second-order Ω µ ˆ A ˆ B accordingly. Theleading order terms of Ω µAA ′ and Ω µA ′ B ′ in these expansions appear at one c -order higher thanthe order dictated by the rescalings of Ω µAA ′ and Ω µA ′ B ′ in the first-order limit. Naively, thisleads to terms that would diverge in the c → ∞ limit (compared to the first-order limit). However,we noticed that such a divergence is absent, due to a fine-tuned cancellation between the kineticterms of the metric and of the KR field.In contrast to the first-order limit, the second-order NR limit of this paper did not lead tothe zero torsion conditions (57). Indeed, like the NR string worldsheet actions of section 2, theNR limits of both the action and equations of motion of NS-NS gravity are invariant under theemergent dilatation symmetry (26). The constraints (57) break this dilatation symmetry and canthus not result from the NR limit considered here. It is worth mentioning that the constraints (57)can be relaxed to a dilatation invariant set of torsion constraints. The maximal such set, whichdefines what we call Dilatation invariant SNC (DSNC) geometry, is given by τ A ′ { AB } = 0 , and τ A ′ B ′ A = 0 . (58)Compared to the SNC constraints (57), we have that b A ′ ≡ τ A ′ AA , (59)which acts like the (transverse components of the) gauge field of local dilatations, is non-zero.Let us now see in how far the equations of motion (49)/(54) obtained here are in agreementwith the results from beta function calculations [16, 17]. We want to stress that our starting pointis different from the one taken in the above references. Here, we have made no a priori assumptionson torsion components—i.e., we work with TSNC geometry. The string sigma model in [16, 17],however, is defined on an SNC geometry with the geometric constraints (57). Since the startingpoint is different, we can not compare the results directly. In particular, we find three additionalequations that are absent in the beta function analysis h S − i = 0 , h G i { AB } = 0 , h V + i AA ′ = 0 . (60)15hese impose constraints only on the torsion components and are thus identically satisfied whenworking with SNC geometry. Here, however, we consider SNC as a solution of eqs. (60) ratherthan an a priori constraint. There are more general solutions of (60), of which DSNC geometry(58) is an example. For the moment, however, let us impose the zero torsion constraints (57) inorder to compare with the beta function calculations. The remaining non-trivial equations h S + i = 0 , h V − i AA ′ = 0 , h G i A ′ B ′ = 0 , h b i A ′ B ′ = 0 , h Φ i = 0 , (61)impose constraints on SNC geometry, the dilaton, and the KR field. We find that these equationsare in agreement with the results of [16,17]. In other words, the 101 equations (49)/(54) encompassthe beta functions upon restricting to SNC geometry.Observe that the Poisson equation h S + i = 0 was found as a beta function of the SNC NRstring. It thus does not suffice to take the NR limit of the NS-NS gravity action to retrieve thecorrect background dynamics of NR string theory. It shows that the full set of beta functions ofnon-relativistic string theory does not follow from an action principle. This is different from theanalogous situation in relativistic string theory, where all the beta function constraints follow froma low-energy effective action.The emerging dilatation invariance that is present in the NR string worldsheet actions and thatwe have shown to be preserved in the NR limit of NS-NS gravity, hints that DSNC geometry is anatural target space geometry of NR string theory. In this regard, it is highly suggestive that thezero torsion beta function calculation of [16,17] can be generalized to worldsheet actions for stringsin DSNC backgrounds. It would be interesting to calculate these beta functions and comparethem with the NR limit of the equations of motion of NS-NS gravity, obtained in this paper.
In this paper we showed that a NR limit of the NS-NS gravity action can be defined which isbased upon a crucial cancellation of the quadratic divergences originating from the spin-connectionsquared terms in the Einstein-Hilbert term with those arising from the kinetic term of the KR 2-form field. These cancellations are the target space version of a similar collaboration of divergencesthat takes place when defining the NR limit of the relativistic string sigma model. They enable usto define a finite NR NS-NS gravity action without imposing any geometric constraint such as thezero torsion constraint which we imposed in our earlier work. Both the NR string sigma modeland the NR NS-NS gravity action exhibit an emergent local dilatation symmetry. This emergentsymmetry has the effect that taking the limit of the relativistic NS-NS gravity action produces aNR action that leads to one equation of motion less than the ones that one obtains by taking theNR limit of the relativistic NS-NS gravity equations of motion. This missing equation of motion isprecisely the Poisson equation of the Newton potential. This is consistent with the fact that it isnot known how to obtain an action for NC gravity including the Poisson equation by taking a limitof General Relativity. In this paper we only consider NR limits. If one would consider expansions(without taking the limit that c goes to infinity) and/or more symmetries than (a string extensionof) the Bargmann algebra with more fields than the standard formulation of NC gravity, thereare other ways to construct actions for non-relativistic gravity, see, e.g., [26, 28–30]. Our resultsimply that the NR equations of motion form a reducible, indecomposable representation of theString Galilei symmetries: it is consistent to delete the Poisson equation and obtain the samerepresentation of equations of motion as the one that follows from varying the NR NS-NS gravityaction.In our approach, no geometric constraints are imposed by hand, but instead, the allowed geom-etry follows by solving the equations of motion. One natural geometry that in particular solves thenonlinear constraint equation (49f) is given by a dilatation-invariant extension of the zero torsionconstraint (57) which we called Dilatation invariant String Newton-Cartan (DSNC) geometry. Theconstraints defining this geometry are given in eq. (58). We thank Ziqi Yan for discussions on this point that indicate that such a generalization could indeed be possible. This scale invariance only works in the directions longitudinal to the string. This is reminiscent of the reducedconformal symmetry recently discussed in [27].
16t would be interesting to see whether the same equations of motion that we obtained in thiswork by taking a NR limit can be obtained by redoing the beta function calculations of [16,17] in theabsence of the zero torsion constraint, which breaks the local dilatation invariance. We comparedour NR equations of motion with the beta function calculations of [16,17] for zero torsion and founda one-to-one correspondence except that we have one equation more that does not correspond toany beta function. It is given by the nonlinear constraint (49f). Remarkably, this constraint occursas the coefficient in front a λ ¯ λ operator that is generated by quantum corrections in the sigmamodel effective action. Such a term would deform the theory towards relativistic string theory asshown in [17]. Equation (49f) thus provides an ad-hoc obstruction to generating such a term in thequantum effective action. It would be interesting to understand these constraints from symmetryarguments. It is of interest to also compare our results with the beta function calculations of [18]. Thestarting point of [18] is the relativistic Polyakov sigma model in a background geometry witha null isometry direction. The beta function calculations were performed by first rewriting thestring sigma model in terms of NC variables in one dimension lower. In the light of this paper, itseems plausible that the beta functions calculated in [18], except for the Poisson equation, can beidentified with the equations that follow from the null reduction of the relativistic NS-NS gravityaction. Usually, this null reduction is only performed at the level of the equations of motion withthe argument that otherwise one misses the equation of motion that follows from varying theKaluza-Klein scalar, which by the null isometry direction is set to zero. But this is precisely thePoisson equation, which, as we discussed in this paper, is not expected to follow from an action butdoes follow from a null reduction of the relativistic equations of motion. The picture that arisesis that the double dimensional reduction of the NR NS-NS gravity action we constructed in thispaper is precisely the same action that one obtains from a null reduction of the relativistic NS-NSgravity action. Neither action gives rise to the Poisson equation of the Newton potential butfor different reasons: local dilatation symmetry versus null isometry. For more details about thispicture from a target space point of view, see [31].In the context of string theory, it is well-known that there exists a so-called dual formulation ofthe relativistic NS-NS action where the 2-form KR field has been replaced, via an on-shell dualityrelation, by a 6-form potential that couples to an NS-NS 5-brane. The dual action requires adifferent limit, which is characterized by a 5-brane foliation instead of the string foliation we usedin this work. Whereas the string limit leads to a decoupling of all states that are not criticallycharged under the KR field, the 5-brane limit leads to a different decoupling where all states thatare not critically charged with respect to the 6-form potential are decoupled [1]. We can considera 5-brane limit of the relativistic NS-NS gravity action in the same way that we took the stringlimit of the same action in this paper. We checked that the same crucial cancellation of infinitiesas in the case of the string limit takes place but now between the spin-connection squared termand the kinetic term for the 6-form potential. It would be interesting to see whether one couldmap the string and 5-brane actions into each other thereby establishing a NR duality relation thatmaps a solution of the NR string action to a dual solution of the NR 5-brane action, possibly viadimensional reduction to six dimensions.The expression for the NR NS-NS gravity action that we derived in this work by taking a NRlimit seems identical to the action recently derived in [32] from a Double Field Theory (DFT)point of view. Also there, the Poisson equation takes a special status. In fact, this is a generalphenomenon as discussed in [33]. This relation between DFT and NR string theory was alreadyadvocated some time ago [34, 35]. It is based upon the observation that in DFT it is natural touse a degenerate geometry with a null isometry. What is intriguing and what we learned in thiswork is that imposing a null isometry from one point of view is the same as taking a NR limit froma T-dual point of view. It suggests that one should perhaps also be able to understand the resultsof this paper by defining a proper NR limit of DFT itself.The present work grew out of an effort to define the NR limit of heterotic supergravity. Now a Z. Yan, private communication. Note that one can also perform a direct dimensional reduction of the NR NS-NS gravity action along a transversalspatial direction leading to a sector of NR string theory that does not follow from a null reduction of the relativisticNS-NS gravity action. For other recent work on the NR string theory in DFT, see [36, 37]. µ ˆ A ˆ B ( E, H ) = Ω µ ˆ A ˆ B ( E ) + H µ ˆ A ˆ B , (62)which leads to a linear divergence that is proportional to a self-dual projection of the torsion com-ponents that define the DSNC geometry given in eq. (58). In order to obtain finite supersymmetryrules, we have to set these self-dual projections to zero by hand: τ A ′ + − = τ A ′ B ′ − = 0 . (63)This leads to a so-called self-dual DSNC geometry that seems to play a role in the supersymmetriccase. One attractive feature of the self-dual DSNC geometric constraints (63), not shared by thefull DSNC geometric constraints (58), is that the constraint equations (63) are invariant undersupersymmetry. We hope to give more details about this interesting case soon.The results of this paper can be used as a starting point for exploring many generalizations ofNR string theory. For instance, one could investigate whether one can give a meaning to the NRlimit of IIA and IIB supergravity and even M-theory. Having supersymmetry under control onecould investigate the presence of half-supersymmetric fundamental string and other brane solutionsto the equations of motion [31]. Last but not least, one could consider taking a Carroll limit of NS-NS gravity and investigate whether an action for Carroll NS-NS gravity can be defined. We hopeto come back to this and many other extensions, generalizations and applications in the nearbyfuture. Acknowledgements
We thank Ziqi Yan, Jaume Gomis, Quim Gomis, Jelle Hartong, Niels Obers, Gerben Oling,Umut G¨ursoy, Natale Zinnato, Domingo Gallegos and Toine Van Proeyen for useful comments anddiscussions. JR also thanks the organizers of the NL Zoom Meeting on Non-Lorentzian geometry forthe opportunity to present parts of the results of this paper which has led to useful discussions andcomments. JR especially acknowledges Jeong-Hyuck Park for a useful email exchange regardingthe DFT approach. We would also like to thank Microsoft Skype, Zoom, and our internet providerswho made it possible to collaborate across borders during the COVID-19 pandemic. The work of CS¸is part of the research programme of the Foundation for Fundamental Research on Matter (FOM),which is financially supported by the Netherlands Organisation for Science Research (NWO). Thework of LR is supported by the FOM/NWO free program
Scanning New Horizons . A Conventions
A.1 Index Conventions
In this paper, ten-dimensional curved indices are denoted by lowercase Greek letters. Ten-dimensional flat indices are denoted by ˆ A , where ˆ A = 0 , · · · ,
9. The index ˆ A is split into a‘longitudinal’ index A , where A = 0 ,
1, and a ‘transversal’ index A ′ , where A ′ = 2 , · · · ,
9. Weadopt the ‘mostly plus’ convention for the ten-dimensional Minkowski metric η ˆ A ˆ B , i.e., η ˆ A ˆ B =diag( − , , · · · , A are freely raised and lowered with η AB , while transver-sal indices A ′ are raised and lowered with a Kronecker delta δ A ′ B ′ . The ten-dimensional epsilonsymbols ǫ ˆ A ··· ˆ A and ǫ ˆ A ··· ˆ A are normalized as ǫ ··· = +1 and ǫ ··· = −
1. We also use longitu-dinal epsilon symbols ǫ AB and ǫ AB that are normalized as ǫ = +1 and ǫ = −
1. We then have18he following useful identities: ǫ AC ǫ BD = − η AB η CD + η AD η BC , ǫ AC ǫ CB = η AB . (64)Symmetrization and antisymmetrization is defined with weight one, e.g., A [ µν ] = 12 ( A µν − A νµ ) , A ( µν ) = 12 ( A µν + A νµ ) . (65)We furthermore use curly parentheses to denote traceless symmetric parts, e.g., S { AB } = S ( AB ) − η AB S CC with S C C = η CD S CD . (66) A.2 Lorentzian Geometry Conventions
We denote the Vielbein of ten-dimensional Lorentzian geometry by E µ ˆ A and its inverse by E ˆ Aµ : E µ ˆ A E ˆ Bµ = δ ˆ A ˆ B , E µ ˆ A E ˆ Aν = δ νµ . (67)The fields E µ ˆ A and E ˆ Aµ transform as a one-form, resp. vector under general coordinate transfor-mations and as follows under local SO(1 ,
9) Lorentz transformations with parameter Λ ˆ A ˆ B = − Λ ˆ B ˆ A : δE µ ˆ A = Λ ˆ A ˆ B E µ ˆ B , δE ˆ Aµ = Λ ˆ A ˆ B E ˆ Bµ . (68)Upon splitting the ten-dimensional frame indices into a longitudinal and a transversal part, we findthat δ E µA = Λ AB E µB + Λ AA ′ E µA ′ , δ E µA ′ = Λ A ′ B ′ E µB ′ − Λ AA ′ E µA , (69)and similarly for the inverse Vielbein.We denote the torsionless spin connection of Lorentzian geometry by Ω µ ˆ A ˆ B . In this paper, wework in the second-order formalism and define Ω µ ˆ A ˆ B as the solution of the following zero torsionconstraint: R µν ( P ˆ A ) = 2 ∂ [ µ E ν ] ˆ A − [ µ ˆ A ˆ B E ν ] ˆ B = 0 . (70)Explicitly, one hasΩ µ ˆ A ˆ B = E µ ˆ C E ˆ A ˆ B ˆ C − E µ [ ˆ A ˆ B ] , where E µν ˆ A = ∂ [ µ E ν ] ˆ A . (71)The spin connection then transforms under local Lorentz transformations as an SO(1 ,
9) connection: δ Ω µ ˆ A ˆ B = ∂ µ Λ ˆ A ˆ B − µ ˆ C [ ˆ A Λ ˆ B ] ˆ C . (72)The covariant curvature 2-form, associated to Ω µ ˆ A ˆ B is defined by: R µν ˆ A ˆ B = 2 ∂ [ µ Ω ν ] ˆ A ˆ B + 2 Ω [ µ ˆ A ˆ C Ω ν ] ˆ B ˆ C . (73)These quantities are related to the Christoffel connection Γ µν ρ and the Riemann tensor R µν ρσ asfollows Γ µν ρ = E ˆ Aρ (cid:16) ∂ ( µ E ν ) ˆ A − Ω ( µ ˆ A ˆ B E ν ) ˆ B (cid:17) and R ρσµν = − E ρ ˆ A E σ ˆ B R µν ˆ A ˆ B . (74)The Ricci tensor and scalar are then expressed in terms of the curvature 2-form R µν ˆ A ˆ B as: R µν = R ρµρν = − E ρ ˆ A R ρµ ˆ A ˆ B E ν ˆ B and R = − E µ ˆ A E ν ˆ B R µν ˆ A ˆ B . (75)19 .3 Conversion of Curved to Flat Indices The curved indices on a tensor X µ ··· µ r µ r +1 ··· µ p in the relativistic theory are turned into flatones, using the relativistic (inverse) Vielbein E µ ˆ A ( E ˆ Aµ ) in the usual fashion X ˆ A ··· ˆ A r ˆ A r +1 ··· ˆ A p = E µ ˆ A · · · E µ r ˆ A r E ˆ A r +1 µ r +1 · · · E ˆ A p µ p X µ ··· µ r µ r +1 ··· µ p . (76)Note that the ˆ A index will often be split into a longitudinal index A and a transversal index A ′ .As an example, if X µν is a tensor in the relativistic theory, the quantities X AB , X AB ′ , X A ′ B and X A ′ B ′ are to be understood as X AB = E Aµ E Bν X µν , X AB ′ = E Aµ E B ′ ν X µν , X A ′ B = E A ′ µ E Bν X µν ,X A ′ B ′ = E A ′ µ E B ′ ν X µν . (77)The curved indices on tensors in the non-relativistic theory are turned into flat ones, usingthe longitudinal and transverse Vielbeine τ µA , e µA ′ . For example, if Y µν is a tensor in the non-relativistic theory, the objects Y AB , Y AA ′ , Y A ′ A and Y A ′ B ′ are defined as Y AB = τ µA τ Bν Y µν , Y AA ′ = τ µA e A ′ ν Y µν , Y A ′ A = e µA ′ τ Aν Y µν ,Y A ′ B ′ = e µA ′ e B ′ ν Y µν . (78) B Torsional String Newton Cartan Geometry
In this section, we give details on the non-Lorentzian geometry that appears in the NR limitof NS-NS gravity, discussed in this paper. We refer to this geometry as ‘torsional string NewtonCartan geometry’ (TSNC). The basic geometric fields are the longitudinal Vielbein τ µA ( A = 0 , e µA ′ ( A ′ = 2 , · · · , b µν and the dilaton φ . These fieldstransform under local String Galilei symmetries (longitudinal SO(1 ,
1) Lorentz transformations,transverse SO(8) rotations and Galilean boosts) and local dilatations, according to: δτ µA = λ M ǫ AB τ µB + λ D τ µA , δe µA ′ = λ A ′ B ′ e µB ′ − λ AA ′ τ µA ,δb µν = − ǫ AB λ AA ′ τ [ µB e ν ] A ′ , δφ = λ D , (79)where λ M , λ A ′ B ′ , λ AA ′ , λ D are the parameters of SO(1 , δb µν = 2 ∂ [ µ θ ν ] . (80)Projective inverses τ Aµ and e A ′ µ are introduced via (29). They transform as δτ Aµ = λ M ǫ AB τ Bµ + λ AA ′ e A ′ µ − λ D τ Aµ , δe A ′ µ = λ A ′ B ′ e B ′ µ . (81)Note that the KR field transforms non-trivially to the longitudinal and transverse Vielbeine underGalilean boosts, while the dilaton acquires a shift under dilatations. For this reason, we treat b µν and φ as part of the geometric data. This should be contrasted with relativistic string theory/NS-NS gravity, in which the KR field and dilaton are treated as matter fields, instead of geometricfields.In the following, we will discuss how these fields can be used to define connections and curvaturesfor local SO(1 , × SO(8) transformations, Galilean boosts and dilatations. In Appendix C we willthen describe how the geometric structures, described here, appear in the limit of the action andequations of motion of NS-NS gravity.
B.1 String Galilei and Dilatation Connections
Here, we introduce String Galilei spin connections ω µ , ω µAA ′ and ω µA ′ B ′ for SO(1 ,
1) Lorentztransformations, SO(8) rotations and Galilean boosts resp., as well as a dilatation connection b µ .20n analogy to the spin connection Ω µ ˆ A ˆ B of Lorentzian geometry, we will define these connections asexpressions that depend on the geometric data τ µA , e µA ′ , b µν and φ , with correct transformationproperties. In particular, we seek to define ω µ , ω µAA ′ , ω µA ′ B ′ and b µ as dependent expressionsthat transform under String Galilei transformations and dilatations as δb µ = ∂ µ λ D + · · · , δω µ = ∂ µ λ M + · · · , (82a) δω µAA ′ = ∂ µ λ AA ′ + · · · , δω µA ′ B ′ = ∂ µ λ A ′ B ′ + · · · , (82b)where the ellipses denote terms that do not involve derivatives of a parameter. To do this, weconsider the following ‘covariant’ quantities ∇ µ φ ≡ ∂ µ φ − b µ , (83a)R µν ( H A ) ≡ τ µνA − ǫ AB ω [ µ + δ AB b [ µ ) τ ν ] B , (83b)R µν ( P A ′ ) ≡ e µνA ′ − ω [ µA ′ B ′ e ν ] B ′ + 2 ω [ µAA ′ τ ν ] A , (83c) H µνρ ≡ h µνρ + 6 ǫ AB ω [ µAB ′ τ ν B e ρ ] B ′ . (83d)where we have defined τ µν A = ∂ [ µ τ ν ] A , e µνA ′ = ∂ [ µ e ν ] A ′ , h µνρ = 3 ∂ [ µ b νρ ] . (84)The quantities defined in (83) are covariant in the sense that they transform without derivativesof a parameter, if the transformation rules (82) hold.Similarly to how one defines the relativistic spin connection (71), we can then try to express theString Galilei spin connections and the dilatation connection as dependent fields that solve conven-tional constraints, that are obtained by putting certain components of the covariant quantities(83) equal to zero. Note that we should only constrain those parts of (83) that contain componentsof ω µ , ω µAA ′ , ω µA ′ B ′ and b µ . In particular, the following components of (83)R A ′ B ′ ( H C ) = 2 τ A ′ B ′ C , R A ′ { A ( H B } ) = 2 τ A ′ { AB } , H A ′ B ′ C ′ = h A ′ B ′ C ′ , (85)are independent of the String Galilei spin connections and b µ and are not set to zero as conventionalconstraints. Note that R A ′ B ′ ( H C ) and R A ′ { A ( H B } ) contain information about the intrinsic torsionof the geometry [38]. For the remaining components of (83), we then adopt the following constraints ∇ A φ = 0 , η AB R A ′ A ( H B ) = 0 , (86a) ǫ AB R A ′ A ( H B ) = 0 , ǫ AB R AB ( H C ) = 0 , (86b)R µν ( P A ′ ) = 0 , (86c) H AA ′ B ′ = 0 , H ABA ′ = 0 . (86d)These can be viewed as 444 algebraic equations for the 460 components of b µ , ω µ , ω µAA ′ , ω µA ′ B ′ .These equations are thus not able to determine all components of the String Galilei spin connectionsand dilatation field . The 16 undetermined components reside in ω { AB } A ′ and will in the followingbe denoted by W µAA ′ = τ µB ω { AB } A ′ . We can then express the most general solution of theconventional constraints (86) by b µ = e µA ′ τ A ′ AA + τ µA ∂ A φ , (87a) ω µ = ( τ µAB − τ µC τ ABC ) ǫ AB − τ µA ǫ AB ∂ B φ , (87b) ω µAA ′ = − e µAA ′ + e µB ′ e AA ′ B ′ + 12 ǫ AB h µBA ′ + W µAA ′ , (87c) By conventional constraints we mean constraints that reduce the number of independent fields in a theory, i.e.,constraints that contain certain fields algebraically and that can be used to solve those fields in terms of other fields. A similar phenomenon has been encountered in [5] µA ′ B ′ = − e µ [ A ′ B ′ ] + e µC ′ e A ′ B ′ C ′ − τ µA ǫ AB h BA ′ B ′ . (87d)The transformation rules of (87) under String Galilei symmetries and dilatations can be obtainedby requiring that the set of constraints (86) does not transform under these symmetries. Writingdown these requirements, using (79), one obtains equations for δb µ , δω µ , δω µAA ′ and δω µA ′ B ′ thatcan be solved to give δ b µ = ∂ µ λ D , δ ω µ = ∂ µ λ M , (89a) δ ω µA ′ B ′ = ∂ µ λ A ′ B ′ − ω µC ′ [ A ′ λ B ′ ] C ′ , (89b) δ ω µAA ′ = λ M ǫ AB ω µBA ′ + λ A ′ B ′ ω µAB ′ − λ D ω µAA ′ , (89c)for the transformations under SO(1 , × SO(8) and dilatations and δ b µ = e µA ′ τ A ′ B ′ B λ BB ′ + τ µA λ AA ′ ∇ A ′ φ , (90a) δ ω µ = − e µA ′ ǫ AB τ A ′ B ′ A λ BB ′ − τ µA ( ǫ BC λ BB ′ τ B ′ { AC } + 12 ǫ AB λ BB ′ ∇ B ′ φ ) , (90b) δ ω µAA ′ = ∇ µ λ AA ′ + 2 e µB ′ ( λ BB ′ τ A ′ { AB } + 14 ǫ AB λ BC ′ h A ′ B ′ C ′ ) , (90c) δ ω µA ′ B ′ = 4 τ µA ( λ B [ A ′ τ B ′ ] { AB } − ǫ AB λ BC ′ h A ′ B ′ C ′ ) − e µC ′ ( λ CC ′ τ A ′ B ′ C − λ C [ A ′ τ B ′ ] C ′ C ) , (90d)under Galilean boosts, where we have defined ∇ µ λ AA ′ = ∂ µ λ AA ′ − ω µ ǫ AB λ BA ′ − ω µA ′ B ′ λ AB ′ + b µ λ AA ′ . (91) B.2 Affine Connection
Using the String Galilei spin connections and dilatation connection, we can then also introducean affine, metric compatible connection Γ ρµν , by imposing the following Vielbein postulates ∇ µ τ ν A = ∂ µ τ ν A − ω µ ǫ AB τ νB − b µ τ ν A − Γ ρµν τ ρA = 0 , (92a) ∇ µ e νA ′ = ∂ µ e νA ′ − ω µA ′ B ′ e νB ′ + ω µAA ′ τ νA − Γ ρµν e ρA ′ = 0 . (92b)This connection has intrinsic torsion T ρµν = 2 Γ ρ [ µν ] = R µν ( H A ) τ Aρ . (93)As a corollary of (92), we derive the following identities: ∂ µ ( e e A ′ µ ) = e e B ′ µ ω µA ′ B ′ + 2 e b A ′ , (94a) ∂ µ ( e τ Aµ ) = e τ Bµ ( ǫ AB ω µ + δ AB b µ ) + e e A ′ µ ω µAA ′ , (94b)which are used to derive the action (43) and the equations of motion (49). B.3 Curvatures
Using the transformation rules for the spin connections (89) and (90) we can define covariantcurvature 2 − formsR µν ( D ) = 2 ∂ [ µ b ν ] +2 e [ µA ′ ω ν ] BB ′ τ A ′ B ′ B + 2 τ [ µA ω ν ] AA ′ ∇ A ′ φ , (95a) The undetermined components in the boost spin connection transform as follows δW µAA ′ = τ µB ( ∇ { A λ B } A ′ − ω B ′ A ′ { A λ B } B ′ )+ λ M ǫ AB W µBA ′ + λ A ′ B ′ W µAB ′ − λ D W µAA ′ . (88) µν ( M ) = 2 ∂ [ µ ω ν ] + 2 ǫ AB e [ µA ′ ω ν ] AB ′ τ A ′ B ′ B − τ [ µA ω ν ] BB ′ ǫ BC τ B ′ { AC } + 2 ǫ AB τ [ µA ω ν ] BB ′ ∇ B ′ φ , (95b)R µν ( G ) AA ′ = 2 ∂ [ µ ω ν ] AA ′ − ǫ AB ω [ µ ω ν ] BA ′ − ω [ µA ′ B ′ ω ν ] AB ′ + 2 b [ µ ω ν ] AA ′ − e [ µB ′ ( ω ν ] B [ A ′ τ B ′ ] { AB } − ǫ AB ω ν ] BC ′ h A ′ B ′ C ′ ) , (95c)R µν ( J ) A ′ B ′ = 2 ∂ [ µ ω ν ] A ′ B ′ + 2 ω [ µA ′ C ′ ω ν ] B ′ C ′ + 2 e [ µC ′ (2 ω ν ] C [ A ′ τ B ′ ] C ′ C − ω ν ] CC ′ τ A ′ B ′ C )+ 8 τ [ µA ( ω ν ] B [ A ′ τ B ′ ] { AB } − ǫ AB ω ν ] BC ′ h A ′ B ′ C ′ ) . (95d)The Ricci scalar built from the curvature of transverse rotations readsR( J ) = − R A ′ B ′ ( J ) A ′ B ′ = − e A ′ µ e B ′ ν ( ∂ [ µ ω ν ] A ′ B ′ + ω [ µA ′ C ′ ω ν ] B ′ C ′ )( e A , e B ′ ) − ω A ′ BB ′ τ A ′ B ′ B . (96)Substituting the conventional constraints (86) into the Bianchi identities, one can derive the fol-lowing relations:R [ B ′ C ′ ( J ) A ′ D ′ ] = 0 −→ R C ′ [ A ′ ( J ) B ′ ] C ′ = 0 , (97a)2 R [ B ′ A ( J ) A ′ C ′ ] = − R B ′ C ′ ( G ) AA ′ −→ R C ′ A ( J ) A ′ C ′ = R C ′ A ′ ( G ) AC ′ , (97b) ∇ A τ A ′ B ′ A = − R A ′ B ′ ( D ) −→ ∇ A τ A ′ B ′ A = 2 ∇ [ A ′ ∇ B ′ ] φ . (97c) C Details on the NR Limit of NS-NS Gravity
Here, we provide details on how the NR limit of NS-NS gravity is taken in section 4. Inparticular, we give the results of expanding the geometric objects of Lorentzian geometry of A.2in powers of c − , after applying the redefinitions (27). We also show how these results can berewritten in terms of geometric quantities of TSNC geometry, discussed in the previous AppendixB. We refer to Appendices A and B for the notation used in this section.Let us start by applying the redefinitions (27) to the relativistic spin connection (71) andexpanding the result into leading and subleading orders of powers of c − . This leads toΩ µ = (0) ω µ + 1 c ( − ω µ , (98a)Ω µAB ′ = c (1) ω µAB ′ + 1 c ( − ω µAB ′ = − Ω µB ′ A , (98b)Ω µA ′ B ′ = c (2) ω µA ′ B ′ + (0) ω µA ′ B ′ , (98c)where we have set Ω µAB = Ω µ ǫ AB and we explicitly have (0) ω µ = ǫ AB ( e µC ′ τ C ′ AB − τ µC τ ABC ) , ( − ω µ = − ǫ AB e µC ′ e ABC ′ , (99a) (1) ω µAA ′ = e µB ′ τ B ′ A ′ A − τ µB τ A ′ ( BA ) , ( − ω µAA ′ = 2 e µB ′ e A ( A ′ B ′ ) − τ µB e BAA ′ , (99b) (2) ω µA ′ B ′ = τ µC τ A ′ B ′ C , (0) ω µA ′ B ′ = e µC ′ e A ′ B ′ C ′ − e µ [ A ′ B ′ ] . (99c)It is useful to note that (1) ω ABA ′ = − τ A ′ ( AB ) , (1) ω CCC ′ = − b C ′ , (1) ω B ′ AA ′ = τ B ′ A ′ A , (1) ω A ′ AA ′ = 0 , (100) (2) ω AB ′ C ′ = τ B ′ C ′ A , (2) ω A ′ B ′ C ′ = 0 . None of the expressions in (99) correspond to String Galilei spin connections as they stand; rather (0) ω µ , ( − ω µAA ′ and (0) ω µA ′ B ′ are related to the String Galilei spin connections ω µ , ω µAA ′ and ω µA ′ B ′ via (0) ω µ = ω µ + τ µA ǫ AB ∂ B φ , (101a)23 − ω µAA ′ = ω µAA ′ − ǫ AB h µBA ′ − W µAA ′ , (101b) (0) ω µA ′ B ′ = ω µA ′ B ′ + 12 τ µA ǫ AB h BA ′ B ′ . (101c)Expanding H µνρ = 3 ∂ [ µ B νρ ] in powers of c − , using (27), we have H µνρ = c (2) H µνρ + (0) H µνρ , (102)with (2) H µνρ = 6 ǫ AB τ [ µA τ νρ ] B , (0) H µνρ = h µνρ = 3 ∂ [ µ b νρ ] , (2) H AA ′ B ′ = 2 ǫ AB τ A ′ B ′ B , (2) H ABC ′ = − ǫ AB b C ′ , (2) H A ′ B ′ C ′ = 0 , and (2) H ABC = 0 . Similarly, we can also expand the Ricci scalar (75) as R = c (2) R + (0) R + O ( c − ) , (103)where : (2) R = − η AB τ A ′ B ′ A τ A ′ B ′ B , (104) (0) R = − (0) R A ′ B ′ A ′ B ′ − e A ′ µ (cid:18) ∂ µ b A ′ − (0) ω µA ′ B ′ b B ′ + 32 b µ b A ′ (cid:19) − τ A ′ { AB } τ A ′ { AB } , (105)where (0) R A ′ B ′ A ′ B ′ = 2 e A ′ µ e B ′ ν ( ∂ [ µ (0) ω ν ] A ′ B ′ + (0) ω [ µC ′ A ′ (0) ω ν ] B ′ C ′ ) .The quantity (0) R can be expressed in terms of TSNC geometric variables, using the relations(101) (0) R = R( J ) − D A ′ b A ′ − b A ′ b A ′ +2 ω A ′ BB ′ τ A ′ B ′ B − τ A ′ { AB } τ A ′ { AB } , (106)where the derivative D µ is defined in (41b). References [1] J. Gomis and H. Ooguri,
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