Light-Cone Gauge in Non-Relativistic AdS_5\timesS^5 String Theory
Andrea Fontanella, Juan Miguel Nieto García, Alessandro Torrielli
aa r X i v : . [ h e p - t h ] J a n HU-EP-21/02, DMUS-MP-21/01
Light-Cone Gauge inNon-Relativistic AdS × S String Theory
Andrea Fontanella ♭ , Juan Miguel Nieto Garc´ıa ♯ and Alessandro Torrielli ♯♭ Institut f¨ur Physik, Humboldt-Universit¨at zu Berlin,IRIS Geb¨aude, Zum Gro β en Windkanal 2, 12489 Berlin, Germany ♯ Department of Mathematics, University of SurreyGuildford, GU2 7XH, UK.
Abstract
We consider the problem of fixing uniform light-cone gauge in the bosonic sectorof non-relativistic AdS × S string theory found by J. Gomis, J. Gomis and K.Kamimura. We show that if the common AdS and S radius is kept large and thestring has a non-zero winding mode around the non-relativistic longitudinal spatialdirection, the light-cone gauge fixed model describes at leading order in the largestring tension expansion the dynamics of 8 bosonic free massless scalars in Mink .We discuss limitations and potential issues of fixing the light-cone gauge in the casewhere one evades the large radius assumption. [email protected] [email protected] [email protected] ontents Introduction 11 Non-relativistic AdS × S string theory 4 m µA field . . . . . . . . . . . . . . . . . . . 7 R and T limit 93 Comments on finite R
154 Conclusions 16Appendices 18A Newton-Cartan data 18
A.1 so (4 , ⊕ so (6) algebra contraction . . . . . . . . . . . . . . . . . . . . . . 19A.2 Cartesian global coordinates . . . . . . . . . . . . . . . . . . . . . . . . . . 20A.3 Polar global coordinates . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21A.4 GGK coordinates . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23 B Cubic and Quartic Lagrangians 24C Different winding assumptions 25
Introduction
Recently we experienced in the community of theoretical high-energy physics a growinginterest regarding the topic of non-relativistic string theory. The first theory of non-relativistic strings was formulated in flat spacetime [1] and later it was also found inAdS × S [2], but it is only recently that we gained a better understanding of thesetype of theories. In [3, 4] it was found that non-relativistic string theories (in flat orcurved backgrounds) are theories secretly defined on a background which is String Newton-Cartan . This is quite different from the usual relativistic string theory, where the stringis free to move on a Riemannian manifold. In these papers, T-duality and symmetries fornon-relativistic string theories are also discussed . The approach used in [3, 4] consists in In these papers only closed strings have been considered. For recent work on non-relativistic open strings, see [5, 6]. × S string theory and N = 4 SYM theory in 4d [11].To begin the program of testing the non-relativistic version of the AdS/CFT duality,in the string theory side one needs to compute the spectrum of the string excitations. Thishas been done with great success in relativistic AdS × S string theory, where integrabil-ity in the planar limit has played an important rˆole in terms of making computationsmanageable [12].We remark that at the time of writing, classical integrability of non-relativistic AdS × S string theory is still lacking. A classical integrable extension of the non-relativisticAdS × S superstring theory has been generated via Lie algebra expansion [13]. How-ever it is still unclear if the truncation that takes this extended non-relativistic theory tothe theory of Gomis, Gomis and Kamimura spoils integrability. Work in this direction isin progress [14].The spirit of this paper is to start applying ideas developed in relativistic AdS × S string theory [15–17], which are useful in order to test the AdS/CFT duality, also tothe non-relativistic theory. In order to find the full non-relativistic spectrum of stringexcitations one would need to know the S-matrix of the theory and then apply Betheansatz techniques. To determine the S-matrix, first we need to fix a gauge in such a waythat we get rid of the redundant string degrees of freedom. In the context of relativisticAdS × S string theory, the light-cone gauge is particularly useful, since in this gauge andin the large string tension limit the action expands around the action of free scalar fields inMink and this is particularly convenient for quantisation and the S-matrix computationbetween world-sheet excitations.The purpose of this paper is to apply the uniform light-cone gauge to non-relativisticAdS × S string theory and to study its expansion in the large string tension limit. So2ar, non-relativistic AdS × S string theory has been studied only in the static gauge , seee.g. [2]. In the static gauge, this theory describes the dynamics of 3 massive and 5 masslessfree scalar fields propagating in AdS . Although the dynamics of these fields is “free”,there is still the important complication due to the fact that they propagate in a curvedbackground , and finding the S-matrix of scalar fields moving in AdS is notoriously difficult,see e.g. [18]. Furthermore, if one were to assume the radius of AdS to be large and expandthe non-relativistic action in static gauge perturbatively at large radius around 2d flatspacetime, one would find that the perturbation is untamed at the asymptotic infinities,and this prevents from defining asymptotic scattering states as one would usually do inflat spacetime.The advantage of applying light-cone gauge, instead of static gauge, is that the stringaction typically expands around free scalar fields in 2d flat spacetime with tamed correc-tion terms, such that asymptotic scattering states are well defined. This is suitable thenfor perturbative S-matrix computations based on Feynman diagrams. As we remarkedabove, light-cone gauge has proven to be a very powerful tool in the context of relativisticAdS × S string theory [12, 15–17].The main objective of this paper is to show that the theory found in [2] admits awell defined strong coupling expansion (i.e. large string tension expansion) around thetheory of 8 free scalar fields propagating in flat 2d spacetime. We perform this expansionin the uniform light-cone gauge . However we can show this result only in the restrictedsetting where the common radius of AdS and S is large, where the 8 scalar fields becomemassless. The difficulties to generalise this result to the case of finite radius will also beexplained.A feature which is peculiar of the non-relativistic theory, but not present in the rela-tivistic one, is the following. As it was pointed out in [9, 10], based on the original workin flat spacetime [1, 21], in order to obtain a meaningful spectrum in a non-relativisticstring theory, one needs to wrap the string around the longitudinal spatial direction of thebackground geometry . We find that this statement is true for non-relativistic AdS × S We point out that the Hamiltonian formulation for actions based on string Newton-Cartan back-grounds has been studied in [19]. Moreover light-cone gauge has been fixed for a theory obtained fromrescaling t in AdS and φ in S [20]. However this rescaling of coordinates does not lead to the non-relativistic AdS × S string theory discussed in [2], since it does not single out an AdS subspace insideAdS . It is actually possible in the relativistic AdS × S theory to obtain a meaningful expansion of theaction also when winding around the isometric φ direction in S is considered. This is discussed inthe near-BMN limit in [22]. We thank G. Arutyunov for pointing this out. However for this type ofwinding the quadratic Hamiltonian takes large eigenvalues once the level-matching condition is imposedon physical states. This does not happen in the non-relativistic theory. We stress once more that if havinga winding mode seems to be an additional possibility in the relativistic theory, it becomes a necessary Plan of this paper.
In section 1 we introduce the notion of non-relativistic string theorybased on a general Newton-Cartan geometry, although in some points our discussion willbe closely referring to the case of AdS × S . In this section we also discuss what happenswhen one takes different sets of coordinates for the initial relativistic theory and thenextract the Newton-Cartan data. In section 2 we fix uniform light-cone gauge for thebosonic sector of non-relativistic AdS × S string theory, and we show that the actionexpands around the free scalars action in Mink in the large string tension and radiuslimits. In section 3 we discuss limitations of applying our procedure when the largeradius assumption is removed. This paper ends with several appendices. In Appendix Awe define the algebra contraction, the sets of coordinates relevant in this paper (Cartesian,polar and GGK) and we write down the Newton-Cartan data for all of them. In AppendixB we list the cubic and quartic interactions obtained from the expansion of the action inthe large string tension and radius limits. In Appendix C we fix the light-cone gauge withdifferent winding assumptions and at finite radius, and we provide the action at leadingorder in the large string tension limit. × S string theory Our starting point is to consider the usual relativistic string action in AdS × S . Weshall focus only on the bosonic sector. We choose to describe AdS × S with the set ofCartesian global coordinates given in Appendix A. The relativistic action is then S = − T Z d σ γ αβ ∂ α X µ ∂ β X ν g µν , (1.1)where T is the string tension, σ α = ( τ, σ ), with α = 0 ,
1, are the string world-sheet coordi-nates, γ αβ ≡ √− hh αβ is the Weyl invariant combination of the inverse world-sheet metric h αβ and h = det( h αβ ), X µ = ( t, z i , φ, y i ), where i = 1 , ...,
4, are the string embeddingcoordinates, which are bosonic fields depending on ( τ, σ ), and g µν is the AdS × S metric condition in the non-relativistic theory in order to make the large string tension expansion of its actionmeaningful.
4n Cartesian global coordinates, g µν d X µ d X ν = − (cid:18) z R − z R (cid:19) d t + 1(1 − z R ) d z i d z i + (cid:18) − y R y R (cid:19) d φ + 1(1 + y R ) d y i d y i , (1.2)where R is the common radius of AdS and S . We use the short-hand notation z ≡ z i z i ,where z i = z i , and the same for the y coordinates. In the limit R → ∞ the action (1.1)describes strings moving in flat space.This relativistic action has a so (4 , ⊕ so (6) global symmetry algebra, and the space-time is described by a Riemannian metric which has local Lorentz invariance. To obtainthe non-relativistic AdS × S action found in [2], we need to contract so (4 , ⊕ so (6) as de-scribed in Appendix A, which takes the so (4 ,
2) algebra to the 5d Newton-Hooke algebra.There are four equivalent choices of making such a contraction, which are associated withthe four spatial directions in so (4 , . Here we adopt the same choice made in AppendixA, which consists in choosing the flat direction with index 1 to be different from the otherdirections with index in { , , } .Rescaling the generators of so (4 , ⊕ so (6) with a parameter ω induces a dual rescalingof coordinates, via the coset representative of Cartesian coordinates given in eqn. (A.7).The induced rescaling of coordinates is the following t → t , z → z , z m → ω z m , φ → ω φ , y i → ω y i , (1.3)where m = 2 , ,
4. This has a consequence on the relativistic vielbeine ˆ E µ ˆ A , they willexpand as followsˆ E µA = ωτ µA + 1 ω m µA + O ( ω − ) , ˆ E µa = e µa + O ( ω − ) , (1.4)where ˆ A = 0 , ..., d − A = 0 ,
1, which are the longitudinal directionsin the tangent space, and into a = 2 , ..., d −
1, which are the transverse directions inthe tangent space. Since the relativistic metric is written as g µν = ˆ E µ ˆ A ˆ E µ ˆ B ˆ η ˆ A ˆ B , whereˆ η ˆ A ˆ B = diag( − , , ..., However this is not the case for other types of coordinates. For instance, once the coset representativeis fixed to reproduce polar global coordinates and GGK coordinates, there is only one sensible choice ofalgebra contraction that gives the non-relativistic action [2]. S = − T Z d σ γ αβ (cid:18) ω ∂ α X µ ∂ β X ν τ µν + ∂ α X µ ∂ β X ν H µν (cid:19) + O ( ω − ) , (1.5)where we introduced τ µν ≡ τ µA τ νB η AB , H µν ≡ e µa e νb δ ab + (cid:18) τ µA m νB + τ ν A m µB (cid:19) η AB , (1.6)where η AB = diag( − , , , ..., τ µν is called the longitudinal metric, while H µν is the boost invariant metric [3].In the contraction limit where ω → ∞ , the term in the action containing τ µν diverges.This is cured by tuning a closed B-field which does not affect the dynamics, given asfollows B µν = ω τ µA τ ν B ε AB , (1.7)which couples to the string in the usual fashion S B = − T Z d σ ε αβ ∂ α X µ ∂ β X ν B µν , (1.8)and this needs to be added to the action (1.5). Here we choose the convention where ε = +1 both for ε αβ and ε AB . By doing this, we get S + S B = − T Z d σ γ αβ ∂ α X µ ∂ β X ν H µν + S div + O ( ω − ) , (1.9)where now the divergent part nicely recasts into a perfect square S div = − ω T Z d σ γ F A F B η AB , (1.10)where F A = τ µA ∂ X µ − γ ε AB η BC τ µC ∂ X µ − γ γ τ µA ∂ X µ . (1.11)The divergent action written in the form (1.10) is suitable to be rewritten in terms ofLagrange multipliers λ A as S div = − T Z d σ (cid:18) λ A F A + 14 ω γ λ A λ A (cid:19) , (1.12)which is equivalent to (1.10) by using the equations of motion for λ A . In this way thedivergent action is just the sum of a finite and a subleading term in the limit ω → ∞ .6inally, we take the limit ω → ∞ and we remain with the non-relativistic action S NR = − T Z d σ (cid:18) γ αβ ∂ α X µ ∂ β X ν H µν + λ A F A (cid:19) . (1.13)At this stage, one may want to integrate out the Lagrange multipliers. By imposing theequations of motion for the Lagrange multipliers one gets that h αβ is solved in terms of τ αβ as follows h αβ ∼ τ αβ , (1.14)where ∼ means that they are identified up to a conformal factor, and we denoted τ αβ ≡ τ µν ∂ α X µ ∂ β X ν , which is the pull-back of the τ µν metric into the string world-sheet. Sincein AdS × S the τ µν metric describes an AdS geometry, in [2] the authors imposed staticgauge in order to obtain a 2d theory of free massive and massless bosons propagating inAdS . However we will not follow this approach here.The non-relativistic action (1.13) describes strings propagating in a background whichis not Riemannian, but Newton-Cartan instead. Newton-Cartan geometry is specifiedby the set of vielbeine { τ µA , m µA , e µa } , and their local symmetry is the string Bargmannalgebra as amply discussed in [3, 4, 8–10]. It is interesting to study physical properties ofthis theory as a theory standing on its own, regardless of the limiting procedure to deriveit, and in section 2 we discuss light-cone gauge fixing for the non-relativistic AdS × S action in the large string tension limit. m µA field The general idea is to fix a set of coordinates for AdS × S , apply the rescaling rules ofcoordinates induced by the contraction of so (4 ,
2) to the 5d Newton-Hooke algebra whichsingles out an AdS subspace inside AdS , and read off the Newton-Cartan data from theexpansion of the relativistic vielbeine.The Newton-Cartan m µA field is quite special, because if one applies the expansionprocedure just described above in Cartesian and in GGK coordinates (see Appendix A)one finds that m µA is not zero. However the same procedure applied in polar coordinatesleads to m µA = 0.In the end the non-relativistic action (1.13), regardless of the choice of Cartesian/GGKor polar coordinates, will always give the same physics. This can be seen for instance byfixing static gauge , where the timelike longitudinal direction is identified with τ and the The string Bargmann algebra is a non-central extension of the string Galilei algebra. In Cartesian coordinates, this amounts to fixing t = τ and z = σ (in our conventional choice of σ . In all these cases of different choices of coordinatesone finds that the action (1.13), after integrating out the Lagrange multipliers, reducesto the action of 3 massive and 5 massless free scalar fields propagating in AdS .The fact of having a vanishing m µA in one set of coordinates but not in another onehas also an effect on the boost invariant metric H µν . The most general H µν metric hasfull rank, and this happens to be the case for Cartesian and GGK coordinates. Howeverfor polar coordinates, since m µA = 0, the metric H µν does not have full rank.This means that H µν does not transform under change of coordinates as the usualRiemannian metric g µν , i.e. under a diffeomorphism X µ → ˜ X µ ( X ) with the usual rule˜ H µν ( ˜ X ) = ∂X ρ ∂ ˜ X µ ∂X σ ∂ ˜ X ν H ρσ ( X ) , (1.15)because if this was the case, the rank of H µν would not have changed from one set ofcoordinates to another one.In [9] it has been observed that (for a given choice of set of coordinates) one has thefreedom to remove m µA inside H µν by suitably redefining the Lagrange multipliers andthe B-field. In this way the non-relativistic action (1.13) becomes S NR = − T Z d σ (cid:18) γ αβ ∂ α X µ ∂ β X ν H ⊥ µν + ε αβ ∂ α X µ ∂ β X ν B µν + ˜ λ A F A (cid:19) , (1.16)where H ⊥ µν ≡ e µa e νb δ ab , (1.17)is the transverse metric, ˜ λ A are the new redefined Lagrange multipliers and B µν is thenew B-field which carries the information of the m µA field.In this democratic point of view, H ⊥ µν has always at most rank d − m µA terms missing from H µν .For the specific problem of fixing light-cone gauge, we prefer to work in the picturewhere the action is of the type (1.13) and H µν is invertible. This means that polarcoordinates are not suitable for this type of problem, since they generate a vanishing m µA field in the vielbeine expansion and therefore a non-invertible H µν . s = 1). In polar coordinates, this amounts to fixing t = τ and ρ = σ . In GGK coordinates, this is givenby x = τ and x = σ . Uniform light-cone gauge in the large R and T limit In this section we fix uniform light-cone gauge for the bosonic sector of non-relativisticAdS × S string theory [2]. Computations are given in the attached Mathematica note-book. We consider the non-relativistic action (1.13) where we choose Cartesian globalcoordinates. The boost invariant metric H µν is H µν d X µ d X ν = − H tt d t + H φφ d φ + H IJ d X I d X J , (2.18)where H tt = (1 + ( z R ) ) z m z m R (1 − ( z R ) ) , H φφ = 1 ,H IJ d X I d X J = z m z m R (1 − ( z R ) ) d z + 1(1 − ( z R ) ) d z m d z m + d y i d y i , (2.19)and we denoted by X I ≡ ( z , z m , y i ) the coordinates transversal to the ( t, φ )-coordinates, where z is the non-relativistic longitudinal coordinate, z m and y i , with m = 2 , , i = 1 , ....,
4, are the non-relativistic transverse coordinates. The two vector fields ∂ t and ∂ φ are isometries for the metric H µν .The non-zero longitudinal vielbeine τ µA are τ t = 1 + ( z R ) − z R ) , τ z = 11 − ( z R ) . (2.20)The derivation of this string Newton-Cartan data is described in Apendix A. The depen-dence on the radius R is reinstated via dimensional analysis. In the limit R → ∞ onerecovers the string Newton-Cartan data for the non-relativistic flat space.We consider closed strings, therefore all string embedding coordinate fields are assumedto be periodic in the σ -coordinate, which takes values in 0 ≤ σ ≤ π . Here we shall assumethat the string does not wrap around the φ -direction.Next we shall write the action (1.13) in the first order formalism. The momenta aredefined as p µ ≡ δS NR δ ˙ X µ = − T γ α ∂ α X ν H µν − T λ A τ µA , (2.21) This should not be confused with the non-relativistic transverse coordinates. X µ ≡ ∂ τ X µ and X µ ′ ≡ ∂ σ X µ . The action then takes the form S NR = Z d τ Z π d σ (cid:18) p µ ˙ X µ + γ γ C + 12 T γ C (cid:19) , (2.22)where C = p µ X µ ′ , (2.23)and C = H − µν p µ p ν + T H µν X µ ′ X ν ′ + T λ A τ µA p ν H − µν + T λ A λ B τ µA τ ν B H − µν − T λ A ε AB η BC τ µC X µ ′ , (2.24)are combinations of the Virasoro constraints . Then we introduce light-cone coordinates X + = (1 − a ) t + aφ , X − = φ − t ,p + = (1 − a ) p φ − ap t , p − = p φ + p t , (2.25)where a is a freedom of the rewriting . We fix uniform light-cone gauge by imposing thefollowing conditions X + = τ , p + = K T , (2.26)where K is a constant, real number , fixed as follows K = − am − | m |√ − a + 2 a , (2.27) It is interesting to note that although the non-relativistic action apparently has a different structurefrom the relativistic one, at the end it is still possible to bring it into the form of (2.22), which is typicalof the relativistic case [15]. Precisely, if we define the world-sheet stress-energy tensor as T αβ ≡ − √ h δS NR δh αβ , then C and C are the following linear combinations of T αβ : C = − h (cid:18) γ T − γ T (cid:19) , C = 2 hT T . The parameter a , which parametrises a family of light-cone gauges (or more precisely said, generaliseduniform light-cone gauges ), was first introduced in [23] and it turns out to be useful in order to distinguishgauge-dependent quantities from those gauge-independent. It also allows to keep a control over thecorrectness of the spectrum, as it has to be independent of a . It is real because the argument inside the square root is never negative for 0 ≤ a ≤ a is the gauge parameter and m is the non-zero winding mode along the longitudinalspatial direction introduced later in (2.35). This choice of K avoids the appearance inthe action of imaginary units, and it canonically normalises, up to an overall factor, thequadratic free Hamiltonian.The light-cone momentum P + defined as P + ≡ Z π d σ p + , (2.28)is fixed in terms of the string tension as P + = 2 πK T , (2.29)which is obtained by integrating over σ the light-cone gauge condition (2.26). We areinterested in studying the non-relativistic action in the limit where T → ∞ and P + → ∞ ,keeping P + /T fixed to 2 πK .To proceed , we need to impose the Virasoro constraints C = C ≈
0. The firstVirasoro constraint, after fixing light-cone gauge, can be solved in terms of X ′− C = p + X ′− + p I X I ′ ≈ , = ⇒ X ′− = − K T p I X I ′ . (2.30)The second Virasoro constraint C ≈
0, in the light-cone gauge and after solving the firstVirasoro constraint in terms of X ′− , is an equation which only depends on p − and on X I , X I ′ , p I and λ A . In particular, this is a quadratic equation in p − which can be solved interms of X I , X I ′ , p I and λ A . In this way the action (2.22) becomes S NR = Z d σ ( p I ˙ X I − H ) , (2.31)where H = − p − ( X I , X I ′ , p I , λ A ) , (2.32)and where we neglected the total derivative ˙ X − .Next, we integrate the Lagrange multipliers out. To do so, we point out that theconjugate momenta of λ A is identically zero, since in the non-relativistic action (1.13) Here we solve the constraints in a different order than in [19]. In this article the author first integratesout the Lagrange multipliers (second class constraint) and then the Virasoro constraints (first classconstraints). However we checked explicitly that either our or [19]’s order of solving constraints gives inthe end same results. We thank G. Oling for pointing this out. λ A appearing. Therefore we have identically that p λ A ≈ , (2.33)and this condition must be preserved in the world-sheet time evolution, which means that ∂ τ p λ A = { p λ A , H} ≈ . (2.34)This gives two independent equations that can be solved for ( λ , λ ) in terms of X I , X I ′ , p I . By plugging their solution back into (2.32) one gets a Hamiltonian fully independentof the Lagrange multipliers, i.e. H = H ( X I , X I ′ , p I ).Then, we compactify the longitudinal spatial direction z and we assume that thestring has non-zero winding modes along z , i.e. we make the ansatz z = mσ + ˜ z , (2.35)where m counts how many times the string wraps the longitudinal compact direction z and ˜ z ≡ ˜ z ( τ, σ ) is a periodic function in σ .Next we rescale fields and momenta with the following canonical transformation ˜ z → √ T ˜ z , z m → √ T z m , y i → √ T y i ,p z → √ T p z , p z m → √ T p z m , p y i → √ T p y i , (2.36)and we choose winding number m = 1 and gauge parameter a = 1. For this choice itcorresponds a value of K = − . A comment about different choices of a and m is given below. In the large T and R limit the gauge fixed non-relativistic action expandsas S NRgf = Z d τ d σ (cid:18) √ TR ˆ L + L + 1 R ˆ L + 1 √ T L + 1 R √ T ˆ L + 1 T L + 1 R T ˆ L + .... (cid:19) , (2.37) There is also a non-canonical expansion that gives same results as the canonical one. In this case, oneneeds to fix the light-cone gauge condition p + = K instead of (2.26), to scale σ → T σ and to rescale bothcoordinates and momenta with 1 / √ T . It turns out that the non-canonical expansion is more efficientcomputationally speaking. We thank G. Arutyunov for discussion. If instead of K = − K = 1, then imaginary units would appear as anoverall factor in each term of the expansion (apart from the first one, which is just a constant term thatwe drop from the action). In addition, this would also affect relative signs between the various terms inthe quadratic Hamiltonian. Even if one were to drop the overall imaginary unit in front of the quadraticHamiltonian, the latter would turn out to be neither positive nor negative definite. constant and total derivative terms which scales with positive powersof T . We rescale the world-sheet time as τ → τ in order to make the overall factor ofthe quadratic Hamiltonian canonical . The leading term L in the gauge-fixed action isthe one corresponding to the quadratic Hamiltonian L = p I ˙ X I − (cid:18) p z + p z m + p y i + ˜ z ′ + z ′ m + y ′ i (cid:19) , (2.38)which describes the dynamics of 8 massless scalars in 2d Minkowski space. The ˆ L termis a pure radius correction, quadratic in the fieldsˆ L = − (cid:20) z + 12 z m + σ (3 p z − p z m + p y i ) + 8 σ ˜ z ˜ z ′ − σ ˜ z ′ + σ (5 z ′ m + y ′ i ) (cid:21) . (2.39)From this term we see that the ˜ z and z m modes gain a mass term as a 1 /R correction.The ˆ L term is ˆ L = − σ ˜ z , (2.40)which is also subleading if we send R to infinity faster than T . The cubic and quarticLagrangians are given in Appendix B, and they do not show any pathological behaviour(e.g. non-locality, broken Hermiticity). This hints that this perturbative expansion of thenon-relativistic action is well defined. Moreover the structure of all perturbative termsso far computed shows that the light-cone gauge preserves a manifest SO (3) × SO (4)symmetry.Physical states should satisfy the level-matching condition, which is obtained by in-tegrating out in σ the linear Virasoro constraint C ≈
0. In our setting where the stringhas no winding mode around φ , but it has winding mode m around z , the level-matchingcondition reads p ws = m Z π d σ p ˜ z , (2.41)where p ws ≡ − Z π d σ ( p ˜ z ˜ z ′ + p z m z ′ m + p y i y ′ i ) , (2.42)is the total world-sheet momentum of the string. Here we observe two things: The constant divergent term could be understood as the string analogue of the divergent rest massof the non-relativistic particle, which needs to be subtracted from the action. The quadratic Hamiltonian appears in the expansion with an overall factor of 1 /
4, instead of thecanonical 1 / the gauge fixed action is not invariant under shifts of σ , and therefore p ws is not aconserved charge. • in the near-BMN limit of the relativistic theory, it was found that a winding modearound the φ -direction makes p ws to be large [22]. This is in turn responsible ofmaking the quadratic Hamiltonian to be large on physical states. We remark thatthis does not happen in our setting since the level-matching condition implies that p ws is finite. However if we were instead to turn on a winding mode around the φ -direction, then we would find in the non-relativistic theory that p ws is large and thequadratic Hamiltonian is large on physical states, like in the relativistic theory .We comment now on the choices of m and a . We can identify three different scenarios,accordingly to the three choices of a = 0 , , • a = 0 . In this case one can derive an expansion similar to (2.37), but with theonly difference that the sign of the quadratic Hamiltonian is governed by the signof the winding mode m . Only for negative m one has a positive definite quadraticHamiltonian . • a = . In this case we obtain a similar expansion as in (2.37). The positive definitequadratic Hamiltonian exists for every choice of m . Computations are more involvedin this gauge. • a = 1 . In this case computations turn out to be simpler. The positive definitequadratic Hamiltonian exists for every choice of m . We thank G. Arutyunov for discussion. This is because when a = 0 the Hamiltonian obtained by solving the Virasoro constraints is schemat-ically of the form H = Ax ± √ Bx + C where x is the Lagrange multiplier and A, B, C are generic expressions depending on the fields. Tointegrate out the Lagrange multipliers, one has to solve this type of equationd H d x = A ± B √ Bx + C ! = 0 , which implies that H = AB "(cid:18) B A (cid:19) − C − B A .
This result points out that H is clearly independent of the choice one can make between the two solutionsof the second Virasoro constraint, once the Lagrange multipliers are integrated out. This does not happenfor the other choices of gauge parameter a , where one can always find one solution of the second Virasoroconstraint that gives a positive definite quadratic Hamiltonian. Comments on finite R In the previous section we showed that if the radius R is sufficiently large and the stringis winding around the longitudinal spatial direction, the light-cone gauge fixed non-relativistic action admits a large string tension limit around the dynamics of free masslessscalar fields, which is reminiscent of the BMN expansion in the relativistic model. Thenext question we want to ask is the following: what happens if we keep R finite? If one tries to implement the procedure described in the previous section for finite R ,one will find that the action does not expand around the dynamics of free particles, butinstead it will expand around a much more complicated term, where the complicationis due to the introduction of σ dependent terms through the winding modes of the z coordinate, which enters without σ derivatives in H µν and τ µA .The large R limit has the property of taming the σ dependent terms, in such a waythat they disappear from the leading order action and only appear through subleadingcorrections. The large R limit has also the effect of making the ˜ z and z m fields becomingmassless, and their mass only appears as a correction in 1 /R . This is in agreement withthe fact that non-relativistic string theory in flat spacetime (in static gauge) is just atheory of massless scalar fields in Mink .One may wonder whether the problem of getting a complicated leading order actionwith σ dependent terms is just because of a bad choice of coordinates, which makesthe winding modes appearing through the Newton-Cartan data. To analyse further thisquestion, first we remark that the following conditions are necessary in order to apply thelight-cone gauge to our problem:1. There must exist two coordinates, one timelike and one spacelike, which are isome-tries for H µν and τ µA (i.e. the latter do not depend on those coordinates), such thatthey can be used to define light-cone directions X ± .2. The boost invariant metric H µν must be invertible.3. The two non-relativistic longitudinal directions must be coordinates describing anAdS spacetime. The timelike direction must be a non-relativistic longitudinal di-rection.4. The string must wrap around the non-relativistic longitudinal spatial direction.First of all, we make the following observation. The two non-relativistic longitudinaldirections cannot be chosen to be the timelike and spacelike isometries given in point (1).15f this was possible, it would imply that the longitudinal metric τ µν is just proportional tothe 2 × × S given in Appendix A, and wediscuss the strengths and weaknesses of each of them regarding the conditions (1)-(4). Polar global coordinates.
This set of coordinates does not satisfy condition (2). Fur-thermore, the longitudinal spatial direction is not an isometry, and therefore the problemof having σ dependent terms in the leading order action would still be there. GGK coordinates.
In this set of coordinates, the longitudinal spatial coordinate x is an isometry and therefore it looks promising in terms of winding the string around it,since this will not generate any σ dependent term. However the timelike coordinate x is not an isometry (as expected from the argument above), and this does not fulfil point(1). In other words, if one tries to fix light-cone gauge in this set of coordinates, one willend up at leading order with a time-dependent Hamiltonian. Cartesian global coordinates.
To the best of our knowledge, this is the only set ofcoordinates where all conditions (1)-(4) are satisfied . Since the (longitudinal) timelikedirection is an isometry, the longitudinal spatial direction cannot be an isometry as well.This means that when the string wraps the longitudinal spatial direction, it will produce σ dependent terms entering in the action at leading order, and we are able to removethem in the large R expansion. In this paper we studied the large string tension limit of the light-cone gauge fixed non-relativistic AdS × S string theory found in [2], restricted to the bosonic sector. We showedthat in the large string tension limit the action expands around the dynamics of eightfree massless scalar fields in Mink , which makes it suitable for quantisation and for anS-matrix computation. Our result relies on the assumption that the common AdS and S radius is large and the string wraps around the longitudinal spatial direction. We discussedabout limitations and potential issues to extend our procedure to the more general settingwhere the large radius assumption is removed. At the moment these limitations seem toforbid the possibility of applying the light-cone gauge to non-relativistic AdS × S stringtheory at finite R , and further investigation is deserved. In this paper we considered only global set of coordinates for AdS × S . If one allowed to consideralso local set of coordinates, then one could check what happens for instance in the Poincar´e coordinates.We found that also in the Poincar´e coordinates the condition (2) is not satisfied.
16s discussed in the introduction, classical integrability of non-relativistic AdS × S string theory found in [2] is still an open question, and we plan to address this in thefuture [14] . Finding the S-matrix for the light-cone gauge fixed non-relativistic actionpresented in this paper will provide some hints of integrability of non-relativistic AdS × S .This amounts to checking whether the S-matrix satisfies the Yang-Baxter equation andwhether it forbids particle production and annihilation.Despite some recent work [24–26], we are still missing a complete understanding of theset of classical solutions of the equations of motion for non-relativistic strings in AdS × S .In particular it is still unclear around which classical solution we are expanding ournon-relativistic action in powers of the string tension. We expect that a non-relativisticanalogue of the BMN spinning point-like string [27] also exists, although we leave this forfuture investigation.We also remark that the approach taken in this paper is different from [28]. In [28]the non-relativistic AdS × S action is seen as the dynamics arising as the semiclassicalexpansion of the relativistic action in static gauge around a static 1/2-BPS classical stringsolution. In our paper, instead, we considered the non-relativistic AdS × S action as a starting point , to be treated as an independent theory on its own, and then wonderedabout its semiclassical expansion.It would be an interesting question to study the strong coupling expansion of thenon-relativistic action in light-cone gauge when supersymmetry is considered. Howeverprogress in this direction is limited at the moment by the fact that we still miss tounderstand how to generically couple fermionic fields to the string action on a stringNewton-Cartan background.Another interesting open question is the following: what is the holographic dual fieldtheory of non-relativistic AdS × S string theory? Answering this question requires todefine what the boundary geometry of a string Newton-Cartan manifold is, which is stillnot understood . The problem is that it is not yet clear how to define the boundarygeometry of a spacetime which is equipped with a pair of degenerate metrics, namely( τ µν , H ⊥ µν ). In [20], a Lax pair has been found for the theory obtained by rescaling the coordinates t and φ ofrelativistic AdS × S in Cartesian coordinates. However we point out that the theory obtained in sucha way is not the non-relativistic theory constructed in [2] and considered in our paper. This is becauserescaling t in AdS and φ in S does not single out an AdS subspace inside AdS as in [2]. So fromour point of view, classical integrability discussed in [20] was proven for a different theory than the oneconsidered in this paper. For past work on this topic, see [29]. cknowledgements We are extremely grateful to G. Arutyunov, N. Obers and G. Oling for spending theirtime in reading the manuscript and for providing very useful comments and insights. AFalso thanks S. van Tongeren for useful discussions and for collaboration on related topics.AF has been supported by the Deutsche Forschungsgemeinschaft (DFG, German ResearchFoundation) via the Emmy Noether program “Exact Results in Extended Holography”.JMNG and AT are supported by the EPSRC-SFI grant EP/S020888/1
Solving Spins andStrings . Appendices
A Newton-Cartan data
In this section we shall derive the set of Newton-Cartan vielbeine (or “data”) { τ µA , m µA , e µa } for various set of coordinates of AdS × S . The idea consists of the following steps:1. choose a given coset representative g ∈ SO (4 , × SO (6). This determines the choiceof coordinates for AdS × S . In what follows we will choose coset representativescorresponding to three sets of global coordinates: Cartesian, polar and GGK.2. rescale the generators of the isometry algebra so (4 , ⊕ so (6) by a parameter ω .This rescaling is made in such a way that if one takes the limit ω to ∞ , the so (4 , space inside AdS , where the sl (2 , R ) symmetry preserved by the contraction limit can act on.3. compute the left-invariant Maurer-Cartan 1-form g − d g and extract the relativisticvielbeine.4. by expanding the relativistic vielbeine in powers of ω one extracts the Newton-Cartan data. 18 .1 so (4 , ⊕ so (6) algebra contraction The rescaling of the so (4 , ⊕ so (6) generators which makes the contraction of so (4 , so (4 ,
2) are m ˆ i ˆ j , with ˆ i, ˆ j, ... = 0 , ...,
5, where 0 , so (4 ,
2) is acting on, and they satisfy the algebra[ m ˆ i ˆ j , m ˆ k ˆ ℓ ] = η ˆ j ˆ k m ˆ i ˆ ℓ − η ˆ i ˆ k m ˆ j ˆ ℓ + η ˆ i ˆ ℓ m ˆ j ˆ k − η ˆ j ˆ ℓ m ˆ i ˆ k , (A.1)where η = diag( − , , , , , − so (4 ,
1) subalgebra by choosing oneparticular time direction, e.g. ˆ j = 5, such that m i ≡ P i are the momentum generatorson AdS , and m ij ≡ J ij are the angular momentum generators spanning so (4 , i, j, ... = 0 , ..., so (6) are denoted by n ˆ a ˆ b , with ˆ a, ˆ b, ... = 1 , ...,
6, and they satisfy thealgebra [ n ˆ a ˆ b , n ˆ c ˆ d ] = δ ˆ b ˆ c n ˆ a ˆ d − δ ˆ a ˆ c n ˆ b ˆ d + δ ˆ a ˆ d n ˆ b ˆ c − δ ˆ b ˆ d n ˆ a ˆ c . (A.2)One can identify the so (5) subalgebra by choosing one spatial direction, e.g. ˆ b = 6,such that n a ≡ P a are the momentum generators on S , and n ab ≡ J ab are the angularmomentum generators spanning so (5), where a, b, ... = 1 , ..., m ˆ i ˆ j → ω m ˆ i ˆ j if ˆ i ∈ { , , s } and ˆ j ∈ { s , s , s } , (A.3) m ˆ i ˆ j → m ˆ i ˆ j otherwise , (A.4)where s , s , s , s take a fixed value in the set { , , , } , and n a → ω n a , n ab → n ab . (A.5)In the limit ω → ∞ , the so (4 ,
2) algebra contracts to the Newton-Hooke algebra in 5-dimensions. Moreover, the subalgebra generated by m ˆ i ˆ j with ˆ i, ˆ j ∈ { , , s } is the sl (2 , R )algebra, which will act as a global symmetry of the AdS divergent part of the metric.In the spinorial representation, the generators are m ij = 14 [ γ i , γ j ] , m i = 12 γ i , n ab = 14 [ γ a , γ b ] , n a = i γ a , (A.6) We remark that in (A.3), (A.4) we have already taken into account the rescaling of the common AdS and S radius, which is imposed separately as an additional condition in [2]. γ = −
10 0 1 00 1 0 0 − , γ = i i − i − i , γ = ,γ = − i
00 0 0 ii − i , γ = − − , γ = iγ . A.2 Cartesian global coordinates
Here we introduce the AdS × S analogue of the Cartesian coordinates for Minkowskispace.The coset representative of SO (4 , × SO (6) is of the type g =diag( g a , g s ), where g a and g s are coset representatives for SO (4 ,
2) and SO (6) respectively. In the Cartesianglobal coordinates, g a and g s are taken as g a = Λ( t ) · G ( z ) , g s = Λ( φ ) · G ( y ) , (A.7)where Λ( t ) = exp( t m ) , Λ( φ ) = exp( φ m ) ,G ( z ) = 1 q − z ( + z i m i ) , G ( y ) = 1 q y ( + y i n i ) , (A.8)Here we denoted z ≡ z i z i , where z i = z i , and i = 1 , ...,
4. The same applied for the y coordinates. t is the global time in AdS and φ is the angle describing the greatest circlein S .The AdS × S metric in these coordinates becomesd s = − (cid:18) z − z (cid:19) d t + 1(1 − z ) d z i d z i + (cid:18) − y y (cid:19) d φ + 1(1 + y ) d y i d y i . (A.9)In these coordinates, every choice of s ∈ { , , , } entering in the rescaling (A.3)and (A.4) will give a sensible non-relativistic model based on Newton-Cartan geometry.For instance, one can take s = 1, and the rescaling of coordinates that induce the algebracontraction (A.3) and (A.4) is t → t , z → z , z m → ω z m , φ → ω φ , y i → ω y i , (A.10)20here m = 2 , ,
4. By taking this choice of decomposition, from the vielbeine expansionin powers of ω we read off the following Newton-Cartan data τ µA = diag (cid:18) z ) − z ) , − ( z ) , , ..., (cid:19) ,m µA = diag (cid:18) − z m z m − ( z ) ) , z m z m − ( z ) ) , , ..., (cid:19) , (A.11) e µa = − ( z ) − ( z ) − ( z ) , where the ordering of the coordinates is X µ = { t, z , z , z , z , φ, y , y , y , y } .The longitudinal metric τ µν is τ µν d X µ d X ν = − (cid:18) z ) − z ) (cid:19) d t + 1(1 − ( z ) ) d z , (A.12)and the boost invariant metric H µν is H µν d X µ d X ν = − (1 + ( z ) ) z m z m (1 − ( z ) ) d t + z m z m − ( z ) ) d z + 1(1 − ( z ) ) d z m d z m + d y i d y i . (A.13)The closed B-field compensating the divergent part of the metric is B µν d X µ ∧ d X ν = ω z ) (1 − ( z ) ) d t ∧ d z . (A.14) A.3 Polar global coordinates
The set of polar global coordinates is obtained via the coset representative g a = Λ a ( t, ψ , ψ ) · Θ a ( x ) · G a ( ρ ) , g s = Λ s ( φ, χ , χ ) · Θ s ( w ) · G s ( r ) , (A.15)21here Λ a ( t, ψ , ψ ) = exp( t m − ψ m − ψ m ) , Θ a ( x ) = exp(arcsin( x ) m ) ,G a ( ρ ) = exp(arcsinh( ρ ) m ) , (A.16)and Λ s ( φ, χ , χ ) = exp( φ n − χ n − χ n ) , Θ s ( w ) = exp(arcsin( w ) n ) ,G s ( r ) = exp(arcsin( r ) n ) . (A.17)In this set of coordinates the AdS × S metric reads asd s = d s a + d s s , (A.18)whered s a = − (1 + ρ )d t + 11 + ρ d ρ + ρ − x d x + ρ (1 − x )d ψ + ρ x d ψ , d s s = (1 − r )d φ + 11 − r d r + r − w d w + r (1 − w )d χ + r w d χ (A.19)In these coordinates, there is only one choice of s entering in (A.3) and (A.4) whichproduces a sensible non-relativistic action. This choice is s = 1, and it produces an AdS divergent part of the metric where the sl (2 , R ) algebra coming from the contraction of so (4 ,
2) is acting on. All other choices of s will produce a divergent part of the metricwhich is the R metric, and this is not the non-relativistic model described in [2].Therefore, for the unique sensible choice s = 1, the coordinate rescaling induced bythe algebra contraction (A.3) and (A.4) is t → t , ρ → ρ ψ → ψ , ψ → ω ψ , x → ω x ,χ → χ , χ → χ , w → w , φ → ω φ , r → ω r , (A.20)and from the vielbeine expansion we read off the Newton-Cartan data τ µA = diag( − p ρ , p ρ , , ..., , µA = 0 , (A.21) e µa = ρ − ρ − ρx r √ − w
00 0 0 0 0 0 0 − r √ − w − rw , where the ordering of the coordinates is X µ = { t, ρ, x, ψ , ψ , φ, r, y, χ , χ } .The longitudinal metric τ µν is τ µν d X µ d X ν = − (1 + ρ ) d t + 11 + ρ d ρ , (A.22)and the boost invariant metric H µν is H µν d X µ d X ν = ρ (d x +d ψ + x d ψ )+d φ +d r + r (cid:18) − w d w +(1 − w )d χ + w d χ (cid:19) . (A.23)We remark that in this set of coordinates, the boost invariant metric has not full rank andtherefore it is not invertible. This makes it not suitable for the light-cone gauge fixing.The closed B-field that compensates the divergent part of the metric is B µν d X µ ∧ d X ν = ω d t ∧ d ρ . (A.24) A.4 GGK coordinates
The set of coordinates used by Gomis, Gomis and Kamimura [2] is, in our convention,given by the following choice of coset representative g a = exp( x m ) exp( x m ) exp( x a m a ) ,g s = exp(˜ x m ′ n m ′ ) , (A.25)where a = 2 , ,
4, and m ′ = 1 , ...,
5. In this set of coordinates, the spacetime metric takesa complicated form given in [2], which we do not report here.23lso in this set of coordinates, there is only one sensible choice of s entering in (A.3)and (A.4). This is s = 1, and intuitively it is because the choice of coset representa-tive made here treats x differently from the other AdS coordinates. The associatedcoordinates rescaling is x → x x → x x a → ω x a x m ′ → ω x m ′ (A.26)The Newton-Cartan data is τ µA = diag( − , cos x , , ..., ,m µA = x a x a − , , , ...., , (A.27) e µa = diag(0 , , , ...., , where the ordering of the coordinates is X µ = { x , x , x a , x m ′ } .The longitudinal metric τ µν is τ µν d X µ d X ν = − d x + cos ( x ) d x , (A.28)and the boost invariant metric H µν is H µν d X µ d X ν = x a x a ( − d t + cos ( x ) d x ) + d x a d x a + d˜ x m ′ d˜ x m ′ . (A.29)The closed B-field compensating the divergent part of the metric is B µν d X µ ∧ d X ν = ω cos x d x ∧ d x . (A.30) B Cubic and Quartic Lagrangians
Here we list the cubic and quartic Lagrangians entering in the expansion (2.37). Thisshows that the perturbative expansion of the action in the large string tension parameteris well defined. They all show a manifest SO (3) × SO (4) symmetry. L = − (cid:20) p ˜ z ( p ˜ z ˜ z ′ + p z m z ′ m + p y i y ′ i ) − ˜ z ′ ( p z + p z m + p y i + ˜ z ′ + z ′ m + y ′ i ) (cid:21) , (B.31)24 L = − (cid:20) σ ˜ z ′ (3 p z + 3 p z m − p y i ) + 8˜ z ′ ˜ z + 36˜ z ′ z m − σ ˜ z ′ (5 z ′ m + y ′ i )+ 4 σ ˜ z (3 p z − p z m + p y i ) + 6 σ p ˜ z p z m z ′ m + 6 σ p ˜ z p y i y ′ i + 20 σ ˜ zz ′ m + 4 σ ˜ zy ′ i + σ ˜ z ′ − σ ˜ z ˜ z ′ (cid:21) , (B.32) L = − (cid:20) ˜ z ′ ( − p z + 6 p z m + 6 p y i + 6 z ′ m + 6 y ′ i ) + ( p z + p z m + p y i + z ′ m + y ′ i ) − p ˜ z ˜ z ′ ( p z m z ′ m + p y i y ′ i ) + 5˜ z ′ (cid:21) , (B.33)ˆ L = − (cid:20) σ (cid:0) p ˜ z ˜ z ′ ( p z m z ′ m + p y i y ′ i ) − z ′ + 4˜ z ′ ( p z − z ′ m − y ′ i − p z m − p y i ) (cid:1) − σ p z + p z m + p y i + z ′ m + y ′ i ) + 10 σ ˜ z ˜ z ′ + 8˜ z ˜ z ′ + 120 z m ˜ z ′ − z ′ (cid:0) z + σ (2 p z + p z m − p y i ) + 18 z m − σ (3 z ′ m + y ′ i ) (cid:1) + 8 σ ˜ z ˜ z ′ z ′ m − σ ˜ z ˜ z ′ p z m − σ ˜ z ˜ z ′ ( p z + p z m + p y i + ˜ z ′ + z ′ m + y ′ i ) − σ ˜ z ′ (cid:0) zz ′ m + ˜ zy ′ i + ˜ z (3 p z − p z m + p y i ) + 2 σp ˜ z p z m z ′ m + 2 σp ˜ z p y i y ′ i (cid:1) + 2 σ ˜ z ′ p ˜ z (cid:0) p ˜ z ˜ z + σ ( p ˜ z ˜ z ′ + p z m z ′ m + p y i y ′ i ) (cid:1) + 2 σp ˜ z ( σ ˜ z ′ − z )( p ˜ z ˜ z ′ + p z m z ′ m + p y i y ′ i )+ 16 p z ˜ z − p z m ˜ z + 32˜ z z ′ m − p z z m + 52 σp ˜ z ˜ z ( p ˜ z ˜ z ′ + p z m z ′ m + p y i y ′ i )+ ( p z + p z m + p y i + z ′ m + y ′ i ) (cid:0) z + 4 σ ˜ z ˜ z ′ + σ (2 p z − p z m + p y i ) + 6 z m + σ (3 z ′ m + y ′ i ) (cid:1) + ( p z + p z m + p y i + ˜ z ′ + z ′ m + y ′ i ) (cid:0) z + 8 σ ˜ z ˜ z ′ + σ (2 p z − p z m + p y i ) + 6 z m + σ (3 z m + y ′ i ) (cid:1)(cid:21) . (B.34) C Different winding assumptions
In this Appendix we fix light-cone gauge by following the procedure described in section2, but with different winding assumptions. In what follows, we keep R finite. No winding.
If we do not require that the string wraps around any spacetimecoordinate, the procedure described in section 2 without imposing eqn. (2.35) and withthe gauge fixing: X + = τ , p + = T , (C.35)25ives the following expansion of the action (for a = 1) S NRgf = Z d τ d σ (cid:18) T / L / + T / L / + L + O ( T − ) (cid:19) , (C.36)where L / = i √ p z ′ , (C.37) L / = i / p z ′ (cid:18) p z m + p y i + z ′ m + y ′ i (cid:19) , (C.38) L = p I ˙ X I . (C.39)It is quite convincing that this expansion of the non-relativistic action is not meaningful. Winding around the transverse direction φ . This type of winding is achievede.g. by imposing, instead of (2.26), the following light-cone gauge fixing (for a = 1) X + = τ + m σ , p + = T , (C.40)which means that the string wraps the coordinate φ which parametrise the big circleinside S . We recall that here we are not imposing winding along the longitudinal spatialdirection, but only around φ . In this gauge the action becomes (neglecting constant andtotal derivative terms) S NRgf = Z d τ d σ (cid:18) √ T L + L + O ( T − ) (cid:19) . (C.41)where L = m p z , (C.42) L = p I ˙ X I − i m (cid:18) − m p z + p z m + p y i + 32 m z m − m z ′ + z ′ m + y ′ i (cid:19) . (C.43)We point out that if the winding number m is chosen to be purely imaginary , e.g. m = i ,then the quadratic Lagrangian L is in the canonical form, and with a positive mass termin the SO (3) sector. The price to pay is that every term which depends linearly in themomenta p I (except of p I ˙ X I ) is now imaginary. This happens in L and in the higherorder corrections in T , and can be compensated by choosing the momenta also to beimaginary, i.e. p I → ip I , together with τ → iτ and S NRgf → iS NRgf . However, in this caseone would find the same issue of having large quadratic Hamiltonian as in [22] once the26evel-matching condition is imposed on physical states, because of the non-zero windingmode along φ .We remark that for the moment these are just purely observations, and it would beinteresting to further understand the physical interpretation, if any. References [1] J. Gomis and H. Ooguri, “Nonrelativistic closed string theory,” J. Math. Phys. (2001), 3127-3151 [arXiv:hep-th/0009181 [hep-th]].[2] J. Gomis, J. Gomis and K. Kamimura, “Non-relativistic superstrings: A New solublesector of AdS × S ,” JHEP (2005), 024 [arXiv:hep-th/0507036 [hep-th]].[3] E. Bergshoeff, J. Gomis and Z. Yan, “Nonrelativistic String Theory and T-Duality,”JHEP (2018), 133 [arXiv:1806.06071 [hep-th]].[4] E. A. Bergshoeff, J. Gomis, J. Rosseel, C. S¸im¸sek and Z. Yan, “String The-ory and String Newton-Cartan Geometry,” J. Phys. A (2020) no.1, 014001[arXiv:1907.10668 [hep-th]].[5] J. Gomis, Z. Yan and M. Yu, “Nonrelativistic Open String and Yang-Mills Theory,”[arXiv:2007.01886 [hep-th]].[6] J. Gomis, Z. Yan and M. Yu, “T-Duality in Nonrelativistic Open String Theory,”[arXiv:2008.05493 [hep-th]].[7] T. Harmark, J. Hartong and N. A. Obers, “Nonrelativistic strings and limits of theAdS/CFT correspondence,” Phys. Rev. D (2017) no.8, 086019 [arXiv:1705.03535[hep-th]].[8] T. Harmark, J. Hartong, L. Menculini, N. A. Obers and Z. Yan, “Strings with Non-Relativistic Conformal Symmetry and Limits of the AdS/CFT Correspondence,”JHEP (2018), 190 [arXiv:1810.05560 [hep-th]].[9] T. Harmark, J. Hartong, L. Menculini, N. A. Obers and G. Oling, “Relating non-relativistic string theories,” JHEP (2019), 071 [arXiv:1907.01663 [hep-th]].[10] J. Gomis, J. Oh and Z. Yan, “Nonrelativistic String Theory in Background Fields,”JHEP (2019), 101 [arXiv:1905.07315 [hep-th]].2711] O. Aharony, S. S. Gubser, J. M. Maldacena, H. Ooguri and Y. Oz, “Large N fieldtheories, string theory and gravity,” Phys. Rept. (2000), 183-386 [arXiv:hep-th/9905111 [hep-th]].[12] N. Beisert, C. Ahn, L. F. Alday, Z. Bajnok, J. M. Drummond, L. Freyhult, N. Gro-mov, R. A. Janik, V. Kazakov and T. Klose, et al. “Review of AdS/CFT Integrability:An Overview,” Lett. Math. Phys. (2012), 3-32 [arXiv:1012.3982 [hep-th]].[13] A. Fontanella and L. Romano, “Lie Algebra Expansion and Integrability in Super-string Sigma-Models,” JHEP (2020), 083 [arXiv:2005.01736 [hep-th]].[14] A. Fontanella and S. van Tongeren, in progress .[15] G. Arutyunov and S. Frolov, “Foundations of the AdS × S Superstring. Part I,” J.Phys. A (2009), 254003 [arXiv:0901.4937 [hep-th]].[16] G. Arutyunov and S. Frolov, “Uniform light-cone gauge for strings in AdS × S :Solving SU (1 |
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