New formulation of non-relativistic string in AdS 5 × S 5

: We study non-relativistic limit of AdS 5 × S 5 background and determine corresponding Newton-Cartan ﬁelds. We also ﬁnd canonical form of this new formulation of non-relativistic string in this background and discuss its formulation in the uniform light-cone gauge.


Introduction and summary
In the past few years it is observed renewed interest in the study of non-relativistic string theories in Newton-Cartan (NC) formulation [1,2]. Basically, NC gravity provides covariant description of Newton's law. However it is very remarkable that NC description can be extended also into more broader class of theories, as for example field theory and string theory. In fact, non-relativistic string theory was originally introduced in 2000 in two papers [3,4]. These theories were defined without Newton-Cartan formalism which was firstly introduced in the context of string theory in the remarkable paper [5], for related works, see for example [7][8][9][10][11][12][13][14][15][16][17][18][19][20][21][22]34]. Different formulation of non-relativistic string was presented in [6] that was based on null reduction of string theory in the background with null isometry. 1 In [34] new reformulation of non-relativistic string theory was proposed that has interesting property that non-relativistic string naturally couples to two form field m M N in the similar way as non-relativistic particle couples to mass form m M , where M, N label space-time indices that vary from 0 to 9. 2 This new formulation of non-relativistic string was further studied in [40] where its canonical form of this theory was found. We also analysed the possibility to impose uniform light-cone gauge on this theory and found Hamiltonian on reduced phase space. Uniform light-cone gauge was used in [35,[37][38][39] 3 where it was shown that it is very efficient for the study of dynamics of the relativistic string in AdS 5 × S 5 . We showed in [40] that such an uniform light cone gauge fixing can be imposed in case of non-relativistic string as well at least at the formal level.

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In this paper we continue the analysis of this new reformulation of non-relativistic string when we focus on its explicit form as non-relativistic limit of AdS 5 × S 5 background. Our starting point is an important paper [41] where non-relativistic strings in AdS 5 × S 5 was defined by specific limiting procedure. We combine this procedure with the definition of Newton-Cartan fields as was given in [34] and we will be able to find Newton-Cartan background fields for non-relativistic limit of AdS 5 × S 5 . Explicitly, we find 2 × 2 twobein τ A M together with field π A M that was introduced in [34]. Then we will be able to determine two form m M N and hence corresponding Hamiltonian. As the next step we study gauge fixed form of the theory when we impose uniform light-cone gauge. Solving Hamiltonian constraint we determine Hamiltonian on the reduced phase space. We find that the structure of this Hamiltonian depends on the free parameter that defines generalized uniform light cone gauge [35,[37][38][39]. Then we study equations of motion on the reduced phase space. We show that it is possible to have configuration with all free fields to be equal to zero and concentrate on the dynamics of the mode z 1 . 4 However then we find that the equation of motion for z 1 is solved by arbitrary function and hence does not determine dynamics of z 1 at all. We mean that this is a sign that the present form of uniform light cone gauge is not suitable for the specific form of non-relativistic string studied in this paper and for the configurations when all free fields have trivial dynamics. In other words it is possible that more general ansatz, when remaining world-sheet fields depend on σ α , where σ α , α = 0, 1 label string world-sheet, could lead to more interesting dynamaics of z 1 coordinate.
Then in order to study properties of non-relativistic string in more details we focus on its Lagrangian equations of motion. We determine their form for general background. Since these equations of motion are rather complicated in the full generality we restrict ourselves to an analysis of the dynamics of single coordinate z 1 . We find solution that has formally the same form as solution found recently in [42] however there is a crucial difference since we consider extended string along non-compact coordinate and hence we should interpret this solution as the string with infinite number of spikes.
Let us outline our results. The main goal of this paper was to find Hamiltonian for new formulation of non-relativistic string in AdS 5 ×S 5 background. Performing appropriate limit we determined components of Newton-Cartan fields and then we obtained corresponding Hamiltonian. Then we fixed gauge symmetries introducing uniform light cone gauge in order to see whether this gauge fixing choose, that was such useful in case of relativistic string in AdS 5 × S 5 , could be helpful too in case of non-relativistic string. We showed that when we restrict to the case of excited single world-sheet mode z 1 then its dynamics is trivial that suggests that more general ansatz when more world-sheet fields have non-trivial profile could bring more interesting result. In fact, it is possible that different gauge fixing procedure, as for example static gauge, could lead to non-trivial dynamics of free world-sheet fields on AdS 2 background as was shown in [41]. We also studied Lagrangian equations of motion and we found solution corresponding to string extended along one free spatial coordinate and non-trivial dynamics along of z 1 coordinate that agrees with solution found in [42].

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This paper is organized as follows. In the next section 2 we review basics facts about new formulation of non-relativistic string and its canonical form. Then in section 3 we find its form in non-relativistic limit of AdS 5 × S 5 . In section 4 we study properties of this string in uniform light-cone gauge. In section 5 we determine Lagrangian equations of motion and study correspoding solution. Finally in appendix A we perform non-relativistic limit in coordinates that were used [41] and find corresponding Hamiltonian.

Review of new formulation of non-relativistic string and its canonical formulation
In this section we review basic facts about new form of non-relativistic string action as was proposed in [34] We firstly describe derivation of this action, following [34]. Let us introduce relativistic vielbein e a M so that target space metric has the form where we use the similar notation as in [34] so that frame indices are a, b = 0, . . . , 9 and where η ab = diag(−1, 1, . . . , 1). Note that space-time indices are M, N = 0, 1, . . . , 9. Following [34] we also introduce parametrization of NSNS two form B M N as To begin with let us write Nambu-Goto form of the action for relativistic string in general background and where 01 = 1 = − 01 and T F is string tension. As in [34] we introduce indices a = (A, a) corresponding to directions longitudinal and transverse to string world-sheet where A = 0, 1 are longitudinal and a = 2, . . . , 9 are transverse. Then we have We further parametrize longitudinal components in the following way Then it can be shown that the resulting non-relativistic action has the form where we introduced rescaled tension cT F = T and we have taken the limit c → ∞. Finally we also introduced matrix The canonical form of the action (2.10) was recently analysed in [40] where it was shown that the Hamiltonian is sum of two first class constraints where where Π M is defined as After the review of main properties of the action (2.10) we proceed to its explicit form when we consider non-relativistic limit of AdS 5 × S 5 .

Non-relativistic AdS × S 5 background
Following general prescription reviewed in the previous section we would like to find Newton-Cartan fields for non-relativistic limit of AdS 5 × S 5 . Let us now consider AdS 5 × S 5 background in Cartesian global coordinates where line element has the form where i, j = 1, 2, 3, 4 and where R is common radius of AdS 5 and S 5 . It is convenient to write this line element as ds 2 = eāebηāb. Then, following [41], we define non-relativistic limit as where m = 2, 3, 4 and where non-relativistic limit corresponds to c → ∞. We start with the vielbein e 0 that after rescaling (3.3) takes the form Then comparing this expression with (2.5) and (2.7) we can identify In the same way we proceed with e 1 e 1 = 1 so that comparing with (2.7) we get In case of the e m the situation is simpler and hence e n zm = In the same way we obtain e Φ φ = 1 , e j y i = δ j i . (3.10)

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It is important to stress that due to the fact that R = cR 0 → ∞ the φ coordinate is effectively non-compact since original variable Φ was periodic with period 2πR. Now we are ready to proceed to find corresponding Hamiltonian. We firstly determine components of m M N that, using (3.5) and (3.7) have following non-zero elements we obtain that there are non-zero components of matrix inverse τ M A equal to Taking all these results into account we obtain that the Hamiltonian constraint of nonrelativistic string is equal to where (3.14) As the next step we would like to find uniform light-cone gauge fixing form of non-relativistic string. Discussion of the general case was performed in [40] and here we focus on the nonrelativistic limit of AdS 5 × S 5 . In order to impose uniform light-cone gauge the background should possesses two abelian isometries where one of them is t. From the form of the non-relativistic background it is clear that the second one can be either φ or y i where now φ is non-compact. Without lost of generality we select φ as the second coordinate with isometry.

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where α is free parameter. Let us insert these relations to the Hamiltonian constraint given above and we get Now we are ready to impose uniform light-cone gauge by introducing following gauge fixing functions [35] Clearly G + , G − have non-zero Poisson brackets with H τ , H σ and hence together form set of second class constraints that vanish strongly. As a result constraints H τ = 0, H σ = 0 can be explicitly solved. We firstly solve H σ = 0 for ∂ σ x − and we get Further, Hamiltonian constraint H τ = 0 can be solved for p + which is related to the Hamiltonian on the reduced phase space as H red = −p + . To do this we insert ∂ σ x − given in (3.19) into (3.17) and using also (3.18) we get quadratic equation that can be solved for p + as where we defined K as

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Previous form of the Hamiltonian density on the reduced phase space is valid for a = 0. Explicitly, it is not valid for α = − 1 2 , that, according to [35][36][37], defines temporal gauge In this case we should start again with the Hamiltonian constraint (3.17) that for a = 0 has the form that can be solved for p + as Let us return to (3.17) and determine its explicit form for some special cases. For α = 0 we get uniform light-cone gauge when where p + is equal to (3.27) Finally we can consider special case when α = 1 2 corresponding to a = 1. In this case we find

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We see that the case α = − 1 2 is exceptional since in this case the Hamiltonian density on the reduced phase space is quadratic in momenta while generally the Hamiltonian has square root structure. In the next section we will analyse some classical solutions of the equations of motion on the reduced phase space.

Properties of non-relativistic string in uniform light-cone gauge
In this section we will discuss properties of non-relativistic string theory on reduced phase space. First of all we consider equations of motion for y i , p y i , z m , p m that due to the fact that the background fields do not depend on them have the form where H red = dσH red and where (. . . ) mean terms which are not important for us. The equations above can be solved by the ansatz z m = p zm = y i = p y i = 0. In fact, this ansatz also solves equations of motion for p zm and p y i . As a result we can consider Hamiltonian density for z 1 only that has the form In order to find equation of motion for z 1 it is convenient to find Lagrangian from (4.2). To do this we firstly determine canonical equation of motion using (4.2) Then L z 1 red is given by standard formula where we introduced g and B defined as

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Note also that (4.4) is valid for a = 0. Then performing variation of (4.4) we get following equation of motion for z 1 This is rather complicated equation and it is difficult to solve it in the full generality. On the other hand we can certainly gain insight into its form when we consider some simpler ansatz as for example z z = z z (τ ). However it turns out that this is too restrictive since it is easy to see that the equation above is solved for any z 1 (τ ). It is possible that more general ansatz could be desirable but we are not going proceed along this way. we rather proceed to the more interesting situation when a = 0. It is easy to see that as in general case a = 0 we can consistently set p m = z m = y i = p y i = 0 so that the reduced Hamiltonian density has the form Then the first canonical equation has the form so that Lagrangian density has the form It is easy to derive corresponding equation of motion for z 1 Clearly this equation has solution z 1 = z 1 (σ) for any function z 1 . On the other hand let us consider an ansatz z 1 = f (σ − vτ ) so that ∂ τ z 1 = −vf , ∂ σ z 1 = f and hence the equation of motion has the form In other words this equation is obeyed by any function f (σ − vτ ). This is again interesting property of non-relativistic string in uniform light-cone gauge.

Lagrangian equations of motion
In this section we find equations of motion of new-non-relativistic string with application to the AdS 5 × S 5 case. Recall that the Lagrangian has the form

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A Non-relativistic string in AdS 5 ×S 5 in an alternative set of coordinates In this appendix we present formulation of non-relativistic string in AdS 5 × S 5 background using coordinates that were introduced in [41]. In this formulation the vector indices for AdS 5 are labelled with m = 0, 1, 2, 3, 4 while for S 5 we have m = 1 , 2 , 3 , 4 , 5 with the flat metric η mn = diag(−1, 1, 1, 1, 1) and δ m n = diag (1, 1, 1, 1, 1). Then vielbein has following form e 0 = dT cosh ρ , ρ = X a η ab X b R , e 1 = dX 1 cosh ρ cos T R , Finally we have Now we are ready to proceed to find corresponding Hamiltonian. As the first step we determine following non-zero components of m M N Form of the Hamiltonian constraint implies that it is not possible to impose uniform light-cone gauge due to its explicit dependence on t. Certainly it is possible to study non-relativistic string in the background defined by (A.4) and (A.6) but the Lagrangian formulation is the same as in [41] which is well known and hence we will not study it in this paper.