Coset space actions for nonrelativistic strings

We formulate the stringy nonrelativistic limits of the flat space and AdS$_5\times$S$^5$ string as coset models, based on the string Bargmann and extended string Newton-Hooke algebras respectively. Our construction mimics the typical relativistic one, but differs in several interesting ways. Using our coset formulation we give a Lax representation of the equations of motion of both models.

In this paper we give a coset model formulation for the bosonic sectors of the stringy NR limit of the flat space [7] and AdS 5 ×S 5 strings [9], which from here on out we will call the NR flat space and NR AdS string respectively. We also use this coset formulation to give a Lax representation of the corresponding equations of motion. Our formulation is based on the string Bargmann algebra and the direct sum of the extended string Newton-Hooke and Euclidean algebras, in the flat space and AdS cases, respectively. Related coset models were previously studied in [34,35], however our formulation uses a different coset structure, and we moreover work with the Polyakov form of these strings to remain as close as possible to models such as the AdS 5 ×S 5 string. 2 Integrability of NR strings has previously been investigated in [38,39], and for related sigma models in [40], but not for the NR limit of the AdS 5 ×S 5 string considered in [9]. 3 Until recently the NR string action for general target space geometry was formulated based on a gauging of the string Newton-Cartan (SNC) algebra [20], using a particular torsion-free constraint. 4 Half a year ago this formulation was superceded by the so-called torsional string Newton-Cartan (TSNC) formulation [11], circumventing what may be viewed as downsides of the SNC formulation, such as field redundancy and associated Stückelberg symmetry. 5 As will come back shortly, the torsional string Newton-Cartan formulation uses a gauging procedure based on the so-called F string Galilei (FSG) algebra, 6 which arises as a direct infinite speed of light limit of relativistic string symmetries. At the relativistic level, the string geometry is first formulated via a gauging procedure based on the so-called string Poincaré algebra, which does not impose a torsion constraint, and incorporates the B field. This string Poincaré algebra is an extension of the Poincaré algebra by a second independent set of generators transforming in the fundamental representation of the Lorentz algebra, with the same semi-direct product structure. These extra generators are associated to the B field of the string. The limiting procedure to arrive at the NR string involves mixing up the string Poincaré generators, and as a result there is no longer a semi-direct product structure involving the FSG generators associated to the TSNC two form.
Our goal is to formulate the NR flat space and AdS strings in Polyakov form as coset 2 Similar coset structures were investigated for NR particle geometries in [36,37]. 3 The author of [39] moreover appears to claim integrability for arbitrary (torsion-free) backgrounds, which we do not expect to be able to replicate. 4 There were also previous iterations built on the string Bargmann algebra, see e.g. [10]. 5 The possibility to avoid Stueckelberg type redundancies, as well as to take nonrelativistic limits without imposing torsion constraints mentioned below, were previously observed in NS-NS gravity in [15]. 6 As the name implies, it imposes no constraints on torsion. The role of the torsion constraint was also investigated in [41]. sigma models, mirroring the construction of relativistic symmetric space sigma models such as the AdS 5 ×S 5 string [42], see e.g. [43] for a review. In the relativistic setting we can naturally embed torsion-free geometry in the string Poincaré picture mentioned above, since in essence we can simply drop the additional generators corresponding to the B field. 7 Hence we can use (torsion-free) Maurer-Cartan (MC) geometry as the natural language to describe a relativistic string with a symmetric target space. In contrast, in the NR setting, the torsional nature of the TSNC formulation appears to be at odds with MC type language. Moreover, at least for the NR flat space string, there is a concrete obstacle. A coset formulation is based on the global symmetry algebra of a model, which for the NR flat space string was determined in [44]. However, it appears impossible to embed a sufficiently large subalgebra of the FSG in this global symmetry algebra. 8 In other words, the TSNC analogue of relativistic local Lorentz symmetry has no clear relation to an H we would pick in a coset formulation for the NR flat space string. This obscures the interpretation of the components of a MC type form in terms of the geometric data of the TSNC formulation.
Coming back to our actual coset sigma model construction, we first consider the NR flat space string. The global symmetry algebra of this string is actually infinite dimensional [44], leading us to consider a suitable finite dimensional subalgebra as the basis for our coset model instead. There are two natural candidates for such a subalgebra [20]: the string Bargmann and the SNC algebra. We find that while both choices are admissible, the string Bargmann choice is more economical, and appears naturally adapted to finding a Lax formulation of the equations of motion. Our coset model construction is different from the standard relativistic one however. In a standard G/H coset sigma model, local H invariance follows essentially from H invariance of the bilinear form on the (relevant subspace of) the Lie algebra of G. In our case, this partly works as usual, but also partly relies on a cancellation between gauge transformations of the extra Lagrange multiplier fields that appear in the Polyakov formulation of the NR string, and gauge transformations of the usual MC action arising from a not fully H invariant bilinear form. As it turns out, it proves practical to refer to the SNC formulation of the NR string to see this, if only for reasons of familiarity.
Due to our differently realized gauge symmetry, naively varying the action gives more nontrivial equations of motion than independent fields, but these are appropriately related by corresponding Noether identities. Moreover, based on our coset formulation and the form of the equations of motion, there is a simple argument that allows us to modify the standard Lax connection encoding the equations of motion of a relativistic symmetric space sigma model, and obtain a Lax formulation for the present case. 9 Coming finally to the NR AdS string, we find that while its (infinite dimensional) global symmetry algebra has not been explicitly determined, the extended string Newton-Hooke algebra provides a suitable analogue of the string Bargmann algebra used to construct the flat space coset model, requiring only minimal changes to the derivation of the coset action. Here our Lax representation of the equations of motion still works, now on solution of the constraints enforced by the Lagrange multipliers.
Structure of the paper. In §1 we give a coset formulation of the NR string action in flat space and a Lax representation for its equations of motion. In §2 we repeat the construction for NR strings in AdS. The paper ends with several appendices. In §A we give our conventions and provide useful identities. In §B we review SNC strings, make contact with the coset construction and review their connection with relativistic strings. In §C we show the relation between the equations of motion in the coset and Polyakov languages. In §D we derive the global symmetry algebra of AdS strings. In §E we provide the coset construction of NR strings in flat space based on the SNC algebra.

The NR flat space string as a coset model
The action for the NR flat space string in Polyakov form is given by [7] where T is the string tension, σ α = (τ, σ), with α = 0, 1, are the string world-sheet coordinates, γ αβ ≡ √ −hh αβ is the Weyl invariant combination of the inverse world-sheet metric h αβ and h = det(h αβ ),and e α ± are (the light-cone components of) the worldsheet zweibein, see appendix A for our conventions. The x a , a = 2, . . . , 9 are the transverse string embedding coordinates, and x ± the longitudinal ones. λ + and λ − are non-dynamical scalar Lagrange multiplier fields.
Integrating out the Lagrange multiplier fields makes the worldsheet metric conformally flat, and in static gauge (x ± = τ ± σ) gives the corresponding Nambu-Goto action [7] describing eight free massless bosons.
To write down a G/H coset model corresponding to this string, we need its global symmetry group G, an appropriate denominator H, and a suitable bilinear form on the Lie algebra g of G. As our construction will necessarily differ from the typical relativistic symmetric space sigma model construction, let us briefly recall this for illustrative purposes.

Relativistic coset models
Let us consider a (pseudo-)Riemannian symmetric space M, i.e. Lie group theoretically a manifold that can be represented in the form G/H, where G is the isometry group of M and H is the isotropy group of a point in M, with associated Lie algebras g and h respectively. The symmetric space structure means that g has a Z 2 graded structure, splitting as g (0) ⊕ g (1) = h ⊕ p with respect to the corresponding automorphism. The string action on M is built using a nondegenerate h-invariant grade-compatible symmetric bilinear form , , 10 and a Maurer-Cartan form A = g −1 dg, g ∈ G, as where P projects g onto p. Due to the projector and the h-invariance of the bilinear form, the action has H gauge invariance under right multiplication of g, and we describe a model on G/H. We would now like to give a similar construction for the NR flat space string (1.1), which starts with its global symmetry algebra.

Symmetry algebra
The global symmetry algebra of the flat space NR string was determined in [44], and shown to be a particular infinite dimensional extension of the string Galilei algebra, see also [20,47]. Since it seems unnecessary and undesirable to attempt to use an infinite dimensional group in a coset model construction, we will try to base ourselves on a suitable finite dimensional subalgebra instead. Two such candidate algebras are known [20], the string Bargmann and the SNC algebra. Both algebras can be used for a coset model construction, we focus here on the simpler string Bargmann case, giving details on the SNC case in appendix E. The string Bargmann algebra is spanned by a longitudinal boost M, longitudinal translations H A , transverse rotations J ab , transverse translations P a , string-Galilei boosts G Ab , and non-central extensions Z A and Z, with commutation (1.4) We take G to be the corresponding string Bargmann group.

Gauge group, bilinear form
Next we need to determine our coset denominator H, with Lie algebra h. Here our construction starts to differ from the relativistic case. We want to encode our dynamical degrees of freedom in a MC form. However, we have extra Lagrange multiplier fields that do not naturally fit this language, and require modifying the typical construction. Fortunately, the close link between the string-Bargmann algebra and the SNC formulation of the nonrelativistic string, gives us a clear path to construct our model. First there is the question how to choose H. By analogy to relativistic strings we take H to be generated by everything except the H A and P a , further supported by the association between these generators and the longitudinal and transverse vielbeine in the SNC picture, see appendix B for a review of the SNC formulation. 11 Second, in the SNC formulation [20], the Lagrange multiplier fields transform under Z A transformations, meaning that we should not expect Z A transformations of our dynamical (MC) fields to leave the action invariant separately. Consequently, we should not insist on full h invariance for our bilinear form.
Concretely we take h = span{M, J ab , G Aa , Z A , Z}, and look for a bilinear invariant under the adjoint action ofh The corresponding symmetric bilinear form is 12 where the ω i are arbitrary coefficients. It is degenerate on the full string Bargmann algebra -Z, · = 0 for example -but nondegenerate on the span of {H A , P a , Z A } (for nonzero ω 1 ), which is sufficient. Out bilinear form has further bonus invariance conditions

Action
To give an action for our coset model, we construct a MC form A = g −1 dg with g an element of the string Bargmann group, with algebra components and we collect the Lagrange multiplier fields in a separate current To reproduce the flat space action we need to keep specific components of our MC form. Also here we could appeal to the SNC formulation to arrive at an appropriate choice. This choice turns out to be compatible with a grading that can be put on the string Bargmann algebra (1.4). 13 Namely, this algebra can be split as where g (0) and g (1) are the eigenspaces of a Z 2 automorphism Ω : g → g, corresponding to eigenvalues 1 and −1 respectively. This grading is compatible with our bilinear form: defining P to be the projector from g down to g (1) , we have x, P y = P x, P y , for all x, y ∈ g. Note that g (0) is nothing but theh that our bilinear form is invariant under. As mentioned above, our bilinear form is nondegenerate on g (1) . With this projector, fixing our bilinear form by taking ω 1 = 1 and ω 2 = 0, we can write the action for the NR flat space string as 14 Part of the H gauge invariance of this action follows from the typical considerations for gauge invariance of the relativistic symmetric space model. Namely the action is manifestly invariant under right multiplication of g → gh combined with Λ → h −1 Λh by elements h of the group generated byh = g (0) . This follows from the grading of the algebra, the projector, and grade compatibility and adjoint invariance of the bilinear form.
To have full H gauge invariance, we also need invariance under transformations associated to Z A . This gauge invariance is well known in the SNC formulation of the NR string, see appendix B for details. In our language, the action is invariant under the transformation 15 where D is the covariant derivative where the variations of the kinetic and Wess-Zumino term cancel up to a total derivative, via identities for the worldsheet zweibein given in appendix A, and using flatness of A. Finally, the coset representative and associated MC form explicitly give the coordinate action (1.1) above.

Equations of motion
We can compute the equations of motion associated to the action (1.11) by considering an arbitrary variation of the group element g, δg = gξ, which induces δA We also vary the Lagrange multiplier fields, giving δΛ α = e α − δλ − Z + + e α + δλ + Z − . By splitting the variation of ξ into its components, and using the invariance properties of our bilinear form, we get the following equations of motion We can represent these equations efficiently by introducing It may appear that at this stage that we have more equations of motion than we would expect for ten dynamical fields and two Lagrange multipliers. However, we also have gauge invariance at play. This leads to Noether identities between the equations of motion above, and appropriately reduces their number. 16 The relevant gauge transformations are those assocated to Z A and M.
First we consider the Z ± gauge variation of the action, with gauge parameters σ ± . Gauge invariance of the action implies that which after an integration by parts gives the Noether identities where only the Wess-Zumino term contributes nontrivially, is responsible for the nontrivial Noether identity The Noether identities (1.20) and (1.22) reduce the number of independent equations of motion. We can choose as our independent equations of motion. These are equivalent to the equations of motion derived directly from the component action (1.1), see appendix C for details.

Lax representation
The form of the equations of motion (1.18) is suggestive. Compared to the equations of motion of the relativistic symmetric space sigma model (1.3) the only difference with (1.18) is that we have A instead of J. 18 These equations of motion are known to have a Lax representation, and we can readily adapt it to our case. The standard Lax ansatz in the relativistic case takes the form 19 where the curvature of L, F (L), can be split over the grading of the algebra. Grade zero 17 This is D of eqn. (1.13) with image and domain {H A }. I.e., Df A = (Df C H C )| HA . Given the commutation relations, the same formula applies when we replace H A by Z A , and can hence refer consistently to just the index A. 18 In the relativistic case the equations of motion typically immediately follow by using full g invariance of the bilinear form, in contrast to our situation. 19 See e.g. [43], but note that our A has opposite sign.
flatness of L requires β ] = 0, (1.25) which follows since A itself is flat, provided we take ℓ 0 = 1, and ℓ 2 1 − ℓ 2 2 = 1, leaving a single free (spectral) parameter in the coefficients ℓ 1 and ℓ 2 . In grade one we then find (1. 26) We see that since A is flat, and ℓ 2 a nontrivial function of the spectral parameter, flatness of the Lax connection is equivalent to the equations of motion. In our case, J differs by A only in grade one, and by a term that commutes with everything in grade one. In other words, the modification does not affect grade zero. Any modification of this type can be accounted for in the Lax ansatz simply by replacing A by J in the ℓ 2 term, i.e. by taking If we consider flatness of this Lax connection, in grade zero we find the same as beforei.e. the same solution for the ℓ i admitting a spectral parameter -since our modification does not affect grade zero, and A is still flat. In grade one, we do not modify the ℓ 1 term, which hence still vanishes due to flatness of A, while the ℓ 2 term precisely picks up the desired replacement of A by J to get the equations of motion (1.18). Of course, unlike the relativistic case, our equations of motion still come supplemented with the constraints (1.16g,1.16h).

The NR AdS string
The NR AdS string, as obtained in [9] from the AdS 5 ×S 5 string, has coordinate Polyakov action 20 1) where x 0 , x 1 are longitudinal coordinates originating from AdS 5 , while x a and x a ′ , with a, b, ... = 2, 3, 4 and a ′ , b ′ , ... = 1, ..., 5, are transverse coordinates originating from AdS 5 and S 5 respectively, which are contracted with δ ab . The corresponding NG action in static gauge is given by

2)
where τ αβ = diag(−1, cos 2 τ ) is the AdS 2 metric. The above action describes eight free scalar fields in AdS 2 , five massless and three with mass two. This NR string can be described as a coset model similarly to the flat space case. Here we would like to briefly summarise the relevant modifications.

Symmetry algebra
To start we need to determine our symmetry algebra. Unlike the NR flat space string, the global symmetries of the NR AdS string have not been explicitly determined. It is however readily possible to determine a sufficient finite dimensional (sub)algebra for our construction. One way to do so is to consider the string Bargmann algebra that we used for the NR flat space string, as arising via Lie algebra expansion of the Poincaré symmetry of the relativistic flat space string, see e.g. [19]. Applying the same technique to the so(4, 2) ⊕so (6)  The generators of the five-dimensional Euclidean algebra are the spatial translations P a ′ and the spatial rotations J a ′ b ′ , where a ′ , b ′ , . . . = 1, . . . , 5, with non-trivial commutation relations

4) Our coset description is now based on
where we can give g a Z 2 grading corresponding to

Bilinear form and action
Our g (0) -invariant bilinear form is now which has the same bonus adjoint invariance (1.7) as before, and again is grade compatible. The NR AdS string action is still given by equation (1.11), provided that ω 6 = ω 1 = 1. Gauge invariance of this action follows as before, with the same transformation rules for the Lagrange multipliers. 21 With the coset representative is the longitudinal MC form corresponding to a coset representative for AdS 2 , i.e. e is the AdS 2 vielbein and ω the spin connection. The transverse MC form A t is corresponding to a vielbein for R 8 . Using the explicit commutation relations, we readily find Substituting this in the action we find the coordinate action (2.1) above, the extra x a x a terms arising precisely from A t combining with the extra Z A contribution in (2.12). 22 21 The new terms generated from the transformation of the MC form by the new nonzero commutators (2.3) do not affect the Lagrangian due to the structure of the bilinear form. 22 Here we used the coordinates of [9]. Alternatively, e.g. the coset representative g l = e tH0 e xH1 gives AdS 2 in global coordinates.

Equations of motion and Lax representation
Coming next to the equations of motion, the extended string Newton-Hooke and the string Bargmann algebras differ by the commutation relations (2.3). These relations, however, can only contribute to MC components associated with the generators M, Z, and G Ab . As these components do not appear in the action, there is no extra contribution to the equations of motion, which are again given by eqs. (1.16) or (1.18). 23 Since gauge symmetry is formally the same as in the flat space case, 24  It is interesting to note that the Lax connection (1.27) in the coordinates of eqs. (2.8-2.12), upon solving the constraints in terms of the worldsheet zweibein and imposing static gauge, still depends on the Lagrange multiplier fields in a nontrivial way. As a result, this Lax connection encodes the equations of motion of the NG action (2.2), but only when taking into account the equations of motion for the longitudinal fields x 0 and x 1 of (2.1), that are gauge fixed to arrive at (2.2). 25

Conclusions
In this paper we have given a coset formulation for the nonrelativistic flat space and AdS strings of [7] and [9] respectively, manifesting a finite dimensional part of the symmetries of these models. Using this formulation we were moreover able to give a Lax representation of the associated equations of motion. While NR strings are typically referred to as a limit of relativistic strings, the actual procedure to arrive at them is not a straightforward limit, and leads to the introduction of new Lagrange multiplier fields, see appendix B. For this reason, our coset formulation and in particular Lax representation do not simply follow as a limit of the corresponding relativistic ones.
Our formulation is intended to enable a systematic investigation of integrability of the NR flat space and AdS strings. There are various natural questions to pursue here. For one, here we focussed purely on the bosonic sector of the models, and it is important to extend our formulation to incorporate fermions. Next, while we have found a Lax representation of the equations of motion, this direction certainly deserves more study, in particular in a Hamiltonian framework. Moreover, with a coset formulation of these models, we may be able to define integrable deformations of these models, similar to those for the relativistic AdS 5 ×S 5 string [49], see e.g. [50] for an excellent pedagogical review. Since such deformations are typically built on the symmetries of a model, it would be interesting to see if the infinite dimensional symmetries of the NR flat space (and presumably AdS) string, that are not manifested in our coset approach, can still be involved here. Next, given that NR strings are known to be T dual to relativistic strings [10], and as a canonical transformation T duality preserves integrability, it could be enlightening to contrast these two descriptions from an integrability point of view.
It would also be interesting to go beyond the two models we considered here, and study coset model descriptions and integrability for NR version of other strings. Here the AdS 3 ×S 3 ×S 1 superstring, see e.g. [51], immediately comes to mind, given its d(2|1; α) symmetry which may yield some new structure in the NR limit. Another interesting case would be the AdS 3 string supported by mixed NSNS and RR flux [52], assuming the B field can be nontrivially taken along in the NR limit.
Next, there is an interesting connection between NR strings and relativistic strings expanded near minimal surfaces associated with Wilson loops on the boundary [53], where the NR string action appears at leading order [9]. 26 It would be interesting to explore whether higher order terms may be incorporated in a would-be NR string action and whether the Lax representation can be correspondingly adapted.
Beyond integrability, it is important to understand how our formulation can be incorporated in the general TSNC framework for NR strings.

Appendices A Conventions and Identities
Here we give our conventions, and collect some identities used to show gauge invariance of the action, and derive Noether identities.
For a generic object O A , we define its light-cone combinations as The longitudinal Minkowski metric then has non-vanishing components η +− = −1/2 and η +− = −2. We take ε 01 = −ε 01 = +1 for ε αβ , ε ab and ε AB . In light-cone components We Products of zweibeine satisfy the identities For the SNC gauge parameters σ AB we have B Review of (torsionless) SNC strings The Polyakov action for a string propagating on a (torsionless) SNC background is [10] where the notation is explained below (1. The SNC vielbeine τ µ A and E µ a , although not invertible, satisfy projected invertibility conditions, There are two useful quantities we can construct out of the SNC vielbeine. One is the longitudinal metric τ µν , while the other one is the so-called boost invariant metric H µν , The equations of motion for the fields λ ± are ε αβ e α ± τ µ ± = 0 , (B.5) which admit the solution where φ ± are arbitrary functions of the world-sheet coordinates. This implies that h αβ = φ + φ − τ αβ , where τ αβ = τ µν ∂ α X µ ∂ β X ν , and leads to the NG form of the action, The action (B.1) is locally invariant under the SNC algebra, which is a non-central extension of the string Galilei algebra. Its generators are longitudinal boost M, longitudinal translations H A , transverse rotations J ab , transverse translations P a , string-Galilei boosts G Ab , and non-central extensions Z A and Z AB , with the traceless condition Z A A = 0. The commutation relations are the same as in (1.4), except that the commuta- Infinitesimal transformations of SNC vielbeine (and associated spin connections) can be derived via the gauging procedure of the SNC algebra [20] . For this purpose we define the SNC algebra valued 1-form and the algebra valued gauge parameter We ignore the transformations associated with longitudinal and transverse translations, assuming they are traded for diffeomorphism via the standard procedure, see e.g. [56]. The gauge transformations of SNC vielbeine and spin connections are then given by which explicitly are and we need to take the λ ± fields to transform as

Relation to the coset approach
Although the local symmetry of the action (B.1) is the SNC algebra, to find a coset action and Lax representation it proves useful to restrict to the string Bargmann algebra, which is a subalgebra of the SNC algebra obtained by setting Z ++ = Z −− = 0, and identifying Z +− = 1 2 Z. There is then a clear connection between the gauging procedure described above, and our coset construction.
The Maurer-Cartan form A µ may be regarded as a particular case of the one form Θ µ , and we can think of the various components of A as corresponding to particular SNC fields. Since the MC form is flat by construction, however, we are automatically restricted to the subclass of SNC geometries with zero curvature of all types, including in particular the torsion constraint (B.13) represented by the H A components of the curvature. 27 To make contact between the coset action (1.11) and the SNC string action (B.1), we can write (1.11) explicitly as and identify the components of A in terms of those of Θ.

Connection with the relativistic string action
The action (B.1) can also be derived by a limiting procedure from the relativistic string action, where G µν is the spacetime metric and B µν is the Kalb-Ramond 2-form field, which is assumed to be closed. 28 To consider a NR limit of this action we choose two spacetime coordinates to rescale, one of which is timelike. 29 . For curved backgrounds, this choice in general affects the resulting model, and not all choices lead to an action of the form (B.1). Symmetries of the relativistic background may indicate suitable rescalings, depending on the desired outcome.
Let us suppose that a suitable coordinate rescaling has been identified and that it induces the following rescaling of relativistic vielbeinÊ µÂ for the metric G µν =Ê µÂÊµBηÂB , where c is the scaling dimensionless parameter here assumed to be large. Plugging this into the relativistic action (B.18) gives 28 It is possible to generalise this procedure for a generic 2-form, and also including the dilaton [10]. 29 The NR limit is not unique, and rescaling only the time-like coordinate can also provide meaningful string models [44].
To compensate the divergent c 2 term, we can fine-tune the B-field as so that (B.20) becomes The divergent term c 2 F A F Bη AB can be rewritten as a finite plus a subleading term, by introducing two Lagrange multiplier fields λ A as, If we now take the strict limit c → ∞, we find the NR action This is the form of the action given in [9], there for AdS strings. This is equivalent to the action (B.1) of [10], provided we choose our zweibein such that [27] e 1 e 0 − e 0 e 1 ≥ 0 . Here we demonstrate the correspondence between the equations of motion obtained from the Polyakov form of the NR string action B.1, 30 and those derived from our coset space formulation. First, the equations of motion for the λ ± fields are clearly identical, For the remaining equations, we first use the projected invertibility condition (B.2a) to 30 We work with the SNC form for convenience. It covers both our flat space and AdS case, and holds more generally. split the equations of motion for X µ derived from S Poly in two parts, These parts can be related to the coset formulation equations of motion as Due to the Noether identities (1.20), E H A on the right hand side is not independent, but can be written purely in terms of E λ ± . This shows that the equations of motion derived from the NR Polyakov action match with the independent equations of motion derived of the coset formulation.

D Derivation of the NR AdS symmetry algebra
String Bargmann is a subalgebra of the (infinite dimensional) global symmetry algebra of NR strings in flat spacetime, but its commutation relations can also be derived via Lie algebra expansion from the Poincaré algebra [19]. Here we apply the same method to the so(4, 2) ⊕ so(6) symmetry of the relativistic AdS 5 ×S 5 string, giving us an algebra that we will take to be (a finite dimensional subalgebra of) the symmetry algebra of our NR AdS string. so(4, 2) ⊕ so (6) is generated by the relativistic translations Pâ and rotations Jâb for the AdS 5 part, withâ,b, ... = 0, 1, ..., 4, and spatial translations P a ′ and rotations J a ′ b ′ for the S 5 part, with a ′ , b ′ , ... = 1, ..., 5. As a first step we write down the commutation relations with all powers of c reinstated, We then decompose the indexâ = (A, a), with A = 0, 1 and a = 2, 3, 4, and identify the generators J Ab ≡ c G Ab and P A ≡ 1 c H A , while leaving the other generators unchanged. Next, by tensoring each generator T A with the polynomial ring in the variable ζ = 1 c 2 , i.e. T . (D.2g) This infinite dimensional algebra can be truncated by setting all generators with n ≥ 2 to zero, as they form an ideal. We can make a further truncation by keeping only the n = 0 generators, together with Z A ≡ H

E Coset action based on the SNC algebra
As mentioned in section 1.2, the global symmetry algebra of NR strings in flat spacetime is infinite dimensional, and there are (at least) two natural choices of finite dimensional subalgebra to attempt to base our coset construction on. Here we briefly discuss a coset construction based on the SNC algebra, highlighting the differences with the string Bargmann case discussed in the main text. Although there appears to be no obstruction to using the SNC algebra for the coset action, it is ill suited to find a Lax representation for the equations of motion. For our coset model we now take G to be the group generated by the SNC algebra of eqs. (B.8). This algebra is the same as the string Bargmann algebra, with the exception of Z which is replaced by the traceless Z AB . We again take H to be generated by everything except the H A and P a .
For the bilinear form, we now ask for invariance under the adjoint action ofh ≡ h \ {Z A , Z AB }, 32 , and find for arbitrary ω i . It still has the bonus invariance (1.7), now for x ∈ {Z ± , Z AB }. Gauge invariance of the action works as in the string Bargmann construction, now taking the Lagrange multipliers to transform as in (B.14) instead of as in section 1.4, compensating the new MC gauge transformations. The graded subspaces g (0) , g (1) remain defined as in (1.10), but with the replacement of Z with Z AB . In particular, now g (0) is no longer equal toh. Then, provided that ω 1 = 1 and ω 2 = 0, the action takes again the form (1.11).
The equations of motion are as in (1.16), except for (1.16b) and (1.16c), which now become Moreover, there are two additional equations of motion, The Noether identities discussion applies as before, except that now Z AB gauge invariance, with traceless gauge parameters σ AB , implies which gives the non-trivial Noether identities