Nonrelativistic spinning strings

We construct nonrelativistic spinning string solutions corresponding to $ SU(1,2|3) $ Spin-Matrix theory (SMT) limit of strings in $ AdS_5 \times S^5 $. Considering various nonrelativistic spinning string configurations both in $ AdS_5 $ as well as $ S^5 $ we obtain corresponding dispersion relations in the strong coupling regime of SMT where the strong coupling ($ \sim \sqrt{\mathfrak{g}} $) corrections near the BPS bound have been estimated in the slow spinning limit of strings in $ AdS_5 $. We generalize our results explicitly by constructing three spin folded string configurations that has two of its spins along $ AdS_5 $ and one along $ S^5 $. Our analysis reveals that the correction to the spectrum depends non trivially on the length of the NR string in $ AdS_5 $. The rest of the paper essentially unfolds the underlying connection between $ SU(1,2|3) $ Spin-Matrix theory (SMT) limit of strings in $ AdS_5 \times S^5 $ and the nonrelativistic Neumann-Rosochatius like integrable models in 1D. Taking two specific examples of NR spinning strings in $ R \times S^3 $ as well as in certain sub-sector of $ AdS_5 $ we show that similar reduction is indeed possible where one can estimate the spectrum of the theory using 1D model.


Spin-Matrix theory and NR strings
During last one decade, a series of work [1]- [7] have been put forward which essentially argues about a more tractable limit of the AdS 5 /CFT 4 correspondence [8]- [9] in a regime, where both the descriptions are under control and hence the corresponding spectrum could be subjected to a precise test. Here, ∆ is the energy of a state in N = 4 SYM on R × S 3 and J is a linear sum over Cartan charges. The above limit (1) essentially takes us towards a (nonrelativistic) corner of N = 4 SYM known as the Spin-Matrix theory (SMT) [6] where states in the Hilbert space carry indices both in the spin as well as adjoint representation. On the gauge theory side of the duality, the above limit (1) corresponds to operators with classical/tree level dimension ∆ 0 = J where all the other operators with ∆ 0 > J are essentially decoupled from the rest of the spectrum. As for example, one can think of the SU (2) sector of N = 4 SYM where the one loop correction to the scaling dimension (∆) could be formally expressed as, ∆ = J + λ∆ 1 + O(λ 3/2 ). (2) Combining (1) and (2) the corresponding SMT Hamiltonian turns out to be, where, g is the coupling constant appearing in the SMT limit [6]. Therefore, in the decoupling/SMT limit, the gauge theory side of the duality is fully under control provided one knows the one loop contribution to the anomalous dimension [4]. In the planar (N → ∞) limit, the weakly coupled (g 1) version of the theory recast itself as an integrable spin chain with nearest neighbour interactions [6]. The low energy spectrum of the theory contains excitations like single spin magnons and so on. However, a detailed understanding of similar phenomena from the perspective of strong coupling (g 1) physics is still lacking in the literature. The understanding of strong coupling phenomena requires a dual nonrelativistic (NR) stringy counterpart [10]- [12] living in AdS 5 × S 5 which for our case would correspond to constructing a null reduced sigma model action over torsional Newton-Cartan (TNC) geometry [13]- [24] and thereby taking 1/c limit of the world-sheet degrees of freedom.
On the string theory side of the correspondence, one typically starts with the null reduced form of the metric [13]- [14] (in the presence of a null isometry direction u), corresponding to AdS 5 × S 5 and consider the SMT limit (3) as [14], where, g s = λ 4πN is the string coupling constant. As a matter of fact, in order to construct a finite sigma model action in the above limit (5) one needs to simultaneously rescale the world-sheet vielbein (τ α ) as well as the string tension (T ) with some appropriate factor/power of c = 1 √ 4πgsN which goes to infinity in the limit (5). As a result of this rescaling, the original sigma model action essentially boils down into a NR sigma model action which is conjectured to be dual to the SMT physics as mentioned above [13]- [14].

Summary of results
One of the primary ambitions of the present paper is to construct various NR (semiclassical) spinning string configurations in AdS 5 × S 5 and explore the corresponding dispersion relations in the (large g 1) SU (1, 2|3) SMT limit 1 of N = 4 SYM. In particular, we construct the NR analogue of GKP like solutions [25] by computing the energy as well as spin associated with various folded (spinning) string configurations in AdS 5 × S 5 [26]- [27]. Unlike its relativistic cousins [25], here the results are obtained in the slow spinning (S √ g) limit of NR strings in AdS 5 which has a dispersion relation of the form, where, the coefficients (a n ) of the expansion (6) are non trivial functions of string length.
• Considering spinning strings in S 5 , on the other hand, we notice that the dispersion relation takes the following form, where, γ stands for the leading order correction to the spectrum in the semiclassical limit. Our analysis reveals that the correction becomes large as we move form single spin to multi-spin string configurations.
• We generalize above observations by considering folded three spin solutions that has two of its spins along AdS 5 and the remaining one along S 5 . In the slow frequency limit, this configuration yields the dispersion relation that is of the form, where, Q = S + J is the total spin of the configuration such that, Q √ g < 1. Here, the entities within the parenthesis on the R.H.S. of (8) are some complicated functions of the string length whose detailed expressions have been provided in the following Section 2.
• The rest of the paper essentially unfolds the relationship between SU (1, 2|3) SMT limit of spinning strings in AdS 5 × S 5 and the Neumann-Rosochatius integrable systems [28] in 1D. The 1D model thus obtained are defined as the nonrelativistic analogue of Neumann-Rosochatius like models those constructed previously in the context of type IIB strings propagating in AdS 5 × S 5 [29]- [30]. Eventually, we compute the spectrum of spinning strings using this reduced model. We consider two specific examples.
The first example takes into account spinning strings in R × S 3 and its 1D reduction to nonrelativistic extended Neumann-Rosochatius like systems. The spectrum that one computes from this reduced model typically takes the form, where,J = J √ g is the effective R-charge in the strong coupling regime of SMT. On the other hand, m 1,2 are the windning numbers of the string along two of the azimuthal directions (φ 1,2 ) of S 5 .
A similar analysis for folded spinning strings in AdS 5 reveals, where, k is the winding number of the string along one of the azimuthal directions (ϕ) of S 3 ⊂ AdS 5 and ω is the corresponding spinning frequency of the soliton. While obtaining the above relation (10), one essentially considers the so called short string limit where the string soliton is considered to be sitting near the north pole (ψ ∼ 0) of the three sphere (S 3 ) and is located near the centre of AdS 5 as well. We conclude in Section 5 with some final remarks.
2 SU (1, 2|3) SMT limit and NR strings We start with the null reduced form of the NR sigma model action over AdS 5 × S 5 in the SU (1, 2|3) SMT limit. The NR Nambu-Goto (NG) action could be formally expressed as, where the corresponding Lagrangian density is given by [13]- [14], where η is the compact dual dimension along which the string has a nonzero winding mode. Here {τ α , m α } are the world-sheet one forms and h αβ is the world-sheet two form whose detailed expressions are given below [14], and, Notice that, the angles ψ and ϕ belong to S 3 which is a subset of the full AdS 5 geometry. The information regarding the remaining coordinate of the three sphere is encoded in the angular direction χ which is periodic with a period of 2π. On the other hand, the remaining angular variables {θ 1 , θ 2 , φ 1 , φ 2 } belong to the five sphere (S 5 ) part of the original 10D target space geometry.

Spinning strings in AdS 5
We consider closed NR spinning strings in AdS 5 and construct solutions with single spin in S 3 . To work with such stringy configurations, we choose the following ansatz and switch off all the remaining coordinates. The corresponding NG Lagrangian density (12) takes the following form, where prime corresponds to derivative w.r.t. σ 1 .
The resulting equations of motion may be obtained as,

Short strings
To proceed further, we choose to work with the short string [25] limit where we consider that the (folded) string is not stretched enough and is located near the centre (ρ ∼ 0) of AdS 5 . In other words, this is the limit that essentially describes NR spinning strings in flat space where we ignore the curvature effects of AdS 5 . We also consider that the center of the string soliton is located at the north pole (ψ = 0) of S 3 . Combining all these pieces together we finally obtain, where we retain ourselves only upto quadratic order in the bosonic field fluctuations ρ(σ 1 ).
The solution corresponding to the radial fluctuation may be obtained as, where ρ m ( 1) is the maximum value of the radial coordinate such that the string has four segments in it, each of which is ranging between 0 to ρ m . A straightforward computation further reveals the energy, as well as the spin angular momentum, Combing (22) and (23) we finally obtain,

Extended strings
In order to extract dispersion relation corresponding to NR extended (spinning) strings we again start with the following equation that describes strings sitting at the north pole of S 3 , In order to simplify the problem, we look into the specific sector of the parameter space namely we try to construct solution in the slow spinning limit of the string. In the slow velocity (ω 1/ρ m ) limit, the corresponding solution may be expressed as, The energy of the configuration may be noted down as, which after a straightforward computation yields 2 , and, The spin angular momentum, on the other hand, turns out to be, where we identify each of the individual entities as 3 , and, In the large ρ m ( 1) limit, the above charges (27) and (30) become, where we denote, ∆ 1 ρm 1 = ρ 2 m∆ 1 and so on. Combining (33) and (34) we finally obtain, where we identify, Sϕ √ g (< 1) as the effective spin in the strong coupling regime of SMT.

Single spin solution
The second embedding that we consider is that of a NR folded spinning string on S 5 whose centre of mass is fixed and located at the north pole. The string soliton is considered to be stretched along one of the polar coordinates of S 5 whose end points are spinning along the azimuthal direction φ 1 , while all the remaining coordinates are switched off. The corresponding sigma model Lagrangian turns out to be, The resulting equation of motion, could be integrated once to obtain, The space time energy of the NR string is given by 4 , On the other hand, the R-charge on the stringy side can be computed as, Combining both (40) and (41) we find, As a further remark, we notice that, Using (43) we finally obtain 5 , which clearly reveals that the maximum correction to the anomalous dimension (∆ N R ) in the dual SMT is ∼ √ g 4 which corresponds to spinning strings whose end points are stretched upto the equatorial plane (θ m = π 2 ) of S 5 .

Multi-spin solutions
We now generalize the above result for NR folded (closed) spinning string configurations with two unequal spins (to start with) along two azimuthal directions [26], The corresponding Lagrangian density turns out to be, which yields the equation of motion of the form, where,ω = ω 1 + ω 2 is the total angular frequency of the string. The corresponding conserved charges are given by, A straightforward computation reveals the total R-charge of the configuration as, Combining all these results, we finally obtain, Setting, ω 1 = ω 2 = 1 and θ m = π 2 we find the leading order correction to the spectrum, Notice that the value of the correction coefficient (γ) increases from single spin solution (γ = 0.25) to multi spin solution by 14%.

Folded three spin solutions
We now generalize the above construction by considering NR folded spinning strings in AdS 5 × S 5 with two equal spins in AdS 5 and one spin along S 5 , where the string is stretched along the radial coordinate ρ of AdS 5 as well as the angular direction θ 1 of S 5 .
The corresponding Lagrangian density turns out to be, The resulting equations of motion are given by, and,

Perturbative solution
The above set of equations (56)-(57) are indeed difficult to solve analytically. To simplify the problem, we consider a configuration where the string soliton is considered to be sitting at the equatorial plane (θ 1 = π 2 ) of S 5 while having a spin along the azimuthal direction φ 1 . This clearly satisfies (57) and upon substitution into (56) yields, We propose the following perturbative solutions for 6 (58), where the zeroth order solution we note down to be, The leading order equation in ω has a solution of the form, Finally, the leading order solution in ν turns out to be,

Energy-spin relations
Energy of the stringy configuration turns out to be, where the sub-leading corrections in (63) are given by, and, The total spin of the configuration, on the other hand, is given by where, the sub-leading corrections are given by, Finally, the R-charge of the configuration is given by where, Φ(ρ m ) is the leading order correction to the R-charge that goes like ∼ 1 ρ 2 m similar to that of Λ 1 (ρ m ) as given in (31).
Considering the extended string (ρ m 1) limit, the total angular momentum of the NR stringy configuration turns out to be, which finally results in the dispersion relation of the form, 3 Spinning strings in R × S 3 The purpose of this Section is to revisit the NR spinning string configurations and establish a precise mapping between NR sigma models in AdS 5 ×S 5 and one dimensional Neumann-Rosochatius like integrable models [28]. We select a specific sub-sector R×S 3 ⊂ AdS 5 ×S 5 of the full target space geometry where we consider the string soliton to be sitting at the centre of AdS 5 and switch on spin along one of the azimuthal directions (φ 2 ) of S 5 . We choose the string embedding of the following form, and switch off all the remaining coordinates. For simplicity, from now on we set, θ 1 = θ. The resulting sigma model Lagrangian turns out to be, To proceed further, we redefine coordinates as, which upon substitution into (75) yields, The first four terms on the R.H.S of (77) together constitute what we define as the nonrelativistic analogue of 1D Neumann-Rosochatius like integrable models derived directly from 2D sigma models in the SU (1, 2|3) SMT limit of strings on AdS 5 × S 5 . Contrary to its relativistic cousins [29]- [30], the nonrelativistic 1D model (77) is linear in the time derivative/ frequency (ν) and is quadratic in space derivatives. Here, N is the Lagrange multiplier that preserves the constraint condition, 2 i = 1. The last term on the R.H.S. of (77), is identified as the deformation to the nonrelativistic Neumann-Rosochatius model. The phase space of the 1D model (77) is eventually 6 (= 2N ) dimensional. The configuration is integrable as there are Q i (i = 1, 2) conserved charges associated with the dynamical phase space. This follows directly from the fact that the system is constrained.
The canonical momentum densities are given by, Clearly, there are two constants of motion (υ i ∼ Q i ) associated with the configuration. The corresponding canonical Hamiltonian density turns out to be, where we identify, as nonrelativistic analogue of the Uhlenbeck constant [29]- [30].

Constant radii solutions
We first note down the equations of motion that directly result from (77), The above set of equations (84)-(85) could in principle be solved considering a fixed radius [29] ansatz namely, i = a i = constant and choosing an ansatz for the winding modes along the azimuthal direction of S 5 , where m i s are the respective winding numbers. This results in the following set of algebraic equations, Using (87)-(88), we finally obtain, subjected to the constraint condition, a 2 1 + a 2 2 = 1.

Dispersion relation
The energy of the NR stringy configuration is given by, On the other hand, the R-charge of the configuration reads as, Combining (91) and (92) and after some trivial algebra we find, where, we defineJ = J √ g as the effective R-charge in strong coupling limit of SMT.

Spinning strings in AdS 5
We now focus on folded spinning strings in AdS 5 and switch off any dynamics along S 5 . In order to map the corresponding sigma model action into a 1D Neumann-Rosochatius like integrable model we choose to work with the following string embedding, The resulting Lagrangian density turns out to be, We define the following set of coordinates, and consider the dynamics near the centre of AdS 5 . This finally results in the 1D Lagrangian of the following form, where, we introduce the diagonal metric g ab = diag(−1, 1) as well as the Lagrange multiplier G which implies the constraint condition [30], The first two terms (in the parenthesis) on the R.H.S. of (97) are the standard Neumann-Rosochatius piece while the remaining one, serves as an extension to it.

Equations of motion
The equations of motion could be enumerated as, z a − z a β 2 + (ω − G)z a = 0 ; a = 0, 1.

Energy-spin relation
The energy and as well as the spin of the NR configuration are given below, ρm 0 dρ(p cosh 2 ρ + k 2 sinh 2 ρ) Using (103) and (104) we finally obtain, where, Sϕ √ g ∼ ρ 3 m 1 is the effective spin in the strong coupling regime of SMT.

Summary and final remarks
We conclude our paper with a brief summary of the analysis and mentioning some of its possible implications as well. The present paper explores various corners of the SU (1, 2|3) Spin-Matrix theory limit of N = 4 SYM in the limit of strong coupling. Using the dual semiclassical nonrelativistic stringy counterpart in AdS 5 × S 5 , the present paper realizes the spectrum of the theory for various sub-sectors of the full Hilbert space. We also show that the SU (1, 2|3) Spin-Matrix theory limit of the sigma model in AdS 5 × S 5 can be mapped into 1D Neumann-Rosochatius like integrable models. This further allows us to compute the spectrum of the 2D sigma model corresponding to various spinning string configurations in AdS 5 ×S 5 . It would be really nice to reproduce these semiclassical string states (and hence the spectrum) considering a large (g 1) Spin-Matrix theory limit in the SU (1, 2|3) sector of N = 4 SYM. We hope to address these issues in the near future.