NNonrelativistic spinning strings
Dibakar Roychowdhury ∗ Department of Physics, Indian Institute of Technology Roorkee,Roorkee 247667, Uttarakhand, India
Abstract
We construct nonrelativistic spinning string solutions corresponding to SU (1 , | AdS × S . Considering various non-relativistic spinning string configurations both in AdS as well as S we obtaincorresponding dispersion relations in the strong coupling regime of SMT where thestrong coupling ( ∼ √ g ) corrections near the BPS bound have been estimated in theslow spinning limit of strings in AdS . We generalize our results explicitly by con-structing three spin folded string configurations that has two of its spins along AdS and one along S . Our analysis reveals that the correction to the spectrum dependsnon trivially on the length of the NR string in AdS . The rest of the paper es-sentially unfolds the underlying connection between SU (1 , |
3) Spin-Matrix theory(SMT) limit of strings in
AdS × S and the nonrelativistic Neumann-Rosochatiuslike integrable models in 1D. Taking two specific examples of NR spinning stringsin R × S as well as in certain sub-sector of AdS we show that similar reduction isindeed possible where one can estimate the spectrum of the theory using 1D model. During last one decade, a series of work [1]-[7] have been put forward which essentiallyargues about a more tractable limit of the AdS /CFT correspondence [8]-[9] in a regime, H = lim λ → ∆ − Jλ = f ixed ; N = f ixed (1)where both the descriptions are under control and hence the corresponding spectrumcould be subjected to a precise test. Here, ∆ is the energy of a state in N = 4 SYMon R × S and J is a linear sum over Cartan charges. The above limit (1) essentiallytakes us towards a (nonrelativistic) corner of N = 4 SYM known as the Spin-Matrixtheory (SMT) [6] where states in the Hilbert space carry indices both in the spin as wellas adjoint representation.On the gauge theory side of the duality, the above limit (1) corresponds to operatorswith classical/tree level dimension ∆ = J where all the other operators with ∆ > J are ∗ E-mail: [email protected], [email protected] a r X i v : . [ h e p - t h ] O c t ssentially 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 ∆ + O ( λ / ) . (2)Combining (1) and (2) the corresponding SMT Hamiltonian turns out to be, H SMT = J + g lim λ → ∆ − Jλ = J + g ∆ (3)where, g is the coupling constant appearing in the SMT limit [6]. Therefore, in thedecoupling/SMT limit, the gauge theory side of the duality is fully under control providedone knows the one loop contribution to the anomalous dimension [4].In the planar ( N → ∞ ) limit, the weakly coupled ( g (cid:28)
1) version of the theory recastitself as an integrable spin chain with nearest neighbour interactions [6]. The low energyspectrum 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 (cid:29)
1) physics is still lacking in the literature. The understanding of strong couplingphenomena requires a dual nonrelativistic (NR) stringy counterpart [10]-[12] living in
AdS × S which for our case would correspond to constructing a null reduced sigmamodel action over torsional Newton-Cartan (TNC) geometry [13]-[24] and thereby taking1 /c limit of the world-sheet degrees of freedom.On the string theory side of the correspondence, one typically starts with the nullreduced form of the metric [13]-[14] (in the presence of a null isometry direction u ), ds = 2 τ ( d u − m ) + h µν dX µ dX ν (4)corresponding to AdS × S and consider the SMT limit (3) as [14], H string = lim g s → E − Qg s = f ixed ; N = f ixed ; Q = S + J (5)where, g s = λ πN is the string coupling constant. As a matter of fact, in order to constructa finite sigma model action in the above limit (5) one needs to simultaneously rescalethe world-sheet vielbein ( τ α ) as well as the string tension ( T ) with some appropriatefactor/power of c = √ πg s N which goes to infinity in the limit (5). As a result of thisrescaling, the original sigma model action essentially boils down into a NR sigma modelaction which is conjectured to be dual to the SMT physics as mentioned above [13]-[14]. One of the primary ambitions of the present paper is to construct various NR (semi-classical) spinning string configurations in
AdS × S and explore the corresponding dis-persion relations in the (large g (cid:29) SU (1 , |
3) SMT limit of N = 4 SYM. In particular,we construct the NR analogue of GKP like solutions [25] by computing the energy as well SU (1 , |
3) SMT limit of N = 4 SYM is of particular interest as this contains 1 /
16 BPS supersym-metric states. In the limit of strong ( g → ∞ ) coupling and large temperatures, these supersymmetricstates are supposed be describing supersymmetric black holes on the stringy side of the correspondence.
2s spin associated with various folded (spinning) string configurations in
AdS × S [28]-[29]. Unlike its relativistic cousins [25], here the results are obtained in the slow spinning ( S (cid:28) √ g ) limit of NR strings in AdS which has a dispersion relation of the form, E NR ∼ S + √ g (cid:88) n ≥ a n (cid:18) S √ g (cid:19) n (6)where, the coefficients ( a n ) of the expansion (6) are non trivial functions of string length. • Considering spinning strings in S , on the other hand, we notice that the dispersionrelation takes the following form, E NR ∼ J + γ √ g (7)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 tomulti-spin string configurations. • We generalize above observations by considering folded three spin solutions that hastwo of its spins along
AdS and the remaining one along S . In the slow frequency limit,this configuration yields the dispersion relation that is of the form, E NR ∼ Q + √ g ρ m π ( ˜∆ ( ρ m ) + ω Σ ( ω ) ( ρ m ) + ν Σ ( ν ) ( ρ m ) + · · · ) + O ( √ g /ρ m ) (8)where, Q = S + J is the total spin of the configuration such that, Q √ g <
1. Here, theentities within the parenthesis on the R.H.S. of (8) are some complicated functions of thestring length whose detailed expressions have been provided in the following Section 2. • The rest of the paper essentially unfolds the relationship between SU (1 , |
3) SMTlimit of spinning strings in
AdS × S and the Neumann-Rosochatius integrable systems[30] in 1D. The 1D model thus obtained are defined as the nonrelativistic analogue ofNeumann-Rosochatius like models those constructed previously in the context of typeIIB strings propagating in AdS × S [31]-[32]. Eventually, we compute the spectrum ofspinning strings using this reduced model. We consider two specific examples.The first example takes into account spinning strings in R × S and its 1D reductionto nonrelativistic extended Neumann-Rosochatius like systems. The spectrum that onecomputes from this reduced model typically takes the form, E NR ∼ (cid:18) m m (cid:19) J + √ g (cid:18) m + 23 m − m ( ˜ J + 6 ˜ J ) (cid:19) (9)where, ˜ J = J √ g is the effective R- charge in the strong coupling regime of SMT. Onthe other hand, m , are the windning numbers of the string along two of the azimuthaldirections ( φ , ) of S .A similar analysis for folded spinning strings in AdS reveals, E NR ∼ (2 k − ω ) S ϕ + √ g (cid:18) π ( k − ω ) (cid:19) / (cid:18) S ϕ √ g (cid:19) / + · · · (10)3here, k is the winding number of the string along one of the azimuthal directions ( ϕ ) of S ⊂ AdS and ω is the corresponding spinning frequency of the soliton. While obtainingthe above relation (10), one essentially considers the so called short string limit where thestring soliton is considered to be sitting near the north pole ( ψ ∼
0) of the three sphere( S ) and is located near the centre of AdS as well.Finally, we conclude in Section 5 where we briefly outline possible interpretation forthese nonrelativisitc (spinning) string states in terms of dual SMT degrees of freedom.In particular, we discuss various decoupling limits associated to N = 4 SYM those maybe interpreted as the degrees of freedom of a nonrelativistic string in the limit of strongcoupling. This finally provides a platform for several non trivial checks in nonrelativisticholographic correspondence using quantum mechanical degrees of freedom. SU (1 , | SMT limit and NR strings
We start with the null reduced form of the NR sigma model action over
AdS × S in the SU (1 , |
3) SMT limit. The NR Nambu-Goto (NG) action could be formally expressed as, S NG = √ g π (cid:90) d σ L NG ; σ α = { σ , σ } (11)where the corresponding Lagrangian density is given by [13]-[14], L NG = (cid:15) αβ m α ∂ β η + (cid:15) αα (cid:48) (cid:15) ββ (cid:48) τ α (cid:48) τ β (cid:48) (cid:15) γγ (cid:48) τ γ ∂ γ (cid:48) η h αβ (12)where η is the compact dual dimension along which the string has a nonzero windingmode. Here { τ α , m α } are the world-sheet one forms and h αβ is the world-sheet two formwhose detailed expressions are given below [14], τ α = cosh ρ∂ α t (13) m α = − tanh ρ (cid:18) ∂ α χ + 12 cos ψ∂ α ϕ (cid:19) + cosh − ρ (cid:18)
12 cos θ sin θ ∂ α φ − (cid:18) −
12 sin θ (cid:19) ∂ α φ (cid:19) (14)4nd, h αβ = tanh ρ∂ α χ∂ β χ + ∂ α ρ∂ β ρ + 14 sinh ρ ( ∂ α ψ∂ β ψ + sin ψ∂ α ϕ∂ β ϕ )+ tanh ρ∂ α χ (cid:18)
12 cos ψ∂ β ϕ + 12 cos θ sin θ ∂ β φ − (cid:18) −
12 sin θ (cid:19) ∂ β φ (cid:19) + ( α ↔ β )+ tanh ρ ψ∂ α ϕ∂ β ϕ + ∂ α θ ∂ β θ + 14 sin θ ( ∂ α θ ∂ β θ + sin θ ∂ α φ ∂ β φ )+ 14 sin θ cos θ ( ∂ α φ ∂ β φ + cos θ ( ∂ α φ ∂ β φ + ∂ β φ ∂ α φ ) + cos θ ∂ α φ ∂ β φ )+ tanh ρ (cid:18) cos ψ (cid:18)
12 cos θ sin θ ∂ α φ − (cid:18) −
12 sin θ (cid:19) ∂ α φ (cid:19) ∂ β ϕ + ( α ↔ β ) (cid:19) + tanh ρ (cid:18)
12 cos θ sin θ ∂ α φ − (cid:18) −
12 sin θ (cid:19) ∂ α φ (cid:19) × (cid:18)
12 cos θ sin θ ∂ β φ − (cid:18) −
12 sin θ (cid:19) ∂ β φ (cid:19) . (15)Notice that, the angles ψ and ϕ belong to S which is a subset of the full AdS geometry. The information regarding the remaining coordinate of the three sphere isencoded in the angular direction χ which is periodic with a period of 2 π . On the otherhand, the remaining angular variables { θ , θ , φ , φ } belong to the five sphere ( S ) partof the original 10D target space geometry. AdS We consider closed NR spinning strings in
AdS and construct solutions with single spinin S . To work with such stringy configurations, we choose the following ansatz t = σ ; ρ = ρ ( σ ) ; ϕ = ωσ ; η = σ ; ψ = ψ ( σ ) (16)and switch off all the remaining coordinates.The corresponding NG Lagrangian density (12) takes the following form, L NG = − ω ρ cos ψ + 12 cosh ρ (cid:18) ρ (cid:48) + ψ (cid:48) ρ (cid:19) (17)where prime corresponds to derivative w.r.t. σ .The resulting equations of motion may be obtained as, ρ (cid:48)(cid:48) cosh ρ + ρ (cid:48) cosh ρ sinh ρ + ω tanh ρ cosh − ρ cos ψ − ψ (cid:48)
16 sinh 4 ρ = 0 (18) ψ (cid:48)(cid:48) cosh ρ sinh ρ + ψ (cid:48) ρ (cid:48) ρ − ω tanh ρ sin ψ = 0 . (19) To proceed further, we choose to work with the short string [25] limit where we considerthat the (folded) string is not stretched enough and is located near the centre ( ρ ∼
0) of
AdS . In other words, this is the limit that essentially describes NR spinning strings in5 at space where we ignore the curvature effects of AdS . We also consider that the centerof the string soliton is located at the north pole ( ψ = 0) of S . Combining all these piecestogether we finally obtain, ρ (cid:48)(cid:48) + ωρ ≈ ρ ( σ ).The solution corresponding to the radial fluctuation may be obtained as, ρ ( σ ) ∼ ρ m sin (cid:0) σ √ ω (cid:1) (21)where ρ m ( (cid:28)
1) is the maximum value of the radial coordinate such that the string hasfour segments in it, each of which is ranging between 0 to ρ m .A straightforward computation further reveals the energy, E NR = √ g π (cid:90) π dσ δ L NG δ ˙ t ≈ √ g π (cid:90) ρ m dρρ (cid:48) = √ g √ ωρ m (22)as well as the spin angular momentum, S ϕ = √ g π (cid:90) π dσ δ L NG δ ˙ ϕ ≈ √ g π (cid:90) ρ m dρ ρ ρ (cid:48) = √ g ρ m √ ω . (23)Combing (22) and (23) we finally obtain, E NR ∼ ωS ϕ . (24) In order to extract dispersion relation corresponding to NR extended (spinning) stringswe again start with the following equation that describes strings sitting at the north poleof S , ρ (cid:48)(cid:48) cosh ρ + ρ (cid:48) cosh ρ sinh ρ + ω tanh ρ cosh − ρ = 0 . (25)In order to simplify the problem, we look into the specific sector of the parameterspace namely we try to construct solution in the slow spinning limit of the string. In theslow velocity ( ω (cid:28) /ρ m ) limit, the corresponding solution may be expressed as, ρ ( σ ) = ρ m sinh − σ (cid:32) ω (2 σ + tan − σ )2 (cid:112) ( σ ) + 1 sinh − σ + · · · (cid:33) . (26)The energy of the configuration may be noted down as, E NR = √ g π (cid:90) π dσ ρ (cid:48) cosh ρ (cid:39) √ g π (∆ ( ρ m ) + ωρ m ∆ ( ρ m ) + · · · ) (27)6hich after a straightforward computation yields ,∆ ( ρ m ) = ˜ F (cid:18) , − ρ m ; 32 − ρ m ; − π (cid:16) √ π + 2 π (cid:17) − (cid:19) × e (1 − ρ m ) sinh − (2 π ) Γ (cid:18) − ρ m (cid:19) ρ m ˜ F (cid:18) , ρ m + 12 ; ρ m + 32 ; − π (cid:16) √ π + 2 π (cid:17) − (cid:19) × e (2 ρ m +1) sinh − (2 π ) Γ (cid:18) ρ m + 12 (cid:19) ρ m ρ m (cid:0) − (2 π ) − π sec( πρ m ) (cid:1) (28)and, ∆ ( ρ m ) = (cid:90) π dσ (2 σ + tan − σ )2 (( σ ) + 1) / sinh (cid:0) ρ m sinh − σ (cid:1) . (29)The spin angular momentum, on the other hand, turns out to be, S ϕ = √ g π (cid:90) π dσ tanh ρ = √ g π (Λ ( ρ m ) + ωρ m Λ ( ρ m ) + · · · ) (30)where we identify each of the individual entities as ,Λ ( ρ m ) = (cid:20) B − e ρm sinh − π ) (cid:18) − ρ m , (cid:19) + B − e ρm sinh − π ) (cid:18) − ρ m , (cid:19)(cid:21) × (cid:16) √ π − π (cid:17) (cid:16) − e ρ m sinh − (2 π ) (cid:17) ρm ρ m − (cid:20) B − e ρm sinh − π ) (cid:18) ρ m , (cid:19) + B − e ρm sinh − π ) (cid:18) ρ m , (cid:19)(cid:21) × (cid:16) √ π + 2 π (cid:17) (cid:16) − e ρ m sinh − (2 π ) (cid:17) − ρm ρ m + 14 ρ m (cid:18) πρ m + π tan (cid:18) π ρ m (cid:19) + π cot (cid:18) π ρ m (cid:19)(cid:19) − √ π ρ m tanh (cid:0) ρ m sinh − (2 π ) (cid:1) (31)and, Λ ( ρ m ) = (cid:90) π dσ tanh (cid:0) ρ m sinh − σ (cid:1)(cid:112) ( σ ) + 1 × (cid:0) tan − σ − σ tanh (cid:0) ρ m sinh − σ (cid:1) − tan − σ tanh (cid:0) ρ m sinh − σ (cid:1)(cid:1) + O (1 /ρ m ) . (32) Here, ˜ F ( a, b ; c ; d ) is the regularised hypergeometric function [33]. Here B z ( a, b ) is the incomplete beta function which becomes the usual beta function for z = 1 [33].
7n the large ρ m ( (cid:29)
1) limit, the above charges (27) and (30) become, E NR ≈ √ g π ρ m ( ˜∆ + ωρ m ˜∆ + · · · ) (33) S ϕ ≈ √ g π ωρ m ˜Λ (34)where we denote, ∆ (cid:12)(cid:12)(cid:12) ρ m (cid:29) = ρ m ˜∆ and so on.Combining (33) and (34) we finally obtain, E NR ∼ S ϕ + (8 π ˜Λ − ) √ g (cid:34) ˜∆ (cid:18) S ϕ √ g (cid:19) + (8 π ˜∆ ˜Λ − ) (cid:18) S ϕ √ g (cid:19) + · · · (cid:35) (35)where we identify, S ϕ √ g ( <
1) as the effective spin in the strong coupling regime of SMT. S The second embedding that we consider is that of a NR folded spinning string on S whosecentre of mass is fixed and located at the north pole. The string soliton is considered tobe stretched along one of the polar coordinates of S whose end points are spinning alongthe azimuthal direction φ , t = σ ; η = σ ; ρ = 0 ; θ = θ ( σ ) ; φ = ωσ (36)while all the remaining coordinates are switched off.The corresponding sigma model Lagrangian turns out to be, L NG = ω θ + θ (cid:48) . (37)The resulting equation of motion, θ (cid:48)(cid:48) − ω θ = 0 (38)could be integrated once to obtain, θ (cid:48) = ω (cos θ m − cos θ ) . (39)The space time energy of the NR string is given by , E NR = √ g π √ ω (cid:90) θ m dθ (cid:112) (cos θ m − cos θ )= √ g π √ ω sin θ m E (cid:0) θ m (cid:12)(cid:12) csc θ m (cid:1) . (40) Here, F ( ϕ | k ) and E ( ϕ | k ) are respectively the incomplete elliptic integrals of the first and secondkind [33].
8n the other hand, the R- charge on the stringy side can be computed as, J = √ g π (cid:90) θ m dθ θ (cid:48) sin θ = √ g π √ ω sin θ m ( E (cid:0) θ m (cid:12)(cid:12) csc θ m (cid:1) − F (cid:0) θ m (cid:12)(cid:12) csc θ m (cid:1) ) . (41)Combining both (40) and (41) we find, E NR = ωJ + √ g π √ ω sin θ m F (cid:0) θ m (cid:12)(cid:12) csc θ m (cid:1) . (42)As a further remark, we notice that,2 π = 4 (cid:90) θ m dθ θ (cid:48) = 4 √ ω θ m F (cid:0) θ m (cid:12)(cid:12) csc θ m (cid:1) . (43)Using (43) we finally obtain , E NR ∼ J + √ g θ m (44)which clearly reveals that the maximum correction to the anomalous dimension (∆ NR )in the dual SMT is ∼ √ g which corresponds to spinning strings whose end points arestretched upto the equatorial plane ( θ m = π ) of S . We now generalize the above result for NR folded (closed) spinning string configurationswith two unequal spins (to start with) along two azimuthal directions [28], t = σ ; η = σ ; ρ = 0 ; θ = 0 ; θ = θ ( σ ) ; φ = ω σ ; φ = ω σ . (45)The corresponding Lagrangian density turns out to be, L NG = ω θ − ω (cid:18) −
12 sin θ (cid:19) + θ (cid:48) θ (cid:48)(cid:48) − ˜ ω θ = 0 (47)where, ˜ ω = ω + ω is the total angular frequency of the string.The corresponding conserved charges are given by, E NR = √ g π √ ˜ ω sin θ m E (cid:0) θ m (cid:12)(cid:12) csc θ m (cid:1) (48) J = √ g sin θ m π √ ˜ ω ( E (cid:0) θ m (cid:12)(cid:12) csc θ m (cid:1) − F (cid:0) θ m (cid:12)(cid:12) csc θ m (cid:1) ) (49) J = √ g sin θ m π √ ˜ ω (cid:0)(cid:0) − θ m (cid:1) F (cid:0) θ m (cid:12)(cid:12) csc θ m (cid:1) − E (cid:0) θ m (cid:12)(cid:12) csc θ m (cid:1)(cid:1) . (50) We set the overall scale factor, ω = 1. | J | = | J + J | = √ g π √ ˜ ω (sin θ m ) − F (cid:0) θ m (cid:12)(cid:12) csc θ m (cid:1) = √ g . (51)Combining all these results, we finally obtain, E NR = J + √ g π (cid:16) √ ˜ ω sin θ m E (cid:0) θ m (cid:12)(cid:12) csc θ m (cid:1) + π (cid:17) . (52)Setting, ω = ω = 1 and θ m = π we find the leading order correction to the spectrum, E NR = J + γ √ g ; γ = 0 . γ ) increases from single spin solution( γ = 0 .
25) to multi spin solution by 14%.
We now generalize the above construction by considering NR folded spinning strings in
AdS × S with two equal spins in AdS and one spin along S , t = σ ; η = σ ; ρ = ρ ( σ ) ; ψ = 0 ; θ = θ ( σ ) ; χ = ϕ = ωσ ; φ = νσ (54)where the string is stretched along the radial coordinate ρ of AdS as well as the angulardirection θ of S .The corresponding Lagrangian density turns out to be, L NG = − ω ρ + ν − ρ sin θ + 12 cosh ρ ( ρ (cid:48) + θ (cid:48) ) . (55)The resulting equations of motion are given by, ρ (cid:48)(cid:48) cosh ρ + ( ρ (cid:48) − θ (cid:48) ) cosh ρ sinh ρ +3 ω tanh ρ cosh − ρ + ν cosh − ρ sinh ρ sin θ = 0 (56)and, θ (cid:48)(cid:48) cosh ρ + θ (cid:48) ρ (cid:48) sinh 2 ρ − ν − ρ sin 2 θ = 0 . (57) The above set of equations (56)-(57) are indeed difficult to solve analytically. To simplifythe problem, we consider a configuration where the string soliton is considered to besitting at the equatorial plane ( θ = π ) of S while having a spin along the azimuthaldirection φ . This clearly satisfies (57) and upon substitution into (56) yields, ρ (cid:48)(cid:48) cosh ρ + ρ (cid:48) cosh ρ sinh ρ + 3 ω tanh ρ cosh − ρ + ν cosh − ρ sinh ρ = 0 . (58)We propose the following perturbative solutions for (58), ρ = ρ (0) (1 + ωρ ( ω ) + νρ ( ν ) + · · · ) (59) Like before, we take into account the slow frequency limit for strings which amounts of considering, ωρ m (cid:28) νρ m (cid:28) ρ (0) ( σ ) = ρ m sinh − σ . (60)The leading order equation in ω has a solution of the form, ρ ( ω ) ( σ ) = 2 σ + 3 tan − σ (cid:112) ( σ ) + 1 sinh − σ . (61)Finally, the leading order solution in ν turns out to be, ρ ( ν ) ( σ ) = 2 σ + tan − σ (cid:112) ( σ ) + 1 sinh − σ . (62) Energy of the stringy configuration turns out to be, E NR = √ g π (cid:90) π dσ ρ (cid:48) cosh ρ (cid:39) √ g π (∆ ( ρ m ) + ωρ m Σ ( ω ) ( ρ m ) + νρ m Σ ( ν ) ( ρ m ) + · · · ) (63)where the sub-leading corrections in (63) are given by,Σ ( ν ) ( ρ m ) = (cid:90) π dσ cosh (cid:0) ρ m sinh − σ (cid:1) (( σ ) + 1) / × I ( σ , ρ m ) (64) I ( σ , ρ m ) = (cid:112) ( σ ) + 1 (cid:0) − σ tan − σ (cid:1) cosh (cid:0) ρ m sinh − σ (cid:1) + ρ m (cid:0) ( σ ) + 1 (cid:1) (cid:0) σ + tan − σ (cid:1) sinh (cid:0) ρ m sinh − σ (cid:1) (65)and, Σ ( ω ) ( ρ m ) = (cid:90) π dσ cosh (cid:0) ρ m sinh − σ (cid:1) × I ( σ , ρ m ) (66) I ( σ , ρ m ) = (5 − σ tan − σ ) cosh (cid:0) ρ m sinh − σ (cid:1) (( σ ) + 1) + ρ m (2 σ + 3 tan − σ ) sinh (cid:0) ρ m sinh − σ (cid:1) (( σ ) + 1) / . (67)The total spin of the configuration, on the other hand, is given by S = − √ g π (cid:90) π dσ tanh ρ (cid:39) − √ g π (Λ ( ρ m ) + ωρ m Ψ ( ω ) ( ρ m ) + νρ m Ψ ( ν ) ( ρ m ) + · · · ) (68)where, the sub-leading corrections are given by,Ψ ( ω ) ( ρ m ) = (cid:90) π (2 σ + 3 tan − σ ) tanh (cid:0) ρ m sinh − σ (cid:1) sech (cid:0) ρ m sinh − σ (cid:1)(cid:112) ( σ ) + 1 dσ (69)Ψ ( ν ) ( ρ m ) = (cid:90) π (2 σ + tan − σ ) tanh (cid:0) ρ m sinh − σ (cid:1) sech (cid:0) ρ m sinh − σ (cid:1)(cid:112) ( σ ) + 1 dσ . (70)11inally, the R-charge of the configuration is given by J = √ g π (cid:90) π dσ cosh − ρ (cid:39) √ g π (Φ( ρ m ) − ωρ m Ψ ( ω ) ( ρ m ) − νρ m Ψ ( ν ) ( ρ m ) + · · · ) (71)where, Φ( ρ m ) is the leading order correction to the R-charge that goes like ∼ ρ m similarto that of Λ ( ρ m ) as given in (31).Considering the extended string ( ρ m (cid:29)
1) limit, the total angular momentum of theNR stringy configuration turns out to be, Q = | S + J | ≈ √ g π ( ωρ m Ψ ( ω ) ( ρ m ) + νρ m Ψ ( ν ) ( ρ m ) + · · · ) (72)which finally results in the dispersion relation of the form, E NR ∼ Q + √ g ρ m π ( ˜∆ ( ρ m ) + ω Σ ( ω ) ( ρ m ) + ν Σ ( ν ) ( ρ m ) + · · · ) + O ( √ g /ρ m ) . (73) R × S AdS × S and one dimensional Neumann-Rosochatius like integrable models [30]. We select a specific sub-sector R × S ⊂ AdS × S of the full target space geometry where we consider the string soliton to be sitting at thecentre of AdS and switch on spin along one of the azimuthal directions ( φ ) of S .We choose the string embedding of the following form, t = σ ; η = σ ; ρ = 0 ; θ = π/ θ = θ ( σ ) ; φ i = φ i ( σ α ) (74)and switch off all the remaining coordinates. For simplicity, from now on we set, θ = θ .The resulting sigma model Lagrangian turns out to be, L NG = (cid:18)
12 sin θ − (cid:19) ˙ φ + θ (cid:48) −
18 cos θφ (cid:48) + 18 (cid:0) sin θφ (cid:48) + cos θφ (cid:48) (cid:1) . (75)To proceed further, we redefine coordinates as, (cid:96) = 1 √ θ ; (cid:96) = 1 √ θ ; φ = √ ξ ( σ ) ; φ = − νσ + √ ξ ( σ ) (76)which upon substitution into (75) yields, L D = (cid:96) (cid:48) i + (cid:96) i ξ (cid:48) i − (cid:96) i νδ i − N ( (cid:96) i − /
2) + ∆ S . (77)The first four terms on the R.H.S of (77) together constitute what we define as the non-relativistic analogue of 1D Neumann-Rosochatius like integrable models derived directlyfrom 2D sigma models in the SU (1 , |
3) SMT limit of strings on
AdS × S . Contrary12o its relativistic cousins [31]-[32], the nonrelativistic 1D model (77) is linear in the timederivative/ frequency ( ν ) and is quadratic in space derivatives. Here, N is the Lagrangemultiplier that preserves the constraint condition, (cid:96) i = 1.The last term on the R.H.S. of (77),∆ S = ν − (cid:96) i ξ (cid:48) i δ i (78)is identified as the deformation to the nonrelativistic Neumann-Rosochatius model.The phase space of the 1D model (77) is eventually 6 (= 2 N ) dimensional. Theconfiguration is integrable as there are Q i ( i = 1 ,
2) conserved charges associated with thedynamical phase space. This follows directly from the fact that the system is constrained.The canonical momentum densities are given by,Π (cid:96) i = 2 (cid:96) (cid:48) i (79)Π ξ = 2 (cid:96) ξ (cid:48) = υ (80)Π ξ = 2 (cid:96) ξ (cid:48) (1 − (cid:96) ) = υ . (81)Clearly, there are two constants of motion ( υ i ∼ Q i ) associated with the configuration.The corresponding canonical Hamiltonian density turns out to be, H D = ν I (82)where we identify, I = (cid:96) −
13 + 14 ν (cid:18) υ (cid:96) + υ (cid:96) (1 − (cid:96) ) + ( (cid:96) (cid:96) (cid:48) − (cid:96) (cid:96) (cid:48) ) (cid:19) (83)as nonrelativistic analogue of the Uhlenbeck constant [31]-[32]. We first note down the equations of motion that directly result from (77), (cid:96) (cid:48)(cid:48) − (cid:96) ξ (cid:48) + (cid:96) ( ν + N ) = 0 (84) (cid:96) (cid:48)(cid:48) − (cid:96) ξ (cid:48) (1 − (cid:96) ) + N (cid:96) = 0 . (85)The above set of equations (84)-(85) could in principle be solved considering a fixedradius [31] ansatz namely, (cid:96) i = a i = constant and choosing an ansatz for the windingmodes along the azimuthal direction of S , ξ i ( σ ) = m i σ (86)where m i s are the respective winding numbers.This results in the following set of algebraic equations, m − ν − N = 0 (87) m (1 − a ) − N = 0 . (88)Using (87)-(88), we finally obtain, a = 7 m + m − ν m (89) a = m − m + ν m (90)subjected to the constraint condition, a + a = 1.13 .2 Dispersion relation The energy of the NR stringy configuration is given by, E NR = √ g a ( m + a m ) . (91)On the other hand, the R- charge of the configuration reads as, J = √ g a −
13 ) . (92)Combining (91) and (92) and after some trivial algebra we find, E NR ∼ (cid:18) m m (cid:19) J + √ g (cid:18) m + 23 m − m ( ˜ J + 6 ˜ J ) (cid:19) (93)where, we define ˜ J = J √ g as the effective R- charge in strong coupling limit of SMT. AdS folded spinning strings in
AdS and switch off any dynamics along S .In order to map the corresponding sigma model action into a 1D Neumann-Rosochatiuslike integrable model we choose to work with the following string embedding, t = σ ; η = σ ; ψ ∼ ρ = ρ ( σ ) ; χ = const. ; ϕ = ϕ ( σ α ) . (94)The resulting Lagrangian density turns out to be, L NG (cid:39) − tanh ρ ˙ ϕ + cosh ρρ (cid:48) + 14 sinh ρϕ (cid:48) . (95)We define the following set of coordinates, z = cosh ρ ; z = sinh ρ ; ϕ = ωσ + 2 β ( σ ) (96)and consider the dynamics near the centre of AdS .This finally results in the 1D Lagrangian of the following form, L D ∼ g ab ( z (cid:48) a z (cid:48) b + z a z b β (cid:48) − ωz a z b ) + G ( g ab z a z b + 1) + ∆ AdS (97)where, we introduce the diagonal metric g ab = diag ( − ,
1) as well as the Lagrange multi-plier G which implies the constraint condition [32], g ab z a z b = − . (98)The first two terms (in the parenthesis) on the R.H.S. of (97) are the standardNeumann-Rosochatius piece while the remaining one,∆ AdS = − ω + β (cid:48) (99)serves as an extension to it. 14 .1 Equations of motion The equations of motion could be enumerated as, z (cid:48)(cid:48) a − z a β (cid:48) + ( ω − G ) z a = 0 ; a = 0 , . (100)To obtain solutions, we first set, β ( σ ) = kσ where k is the winding number of thestring along ϕ . The corresponding solutions turn out to be, z = cosh √ pσ ; z = sinh √ pσ ; p = k − ω > ρ ( σ ) = √ p σ . (102) The energy and as well as the spin of the NR configuration are given below, E NR = √ g π √ p (cid:90) ρ m dρ ( p cosh ρ + k sinh ρ )= √ g π √ k − ω (cid:0) − ωρ m + (cid:0) k − ω (cid:1) sinh(2 ρ m ) (cid:1) ≈ √ g π √ k − ω (cid:18) ( k − ω ) ρ m + ρ m (cid:0) k − ω (cid:1)(cid:19) , (103) S ϕ = √ g π √ p (cid:90) ρ m dρ tanh ρ = √ g π √ k − ω ( ρ m − tanh ρ m ) ≈ √ g ρ m π √ k − ω . (104)Using (103) and (104) we finally obtain, E NR ∼ (2 k − ω ) S ϕ + √ g (cid:18) π ( k − ω ) (cid:19) / (cid:18) S ϕ √ g (cid:19) / + · · · (105)where, S ϕ √ g ∼ ρ m (cid:28) We conclude our paper with a brief summary of the analysis and mentioning some of itspossible implications as well. The present paper explores various corners of the SU (1 , | N = 4 SYM in the limit of strong coupling. Usingthe dual semiclassical nonrelativistic stringy counterpart in AdS × S , the present paper15ealizes the spectrum of the theory for various sub-sectors of the full Hilbert space. Wealso show that the SU (1 , |
3) Spin-Matrix theory (SMT) limit of the sigma model in
AdS × S can be mapped into 1D Neumann-Rosochatius like integrable models. Thisfurther allows us to compute the spectrum of the 2D sigma model corresponding to variousspinning string configurations in AdS × S .It would be really nice to reproduce these semiclassical string states (and hence thespectrum) considering a large g ( (cid:29)
1) Spin-Matrix theory limit in the SU (1 , |
3) sectorof N = 4 SYM. Below we outline some possible steps in order to achieve this goal inthe limit N → ∞ . The nonrelativistic spinning strings considered in this paper probevarious subsectors of SMT (with largest possible Hilbert space) that can be obtained asa decoupling limit [1], [3], [6] of N = 4 SYM on R × S . For example, nonrelativisticspinning string states in AdS correspond to the decoupling limit in the sector spannedby the single trace operators of the form O S ∼ tr (Φ Z ( n µ D µ ) S Φ Z ) in N = 4 SYM. Here,Φ Z s are the complex scalars built out of two of the six real scalars in N = 4 SYM, D µ isthe gauge covariant derivative and n µ is the constant null vector [26]. On the other hand,nonrelativistic multispin string states on S can be realised as the decoupling limit of thesector (in N = 4 SYM) spanned by the gauge invariant (single trace) operators of theform, O J ,J ∼ tr (Φ J Z Φ J X )+ (permutations). Here, for example, the decoupling/quantummechanical limit corresponds to setting ( T, ω , ω , Ω , Ω , Ω ) = (0 , , , , ,
0) while keep-ing the ratios ˜ T = T − Ω and ˜ λ = λ − Ω fixed [1]. Here, Ω and Ω are the chemicalpotentials corresponding to the R-charges J and J . The resulting states belong to SU (2) ⊂ SU (1 , |
3) representation of the underlying spin group in SMT [6]. Finally,nonrelativistic spinning string states with two spins along S ⊂ AdS and one spin along S can be realised as a quantum mechanical limit of the sector spanned by the gaugeinvariant operators of the form, O S ,S ,J ∼ tr (Φ Z ( n µ D µ ) S ( n ν D ν ) S Φ Z Φ J − X ) + · · · withclassical dimension ∆ ∼ Q ∼ S + S + J . The quantum mechanical limit for theseoperators corresponds to setting ( T, ω , ω , Ω , Ω , Ω ) = (0 , , , , ,
0) which representsstates in the SU (1 , | ⊂ SU (1 , |
3) representation of the underlying spin group [6].The one loop correction to the respective dilatation operators would essentially serveas the interaction Hamiltonian ( H int ) [6] for the corresponding quantum mechanical the-ory in the near critical region. Considering the planar ( N → ∞ ) limit, the final stepwould be to map this quantum mechanical Hamiltonian ( H int ) into a one dimensionalperiodic spin-chain [1], [4], [6]. In the regime of strong ( g (cid:29)
1) coupling and consider-ing large values of the corresponding Cartan/R- symmetry generators ( Q ), the spectrumassociated with this interaction Hamiltonian ( H int ) may be obtained as an expansion inthe ratio involving both the SMT coupling ( g ) and Q [6] and thereby can be comparedwith nonrelativistic semiclassical string spectrum obtained in this paper. Going back tothe periodic spin chain description (with added impurities ), it may also be worthwhile intrying to diagonalize the interaction Hamiltonian ( H int ) using the powerful techniques ofintegrability [27] and thereby comparing the spectrum on both sides of the duality. Wehope to carry out a detailed analysis on some of these issues in the near future. Acknowledgements :
The author is indebted to the authorities of IIT Roorkee fortheir unconditional support towards researches in basic sciences.16 eferences [1] T. Harmark and M. Orselli, “Quantum mechanical sectors in thermal N=4super Yang-Mills on R x S**3,” Nucl. Phys. B , 117-145 (2006)doi:10.1016/j.nuclphysb.2006.08.022 [arXiv:hep-th/0605234 [hep-th]].[2] T. Harmark and M. Orselli, “Matching the Hagedorn temperature in AdS/CFT,”Phys. Rev. D , 126009 (2006) doi:10.1103/PhysRevD.74.126009 [arXiv:hep-th/0608115 [hep-th]].[3] T. Harmark, K. R. Kristjansson and M. Orselli, “Decoupling limits of N=4 superYang-Mills on R x S**3,” JHEP , 115 (2007) doi:10.1088/1126-6708/2007/09/115[arXiv:0707.1621 [hep-th]].[4] T. Harmark, K. R. Kristjansson and M. Orselli, “Matching gauge theory and stringtheory in a decoupling limit of AdS/CFT,” JHEP , 027 (2009) doi:10.1088/1126-6708/2009/02/027 [arXiv:0806.3370 [hep-th]].[5] T. Harmark, “Interacting Giant Gravitons from Spin Matrix Theory,” Phys. Rev.D , no.6, 066001 (2016) doi:10.1103/PhysRevD.94.066001 [arXiv:1606.06296 [hep-th]].[6] T. Harmark and M. Orselli, “Spin Matrix Theory: A quantum mechanical model ofthe AdS/CFT correspondence,” JHEP , 134 (2014) doi:10.1007/JHEP11(2014)134[arXiv:1409.4417 [hep-th]].[7] T. Harmark and N. Wintergerst, “Nonrelativistic Corners of N = 4 Super-symmetric Yang–Mills Theory,” Phys. Rev. Lett. , no.17, 171602 (2020)doi:10.1103/PhysRevLett.124.171602 [arXiv:1912.05554 [hep-th]].[8] J. M. Maldacena, “The Large N limit of superconformal field theories and super-gravity,” Int. J. Theor. Phys. , 1113-1133 (1999) doi:10.1023/A:1026654312961[arXiv:hep-th/9711200 [hep-th]].[9] E. Witten, “Anti-de Sitter space and holography,” Adv. Theor. Math. Phys. , 253-291 (1998) doi:10.4310/ATMP.1998.v2.n2.a2 [arXiv:hep-th/9802150 [hep-th]].[10] J. Gomis and H. Ooguri, “Nonrelativistic closed string theory,” J. Math. Phys. ,3127-3151 (2001) doi:10.1063/1.1372697 [arXiv:hep-th/0009181 [hep-th]].[11] J. Gomis, J. Gomis and K. Kamimura, “Non-relativistic superstrings: A New solublesector of AdS(5) x S**5,” JHEP , 024 (2005) doi:10.1088/1126-6708/2005/12/024[arXiv:hep-th/0507036 [hep-th]].[12] E. A. Bergshoeff, J. Gomis, J. Rosseel, C. [ ? ]im[ ? ]ek and Z. Yan, “String The-ory and String Newton-Cartan Geometry,” J. Phys. A , no.1, 014001 (2020)doi:10.1088/1751-8121/ab56e9 [arXiv:1907.10668 [hep-th]].[13] T. Harmark, J. Hartong and N. A. Obers, “Nonrelativistic strings and lim-its of the AdS/CFT correspondence,” Phys. Rev. D , no.8, 086019 (2017)doi:10.1103/PhysRevD.96.086019 [arXiv:1705.03535 [hep-th]].1714] 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 , 190 (2018) doi:10.1007/JHEP11(2018)190 [arXiv:1810.05560 [hep-th]].[15] K. T. Grosvenor, J. Hartong, C. Keeler and N. A. Obers, “Homogeneous Nonrela-tivistic Geometries as Coset Spaces,” Class. Quant. Grav. , no.17, 175007 (2018)doi:10.1088/1361-6382/aad0f9 [arXiv:1712.03980 [hep-th]].[16] M. H. Christensen, J. Hartong, N. A. Obers and B. Rollier, “Torsional Newton-Cartan Geometry and Lifshitz Holography,” Phys. Rev. D , 061901 (2014)doi:10.1103/PhysRevD.89.061901 [arXiv:1311.4794 [hep-th]].[17] J. Hartong, E. Kiritsis and N. A. Obers, “Schr¨odinger Invariance from LifshitzIsometries in Holography and Field Theory,” Phys. Rev. D , 066003 (2015)doi:10.1103/PhysRevD.92.066003 [arXiv:1409.1522 [hep-th]].[18] J. Hartong, E. Kiritsis and N. A. Obers, “Lifshitz space–times for Schr¨odingerholography,” Phys. Lett. B , 318-324 (2015) doi:10.1016/j.physletb.2015.05.010[arXiv:1409.1519 [hep-th]].[19] J. Hartong and N. A. Obers, “Hoˇrava-Lifshitz gravity from dynamical Newton-Cartangeometry,” JHEP , 155 (2015) doi:10.1007/JHEP07(2015)155 [arXiv:1504.07461[hep-th]].[20] M. H. Christensen, J. Hartong, N. A. Obers and B. Rollier, “Boundary Stress-EnergyTensor and Newton-Cartan Geometry in Lifshitz Holography,” JHEP , 057 (2014)doi:10.1007/JHEP01(2014)057 [arXiv:1311.6471 [hep-th]].[21] D. Roychowdhury, “Semiclassical dynamics for torsional Newton-Cartan strings,”doi:10.1016/j.nuclphysb.2020.115132 [arXiv:1911.10473 [hep-th]].[22] D. Roychowdhury, “Nonrelativistic pulsating strings,” JHEP , 002 (2019)doi:10.1007/JHEP09(2019)002 [arXiv:1907.00584 [hep-th]].[23] T. Harmark, J. Hartong, L. Menculini, N. A. Obers and G. Oling, “Relating non-relativistic string theories,” JHEP , 071 (2019) doi:10.1007/JHEP11(2019)071[arXiv:1907.01663 [hep-th]].[24] D. Roychowdhury, “Nonrelativistic giant magnons from Newton Cartan strings,”JHEP , 109 (2020) doi:10.1007/JHEP02(2020)109 [arXiv:2001.01061 [hep-th]].[25] S. S. Gubser, I. R. Klebanov and A. M. Polyakov, “A Semiclassical limit of thegauge / string correspondence,” Nucl. Phys. B , 99-114 (2002) doi:10.1016/S0550-3213(02)00373-5 [arXiv:hep-th/0204051 [hep-th]].[26] S. Frolov and A. A. Tseytlin, “Multispin string solutions in AdS(5) x S**5,”Nucl. Phys. B , 77-110 (2003) doi:10.1016/S0550-3213(03)00580-7 [arXiv:hep-th/0304255 [hep-th]].[27] N. Beisert, J. A. Minahan, M. Staudacher and K. Zarembo, “Stringing spins and spin-ning strings,” JHEP , 010 (2003) doi:10.1088/1126-6708/2003/09/010 [arXiv:hep-th/0306139 [hep-th]]. 1828] S. Frolov and A. A. Tseytlin, “Rotating string solutions: AdS / CFTduality in nonsupersymmetric sectors,” Phys. Lett. B , 96-104 (2003)doi:10.1016/j.physletb.2003.07.022 [arXiv:hep-th/0306143 [hep-th]].[29] S. Ryang, “Folded three spin string solutions in AdS(5) x S**5,” JHEP , 053(2004) doi:10.1088/1126-6708/2004/04/053 [arXiv:hep-th/0403180 [hep-th]].[30] O. Babelon and M. Talon, “Separation of variables for the classical and quantumNeumann model,” Nucl. Phys. B , 321 (1992) doi:10.1016/0550-3213(92)90599-7[hep-th/9201035].[31] G. Arutyunov, S. Frolov, J. Russo and A. A. Tseytlin, “Spinning stringsin AdS(5) x S**5 and integrable systems,” Nucl. Phys. B , 3 (2003)doi:10.1016/j.nuclphysb.2003.08.036 [hep-th/0307191].[32] G. Arutyunov, J. Russo and A. A. Tseytlin, “Spinning strings in AdS(5) xS**5: New integrable system relations,” Phys. Rev. D69