Neumann-Rosochatius system for (m,n) string in $AdS_3 \times S^3$ with mixed flux

$(m,n)$ string in $AdS_3\times S^3\times T^4$ with mixed three-form fluxes can be described by an integrable deformation of an one-dimensional Neumann-Rosochatius (NR) system. We study general class of rotating and pulsating solutions in Lagrangian and Hamiltonian formulation. For the rotating string, the explicit solutions can be expressed in terms of elliptic functions. We compute integrals of motion and find out the scaling relation among conserved charges for the particular case of constant radii solutions. Then we study the closed $(m,n)$ string pulsating in $R\times S^3$. We find the string profile and calculate the total energy of such pulsating sting in terms of oscillation number $(\cal{N})$ and angular momentum $(\cal{J})$.


Introduction
Planar integrability on both sides of the famous AdS/CF T correspondence [1][2][3] has been proved to be very promising technique to unravel a deeper understanding in the study of string spectrum in different semisymmetric superspaces [4]. Minahan and Zarembo in [5] first established the matching between the one-loop dilatation operators of the SU (2) sector of N = 4 SYM theory with the Hamiltonian of integrable SO(6) spin chain model [6,7]. Over the last few years, AdS 3 × S 3 × M 4 geometry supported by mixed three form fluxes (both NS-NS and R-R) has been studied as a nice tool in the context of classical integrability in AdS 3 /CF T 2 correspondence [8][9][10][11][12][13][14]. The appearance of integrability as a symmetry in Green-Schwarz action of type IIB string in compactified AdS 3 × S 3 × M 4 background with mixed R-R and NS-NS three form flux has put forth a renewed interest to analyse the AdS 3 /CF T 2 duality in the presence of mixed fluxes by means of well known classical integrable models.
In proving the AdS/CFT duality in large charge limit, varieties of rigidly rotating strings have been studied in different backgrounds to enhance our understanding of the dual states in the gravity side. Among them special cases on sphere consist of giant magnon [15] and spiky string [16] solutions which are dual to elementary excitation with large momentum p and higher twist operators respectively in the field theory side of the duality. Also in [17] an exact correspondence was proposed between string states and some dual field theory operators by considering spinning as well as pulsating string solutions, the later first developed in [18]. Despite being more stable solutions these types of string state has not been explored enough as compared to the rigidly rotating strings.
A general description of the finite gap solutions of classically integrable string sigma model has been demonstrated in [19,20] in terms of solutions of certain integrable models. In this regard, it is worth emphasizing that a generalised ansatz for string coordinates [21][22][23] provides a method to reformulate a large class of string configurations in terms of solutions of very well-known one dimensional integrable Neumann or Neumann-Rosochatius (NR) system. The Neumann integrable model describes harmonic oscillator restricted to move on a sphere whereas NR system is an integrable extension of the former with an additional centrifugal potential barrier of the form 1 r 2 . These mechanical models have been proved to be very effective to deal with the problem of geodesics on ellipsoid or equivariant harmonic maps into sphere [24][25][26][27]. Earlier, in the context of classical integrability, classical giant magnon [21][22][23] and spiky string [28][29][30] solutions on AdS 5 × S 5 was obtained by using NR model approach. Moreover, there are several recent works [31][32][33][34][35][36][37] which reveals that general solutions corresponding to closed rotating and pulsating string in various 10D AdS/S compact spaces with pure or mixed flux may be constructed from systematic analysis of one dimensional Lagrangian of NR system with integrable deformation. Closed circular type solutions for rotating string with constant radii [9] and their finite gap effects [32] has been confirmed to be possible to achieve with this method in hand. Also there are very recent observations [37] on energy-angular momenta relations for pulsating string in terms of oscillation number, which is credibly supported by NR model.
In a seminal paper, J. H. Schwarz [38] had constructed an SL(2, Z) multiplet of string-like solutions in type IIB supergravity starting from the fundamental string solution. It is also known that the equations of motion of type IIB theory are invariant under an SL(2, R) rotation group. This enables one to generate new supergravity solutions in type IIB theory by applying this rotation to known solutions in supergravity. A discrete subgroup SL(2, Z) of this SL(2, R) group has been conjectured later to be the exact symmetry group of the type IIB string theory. Till date, a lot of work has been devoted for constructing various string-like as well as five-brane solutions of type IIB supergravity equations using SL(2, Z) invariance of low energy effective action of this theory [38][39][40][41][42][43][44]. In the near horizon limit, an SL(2, Z) transformed bound state solution of Q 5 NS5-branes and Q 1 F-strings, gives rise to AdS 3 × S 3 background with mixed three form fluxes with integer charges. It has also been recently shown that the SL(2, Z)-transformation and the near horizon limit commute. This allows us to map the (m, n) string action in AdS 3 × S 3 background with mixed three form fluxes to (m , n ) string action in AdS 3 × S 3 background with NS-NS two form flux using the relation where a, b, c and d are the integer entries of SL(2, Z) matrix with ad − bc = 1.
Interesting analysis of this fact has so far been put forth with (m, n) string action as natural probe in various backgrounds [43][44][45][46][47]. In [46], some fascinating observations on giant magnon and single spike solutions for rigidly rotating string along with the analysis of pulsating string as probe (m, n) string in mixed flux background have been performed by mapping it into (m , n ) string in AdS 3 × S 3 background with NSNS two form flux.
Motivated by such analysis, we cast our focus on the manifest SL(2, Z) covariant action of probe (m, n) string in AdS 3 × S 3 × T 4 background considering the presence of both the three form NSNS and RR fluxes using general rotating and pulsating string ansatz in the light of classical integrability of both sides of the correspondence. In this article we wish to construct integrable deformed NR system and utilize the solutions of integrable equations of motion of that system to obtain the relations between conserved charges for both rotating and pulsating (m, n) string. The rest of the paper is organized as follows. In section 2, we present the construction of deformed Lagrangian and Hamiltonian of integrable NR model for closed (m, n)string rotating in R t × S 3 background in the presence of mixed three form fluxes. Also we discuss for corresponding Uhlenbeck integrals of motion for such system. We derive elliptic solutions for rotating (m, n) string which resembles the circular-type string solutions. We further study of conserved charges and energy-angular momenta dispersion relation for constant radii case for the closed rotating string. Section 4 is devoted to the study of (m, n) type pulsating string in mixed flux background from the solutions of the modified NR model. We study the string profile and compute small energy correction to the scaling relation in terms of oscillation number in short string limit. Finally, we conclude our results in section 4. 2 Modified NR model for rotating (m, n) string in AdS 3 × S 3 with mixed flux We start by writing down the background metric and associated NS-NS and R-R fluxes for AdS 3 × S 3 background is the line element of the corresponding background in terms of dimensionless variables. In this section, we wish to study the dynamics of (m, n) string in general background with mixed three form flux . On the other hand, the action of (m, n) string in general background with metric g M N and flux B M N is given by with g M N to be the Einstein frame metric which is invariant under SL(2, Z) transformations. This action was derived in [44] by applying SL(2, Z) covariance of the action of n coincident D1-branes in general background. The vector B is defined as where B (2) and C (2) are NS-NS and R-R two form fluxes respectively. It transforms under SL(2, Z) transformation asB where Λ is given by It is convenient to introduce a complex field τ = χ + ie −φ containing both the axion scalar χ and dilaton φ corresponding to NS-NS and R-R sectors respectively. With this complex field, we introduce a matrix in the following form: which transforms under SL(2, Z) transformation as, Here m, n are the numbers of fundamental string coupled to the NS-NS flux and D1 branes coupled to the R-R flux respectively. Therefore substitution of equation (2.9) reduces the action as It is to be noted that this action is taken in string frame with metric G M N given by To avoid such discrepancy it may be modified by introducing an auxiliary metric h αβ . With such consideration, the action becomes where, where tensions of D1-string and fundamental string, respectively, are with g s being the string coupling. From this action equation of motion for h αβ may be obtained as which gives nothing but tracelessness of the stress-energy tensor and can reproduce the action (2.10) from (2.11) by inserting h αβ =G αβ . Equation of motion for x M can be easily found as where, In what follows, we will use the metric for AdS 3 × S 3 background in terms of the global background coordinates, The accompanying flux components are given by where q and 1 − q 2 =q, say, are the parameters associated to field strengths of NS-NS and R-R fluxes respectively. It satisfies, 0 ≤ q ≤ 1 and q 2 + 1 − q 2 = 1. For q = 0, it is a case of pure RR flux and the worldsheet theory can be described in terms of a Green-Schwarz coset [48][49][50]. On the other hand, for q = 1 it corresponds to pure NS-NS background, where the theory can be described in terms of a class of supersymmetric WZW model. For intermediate value of q the theory has not been completely understood till date.

Lagrangian and Hamiltonian formulation
To specify the geometry of AdS 3 and S 3 it is convenient to consider the embedding coordinates to be Y i 's and W i 's respectively. These are related to the global coordinates as The which precisely define the geometry of AdS 3 and S 3 respectively. As we are interested in the dynamics in R t × S 3 , we take Y 1 = Y 2 = 0 so that Y 3 + iY 0 = e it . Taking r 1 (ξ) = sin θ, r 2 (ξ) = cos θ, Φ 1 (ξ) = φ 1 and Φ 2 (ξ) = φ 2 with ξ = ασ + βτ , the Lagrangian of (m, n) string in R t ×S 3 geometry with mixed three form fluxes becomes where Λ being the Lagrange multipliers. Let us consider the following parametrization where κ, ω a , α and β are constants. For closed strings r a and f a should follow periodic conditions m a being integer winding numbers. Using such parametrization for the string rotating in R t × S 3 , the Lagrangian becomes, where derivatives with respect to ξ is denoted by prime. Equations of motion for f a 's can be calculated as (2.27) where ba = 1 and C a 's are suitable integration constants. Also where we have used (2.12) and (2.13). Putting this expression of f a in the Lagrangian (2.26) we achieve, The Lagrangian contains both r 2 a and 1 r 2 a -terms representing harmonic oscillator type potential and centrifugal potential barrier respectively. Hence it can be identified as the Lagrangian of one dimensional integrable Neumann-Rosochatius system with extra terms due to presence of mixed flux. We can get the equations of motion for r 1 and r 2 as and Putting the expressions of f a 's from (2.27) in equations (2.30) and (2.31) we achieve, and Above two equations can also be obtained from the Eular-Lagrange equations of the following equivalent Lagrangian of an integrable NR system for general coordinates r 1 and r 2 . The Virasoro constraints may be expressed as With the embedding (2.21) and (2.22) and using the ansatz (2.24), the constraints reduce to (α 2 +β 2 )(r 2 1 +r 2 2 )+r 2 1 α 2 f 2 1 + (ω 1 + βf 1 ) 2 +r 2 2 α 2 f 2 2 + (ω 2 + βf 2 ) 2 = κ 2 (2.36) and αβ(r 2 1 + r 2 2 ) + α ω 1 r 2 respectively. The Hamiltonian of such a system can be written as This again takes the form of the Hamiltonian of integrable one dimensional NR model. Hence it is obvious that classically integrable NR model may represent a nice tool to study for probe (m, n) string in the desired background, even in the presence of flux.

Integrals of Motion
Any classically integrable system contains infinite number of conserved quantities, also known as integrals of motion, in involution, i.e., they must Poisson commute each other. The integrability of Neumann-Rosochatius system demands the existence of a set of integrals of motion, named as Uhlenbeck constants, in involution. These were first introduced by K. Uhlenbeck [24]. In the case of closed rotating string in S 3 there are two integrals of motion I 1 and I 2 constrained by the relation I 1 + I 2 = 1. A general form of Uhlenbeck integrals of motion for the case of closed rotating string can be written as which eventually gives for some arbitrary values of constants C a 's. Here we have used the embeddings (2.21), (2.22) and ansatz (2.24) for closed string rotating on S 3 with x a (ξ) = r a e fa(ξ) [21].
To find out the integrals of motion in the deformed background, let us proceed with the methodology described in [32,33] where a term g = g(r 1 , r 2 , Q) representing the deformation due to fluxes is added without affecting the integrable properties of I a such that, (2.41) The function g may be derived by settingĪ a = 0. This will yield, Now, by using the constraints r 2 1 + r 2 2 = 1, r 1 r 1 + r 2 r 2 = 0, r 1 r 1 + r 2 1 + r 2 2 + r 2 r 2 = 0 (2.43) and the equations (2.30), (2.31) and (2.27), we get the integral of motion in the following form: (2.44) It is henceforth clear that in the absence of flux, the deformed constants satisfy the relation 2 a=1Ī a = 1.

Elliptic solutions for r 1 and r 2
In this section we present the solutions for such system for (m, n) string rotating in S 3 with two different angular momenta. In this context we can introduce three parameters on a sphere like ζ 1 , ζ 2 and ζ 3 to express the roots of the equation in ellipsoidal coordinate ζ r 2 With angular frequencies ω 1 < ω 2 , this equation is defined on a sphere while being invariant under r a → λr a [52]. The range of ellipsoidal coordinate is ω 2 1 ≤ ζ ≤ ω 2 2 for which ζ covers 1 4 th of a sphere corresponding to r i ≥ 0 and the whole sphere may be thought of as a covering of the domain of ζ having branches along its boundary. If we enter into the equations of motion we may get a set of second order differential equations for ζ. But for the sake of simplicity, we inject this ellipsoidal coordinates ζ into the Uhlenbeck integrals of motion [22,23] and can have a first order differential equation for ζ. Differentiating equation (2.45) we find . (2.46) The integral of motion (2.44) in terms of the ellipsoidal coordinate takes the following formĪ (2.47) Solving for ζ 2 from this expression of deformed Uhlenbeck constant we explore that Here P 3 (ζ) is evidently a third order polynomial defining an elliptic curve s 2 + P 3 (ζ) = 0. Now changing the variables as and substituting for η in equation (2.48) we get where k = ζ 3 −ζ 2 ζ 3 −ζ 1 is the elliptic modulus ξ 0 is the integration constant which may be set to zero by a rotation. This gives the expression for r 1 from equation (2.45) as and similarly the expression for r 2 is The range of elliptic modulus k should be 0 < k < 1 which eventually needs ζ 1 < ζ 2 < ζ 3 . Also we can achieve circular type solution for ω 2 1 ≤ ζ 2,3 ≤ ω 2 2 in the codomain of equations (2.51) and (2.52) between 0 and 1 provided that no such restriction is needed for ζ 1 .

Conserved charges and dispersion relation
In this section we are interested in finding out the scaling relation among various conserved charges, The energy and angular momenta for this system are expressed as , with a = 1, 2 Therefore using the Lagrangian (2.23), conserved quantities may be expressed as

Constant radii solutions
We are interested in deriving the constant radii solutions for the conserved energy and momenta which may be obtained by taking the limit ζ 2 → ζ 3 in the equations (2.30) and (2.31). These type of solutions have been constructed by using Neumann-Rosochatius integrable system in [32] for circular strings. With this limit, the radial coordinates become constant such as r 1 = where the integration constants f 0a can be chosen to be zero through a rotation andm a = 1 α 2 −β 2 Ca a 2 a + βω a + Qα 2 a 2 2 ω b a 2 a ba are assumed to be the constant integer windings of the string satisfying the closed string periodicity conditions. Also the Virasoro constraints (2.36), (2.37) and currents (2.54b), (2.54c) yield , a 2 2 =m , (2.57) These above relations along with the conserved quantities yield from (2.36), where we have used T (m,n) = 2πτ (m,n) L 2 and w = ω 1 +βm 1 ω 2 +βm 2 . To derive the dispersion relation it would be convenient to find out w in terms of windingsm a and angular momenta J a . To do this let us write the reduced equations of motion for constant radii solutions asm (2.60) Now adding equations (2.54b) and (2.54c), subtracting equation (2.60) from (2.59) and solving the resulting system of equations we get the relation where ω 1 is expressed as where J = J 1 + J 2 as the total angular momentum. Eliminating ω 1 from these two relations we are left with a quartic equation of w given by (2.63) Instead of solving this equation explicitly, it is convenient to get an approximate solution as a power series expansion in large J T (m,n) which is given by where we have neglected the other higher order terms in the series expansion. Substituting equation (2.64) in equation (2.58) we get a relation between energy and angular momenta as follows where we have considered α = 1. For two equal angular momentum J 1 = J 2 (which setsm 1 = −m 2 =m) , one gets where T (m,n) = 2πτ (m,n) L 2 . For large J, the above expression reduces to E ≈ J. Such linear dependence of E on J stems for the identification of the string state with the gauge theory operator corresponding to the maximal-spin state of XXX spin chain.

NR system for pulsating (m,n) strings in AdS × S with mixed flux
In this section, we study the circular closed string pulsating in R t × S 3 by using a deformed integrable one dimensional Neumann-Rosochatius model. We use the following embedding [36] Y 3 + iY 0 = cosh ρe it = z 0 (τ )e ih 0 (τ ) , (3.1a) These will follow the relations between global and local coordinates as Winding numbersm a 's are assumed along the σ direction only to make the timedirection single-valued. With this embedding, the Lagrangian can now be expressed as, where the derivative with respect to τ is denoted by dots. Λ,Λ are suitable Lagrange multipliers.

Lagrangian and Hamiltonian formulation
Euler-Lagrange equations of motion for z 0 and f a 's can be derived from the Lagrangian (3.3) asz Putting the expressions forḟ 1 andḟ 2 in terms of r 1 and r 2 in the Lagrangian (3.3) we achieve, We can get the equations of motion for r 1 and r 2 as and These three equations of motion for z 0 , r 1 and r 2 can also be obtained from the following Lagrangian (3.8) Therefore the Hamiltonian of the system is From these two expressions it is obvious that the forms of both the Lagrangian and Hamiltonian are in agreement with those of the one-dimensional Neumann-Rosochatius system, only with extra terms due to the presence of mixed flux in the background. The Uhlenbeck integrals of motion may be obtained in a similar way as used in the case of the spinning string ansatz. These yield the expressions as (3.10) The two Virasoro constraints G τ τ + G σσ = 0 and G τ σ = G στ = 0 can be calculated asṙ 2 1 +ṙ 2 2 +ḟ 2 1 r 2 1 +ḟ 2 2 r 2 2 +m 2 1 r 2 1 +m 2 2 r 2 2 =ż 2 0 + z 2 0ḣ 2 0 , (3.11a) m 1 r 2 1ḟ 1 +m 2 r 2ḟ2 = 0 (3.11b) respectively.

Solutions for r 1 and r 2 and string profile
Here we use the same procedure for finding the pulsating solutions to this integrable system by choosing ellipsoidal coordinates keeping the conditions as mentioned in section 3.3 intact. In case of pulsating string the ellipsoidal coordinates are taken to be the functions of τ only and the roots of the equation . (3.13) and also gives r 2 1 and r 2 2 as (3.14) Solving equation (3.10) forζ 2 we can get a similar solution containing 3rd order polynomial exactly like the case for rotating string and it is given aṡ Using the same change of variables as given in equation (2.49) and with η taken to be a function of τ only we can get We may choose integration constant τ 0 and the elliptic modulus k = ζ 3 −ζ 2 ζ 3 −ζ 1 . From equation (3.12) we get the expressions for r 2 1 and r 2 2 as and To restrict the elliptic modulus k within the fundamental domain, i.e, 0 < k < 1, we should choose the order of the ζ i 's as ζ 1 < ζ 2 < ζ 3 . Similar to the case for rotating string, here also ζ is assumed to cover 1 4 th of the sphere with r 2 i ≥ 0. This consequently binds the range of ζ asm 2 1 ≤ ζ ≤m 2 2 .

Conserved Charges and Dispersion Relation
The conserved quantities, in the case of pulsating string may be found from the target space Lagrangian in the given background, as follows = τ (m,n) L 2 r 2 aḟ a + Qr 2 2 m b ab .
(3.19) Here, by taking z 0 = 1 we getż 2 0 + z 2 . We may express the Virasoro constraint (3.11a) as a quartic equation of r 1 by substituting forḣ 0 anḋ f a in terms of conserved energy E = E τ (m,n)L 2 and angular momenta J a = Ja τ (m,n)L 2 . To make it simple, let us consider the case J 1 = J 2 andm 1 = −m 2 . With the above, we get It is obvious from the above equation that the expressionṙ 2 1 assumes infinite values for both r 1 → 0 and r 1 → ∞ and oscillates in between. Therefore it must have some minimum value which can be derived to be [E 2 −m 2 1 (1 + 2Q) + 2Qm 1 J 1 ] at This yields the quantum oscillation number [53] as where a ± are the roots of the biquadratic function under the square root. Taking partial derivative of the above equation with respect to m 1 we get . (3.22) Expressing the integrals in terms of standard elliptic integrals we are left with where we have used = a − a + . The standard expansions of elliptic integrals of first and second kinds are respectively  where we have taken Q = 1 and considered the limit form 1 → 1. With such limit in hand, small angular momenta, i.e., J 1 → 0 yields from equation(3.26) This result resembles the energy-oscillation number relation with small energy limit for fundamental string pulsating in background with pure NSNS flux [37] which can be achieved by assuming m = 1, n = 0 and q = 1 in equation (2.28). Also it matches with the energy and oscillation number expansion upto its leading order term in short string limit in [54]for the string pulsating in one plane.

Conclusion
We have investigated, in this paper, the integrable features of (m, n) string in AdS 3 × S 3 background in the presence of mixed flux. We have computed the La-grangian, Hamiltonian and integrals of motion and shown them similar to those of integrable deformed NR models. First, we have derived scaling relations among various conserved charges for closed circular strings of constant radii. We have also elucidated the periodic circular-type string profile for both the cases of closed strings rotating and pulsating in R t × S 3 as solutions of this deformed NR integrable model in the presence of mixed three-form flux. For the pulsating string, we have computed the small energy correction to the scaling relation in terms of oscillation number in short string limit. It will be interesting to construct an NR model for rigidly rotating and pulsating string in AdS 3 × S 3 × S 3 × S 1 background with various mixed fluxes.
In connection with the integrable model, it will be quite interesting to establish relation between the hyperbolic Calogero-Sutherland integrable model to some string sigma model in AdS background with flux and check whether the integrability of AdS 3 /CF T 2 duality remains intact with the solutions of those integrable models. We wish to return to these problems in future.