Spinning strings in AdS_5 x S^5: new integrable system relations

A general class of rotating closed string solutions in AdS_5 x S^5 is shown to be described by a Neumann-Rosochatius one-dimensional integrable system. The latter represents an oscillator on a sphere or a hyperboloid with an additional ``centrifugal'' potential. We expect that the reduction of the AdS_5 x S^5 sigma model to the Neumann-Rosochatius system should have further generalizations and should be useful for uncovering new relations between integrable structures on the two sides of the AdS/CFT duality. We find, in particular, new circular rotating string solutions with two AdS_5 and three S^5 spins. As in other recently discussed examples, the leading large-spin correction to the classical energy turns out to be proportional to the square of the string tension or the 't Hooft coupling \lambda, suggesting that it can be matched onto the one-loop anomalous dimensions of the corresponding ``long'' operators on the SYM side of the AdS/CFT duality.


Introduction and summary
Integrability of the spin chain Hamiltonian representing the planar one-loop dilatation operator of N = 4 Super Yang-Mills theory [1]- [3] has recently made possible, following a proposal in [4,5], a number of remarkable and striking tests of AdS/CFT duality [6]- [11]. This generalizes the near-BPS correspondence of [12] to non-BPS cases.
The AdS/CFT correspondence predicts that the energy of a given physical string state (in global AdS 5 coordinates) should match the scaling dimension of the corresponding operator in gauge theory. While the full energy spectrum of the quantum string in AdS 5 × S 5 is hard to determine, some of its parts can be probed by considering the semiclassical string configurations [13,14]. In certain cases with large quantum numbers (like angular momenta J i in S 5 ), one finds that the energy of the string solution is given by its classical expression, i.e. quantum sigma model corrections appear to be suppressed [5].
On the gauge theory side, the (one-loop) scaling dimensions of gauge-invariant composite operators can be found by solving the eigenvalue problem for the Hamiltonian of an associated spin chain. This is achieved by means of algebraic Bethe ansatz techniques. In general, the Bethe ansatz leads to a complicated system of algebraic equations. However, in the thermodynamic limit (of large quantum numbers or "long" operators) the algebraic equations turn into integral ones and with some natural assumptions about the density distribution of Bethe roots the explicit solutions can be found. Remarkably, the Bethe solutions obtained in the thermodynamic limit turn out to be related to semiclassical string configurations in a precise way.
In general, one can classify strings moving on S 5 with three "R-charges" (SO(6) spins) defining the highest weight state (J 1 , J 2 , J 3 ) of an SO (6) representation. For a simpler case of two non-vanishing spins (J 1 , J 2 ) the string evolution equations are solved in terms of elliptic functions; the corresponding string configurations can have folded [7] or circular [4,8] profiles, giving rise to two different expressions for the space-time energy. On the gauge theory side, the relevant Bethe solutions and the associated scaling dimensions have been found in [6,9], and shown to agree with their string counterparts for both folded [7,6,9] and circular [6,8,9] type configurations.
Other surprising examples of a perfect agreement between string energies and scaling dimensions of gauge theory operators include [11] a simple circular string solution with three spins [4] and a pulsating string solution [15].
Even more remarkably, in the recent work [10] the entire Bethe resolvent (corresponding either to the circular or to the folded string type thermodynamic density distributions) was reproduced from the classical string sigma model. This agreement goes beyond comparing just the string energies with the scaling dimensions: it involves matching the infinite towers of commuting conserved charges on the gauge and string sides of the AdS/CFT correspondence. In fact, the Bethe resolvent is nothing else but a generating function of local conserved commuting charges in string theory properly restricted to the leading (O(λ) or "one-loop") level! The matching of higher local commuting string charges [10] 1 and recent advances in study of integrability of the dilatation operator at higher loops in N = 4 Super Yang-Mills theory [3,28] (and in its S 3 reduced matrix model version [29]) provide strong support that the same integrable structure should be underlying the two sides of the duality.
Still, our understanding of the gauge/string duality, even in the "semiclassical" (large quantum number) sector of states, is far from complete. More detailed analysis of different physical configurations in both gauge and string theories is required to elucidate how the duality works. While recent papers [9,10,11,25] shed some light on how the integrability of the AdS 5 × S 5 string theory is related to that of the planar SYM theory, many details are missing.
In view of the general problem of establishing correspondence between various integrable subsectors of string and gauge theories it is of interest to obtain a systematic picture of reductions of the two-dimensional integrable O(4,2)×O(6) sigma model describing propagation of the classical string in the AdS 5 × S 5 space-time to various one-dimensional integrable models. In [8] we have shown that for a natural rotating string ansatz the AdS 5 × S 5 string sigma model reduces to an integrable Neumann model [30] describing an oscillator on a 2-sphere.
The aim of the present paper is to make further progress in this direction. We will consider a more general integrable subsector in string theory which arises from a rotating string ansatz extending the one in [8]. In this case the 2-d sigma model reduces to the Neumann-Rosochatius (NR) [31] integrable system describing a particle on a sphere in the i (w 2 i r 2 i + v 2 i r −2 i ) potential (in the previous case [8] we had v i = 0). While, as in [8], the general solutions of this system are given by theta-functions on a genus 2 hyperelliptic curve its new feature is the existence of a very simple new class of solutions corresponding to circular strings with constant radii r i (σ) =const. These solutions generalize the ones of [4] (which had two equal spins) to the case when all 2+3 AdS 5 × S 5 spins may be different. The corresponding energy has a very simple dependence on the spins and winding numbers. Understanding its SYM scaling dimension counterpart should help, in particular, to clarify the issue of how the winding numbers of circular strings are reflected in the Bethe root distributions (cf. [6,8,11]).
Let us now summarize the contents of the paper. In section 2.1 we shall present the generalized rotating string ansatz for a closed string fixed at the origin of AdS 5 and rotating in 3 orthogonal planes in S 5 and explain the reduction of the O(6) invariant sigma model to the NR system for the 3 radial directions of the string. In section 2.2 we will list the corresponding integrals of motion and the Virasoro constraints allowing one to express the AdS 5 energy as a function of the three S 5 spins. In section 2.3 we shall mention that a "2d-dual" version of the rotating string ansatz (with roles of τ and σ interchanged) describes a general pulsating string solution with radii oscillating in time which is thus also described by an NR integrable model (some special cases of pulsating solutions were previously discussed in [21,13,15,32,11]). In section 2.4 we shall clarify how the integrability of the NR system follows from its relation to the integrable O(6) sigma model by deriving its Lax representation. We shall also explain how higher commuting charges can be computed from the sigma-model monodromy function.
In section 3 we shall study a very simple special class of NR solutions on S 5 which has a similarity with rotating string solutions in flat space and generalizes the circular 2-spin and 3-spin rotating string solutions in [4]. As will be shown in section 3.2, the corresponding energy has a regular large-spin expansion in λ J 2 . In section 3.3 we shall find the spectrum of quadratic fluctuations near these circular solutions (extending and simplifying the discussion in [5] for the special solutions of [4]). We shall determine the stability conditions and mention some straightforward applications.
In section 4 we shall study more general solutions of the NR system with a nontrivial dependence on the world-sheet coordinate σ. We shall consider, in particular, a two-spin solution which is expressed in terms of the elliptic functions. The resulting system of equations relating energy and two spins turns out to be more involved that in the previously discussed elliptic (sine-Gordon) limit of the Neumann model [7,8,9], but we expect that it might be possible to directly match an appropriate "one-loop" limit of this system onto the corresponding Bethe ansatz equations on the SYM side (as was done in the Neumann model case in [6,9]).
Finally, in section 5 we shall generalize the discussion of sections 2 and 3 to the case when the string can rotate in both AdS 5 and S 5 . Here we get a combination of the two NR systems (an AdS 2 and S 2 one) coupled by the Virasoro constraints. We again consider the simplest solution with constant radii parametrized by 2+3 spins (S a , J i ) and 2+3 winding numbers. If the string rotates only in AdS 5 the corresponding energy does not have a regular large-spin expansion (section 5.1), but it does if there is at least one large spin in S 5 (section 5.2). For example, the simplest (S, J) string solution which is a circle in both AdS 5 and S 5 is stable, and it should be possible to match the leading large J correction to its energy with a particular anomalous dimension on the SYM side by identifying the corresponding distribution of Bethe roots in the associated XXX −1/2 spin chain [2] (as was done for other folded and circular (S, J) string solutions in [9]). Here we shall generalize the rotation ansatz in [8] which allowed us to reduce the classical string sigma-model equations to those of a 1-d integrable model. That will lead to new interesting simple classes of rotating string solutions. Let us consider the bosonic part of the classical closed string propagating in the AdS 5 × S 5 space-time. The world-sheet action in the conformal gauge is It is convenient to represent (2.1) as an action for the O(6)×SO(4,2) sigma-model (we follow the notation of [4]) Here X M , M = 1, . . . , 6 and Y M , M = 0, . . . , 5 are the the embedding coordinates of R 6 with the Euclidean metric in L S and with η M N = (−1, +1, +1, +1, +1, −1) in L AdS respectively. Λ andΛ are the Lagrange multipliers. The action (2.2) is to be supplemented with the usual conformal gauge constraints. The embedding coordinates of AdS 5 × S 5 can be parametrized in terms of angles of AdS 5 and S 5 as in [4,8] X 1 + iX 2 = sin γ cos ψ e iϕ 1 , X 3 + iX 4 = sin γ sin ψ e iϕ 2 , X 5 + iX 6 = cos γ e iϕ 3 , (2.6) In this section we will be discussing the case when the string is located at the center of AdS 5 and rotating in S 5 , i.e. is trivially embedded in AdS 5 as Y 5 + iY 0 = e it , with the global time of AdS 5 being t = κτ and with Y 1 , ..., Y 4 = 0.
The S 5 metric has three commuting translational isometries in ϕ i in (2.5) which give rise to three global commuting integrals of motion (spins) J i . Since we are interested in a periodic motion with J i = 0 it is natural to choose the following ansatz for X M : (2.7) In contrast to our earlier work [8] here we shall not assume that z i are real, i.e. in general z k = r k (σ) e iα k (σ) , k = 1, 2, 3 . (2.8) In order to find the relevant closed string solutions we need also to impose the periodicity conditions on X M or z i : Thus r k are real periodic functions of σ, while real phases α k are periodic only up to 2πm k shift. Comparing (2.7) to (2.5) we conclude that for this general "complex" ansatz the angles ϕ i depend on both τ and σ, (2.10) The integers m i that will label different solutions thus play the role of "winding numbers" in the linear isometry directions ϕ i . As a consequence of X 2 M = 1, r k must lie on a two-sphere: The space-time energy E of the string (related to the generator of a compact SO(2) "05" subgroup of SO(4,2)) here is simply The spins J 1 = J 12 , J 2 = J 34 , J 3 = J 56 forming a Cartan subalgebra of SO(6) are and thus satisfy (2.14) As discussed in [4], to have a consistent semiclassical string state interpretation of these configurations one should look for solutions for which all other components of the SO(6) angular momentum tensor J M N vanish. This is automatically the case if all w i are different [8], but are to be checked in other cases. The Virasoro constraints that need to be imposed on a sigma model solution of (2.3) are (dot and prime are derivatives over τ and σ)

Integrals of motion and constraints
In general, starting with X i (τ, σ) = r i (τ, σ)e iϕ i (τ,σ) (2.17) we get from (2.3) the Lagrangian One can easily check that the ansatz is indeed consistent with the equations of motion. Substituting the ansatz (2.19) or (2.7) into the SO(6) Lagrangian (2.3) we get the following effective 1-d "mechanical" system for a particle on a 5-d sphere (we change the sign of L since now σ plays the role of 1-d time) If we set z k = x k + ix k+3 , this is recognized as a special case of the standard integrable n = 6 Neumann model (harmonic oscillator on a 5-sphere) where three of the six frequencies are equal to the other three. This relation implies integrability of (2.20) model, i.e. determines integrals of motion. Equivalently, in the "planar" coordinates (2.8) we get from (2.18) Equations for the angles α i can be integrated once where v i are three integrals of motion. Substituting α ′ i back into the action we get the following effective Lagrangian for the 3 real radial coordinates When the new integration constants v i vanish, i.e. α i are constant, we go back to the previously studied [8] example of the n = 3 Neumann model. For non-zero v i the Lagrangian (2.23) describes the so called Neumann-Rosochatius (NR) integrable system (see, e.g., [31]). Its integrability follows already from the fact that it is a special case of the 6-dimensional Neumann system.
Finding the integrals of the "radial" system (2.23) is straightforward using the relation to the Neumann model: the n = 6 Neumann system with coordinates x M has, in general, the following six integrals of motion: However, in our case there are equalities between frequencies (w 1 = w 4 , w 2 = w 5 , w 3 = w 6 ) so one should be careful to avoid singularities. The integrals of the Neumann-Rosochatius model are obtained as the following combinations in which singular terms cancel. Explicitly, we find (using (2.22)) This gives us two independent integrals of motion (which we shall denote b a ) in addition to the three other integrals (v i ) we found already. The constraints (2.15),(2.16) can be written as : As a consequence of (2.27) only two of the three integrals of motion v i are independent of w i .
As discussed in [8], the periodicity condition in (2.9) on r i implies that the integrals of motion b a can be traded for two integers n a labeling different types of solutions. Imposing the periodicity condition in (2.9) on α i gives, in view of (2.22), the following constraint: v It implies that v i should be expressible in terms of the integers m i , frequencies w i and the "radial" integrals b a or n a . 2 The moduli space of solutions will thus be parametrized by (w 1 , w 2 , w 3 ; n 1 , n 2 ; m 1 , m 2 , m 3 ). The constraint (2.27) will give one relation between these 3+2+3 parameters. As a consequence, trading w i for the angular momenta, the energy of the solutions as determined by (2.12),(2.26) will be a function of the SO(6) spins and the "topological" numbers n a and m i The constraint (2.27) will provide one additional relation between J i and n a , m i .
In the following sections of this paper we shall consider several special solutions of the above system (2.21). We shall start in section 3 with a discussion of the simplest possible solution with constant r i (for which n a = 0) and which represent an interesting new class of circular 3-spin solutions generalizing the circular solution of [4].

"2d-dual" NR system for pulsating solutions
It is of interest to consider a "2d-dual" version of the rotation ansatz (2.7),(2.8) where τ and σ are interchanged (but still keeping the AdS 5 time as t = κτ ), i.e.
In this case the radial directions depend on τ instead of σ and the "frequencies" m i must take integer values in order to satisfy the closed string periodicity condition. In general, in order to have the zero non-Cartan components of the O(6) angular momentum tensor one is to assume that m i = m j . This ansatz describes an "oscillating" or "pulsating" S 5 string configurations, special cases of which (with motion in both AdS 5 and S 5 ) were discussed previously in [21,13,15,32,11].
Since the sigma model Lagrangian (2.3) is formally invariant under σ ↔ τ , the resulting 1-d effective Lagrangian will have essentially the same form as (2.20),(2.21) (here we do not invert the sign of the Lagrangian) Solving forα i as in (2.22) we get r 2 iα i = J i =const, where the counterparts of the integration constants v i are, in fact, the angular momenta in (2.13). Then we end up with the following analogue of (2.23) Thus pulsating solutions (carrying also 3 spins J i ) are again described by a special Neumann-Rosochatius integrable system. Since (the S 5 part of) the corresponding conformal gauge constraints are also τ ↔ σ symmetric, they take form similar to (2.15),(2.16) or (2.26),(2.27) One may look for periodic solutions of the above NR system (2.32) subject to the constraint (2.33), i.e. having finite 1-d energy (equal to 1 2 κ 2 ). In the simplest ("elliptic") case reducing to a sine-Gordon type system we may follow [33,21,15] and introduce, as for any periodic solitonic solution, an oscillation "level number" N. This may be achieved by considering a semiclassical (WKB) quantization of the action (2.32).
Here we shall not go into detailed study of the resulting pulsating string solutions. Let us only mention that a special r i =const solution of the above system (when, in fact, there is no oscillation of the radii) is essentially the same as the special circular solution with r i =const of the system (2.23) discussed below in section 3.
In the case of the S 5 pulsating solution in [15,11] the expansion of the energy at large level N ≫ 1 appears to be regular in λ N 2 (this is not the case for pulsating string in AdS 5 [15]) and, indeed, the leading λ N 2 term in E can then be matched onto the SYM anomalous dimensions as was shown in [6,11].

Lax representation for the NR system induced from the O(6) sigma model
Having in mind further generalizations, it is useful to understand how the integrability of the NR system (e.g., the Lax representation) follows from the fact that this system is embedded into a much more general integrable [16] O(6) sigma model. Here we will clarify this issue and point out some related open problems. We start with describing the zero-curvature representation for the O(6) sigmamodel in terms 4 × 4 matrices. Let X i be the 3 complex embedding fields (2.7) of the O(6) model. Let us introduce the following skew-symmetric matrix S: (2.35) The matrix S is also unitary, SS † = 1, provided X iXi = 1. Let us also introduce the su(4)-valued current A with components This current can be used to construct the following matrices U and V [16]: Here ℓ is a spectral parameter, and by construction U and V have simple poles at ℓ = ±1. They obey the zero-curvature condition which is a crucial device for demonstrating the integrability of the sigma models. Quite generally, one can associate to eq.(2.38) the transition matrix T(σ, ℓ) (see, e.g., [34]) defined through the path-ordered exponent: 39) and show that the trace of the monodromy matrix (the parallel transport along the period of the zero-curvature connection) generates (when expanded as Q = ∞ n=0 Q n ℓ n ) an infinite tower of commuting integrals of motion. 3 Consider now the generalized rotation (or "Neumann") ansatz for the sigma model variables X i in (2.7), i.e.
Remarkably, the current (2.36) evaluated on X i of the form (2.41) admits the following factorization while the matrices A τ and A σ are independent of τ and given by As the consequence, one finds where U and V depend only on σ. The zero-curvature condition (2.38) reduces to Next, we note that the diagonal matrix Q † ∂ τ Q is σ-independent and, therefore, one can introduce the following L and M-operators: which furnish the Lax representation for the NR system: This is a new Lax representation for the NR system, the previously known examples include the formulation of the Lax equations in terms of 3 × 3 [36] or 2 × 2 [37] matrices. Thus, the O(6) sigma model indices the Lax pair for the NR system in terms of traceless antihermitian 4 × 4 matrices. An interesting open problem is to construct the classical r-matrix corresponding to the Lax system (2.45), (2.46). As was discussed in the previous subsection, the NR system has the (σ-independent) integrals precisely in number which is required for its Liouville integrability. Regarding now σ as a (periodic) time variable, the integrals of motion of the NR system can be constructed, e.g., as F n = Tr L n . However, being embedded into the more general two-dimensional integrable system it inherits an infinite number of conserved (i.e. τ -independent!) integrals of motion. One possible way to exhibit this infinite commuting family is to compute the monodromy (2.40) for the Neumann connection U(σ, ℓ). In general, this is a difficult problem, but it can be simplified by considering the special (simplest) solutions of the NR system. A significant simplification of the Lax pair occurs if we restrict ourselves to the two-spin solutions, which are obtained by setting X 3 = 0. In this case we have effectively the SO(4) sigma model that is isomorphic to two copies of SU(2) models. Indeed, one can show that by a similarity transformation the matrices A τ and A σ can be brought to the form which exhibits factorization into two SU(2) sectors. It is easy to see that the NR evolution equations arise already from a single SU(2) sector, e.g., from the upper left conner of the Lax matrices. Schematically, the corresponding L-operator reads as Let us recall that the Neumann model admits two different kinds of two-spin solutions corresponding to string configurations of the folded or circular type respectively [8]. For instance, the two-spin circular type solution can be written in terms of the standard Jacobi elliptic functions as follows (z 3 = 0) where w 2 12 = w 2 1 − w 2 2 is related to the elliptic modulus t through the closed string periodicity condition. 4 The modulus t is related to the spins J 1 and J 2 by a transcendental equation (see [8] for details). On this particular solution the matrices U and V projected on the first SU(2) sector are −aℓ dn aσ − i(w 1 + w 2 ) sn aσ cn aσ aℓ dn aσ − i(w 1 + w 2 ) sn aσ cn aσ −i(w 1 sn 2 aσ − w 2 cn 2 aσ) .
Using these matrices and applying the (recurrent) abelianization procedure of Zakharov and Shabat [38] one can compute the monodromy function Q(ℓ) in (2.40) and, as a consequence, the corresponding higher commuting charges.
In a recent work [10] the higher commuting (local) charges were obtained for both the folded and the circular two-spin solutions of the Neumann model and linked with those of the one-loop planar N = 4 SYM theory. The approach of [10] was based on finding the form of Bäcklund transformations, which also provides a way of generating the commuting conserved charges (see, e.g., [18]). It would be of interest to understand better a relation between the Bäcklund transformations and the monodromy approach in our stringy context.

Special circular solutions: constant Λ case
A very simple special class of solutions of the system (2.20) or (2.21) which has a similarity with rotating string solutions in flat space and generalizes the circular rotating string solutions in [4,5] has the property that the Lagrange multiplier is constant, i.e. Λ = const.

Constant radii solution
Let us start with the Lagrangian (2.20) in terms of the complex coordinates z i . Then the equations of motion are Eq. (3.1) can be easily integrated if one assumes that Λ = const, where a i , b i are complex coefficients. The periodicity condition z i (σ + 2π) = z i (σ) implies that m i must be integer. To satisfy the constancy of Λ in (3.2) we need to impose In addition, we need to It may seem that one may get a new solution if two of the windings m i are equal while the third is zero, i.e. (this is, in fact, the circular solution of [4]) if z 1 = a cos mσ , z 2 = a sin mσ , but it can be transformed back into the form (3.6) by a global SU(2) rotation.
It is useful also to rederive the solution (3.6) in slightly different way using real coordinates r i , α i , i.e. starting with (2.23), (2.22). The potential w i r 2 i + v 2 i r 2 i in (2.23) has a minimum, and that suggests that r i =const may be a solution. That needs to be checked since r i are constrained to be on S 2 . The equations of motion that follow from (2.23) are They indeed have a solution if where ν is an arbitrary constant (which may be positive or negative). Then it follows that the Lagrange multiplier in (2.23) is thus constant on this solution As a result, we obtain an interesting 3-spin generalization of the circular string solution found in [4] (where two out of three spins where equal). Eq. (3.10) implies i.e.
where m i must be integers to satisfy the periodicity condition (2.9) and α 0i can be set to zero by SO(2) rotations. Then 14) The constraints (2.15),(2.16) or (2.26),(2.27) give or, equivalently, in terms of the energy and spins (cf. (2.12),(2.13),(2.14)) We shall assume for definiteness that all w i and thus all J i are non-negative. Then (3.18) implies that one of the three m i 's must have opposite sign to the other two.
One can check directly that the only non-vanishing components of the SO(6) angular momentum tensor J M N = √ λ 2π 0 dσ 2π (X MẊN − X NẊM ) on this solution are indeed the Cartan ones J 1 = J 12 , J 2 = J 34 , J 3 = J 56 .
Since our aim is to express E in terms of J i and m i as in (2.29) the strategy is then to first solve the condition (3.17) in terms of ν 2 , determining it as a function of J i and m i and then substitute the result into (3.16). The condition (3.18) may then be imposed at the very end.
Let us first consider the special case of ν 2 = 0 (or Λ = 0) which corresponds to a flat-space solution which can be embedded into S 5 by choosing the free radial parameters of a circular string to satisfy the condition 3 i=1 a 2 i = 1. As follows from (3.14) for ν 2 = 0 we find that all frequencies must be integer w i = |m i |, e.g., so that the solution is a combination of the left and right moving waves (here we use complex combinations of coordinates in (2.7)) 5 In the case of (3.19) we get from (3.16)-(3.18) |m i | = 1) and thus E 2 = 4|m 1 |J 1 . Clearly, the energy of this "flat" solution does not have a regular expansion in 1 J 2 (cf. [4]) and thus it cannot be directly compared to some anomalous dimension on the SYM side.

Energy as function of the spins
Now let us turn to the genuinely "curved" (ν = 0) solutions which will have indeed a regular expansion of the energy for large spins as was in the case of the circular solution of [4].
In the 3-spin case one is first to solve the equation (3.17) to determine ν. The solution of this algebraic equation cannot be written down explicitly for generic J i but one can find it as a power series in large J = 3 i=1 J i expansion as in [4,8] (J ≫ 1) where J ≡ J 1 + J 2 + J 3 ≫ 1, and thus As in the previous examples in [4,7,8], here the energy thus admits a regular expansion in 1 Hence it should be possible to match, as in [6,8,9], the coefficient of the O(λ) term in (3.24) with the 1-loop anomalous dimensions of the corresponding SYM operators determined by a special 3-spin case of the integrable SU(2,2|4) spin chain of [2]. The simplicity of the expression (3.24) suggests that one may be able to establish the correspondence with particular solutions of the Bethe ansatz equations in a relatively direct way, as was the case in [6] for the J 1 = J 2 , J 3 = 0 and in [11] for the J 1 = J 2 , J 3 = 0 circular solutions of [4]. Let us now look at some special cases. If J 2 = J 3 = 0, a 2 = a 3 = 0, i.e. in the one-spin case, we have a solution if w 2 1 = ν 2 , i.e. m 1 = 0 and J 1 = w 1 , and then E = J 1 . This is simply the point-like geodesic case: for m 1 = 0 there is no σ-dependence in X i .
In the two-spin case J 3 = 0, a 3 = 0 the equation (3.17) for ν 2 becomes a quartic equation Its simple explicit solution is found in the case when J 1 = J 2 , i.e. a 1 = a 2 = 1 √ 2 , m 2 = −m 1 ≡ m > 0: This is the same E(J ) relation as for the 2-spin circular solution of [4]. In fact, as was already mentioned above, the two solutions are equivalent: here we have 28) which is related to the solution in [4] by an SO(4) rotation: In the general case of two unequal spins we can again solve (3.25) in the limit of large J 1 , J 2 (for fixed m 1 , m 2 ), getting the special case of (3.24) with m 1 J 1 + m 2 J 2 = 0, In another special case when two out of three non-vanishing spins are equal, e.g., J 2 = J 3 , and with m 1 = 0, m 2 = −m 3 = m we get from (3.24) This is the same as the expression for the circular 3-spin solution (J 1 = 0, J 2 = J 3 ) in [4]. Indeed, for any values of J 1 , J 2 = J 3 the two solutions are related by a global rotation in X 2 , X 3 directions as in (3.29), converting e im 2 σ into cos m 2 σ and sin m 2 σ.
To summarize, we have shown that the constant-radius solutions of the NR system represent a simple generalization of the circular 2-spin and 3-spin solutions of [4]. This opens up a possibility of a direct comparison to SYM one-loop anomalous dimensions in the (i) 2-spin sector with unequal spins (cf. [6]) and (ii) general 3-spin sector (cf. [11]).

Quadratic fluctuations and stability
Let us now study small fluctuations near the solutions of section 3.1. This will generalize (and simplify) the discussion in [5] in the case of the special J 1 = J 2 3-spin solution and will clarify the conditions of stability of our new solutions. One application of this analysis would be to compute the 1-loop sigma-model correction to the classical energy (3.24) and to show that it is indeed suppressed by extra power of 1/J as in the special case considered in [5]. Another would be to find the spectrum of excited string states carrying the same charges as the "ground-state" classical solution as these may be possible to compare to the corresponding spectrum of anomalous dimensions on the SYM side (as was done for the special J 1 = J 2 , J 3 = 0 case in [6]).
It is straightforward to find the quadratic fluctuation Lagrangian by expanding near the solution (3.6) or (3.10)-(3.14). We shall follow the discussion in section 2 of [5] where the special case of circular solution with two equal spins was considered. Using three complex combinations of coordinates in (2.7) and expanding X i → X i +X i the sigma model action (2.3) near the classical solution we find the following Lagrangian for the quadratic fluctuations (see [5]) To solve this constraint we set After integrating by parts, (3.33) takes the form (cf. [5]) To solve (3.36) we may apply a global rotation to g i ,ḡ i = M ij (a)g j , that transforms 3 i=1 a i g i intoḡ 1 and set the latter to zero in the resulting Lagrangian (3.37). Equivalently, we may solve (3.36) for g 1 and substitute the result into (3.37).
For simplicity, let us first consider the 2-spin case when (cf. (3.25)) a 2 1 +a 2 2 = 1 , a 3 = 0 , a 2 1 |m 1 |w 1 −a 2 2 m 2 w 2 = 0 , w 2 We shall assume that m 1 < 0, m 2 > 0. These relations allow us to express a 1 and a 2 in terms of m 1 , m 2 and ν (3.39) In this case the fluctuations in the i = 3 direction decouple, and we find the following Lagrangian for the remaining 3 fluctuations g 2 , f 1 , f 2 (e.g. solving (3.36) for g 1 and rescaling g 2 )L (3.40) Solving the resulting equations of motion for F q = (f 1 , f 2 , g 2 ) using the ansatz (see [5]) F q = s,n A q,s,n e iωsτ +inσ we find the following characteristic equation of the frequencies ω This is a quartic equation for ω, and the stability condition is that all four roots should be real. The solutions are obviously real for n = 0 so instability may appear only for n = ±1, ... . In the special case of the 2-spin circular solution of [4], i.e.
which implies instability when n 2 −4m 2 < 0, i.e. for the modes with n = ±1, ..., ±(2m− 1) [5]. For generic a 1 , a 2 , m 1 , m 2 and small enough n one finds that two of the four roots are complex (with non-zero real part). 7 In spite of the instability it is useful to work out the spectrum of frequencies and the stability condition in the limit of large spins (i.e. large ν) since the resulting energies may be compared to SYM theory. First, let us consider the case of equal spins (−m 1 = m 2 = m). Eq. (3.42) implies that [5] so that the large ν expansion gives (for the lower-energy modes) Then the "one-loop" contribution to the energy of rotating string from (a pair of) such modes is (here κ 2 = ν 2 + 2m 2 , J = J 1 + J 2 = √ λ √ ν 2 + m 2 ; m = k in the notation of [5]) This expression was indeed reproduced in [6] (for m = 1) as the 1-loop anomalous dimension of excited states on the SYM side (corresponding to a particular Bethe root distribution for the Heisenberg spin chain).
One should be able to reproduce the analog of (3.45) in the case of (3.46), i.e. (here we assume |m 1 | > m 2 ) on the gauge theory side.
It is straightforward to extend the above discussion to the 3-spin case, i.e. when a 3 is non-zero. This will give a generalization of the spectrum found in the (J 1 , J 2 = J 3 ) case in [5]; as in that special case, there should then be a range of parameters for which the solution is stable. The generalization of the eq.(3.41) to the 3-spin case is 9 where Setting Ω 3 = 0, a 3 = 0 we indeed go back to (3.41). This equation gives 8 characteristic frequencies. Solving the equations for a 2 , a 3 in terms of a 1 and w i = m 2 i + ν 2 we get the following generalization of (3.39) Concentrating then on those frequencies that scale as we get the following equation for the leading part of eq.(3.48) Stable solutions arise in the range of the parameters m 1 , m 2 , m 3 such that (3.52) has real rootsω for all integer n. The general stability condition on m 1 , m 2 , m 3 and a 2 1 appears to be complicated, but one can find particular values of m 1 , m 2 , m 3 for which the solution is stable. For example, setting m 1 = 0, m 2 = −m 3 = m, so that a 1 ≡ a, a 2 3 = a 2 2 = 1 2 (1−a 2 ), which is the case of the 3-spin solution of [4], J 1 = a 2 ν, J 2 = J 3 = 1 2 (1−a 2 ) √ m 2 + ν 2 , we find, in agreement with [5] 10 The condition of stability, i.e.ω 2 ≥ 0 is obtained by demanding that (q 2 − 4)(q 2 − 4a 2 ) ≥ 0 and (3a 2 − 1) 2 + 4a 2 (q 2 − 1) ≥ 0, where q ≡ n m . The stability condition is satisfied if q 2 ≥ 1 and a 2 ≥ 1 4 , which applies to all modes if m = 1 as in [5]. For m = 2 the potentially unstable mode is n = ±1 having q 2 = 1 4 . Then to have stability we need to demand a 2 ≥ 1 16 as well as 1 16 ≤ a 2 ≤ 1 6 (3 − √ 5) or 1 6 (3 + √ 5)a 2 < 1. Similar conditions on a are found for higher values of m.
If instead we set m 1 = m 2 (or m 1 = m 3 ) in (3.52) we find implying that modes with n 2 < 4|m 2 m 3 | are unstable irrespective of the value of a 1 , just like in the 2-spin case (3.46).
4 More general "non-constant" solutions of the Neumann-Rosochatius system 4

.1 NR equations in ellipsoidal coordinates
Analogously to the case of the Neumann system in [8] we can rewrite the equations of motion following from (2.23) in the ellipsoidal coordinates (ζ 1 , ζ 2 ) which are introduced as Expressing the integrals of motion (2.25) in terms of ζ a one finds the following separable system of equations where P (ζ) is Here b 1,2 are the constants of motion which can be expressed in terms of the original integrals I i in (2.25). The Hamiltonian of the NR system reduces then to As in the Neumann case, P (ζ) is the fifth order polynomial which defines a hyperelliptic curve s 2 + P (ζ) = 0. However, with non-zero v i the positions of the roots get shifted. The general solution of the eqs. (4.2) can be given in terms of theta-functions associated to the Jacobian of the hyperelliptic curve. We will not consider the problem of solving the equations (4.2) in full generality, rather we will treat the simplest case of the vanishing integral v 3 . As one can see, for v 3 = 0 the value ζ = w 2 3 is a root of P (ζ) and then the NR system can be solved in terms of elliptic functions.

Two-spin solution of the NR system
If v 3 = 0 we may set α 3 = 0 and further assume that r 3 = 0 (see (2.22)) which brings us to the two-spin case. In terms of the ellipsoidal coordinates the two-spin solution arises in the limit b 2 → w 2 3 . 11 It is convenient to perform the following change of variables ζ a → ξ a (see [8] for details) Then we find that the first equation in (4.2) reduces to Here t ≡ (w 2 2 − b 1 )/w 2 21 is the modulus of the elliptic curve. The variable v 2 can be eliminated using v 2 2 = v 2 1 w 2 1 /w 2 2 . Thus we get a one-parameter family of solutions (parametrized by the additional parameter v 1 ).
Since for the periodic solutions ϕ 1,2 (σ + 2π) = ϕ 1,2 (σ) + 2πm 1,2 we have also the condition (2.28), we can trade the parameters v 1 , v 2 for the two integers m 1 , m 2 . Using the explicit solution (4.9) one can compute the integrals in eq. (2.28) with the result For given non-zero integers m i these are highly transcendental equations on v 1 , v 2 .
Computing the spins we get (4.14) Finally, the energy is given by (4.15) Note that due to the extra condition (2.27), i.e. v 1 w 1 = −v 2 w 2 , the solution exists only if J 1 and J 2 are related in a certain way.
The above system of equations (4.11)-(4.15), determines the energy E parametrically as a function of the R-charges J 1 = √ λJ 1 , J 2 = √ λJ 2 and winding numbers m 1 , m 2 . This system is rather complicated to allow for an explicit formula for . Nevertheless, we hope that it might be possible to directly match this system (its leading O(λ) or the "one-loop" approximation) onto the corresponding equations governing the algebraic Bethe ansatz (for a particular choice of the Bethe root distribution) for the anomalous dimensions of the corresponding operators on the gauge theory side, as was done in the v i = 0 case in [6,9].
5 Rotating strings in AdS 5 × S 5 Let us now generalize the discussion of sections 2 and 3 to the case when the string can rotate in both AdS 5 and S 5 . For that we need to supplement the S 5 rotating string ansatz (2.7) by the similar AdS 5 one where now (generalizing the ansatz considered in [4,8]) z r = (z 0 , z 1 , z 2 ) are complex, and because of the condition η M N Y M Y N = −1, their real radial parts lie on a hyperboloid (η rs = (−1, 1, 1)) z r = r r e iβr , η rs r r r s ≡ −r 2 0 + r 2 1 + r 2 2 = −1 .
In view of (5.2), they satisfy the relation s,r Substituting the above rotational ansatz into the AdS 5 Lagrangian (and changing overall sign) we find the analog of the 1-d Lagrangian (2.20) in the S 5 casẽ Like its S 5 counterpart (2.20), this 1-d Lagrangian is a special case of an n = 6 Neumann system now with signature (− + + + +−), and thus represents again an integrable system (being related, as in [8], to a special euclidean-signature Neumann model by an analytic continuation). The reduction of the total AdS 5 ×S 5 Lagrangian on the rotation ansatz is then given by the sum of (2.20) and (5.8). Writing (5.8) in terms of r r and β r we find as in (2.22) so that finally we end up with where, as above, we assume summation over r, s. Comparing this to the NR Lagrangian (2.23), we conclude that (5.10) describes a system which is similar to the Neumann-Rosochatius integrable system, but with an indefinite signature, i.e. δ ij replaced by η rs . While the equations for r i and r r following, respectively, from (2.23) and (5.10) are decoupled, the variables of the two NR systems are mixed in the conformal gauge constraints (2.15),(2.16) which now take the form (generalizing (2.26),(2.27) where we had r 0 = 1, u r = 0, r a = 0) where r 2 0 − 2 a=1 r 2 a = 1, and 3 i=1 r 2 i = 1. We should also require the periodicity condition analogous to (2.28) Then k 0 implies that we should set u 0 = 0 as a consequence of single-valuedness of the AdS 5 time t.
One can then repeat the discussion of sections 2,3 and in the present case, classifying general solutions of the resulting NR system. The resulting solutions generalize those of sect.4.2 in [4] where the integrals v i and u r were zero.

Simple circular strings in AdS 5
Let us first assume that the string is not rotating in S 5 (i.e. w i , v i = 0, r i = const) and consider the AdS 5 analog of the simplest circular solution of section 3 by demanding thatΛ = const. The discussion is then exactly the same as (a special case of that) in section 3 with few signs reversed. As in section 3.1, finding solutions withΛ = const turns out to be equivalent to looking for constant radii (r r = const) configurations. Then (cf. (3.10),(3.14)) r r = const , β a = k a σ , k 0 = 0 , u 0 = 0 , u a = r a k a , (5.14) The energy as a function of spins is then obtained by solving the system of the two equations that follow from the definition of the charges (5.6) and the constraints (5.11),(5.12) with κ as a parameter (cf. (3.17)-(3.18)) This implies k 2 1 S 1 k 2 1 + κ 2 + k 2 2 S 2 k 2 2 + κ 2 = 1 2 κ 2 . (5.18) Considering the limit of large spins S a ≫ 1, with k a being fixed, we conclude that κ = (2k 2 1 S 1 + 2k 2 1 S 1 ) 1/3 + ... and then E = S 1 + S 2 + 3 4 (2k 2 1 S 1 + 2k 2 2 S 2 ) 1/3 + ... , (5.19) or, in view of k 1 S 1 = −k 2 S 2 (treating S 1 , S 2 and k 1 as independent data) E = S + 3 4 (2k 2 1 S S 1 S 2 ) 1/3 + ... , S ≡ S 1 + S 2 . The case of k 1 = −k 2 = k when the two spins are equal S 1 = S 2 = 1 2 S is that of the the circular solution found in [4] for which we get E = S + 3 4 (2kS) 1/3 + ... . (5.22) As was shown in [4], this k 1 = −k 2 solution is stable only for small enough S. The "non-perturbative" scaling of the subleading term in (5.21) with λ precludes one from entertaining a possibility of a direct comparison to anomalous dimensions of the corresponding operators [4], i.e.ΦD S 1 1+i2 D S 2 3+i4 Φ, in SYM theory, in contrast to what was found in the S 5 case.
Let us now see how this conclusion changes when we consider "hybrid" solutions where the circular string rotates both in AdS 5 and S 5 . different limit when J ≫ S ≫ 1 (corresponding to k 2 1 ≫ m 2 i ). In this case we get a "BMN-type" (single J rotation type) asymptotics with the leading term still given by (5.27), i.e. ∆E ∼ 1 2J 2 S. The conclusion is that to have a regular (i.e. with analytic λ-dependence) largespin expansion of the energy one needs to have at least one large component of the spin in S 5 direction. This turns out to be the same also in the case of other spinning string solutions with more complicated σ-dependence.