Pulsating strings with mixed threeform flux
JHE
Pulsating strings with mixed threeform flux
Rafael Hern´andez 0 1
Juan Miguel Nieto 0 1
Roberto Ruiz 0 1
0 Madrid , 28040 Spain
1 Departamento de F ́ısica Teo ́rica I, Universidad Complutense de Madrid
Circular strings pulsating in AdS3 × S3 threeform fluxes can be described by an integrable deformation of the onedimensional NeumannRosochatius mechanical model. In this article we find a general class of pulsating solutions to this integrable system that can be expressed in terms of elliptic functions. In the limit of strings moving in AdS3 with pure NSNS threeform flux, where the action reduces to the SL(2, R) WZW model, we find agreement with the analysis of the classical solutions of the system performed using spectral flow by Maldacena and Ooguri. We use our elliptic solutions in AdS3 to extend the dispersion relation beyond the limit of pure NSNS flux. The integrable structure that underlies the AdS3/CFT2 correspondence has provided a deep understanding of numerous aspects of string theory in backgrounds with an AdS3 factor and twodimensional conformal field theories with maximal supersymmetry [1][22]. Integrability has also been shown to remain a symmetry of general string backgrounds that support a mixture of RR and NSNS threeform fluxes [23]. This discovery has led to an insightful view of many features of the AdS3/CFT2 correspondence in the presence of mixed fluxes [2426][42]. One of these developments was the demonstration that the sigmamodel of type IIB closed strings spinning in AdS3 ×S3 with mixed fluxes corresponds to an integrable deformation of the NeumannRosochatius mechanical system [43, 44]. The identification of the Lagrangian that describes spinning strings with mixed fluxes with a deformation of the NeumannRosochatius system allows the use of a systematic approach to the construction of general classes of solutions. In this letter we will extend the problem to the case of an ansatz corresponding to closed strings pulsating in AdS3 × S3. In particular, we will show that pulsating strings with mixed fluxes can also be treated using the integrable
AdSCFT Correspondence; Gaugegravity correspondence

× T 4 with mixed R–R and NS–NS
deformation of the NeumannRosochatius system obtained from the spinning string ansatz.
We will make use of the fluxdeformation of the Uhlenbeck constants of the model to
integrate the equations of motion in terms of Jacobi elliptic functions. We will also study
the problem in the limit of pure NS–NS threeform flux, where the AdS3 piece of the system
reduces to the SL(2, R) WZW model. On the basis of our class of elliptic solutions we will
derive a general form of the dispersion relation valid beyond the WZW point.
In what follows we will first present the NeumannRosochatius system that arises from
the motion of closed strings pulsating in AdS3 × S3
× T 4 with nonvanishing NS–NS flux.
We will consider no dynamics along the torus, and thus the background metric will be
ds2 = − cosh2 ρ dt2 + dρ2 + sinh2 ρ dφ2 + dθ2 + sin2 θdφ12 + cos2 θdφ22 ,
HJEP04(218)7
together with
btφ = q sinh2 ρ ,
bφ1φ2 = −q cos2 θ ,
for the NS–NS Bfield, where 0 ≤ q ≤ 1. The limit q = 0 corresponds to the case of pure
R–R flux, while setting q = 1 we are left with pure NS–NS flux. In the case of pure R–R
flux the sigmamodel for closed strings rotating in AdS3 × S3 reduces to the
NeumannRosochatius system [45, 46]. The presence of the NS–NS flux term leads to an integrable
deformation of the NeumannRosochatius model [43, 44]. In this letter we will extend the
analysis in that reference for the spinning string ansatz to the case of pulsating strings.
In order to introduce the pulsating ansatz it will be convenient to use the embedding
coordinates of AdS3 and S3, which are related to the global angles by
Y1 + iY2 = sinh ρ eiφ ,
X1 + iX2 = sin θ eiφ1 ,
Y3 + iY0 = cosh ρ eit ,
X3 + iX4 = cos θ eiφ2 .
(1)
(2)
(3)
(4)
(5)
(6)
(7)
+ 1 gab z˙az˙b + zazaβ˙b2 − z1 k1 − qk1z12β˙0 − 2
2 2 Λ
2
˜
gabzazb + 1
#
where the dot stands for derivatives with respect to τ , the Lagrange multipliers Λ and Λ˜ are
needed to impose that the solutions lie, respectively, on S3 and AdS3, and we have taken
g = diag(−1, 1), with a = 0, 1. We must note that the flux term in (7) appears with the
– 2 –
In these coordinates the ansatz for a pulsating string is
Y1 + iY2 = z1(τ ) eiβ1(τ)+ik1σ ,
X1 + iX2 = r1(τ ) eiα1(τ)+im1σ ,
Y3 + iY0 = z0(τ ) eiβ0(τ) ,
X3 + iX4 = r2(τ ) eiα2(τ)+im2σ ,
where we have excluded the winding along the time coordinate because the time
direction has to be singlevalued. When we enter this ansatz in the worldsheet action in the
conformal gauge we find
L =
√
2π
λ " X2 1
i=1
2
r˙i2 + ri α˙ i − ri2mi2 + qr22 (m2α˙ 1 − m1α˙ 2) − 2
2 2
(r12 + r22 − 1)
Λ
opposite sign than that in the Lagrangian coming from the spinning string ansatz [43, 44].
The equations of motion for the radial coordinates following from (7) are given by
for the ri coordinates, while for the za coordinates they are
The cyclic nature of the spherical and hyperbolic angular coordinates in the Lagrangian
implies that its equations of motion can be easily integrated once,
z¨0 = z0β˙02 − Λ˜ z0 ,
z¨1 = z1β˙12 − z1k12 − Λ˜ z1 − 2qz1k1β˙0 .
α˙ 1 = v1 − qr22m2
2
,
r
1
β˙0 = −
u0 + qk1z12 ,
z
2
0
α˙ 2 = v2 + qr22m1
2
,
r
2
β˙1 =
u1
z
2
1
,
(8)
(9)
(
10
)
(11)
(12)
(13)
(14)
(15)
(16)
(17)
where vi and ua are some integration constants. We can use equations (12) and (13) to
write the energy, the Lorentzian spin and the two angular momenta of the string as
E = −√λu0 ,
S = √λu1 ,
J1 = √λv1 ,
J2 = √λv2 .
The equations of motion coming from (7) must be supplied with the Virasoro constraints,
which are responsible for the coupling between the AdS3 and the S3 pieces of the system.
They take the form
X r˙i2 + α˙ i2 + mi2 ri2 + Xgii(z˙i2 + zi2β˙i2) + k12z12 = 0 ,
2
i=1
2
i=1
z12k1β˙1 + Xri2miα˙ i = 0 .
2
i=1
– 3 –
We will now move to the construction of general solutions to the above system. In
order to proceed, we will follow [45, 46] and introduce ellipsoidal coordinates ζ and μ for
the sphere and the Antide Sitter factors defined, respectively, as the roots of the equations
r
2
1
ζ − m21 +
r
2
2
ζ − m22 = 0 ,
z
2
1
μ − k12 − μ
z
2
The ranges of the ellipsoidal coordinates are m21 ≤ ζ ≤ m22 and k12 ≤ μ. When we enter
directly ζ and μ into the equations of motion for ri and za we are left with a second
order differential equation for each ellipsoidal coordinate. However, we can also use the
Uhlenbeck constants of the system to reduce the problem to a pair of independent first
order differential equations [45, 46]. The Uhlenbeck constants for the pulsating
NeumannRosochatius system in the presence of the flux deformation can be obtained immediately
from the ones for the spinning NeumannRosochatius system in [43, 44] just by replacing q
by −q, because of the change in the sign of flux terms. Thus the Uhlenbeck constants
corresponding to motion of the pulsating string either on the sphere or on the Antide
Sitter factor are, respectively, given by1
where the third order polynomials P3(ζ) and Q3(ζ) are given by
P3(ζ) = −(1 − q2)(m22 − ζ)(ζ − m12)2 + I¯1(m22 − m12)(m22 − ζ)(ζ − m12)
+(v1 − qm2)2(m22 − ζ)2 + v22(ζ − m12)2 ≡ (1 − q2)Y(ζ − ζi) ,
Q3(μ) = (1 − q2)(μ − k12)2μ − F¯1k12(μ − k12)μ + (u0 − qk1)2(μ − k12)2 + u1μ
2 2
3
i=1
≡ (1 − q2)Y(μ − μi) .
3
i=1
(18)
(19)
(20)
(21)
(22)
(23)
(24)
(25)
(26)
(27)
The solution to equations (20) can be written in terms of the Jacobian elliptic functions
(see [43, 44] for details in the case of the spinning string ansatz). We find
ζ(τ ) = ζ3 + (ζ2 − ζ3) sn2
p(1 − q2) (ζ3 − ζ1) τ + τ0, κ ,
μ(τ ) = μ2 +
′
(μ3 − μ2)(μ2 − μ1) sd2
μ3 − μ1
p(1 − q2) (μ3 − μ1) τ + τ0′ , ν ,
where τ0 and τ0 are integration constants that can be set to zero by performing a rotation,
and the elliptic moduli are
We therefore conclude that2
κ =
ζ3 − ζ2
ζ3 − ζ1
,
ν =
μ3 − μ2 .
μ3 − μ1
r12(τ ) =
1The integrability of the NeumannRosochatius system follows from the existence of a set of integrals
of motion in involution, the Uhlenbeck constants [47]. In the case of strings spinning in S
integrals I1 and I2, constrained to satisfy I1 + I2 = 1. In the presence of flux they are deformed to I¯1
and I¯2, with the condition I¯1 + I¯2 = 1 [43, 44]. A similar set of constants, F0 and F1, with the constraint
3 there are two
F1 − F0 = −1, and their corresponding deformation by the flux term, arises when the string spins in AdS3.
2Pulsating string solutions in AdS3 × S
3 with mixed R–R and NS–NS fluxes have been considered before
in [37]. To contact with the notation in that reference we just need to identify the roots R+ and R− in
there with our choice of roots through R+ = μ3/k12 − 1 and R− = μ1/k12 − 1.
– 4 –
We must stress that we need to order the roots of the ellipsoidal coordinate ζ for the
sphere in such a way that ζ1 < ζ3. On the contrary, the ellipsoidal coordinate μ for the
Antide Sitter factor is unbounded and symmetrical under permutation of the roots μ1 and
μ3 and thus there is no need to choose the roots μ1 and μ3 according to any particular
ordering. Moreover, we should choose μ2 ≥ k
12 because z
12 ≥ 0. Furthermore, we will
restrict the elliptic moduli (25) to their fundamental domains 0 ≤ κ, τ ≤ 1, which implies
that ζ1 < ζ2 < ζ3, and either μ1 < μ2 < μ3 or μ3 < μ2 < μ1.
We will now focus on the analysis of string solutions restricted to pulsate in AdS3 × S1
in the limit of pure NS–NS threeform flux. We will therefore set r1 = α1 = 0, and r2 = 1
and α2 = ω. We must first note that the cubic term in Q3(μ) is dressed with a factor 1−q2.
Thus in the case of pure NS–NS flux the degree of the polynomial reduces to two, and the
solution can be written in terms of trigonometric functions. In order to understand the
reduction of the problem from the point of view of the roots of the polynomial, we will first
present them for general values of q. The Virasoro constraints reduce now to
(u0 − k1)2 − k12F¯1 sin2 q(u0 − k12)2 − k12F¯1 τ .
We can therefore write z12(τ ) = sinh2 ρ(τ ) = sinh2 ρ0 sin(ατ ), so that
cosh ρ0 =
is given by μ0 = k12. The remaining roots are given by
(u0 − qk1)2 ∓
q (u0 − qk1)2 − (1 − q2)f1 2 + 4(1 − q2)k12(u0 − qk1)
If we choose (u0 − k12)2 > k12F¯1, the roots become
If we exclude the case with F¯1 = 0, which corresponds to the trivial limit where the solution
collapses to a point, it is immediate to check that the roots satisfy μ− < μ0 < μ+. It is also
clear that the limit of pure NS–NS flux depends on the sign of the term (u0 − k1) − k12F¯1.
2 2
μ− → −∞ ,
can reach. On the contrary, in the case where k12F¯1 > (u0 − k12)2 the roots become
,
Now the parameter ρ0 does not have the interpretation of the maximum size of the solution.
It is also worth to consider the threshold case, with k12F¯1 = (u0 − k12)2, where the ellipsoidal
coordinate μ displays a parabolic behaviour. In this case both μ− and μ+ diverge and
z12(τ ) = k12F¯1τ 2 .
and the hyperbolic radius is unbounded. This is the long string regime of [48], and the
solution reduces now to
k12F¯1
k12F¯1 − (u0 − k1)2 sinh2 qk12F¯1 − (u0 − k1)2 τ .
In this case we will introduce ρ0 using z12(τ ) = sinh2 ρ(τ ) = cosh2 ρ0 sinh(ατ ). Accordingly,
sinh ρ0 =
.
In the same way as the short and long string regimes are, respectively, constructed by
means of spectral flow on timelike and spacelike geodesics, the parabolic behaviour can
be understood as the result of performing spectral flow on lightlike geodesics.
In order to fix the sign of α in the previous expressions we need to find the dependence
of the time coordinate t on τ . We can obtain that dependence by direct integration of the
equation of motion (13) for β˙0. In the short string regime, we conclude that3
tan t(τ ) =
tan(−k1τ ) + cosh ρ0 tan − sign(u0 − k1)p(u0 − k1)2 − F¯1k12τ
1 − cosh ρ0 tan(−k1τ ) tan − sign(u0 − k1)p(u0 − k1)2 − F¯1k12τ
In the long string regime we can proceed identically, or continue analytically
expression (39), to find that
tan t(τ ) =
tan(−k1τ ) + sinh ρ0 tanh − sign(u0 − k1)pk12F¯1 − (u0 − k1)2τ
1 − sinh ρ0 tan(−k1τ ) tanh − sign(u0 − k1)pk12F¯1 − (u0 − k1)2τ
This is indeed the result in [48] provided we read α = −sign(u0 − k1)p(u0 − k1)2 − F¯1k12
and identify the winding number k1 with −w in that reference. Besides, the threshold case
corresponds simply to
t(τ ) = −k1τ .
We will now use our solutions to extend the dispersion relation to general values of the
flux parameter, beyond the WZW limit of the action. At these point we should emphasize
3We must note that when we move away from the limit of pure NS–NS flux the division of the solutions
in three different regimes breaks down. However, we can still find tan t(τ) in terms of complete elliptic
integrals.
– 6 –
.
.
(35)
(36)
(37)
(38)
(39)
(40)
(41)
that, despite we have chosen F¯1 and ω as constants of motion, any other set of first integrals
is valid as long as they remain independent in phase space. In particular, we can replace
F¯1 by n¯ in the argument of the elliptic sine, sn(n¯τ, ν) = sn p(1 − q2)(μ3 − μ1) τ, ν . Thus
which in the limit of pure NS–NS flux becomes
qn¯4 − (1 − q2)(k12 − ω2)
corresponds to the short string regime, while the lower one is the long string regime, in
accordance with [48]. A natural question arising from this letter is the derivation of (43) by
extending the analysis of the conformal field theory beyond the limit of the WZW model.
It would also be interesting to investigate in more detail the relation between the WZW
model and the reduction of the NeumannRosochatius system with pure NS–NS flux.
Acknowledgments
grant GR3/14A 910770.
The work of R.H. is supported by grant FPA201454154P and by BSCHUCM through
Open Access.
This article is distributed under the terms of the Creative Commons
Attribution License (CCBY 4.0), which permits any use, distribution and reproduction in
any medium, provided the original author(s) and source are credited.
[arXiv:1003.0465] [INSPIRE].
[INSPIRE].
[1] A. Babichenko, B. Stefan´ski Jr. and K. Zarembo, Integrability and the AdS3/CF T2
correspondence, JHEP 03 (2010) 058 [arXiv:0912.1723] [INSPIRE].
[2] K. Zarembo, Strings on Semisymmetric Superspaces, JHEP 05 (2010) 002
[3] K. Zarembo, Algebraic Curves for Integrable String Backgrounds, arXiv:1005.1342
[4] O. Ohlsson Sax and B. Stefan´ski Jr., Integrability, spinchains and the AdS3/CFT2
correspondence, JHEP 08 (2011) 029 [arXiv:1106.2558] [INSPIRE].
superstring, JHEP 07 (2012) 159 [arXiv:1204.4742] [INSPIRE].
[5] N. Rughoonauth, P. Sundin and L. Wulff, Near BMN dynamics of the AdS3 × S × S3 × S1
3
– 7 –
[6] O. Ohlsson Sax, B. Stefan´ski Jr. and A. Torrielli, On the massless modes of the AdS3/CFT2
integrable systems, JHEP 03 (2013) 109 [arXiv:1211.1952] [INSPIRE].
[7] C. Ahn and D. Bombardelli, Exact Smatrices for AdS3/CF T2,
Int. J. Mod. Phys. A 28 (2013) 1350168 [arXiv:1211.4512] [INSPIRE].
JHEP 04 (2013) 113 [arXiv:1211.5119] [INSPIRE].
[9] R. Borsato, O. Ohlsson Sax and A. Sfondrini, Allloop Bethe ansatz equations for
AdS3/CFT2, JHEP 04 (2013) 116 [arXiv:1212.0505] [INSPIRE].
HJEP04(218)7
integrable spinchain for strings on AdS3 × S3 × T 4: the massive sector, JHEP 08 (2013) 043
[arXiv:1303.5995] [INSPIRE].
[arXiv:1403.4543] [INSPIRE].
[11] R. Borsato, O. Ohlsson Sax, A. Sfondrini and B. Stefan´ski Jr., Towards the AllLoop
worldsheet S matrix, JHEP 10 (2014) 066 [arXiv:1406.0453] [INSPIRE].
[12] R. Borsato, O. Ohlsson Sax, A. Sfondrini and B. Stefan´ski Jr., The complete AdS3 × S × T
3
4
[13] M. Beccaria, F. LevkovichMaslyuk, G. Macorini and A.A. Tseytlin, Quantum corrections to
spinning superstrings in AdS3 × S
JHEP 04 (2013) 006 [arXiv:1211.6090] [INSPIRE].
3
× M 4: determining the dressing phase,
[14] M. Beccaria and G. Macorini, Quantum corrections to short folded superstring in
AdS3 × S3 × M 4, JHEP 03 (2013) 040 [arXiv:1212.5672] [INSPIRE].
[15] R. Borsato, O. Ohlsson Sax, A. Sfondrini, B. Stefan´ski Jr. and A. Torrielli, Dressing phases
of AdS3/CFT2, Phys. Rev. D 88 (2013) 066004 [arXiv:1306.2512] [INSPIRE].
J. Phys. A 46 (2013) 445401 [arXiv:1306.5106] [INSPIRE].
[16] M.C. Abbott, The AdS3 × S × S × S
3
3
1 Hern´andezLo´pez phases: a semiclassical derivation,
[17] M.C. Abbott and I. Aniceto, Macroscopic (and Microscopic) Massless Modes,
Nucl. Phys. B 894 (2015) 75 [arXiv:1412.6380] [INSPIRE].
[18] M.C. Abbott and I. Aniceto, An improved AFS phase for AdS3 string integrability,
Phys. Lett. B 743 (2015) 61 [arXiv:1412.6863] [INSPIRE].
[19] P. Sundin and L. Wulff, Worldsheet scattering in AdS3/CF T2, JHEP 07 (2013) 007
[arXiv:1302.5349] [INSPIRE].
[20] T. Lloyd and B. Stefan´ski Jr., AdS3/CF T2, finitegap equations and massless modes,
JHEP 04 (2014) 179 [arXiv:1312.3268] [INSPIRE].
[21] P. Sundin, Worldsheet two and fourpoint functions at one loop in AdS3/CF T2,
Phys. Lett. B 733 (2014) 134 [arXiv:1403.1449] [INSPIRE].
[22] A. Sfondrini, Towards integrability for AdS3/CF T2, J. Phys. A 48 (2015) 023001
[arXiv:1406.2971] [INSPIRE].
[23] A. Cagnazzo and K. Zarembo, Bfield in AdS3/CF T2 Correspondence and Integrability,
JHEP 11 (2012) 133 [Erratum JHEP 04 (2013) 003] [arXiv:1209.4049] [INSPIRE].
[24] B. Hoare and A.A. Tseytlin, On string theory on AdS3 × S × T
3
4 with mixed 3form flux:
treelevel Smatrix, Nucl. Phys. B 873 (2013) 682 [arXiv:1303.1037] [INSPIRE].
– 8 –
[25] B. Hoare and A.A. Tseytlin, Massive Smatrix of AdS3 × S3 × T 4 superstring theory with
mixed 3form flux, Nucl. Phys. B 873 (2013) 395 [arXiv:1304.4099] [INSPIRE].
[26] B. Hoare, A. Stepanchuk and A.A. Tseytlin, Giant magnon solution and dispersion relation
in string theory in AdS3 × S × T
[arXiv:1311.1794] [INSPIRE].
3
4 with mixed flux, Nucl. Phys. B 879 (2014) 318
[27] C. Ahn and P. Bozhilov, String solutions in AdS3 × S
Phys. Rev. D 90 (2014) 066010 [arXiv:1404.7644] [INSPIRE].
3
× T 4 with NS–NS Bfield,
[28] J.R. David and A. Sadhukhan, Spinning strings and minimal surfaces in AdS3 with mixed
3form fluxes, JHEP 10 (2014) 049 [arXiv:1405.2687] [INSPIRE].
[29] A. Banerjee, K.L. Panigrahi and P.M. Pradhan, Spiky strings on AdS3 × S3 with NS–NS
JHEP 08 (2014) 097 [arXiv:1405.7947] [INSPIRE].
[30] A. Babichenko, A. Dekel and O. Ohlsson Sax, Finitegap equations for strings on
AdS3 × S × T
3
4 with mixed 3form flux, JHEP 11 (2014) 122 [arXiv:1405.6087] [INSPIRE].
[31] L. Bianchi and B. Hoare, AdS3 × S
3
× M 4 string Smatrices from unitarity cuts,
[32] R. Roiban, P. Sundin, A. Tseytlin and L. Wulff, The oneloop worldsheet Smatrix for the
AdSn × Sn × T 10−2n superstring, JHEP 08 (2014) 160 [arXiv:1407.7883] [INSPIRE].
[33] T. Lloyd, O. Ohlsson Sax, A. Sfondrini and B. Stefan´ski Jr., The complete worldsheet S
matrix of superstrings on AdS3 × S
× T
Nucl. Phys. B 891 (2015) 570 [arXiv:1410.0866] [INSPIRE].
3
4 with mixed threeform flux,
[34] P. Sundin and L. Wulff, One and twoloop checks for the AdS3 × S
mixed flux, J. Phys. A 48 (2015) 105402 [arXiv:1411.4662] [INSPIRE].
3
× T 4 superstring with
[35] A. Stepanchuk, String theory in AdS3 × S × T
3
4 with mixed flux: semiclassical and 1loop
phase in the Smatrix, J. Phys. A 48 (2015) 195401 [arXiv:1412.4764] [INSPIRE].
[36] R. Borsato, O. Ohlsson Sax, A. Sfondrini and B. Stefan´ski Jr., The AdS3 × S × S × S
worldsheet S matrix, J. Phys. A 48 (2015) 415401 [arXiv:1506.00218] [INSPIRE].
3
3
1
mixed threeform fluxes, JHEP 11 (2015) 133 [arXiv:1508.03430] [INSPIRE].
[37] A. Banerjee, K.L. Panigrahi and M. Samal, A note on oscillating strings in AdS3 × S3 with
[38] A. Banerjee and A. Sadhukhan, Multispike strings in AdS3 with mixed threeform fluxes,
JHEP 05 (2016) 083 [arXiv:1512.01816] [INSPIRE].
[39] A. Banerjee, S. Biswas and R.R. Nayak, D1 string dynamics in curved backgrounds with
fluxes, JHEP 04 (2016) 172 [arXiv:1601.06360] [INSPIRE].
[40] R. Borsato, O. Ohlsson Sax, A. Sfondrini, B. Stefan´ski Jr., A. Torrielli and O. Ohlsson Sax,
On the dressing factors, Bethe equations and Yangian symmetry of strings on
AdS3 × S3 × T 4, J. Phys. A 50 (2017) 024004 [arXiv:1607.00914] [INSPIRE].
[41] S.P. Barik, M. Khouchen, J. Klusonˇ and K.L. Panigrahi, SL(2, Z) invariant rotating (m, n)
3 with mixed flux, Eur. Phys. J. C 77 (2017) 298 [arXiv:1610.03402]
[42] S.P. Barik, K.L. Panigrahi and M. Samal, Perturbations of Pulsating Strings,
strings in AdS3 × S
[10] R. Borsato , O. Ohlsson Sax , A. Sfondrini , B. Stefan´ski Jr. and A. Torrielli , The allloop [43] R. Hern ´andez and J.M. Nieto , Spinning strings in AdS3 × S [44] R. Hern ´andez and J.M. Nieto , Elliptic solutions in the NeumannRosochatius system with mixed flux , Phys. Rev. D 91 ( 2015 ) 126006 [arXiv: 1502 .05203] [INSPIRE].
integrable systems, Nucl. Phys. B 671 ( 2003 ) 3 [ hep th/0307191] [INSPIRE]. [45] G. Arutyunov , S. Frolov , J. Russo and A.A. Tseytlin , Spinning strings in AdS5 × S5 and system relations , Phys. Rev. D 69 ( 2004 ) 086009 [ hep th/0311004] [INSPIRE]. [46] G. Arutyunov , J. Russo and A.A. Tseytlin , Spinning strings in AdS5 × S5: New integrable [47] K. Uhlenbeck , Equivariant harmonic maps into spheres , Lect. Notes Math. 949 ( 1982 ) 39 . [48] J.M. Maldacena and H. Ooguri , Strings in AdS3 and SL(2 , R) WZW model. I: The Spectrum , J. Math. Phys. 42 ( 2001 ) 2929 [ hep th/0001053] [INSPIRE].