Pulsating strings with mixed three-form flux
Pulsating strings with mixed three-form 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 three-form fluxes can be described by an integrable deformation of the one-dimensional Neumann-Rosochatius 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 NS-NS three-form 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 NS-NS 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 two-dimensional conformal field theories with maximal supersymmetry -. Integrability has also been shown to remain a symmetry of general string backgrounds that support a mixture of R-R and NS-NS three-form fluxes . This discovery has led to an insightful view of many features of the AdS3/CFT2 correspondence in the presence of mixed fluxes [24-26]-. One of these developments was the demonstration that the sigma-model of type IIB closed strings spinning in AdS3 ×S3 with mixed fluxes corresponds to an integrable deformation of the Neumann-Rosochatius mechanical system [43, 44]. The identification of the Lagrangian that describes spinning strings with mixed fluxes with a deformation of the Neumann-Rosochatius 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
AdS-CFT Correspondence; Gauge-gravity correspondence
× T 4 with mixed R–R and NS–NS
deformation of the Neumann-Rosochatius system obtained from the spinning string ansatz.
We will make use of the flux-deformation 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 three-form 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 Neumann-Rosochatius system that arises from
the motion of closed strings pulsating in AdS3 × S3
× T 4 with non-vanishing 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 ,
btφ = q sinh2 ρ ,
bφ1φ2 = −q cos2 θ ,
for the NS–NS B-field, 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 sigma-model 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 Neumann-Rosochatius 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 gab z˙az˙b + zazaβ˙b2 − z1 k1 − qk1z12β˙0 − 2
2 2 Λ
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 single-valued. When we enter this ansatz in the world-sheet action in the
conformal gauge we find
λ " X2 1
r˙i2 + ri α˙ i − ri2mi2 + qr22 (m2α˙ 1 − m1α˙ 2) − 2
(r12 + r22 − 1)
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
β˙0 = −
u0 + qk1z12 ,
α˙ 2 = v2 + qr22m1
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 ,
z12k1β˙1 + Xri2miα˙ i = 0 .
– 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 Anti-de Sitter factors defined, respectively, as the roots of the equations
ζ − m21 +
ζ − m22 = 0 ,
μ − k12 − μ
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 Neumann-Rosochatius 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 Anti-de
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
≡ (1 − q2)Y(μ − μi) .
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
ζ3 − ζ2
ζ3 − ζ1
μ3 − μ2 .
μ3 − μ1
r12(τ ) =
1The integrability of the Neumann-Rosochatius system follows from the existence of a set of integrals
of motion in involution, the Uhlenbeck constants . 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 . 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
Anti-de 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 three-form 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.
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 , and the
solution reduces now to
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 time-like and space-like geodesics, the parabolic behaviour can
be understood as the result of performing spectral flow on light-like 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  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
– 6 –
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 . 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 Neumann-Rosochatius system with pure NS–NS flux.
grant GR3/14-A 910770.
The work of R.H. is supported by grant FPA2014-54154-P and by BSCH-UCM through
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