Oneloop quantization of rigid spinning strings in AdS3 × S3 × T 4 with mixed flux
HJE
Oneloop quantization of rigid spinning strings in
Juan Miguel Nieto 0 1
Roberto Ruiz 0 1
0 28040 Madrid , Spain
1 Departamento de F ́ısica Teo ́rica, Universidad Complutense de Madrid
We compute the oneloop correction to the classical dispersion relation of rigid closed spinning strings with two equal angular momenta in the AdS3 × S3 × T 4 background supported with a mixture of RR and NSNS threeform fluxes. This analysis is extended to the case of two arbitrary angular momenta in the pure NSNS limit. We perform this computation by means of two different methods. The first method relies on the EulerLagrange equations for the quadratic fluctuations around the classical solution, while the second one exploits the underlying integrability of the problem through the finitegap equations. We find that the oneloop correction vanishes in the pure NSNS limit.
AdSCFT Correspondence; Integrable Field Theories; Sigma Models

S3 ×
T 4
1 Introduction
2 The R × S3 string with mixed flux
3 Quadratic fluctuations around the classical solution
3.1
3.2
3.3
3.4
Bosonic fluctuations on AdS3
4 Frequencies from the algebraic curve
5 Computation of the oneloop correction
6 A comment about nonrigid strings
7 Conclusions
A Conventions B Computation of the corrections to the quasimomenta of the onecut solution
B.1 Contribution from AdS3 excitations
B.2 Contribution from S3 excitations
B.3 Contribution from fermionic excitations
C Finite gap frequencies for general values of q
× M4 backgrounds. The interest has been brought forth by the
progress made in the application of integrability techniques developed for the AdS5/CFT4
correspondence [3]. Its two best understood realizations involve string theories formulated
on AdS3 × S3 × T 4 [
4–19
] and AdS3 × S3 × S3 × S1 [
7, 8, 12, 14, 20–28
] spaces, which have
proven to be classically integrable and conformally invariant [29, 30]. On the one hand, the
dual CFT2 to the former is believed to reduce to the SymN (T 4) orbifold at certain point
of its moduli space [31], albeit deeper insight remains to be gained. On the other hand,
the dual CFT2 to the latter is still not clear and there exist different proposals [32–35].
– 1 –
M4 = S3
× S1 case, where it has been shown to emerge also in its CFT2 side [36].
Besides, integrability has been shown to be more fruitful in the M4 = T 4 than in the
In spite of its similarities with the AdS5 × S5 scenario, these two backgrounds exhibit
some features that obstruct a straightforward integrability approach.
The presence of
massless excitations play a prominent role among them, as it seems to be responsible
for the mismatch between the computation of the dressing factor for massive excitations
performed via perturbative worldsheet calculations [37, 38] and crossing equations [10, 16].
This link was proposed for the AdS3 × S3
× T 4 space in [39], where it is argued that the
lack of suppression of wrapping corrections involving massless modes associated to the T 4
factor could explain the discrepancy.
Unlike AdS5 × S5, just supported by a RamondRamond (RR) fiveform flux, both
× S1 can be deformed through the addition of a
NeveuSchwarzNeveuSchwarz (NSNS) threeform flux, which mixes with the RR one.
Integrability, conformal invariance and kappa symmetry are not spoiled by the introduction
of the NSNS flux [40]. The mixed flux setup has been also extensively studied in the
literature [12, 41–47].
The aim of this paper is to compute the oneloop correction to the dispersion relation
of rigid spinning string on AdS3 × S3
× T 4 in the presence of mixed flux. To this end, we
use two different methods. The first one extracts the characteristic frequencies from the
Lagrangian of quadratic fluctuations, whose signed sum leads to the oneloop correction.
This procedure has been already applied successfully to different string configurations in
AdS5 × S5 [48–58]. It was also applied to rigid spinning strings on AdS3 × S3
× T 4 with
pure RR flux [37]. The second method starts from the construction of the algebraic curve
associated to the Lax connection of the PSU(1, 12)2/(SU(1, 1) × SU(2)) supercoset sigma
model. The quantization of the algebraic curve is well understood for the AdS5 × S5 case
(see [59] for an indepth explanation). The extension of this procedure to the AdS3×S3×T 4
space is straightforward for the massive excitations with pure RR flux. However, the
construction of the finitegap equations requires some extensions when dealing with the
full spectrum supported by a general mixed flux. First of all, the introduction of an NSNS
flux shifts the poles of the Lax connection away from ±1. This issue has been solved in [60],
where the finitegap equations proposed for AdS3 × S3
× T 4 in the pure RR regime [29]
are generalized. A second problem arises from massless excitations, which require to loosen
the usual implementation of the Virasoro constraints [61]. The oneloop correction to the
BMN string for AdS3 × S3
× T 4 and AdS3 × S3
× S3
× S1 with pure RR flux has been
obtained within this framework in [61], while the respective correction to the short folded
string in such backgrounds has been derived in [38]. The oneloop correction to the BMN
string for AdS3 × S3
× T 4 with mixed flux has been computed in [60].
The outline is as follows. In section 2 we summarize some features of the rigid classical
spinning string solution on R × S3 ⊂ AdS3 × S3
× T 4. The pure NSNS limit and the
restriction to the su(2) sector are then discussed. In section 3 we compute the Lagrangian
of the quadratic fluctuations around the rigid spinning solution and solve its equations of
motion, hence obtaining their characteristic frequencies. In section 4 we check the results
of the previous section by rederiving the characteristic frequencies from the fluxdeformed
– 2 –
finitegap equations. In doing so, we neglect contributions from the massless excitations as
we will have proven in section 3 that their net contribution ultimately vanishes. In section 5
we make use of the characteristic frequencies previously obtained to compute the oneloop
correction to the dispersion relation. In section 6 we present an argument regarding the
extension to nonrigid strings in the NSNS limit. We close the article with summary and
conclusions. In the appendices we have collected conventions and detailed computations
that supplement the main text.
2
The R × S
3 string with mixed flux
In this section we review the dynamics of rigid closed spinning strings on a R × S3 ⊂
× T 4 background in the presence of both RR and NSNS threeform fluxes.
This kind of solution as been studied, for example, in references [
62–65
].1 After doing so,
we restrict ourselves to two regimes of interest, the su(2) sector and the pure NSNS limit,
which are the focus of the reminder of the article.
The AdS3 × S3 × T 4 background metric can be parameterized by
(2.1)
(2.2)
(2.3)
ds2 = −dz02 − z02 dt2 + dz12 + z12 dφ2 + dr12 + r12dϕ12 + dr22 + r22dϕ22 + X dγi2 ,
where the parameter q ∈ [0, 1] measures the mixing of the two fluxes.
In the case of pure RR flux, q = 0, the theory can be formulated in terms of a
MetsaevTseylin action akin to that of the AdS5 × S5 background [72]. The Lagrangian
associated to the bosonic spinning string ansatz in AdS3 × S3 reduces in this case to the
NeumannRosochatius integrable system [73, 74]. Turning on the NSNS flux introduces
an additional term in the latter which does not spoil integrability [63, 64]. In fact, the
complete Lagrangian remains integrable under this deformation [40]. The pure NSNS
flux limit, q = 1, is of particular interest because the action can be reexpressed as a
supersymmetric WZW model [75] (see also the recent paper [76] and references therein),
which leads to several simplifications.
Instead of considering the most general setup, the remainder of this paper is devoted
to bosonic classical spinning strings rotating in S3 at the center of AdS3 with no dynamics
1The general spinning string solution encompasses other well known solutions as particular limits which
are interesting enough to be studied separately. Examples are the giant magnon [
66–69
], spiky strings [66–68]
and multispike strings [70]. Spinning D1strings have also been studied [71].
– 3 –
along T 4. Thus we fix z0 = 1 and γi = 0, and impose the ansatz describing a closed string
rotating with two angular momenta in S3
t(τ, σ) = w0τ ,
ri(τ, σ) = ri(σ) ,
ϕi(τ, σ) = ωiτ + αi(σ) ,
i = 1, 2 .
(2.4)
Dealing with closed string solutions requires periodic boundary conditions of (2.4) on σ,
entailing
ri(σ + 2π) = ri(σ) ,
αi(σ + 2π) = αi(σ) + 2πmi ,
where mi are integer winding numbers.
After substituting the ansatz (2.4) into the
Polyakov action with the Bfield in the conformal gauge, we obtain
,
2
i=1
!
where the prime stands for derivatives with respect to σ and h = h(√λ) is the coupling
constant.2 In addition, we can construct three nonvanishing conserved charges from the
isometries of the metric: the energy
and the two angular momenta
E = 4πhw0 ,
J1 = 2h
J2 = 2h
Z 2π
Z 2π
0
0
dσ r12ω1 − qr22α2′ ,
dσ r22ω2 + qr22α1′ .
The EulerLagrange equations for the radial coordinates are
r1′′ = −r1ω12 + r1α1′2 − Λr1 ,
r2′′ = −r2ω22 + r2α2′2 − Λr2 + 2qr2(ω1α2 − ω2α1′) .
′
On the other hand, the equations of motion for the angular equations can be easily
integrated as the Lagrangian only depends on their derivatives
α1′ =
v1 + qr22ω2
r
2
1
,
α2′ =
v2 − qr22ω1
r
2
2
,
where vi are integration constants which can be understood as the momenta associated
to αi. Replacing α‘i by these momenta in the Lagrangian (2.6) leads to aforementioned
2The relationship between the coupling constant and the string tension is the same as the one for
AdS5 × S5 at first order, i.e. h = √4πλ + . . . , although it might receive corrections (both perturbative and
nonperturbative) in the ‘t Hooft coupling. However, it is known that the first correction O(1) should vanish
for the pure RR case [
25–27, 30, 37
] and it might vanish also for the mixed flux case.
(2.5)
(2.6)
(2.7)
(2.8)
(2.9)
(2.10)
(2.11)
(2.12)
deformation of the NeumannRosochatius system. The equations of motion must be
supplemented with the Virasoro constraints, which read
2
X
i=1
2
i=1
ri′2 + ri2(αi′2 + ωi2) = w02 ,
X ri2ωiαi′ = X viωi = 0 .
2
i=1
ri(σ) = ai .
(2.13)
(2.14)
(2.15)
(2.16)
(2.17)
(2.18)
(2.19)
(2.20)
(2.21)
(2.22)
On these solutions α‘i = mi and (2.10) and (2.11) simplify to
Λ = m12 − ω12 = m22 − ω22 + 2q(ω1m2 − ω2m1) .
Furthermore, the restriction (2.2) and the Virasoro constraint (2.14) determine
or, using the definitions of the angular momenta,
,
,
pure NSNS limit we commented above, while the second one corresponds to a restriction
to a su(2) subsector of the theory, which amounts to set the J1 = J2. Note that these two
regimes are not mutually exclusive.
Let us examine in more depth both cases. In the first limit the equations of motion
can be solved by
dispersion relation
where we have defined
ω1 =
J
4πh
,
J
ω2 =
4πh − (m1 − m2) ,
E = J + 4πhm1 .
where J = J1 + J2. Using the first Virasoro constraint (2.13) and (2.17) we can obtain the
On the other hand, if we restrict ourselves to the su(2) subsector, the second Virasoro
constraint (2.14) implies m1 = −m2 ≡ m, while the equations of motion and the definition
of the angular momenta imposes
J
4πh
ω1 =
= Υ + qm ,
J
ω2 =
4πh − 2qm = Υ − qm ,
stant radii
The most straightforward solutions to the equations of motion are those of
conHJEP07(218)4
Υ =
J
4πh − qm =
J1 + J2
4πh
− qm ,
– 5 –
for later convenience. The expression for the dispersion relation reads
E = pJ 2 − 8πhqmJ + 16π2h2m2 = 4πhpΥ2 + κ2m2 ,
(2.23)
with κ2 = 1 − q2.
3
Quadratic fluctuations around the classical solution
In this section we derive the Lagragian for the quadratic fluctuations around the rigid
spinning stringtype solutions on R × S3. We obtain the equation for the characteristic
frequencies for them, which greatly simplifies in both the case of two equal angular momenta
and the case of pure NSNS flux. The signed sum of the characteristic frequencies provide
the oneloop correction to the classical energy.
We derive this effective Lagrangian by splitting the targetspace fields into the classical
background and fluctuation fields and truncating the latter up to second order in the action.
We treat the bosonic fluctuations on the sphere, on the antide Sitter space, on the torus
and the fermionic fluctuations separately in order to make the section more readable.3
3.1
Bosonic fluctuations on S3
Regarding the fluctuations on the sphere, we take advantage of the spherical symmetry
and perform the following substitution
ri cos ϕi → ai cos(αi + ωiτ ) + r˜i cos(αi + ωiτ ) − ρi sin(αi + ωiτ ) ,
ri sin ϕi → ai sin(αi + ωiτ ) + r˜i sin(αi + ωiτ ) + ρi cos(αi + ωiτ ) .
where r˜i, ρi denote the perturbation fields.4
Introducing (3.1) into the Polyakov action with the Bfield in the conformal gauge, the
Lagrangian for the quadratic fluctuations of the fields turns out to be
( 2
X h
i=1
L˜S3 = h
− r˜i − ρ˙i2 + r˜i′2 + ρ′i2 + (r˜i2 + ρi2)(−ωi2 + αi′2) + 2r˜i(−ωiρ˙i + αi′ρ‘i)
˙2
− 2ρi(−ωir˜˙i + α‘ir˜i) − Λ(r˜i2 + ρi2)i + 2q (r˜22 + ρ22)(ω1α‘2 − α1ω2)
′ ′
+ r˜2(ω1ρ′2 − α2ρ˙2) − ρ2(ω1r˜2 − α1′r˜˙2) + a2
′ ′
− aa222 ρ1(r˜˙1α2′ − r˜1′ω2) + a2 (ρ˙1ρ′2 − ρ′1ρ˙2)
1 a1
a1
)
,
2r˜2 − a1
a2
r˜1 (ρ˙1α2′ − ρ′1ω2)
(3.1)
(3.2)
where the dot stands for derivatives with respect to τ .
3In this section we do not label the origin of the frequencies (S, AdS, T or F ) as no ambiguity arises.
4It is also possible to incorporate the fluctuations as ri → ai + r˜i and ϕi → αi + ωiτ + ϕ˜i instead. Both
choices are equivalent since ρi ≈ aiϕ˜i at first order.
– 6 –
Besides, an orthogonality requirement has to be satisfied between the classical solution
and the fluctuation fields. Such condition can be seen as a consequence of the
perturbation of the Lagrange multiplier in (2.6), promoted as Λ → Λ + Λ˜ . In particular, this
constraint reads
2
i=1
X air˜i = 0 .
Imposing it, the EulerLagrange equations for r˜2, ρ1 and ρ2 become
a1
−ρ¨1 + ρ“1 + 2 a2 (ω1 + qm2)r˜˙2 − (m1 + qω2)r˜2′ = 0 ,
−ρ¨2 + ρ“2 − 2 (ω2 + qm1)r˜˙2 − (m2 + qω1)r˜2′ = 0 ,
We are allowed to decompose r˜2, ρ1 and ρ2 in a base of exponential functions due to the
periodic boundary conditions in σ. Supplementing the decomposition with an expansion
in Fourier modes in τ , we write
ρj =
∞
X
6
n=−∞ k=1
X A(jk,n)eiωk,nτ+inσ ,
r˜2 =
X Bn(k)eiωk,nτ+inσ .
(3.5)
∞
X
6
n=−∞ k=1
The sum over k follows from the existence of six different frequencies associated to the same
mode number n [77]. Employing this ansatz, the EulerLagrange equations (3.4) become
the matrix equation
ωk2,n − n
2
0
M31
0
ωk2,n − n
2
M32
M13
M23
ωk2,n − n2 Bk,n
A1,k,n
A2,k,n ≡ M A2,k,n = 0 .
A1,k,n
Bk,n
where the unspecified matrix elements are explicitly
M13 = 2i [(ω1 + qm2)ωk,n − (m1 + qω2)n] a2
,
M23 = −2i [(ω2 + qm1)ωk,n − (m2 + qω1)n] ,
M31 = −2i [(ω1 + qm2)ωk,n − (m1 + qω2)n] a1a2 ,
M32 = 2i [(ω2 + qm1)ωk,n − (m2 + qω1)n] a12 .
a1
The existence of nontrivial solutions to (3.6) requires det M = 0, which provides the
following characteristic equation for the frequencies
(ωk2,n − n2) (ωk2,n − n2)2 − 4a12 [(ω1 + qm2)ωj,n − (m1 + qω2)n]2
−4a22 [(ω2 + qm1)ωk,n − (m2 + qω1)n]2 = 0 .
This equation has six solution as we commented above, and reduces to the one obtained
in [50] in the limit of a pure RR flux.
– 7 –
(3.3)
(3.4)
(3.6)
(3.7)
(3.8)
We should note that two frequencies corresponding to decoupled massless modes arise
as solutions to the characteristic equation. They emerge as a consequence of the
conformal gaugefixing condition [51]. Solving the linearized Virasoro constraints for the AdS3
decoupled massless field and substituting the solution back in the full Lagrangian remove
these spurious frequencies from (3.8), since their associated field gets cancelled. Therefore,
we can safely ignore them.
In principle, we can find the remaining solutions to the equation (3.8) as a series in
inverse powers of the total angular momentum J . Since expressions thus obtained are not
very enlightening, we focus exclusively on the two regimes we presented at the end of the
previous section.
In the su(2) sector (m1 = −m2 = m) the frequencies, written as a series in Υ, are
HJEP07(218)4
whereas in the pure NSNS limit they read
In the overlap of both regimes the frequencies can be written as follows
n2κ2
2Υ
+
We proceed analogously for AdS3 fluctuations using the parameterization
z0 cos t → (1 + z˜0) cos(κτ ) − χ0 sin(κτ ) ,
z0 sin t → (1 + z˜0) sin(κτ ) + χ0 cos(κτ ) ,
z1 sin φ → z˜1 ,
z1 cos φ → χ1 .
This choice leads to the following Lagrangian density for the fluctuation fields
2 ′ ′
+w02(z˜12 + χ1) − 2qw0(z˜1χ1 − χ1z˜1) .
L˜AdS3 = h z˜0′2 + χ′2
0 − z˜0 − χ˙ 0 − z˜1′2 − χ′12 + z˜˙12 + χ˙ 21 + 2w0 z˜0χ˙ 0 − χ0z˜˙0
˙2 2
– 8 –
(3.9)
(3.10)
(3.11)
(3.12)
(3.13)
This Lagrangian has to be supplemented with an orthogonality constraint between the
background and the fluctuation fields similar to (3.3), which in this case reads z˜0 = 0.
As a consequence, the field χ0 decouples, leading to two massless excitations that can be
ignored in view of the discussion above. The relevant EulerLagrange equations are
−z˜¨1 + z˜“1 − w0z˜1 + 2qw0χ′1 = 0 ,
2
−χ¨1 + χ“1 − w02χ1 − 2qw0z˜1′ = 0 .
(3.14)
An analogous expansion to (3.5) allows us to derive the characteristic equations for the
frequencies. In this case,
whose solutions are
(ωk2,n − n2 − w02)2 − 4q2n2w02 = 0 ,
(3.15)
All of them reduce to already known result in the nondeformed case [37, 50]. Furthermore,
the pure NSNS limit allow us to complete squares, obtaining
with uncorrelated signs.
3.3
Bosonic fluctuations on T 4
Since we consider no classical dynamics on the torus, we are led to a free Lagrangian for
the fluctuations. Therefore, the characteristic frequencies are5
ωk,n = ±n ± w0 ,
ωk,n = ±n ,
with one pair of solutions for each of the four coordinates.
3.4
Fermionic fluctuations
As our background solution is purely bosonic, the Lagrangian for the fermionic fluctuations
reduces to the usual fermionic Lagrangian computed up to quadratic order in the fermionic
fields. For a type IIB theory, the latter is given by
LF = i(ηαβδI˙J˙ − ǫαβ(σ3)I˙J˙)θ¯I˙ρα(Dβ) ˙K˙ θK ,
˜ J ˙
where the covariant derivative is [13, 15]
(Dα)I˙J˙ = δI˙J˙ ∂α − 4 ωαabΓaΓb
1
1
8
1
48
+ (σ1)I˙J˙eaαHabcΓbΓc +
(σ3)I˙J˙FabcΓaΓbΓc . (3.20)
We refer to appendix A for definitions and conventions.
5In the T 4 space one could also consider fluctuations with nontrivial windings. However, after averaging
over these windings, one is left only with the contribution from the zero winding sectors. We want to thank
Tristan McLoughlin for pointing us this issue.
– 9 –
We get the characteristic equation for the frequencies by substituting the classical
solution, fixing the kappa symmetry and expanding the fermions in Fourier modes. In
particular, the kappa gauge condition we choose is θ1 = θ
2
≡ θ as [29, 37]. The Fourier
expansion reads
∞
X
8
n=−∞k=1
θ =
Xθn(k)eiωk,nτ+inσ ,
su(2) for i = 3, 7, ωi = −(m1 + m2) + ωisu(2) for
i
m8
256
m2κ2(4κ2 − 5) + 4n2 + (3 − 4q2)Υ2 + 4ωk,n(w0 − ωk,n)
× m2κ2(4κ2 − 5) + 4n2 + (3 − 4q2)Υ2 − 4ωk,n(w0 + ωk,n)
× m2κ2(4κ2 − 3) − 4n2 + (1 − 4q2)Υ2 + 8qnw0 + 4ωk,n(ωk,n − w0)
× m2κ2(4κ2 − 3) − 4n2 + (1 − 4q2)Υ2 − 8qnw0 + 4ωk,n(ωk,n + w0) = 0 ,
where κ2 = 1 − q2. Note that it is a polynomial of eighth degree, which agrees with the
discussion above.
Solving this equation, we find the following frequencies for the su(2) sector:
where we have summed up to eight frequencies for each mode instead of up to the sixteen
frequencies that would be expected from the number of degrees of freedom [78]. We are
allowed to do so because only six of the ten targetspace coordinates are nontrivially
involved in the equations of motion and hence we can restrict ourselves to sixdimensional
gamma matrices. To recover the full set of frequencies we have to double the multiplicity
of each frequency ωk,n. We should remark that we have imposed periodic boundary
conditions to the fermionic fields as [50, 79, 80], relying on the discussion of the appendix E
of [81]. Imposing the vanishing of the determinant of the differential operator in (3.19),
the resulting expression for the characteristic equation is
(3.21)
(3.22)
(3.23)
(3.24)
ω1,n = n +
ω3,n = −n +
1
q − 2
q +
1
2
w0 ,
w0 ,
ω5,n = qn2 − q2w02 + Υ2 +
ω7,n = −
qn2 − q2w02 + Υ2 +
1
We stress that the massless frequencies remain as such independently from the mixing
parameter, cancelling the contributions from the T 4 modes for all values of q. Performing
the q → 1 limit manifestly simplifies the expressions in (3.23)6
ω1,n = n +
ω5,n = n +
1
1
where we have used that limq→1 w0 = Υ.
i = 4, 8 and ωi = ω
su(2) for the remainder.
i
the following shifts of the frequencies: ωi = (m1 + m2) + ω
Frequencies from the algebraic curve
In order to check the computations performed in the previous section, we derive again the
frequencies associated to the fluctuations using a different method that relies on the
integrability of our problem: the semiclassical quantization of the classical algebraic curve. We
start by constructing the eigenvalues of the monodromy matrix for the classical solution,
whose associated quasimomenta define a Riemann surface. This classical setting is
quantized by adding infinitesimal cuts to this surface, thus modifying the analytical properties
of the quasimomenta, which contain the oneloop correction to the energy.
The Riemann surface we are interested in presents only one cut, analogously to the
gR = g ⊕ 1 ∈ psu(1, 12)2 where7
one studied for the AdS5 × S5 scenario in [81]. Although the presence of the NSNS flux
deforms the construction of the algebraic curve, hence rendering the results from AdS5 ×S5
inapplicable, the procedure remains mostly the same and can be used as guideline. For the
full detailed derivation of the finitegap equations for general values of q we refer to [60],
where the authors apply them to the particular case of the BMN string. Although this
procedure neglects the contribution from the massless excitations, we already know from
the previous section that their contributions cancel each other.
We start the computation of the classical algebraic curve by choosing the gauge gL ⊕
g =
eiw0τ
Using the normalization from [60]8 the Lax connection associated reads
L(x) =
Lˆ(x) =
Lˆ(x)
0
0 !
Lˇ(x)
=
Lˆ(x)
0
0
ˆ 1
L x
!
,
ix
(x − s) x + 1s
w0/κ
0
0
0
−w0/κ
0
0
0
fluxes, appearing now at ±s and ± 1s , where s = s(q) is defined as
Notice that the usual ±1 poles of the Lax connection shift due to the presence of both
7Notice that the choice of representative in [60] was not correct. Even though ai2 = wJii holds for the
undeformed case, this relation gets modified due to the flux becoming the one shown in equation (2.18).
In any case, this problem does not affect the computation of the correction to the BMN spectrum therein.
Our quasimomenta for the rigid spinning string are the same as those obtained there after replacing their
Ω by our Υ.
8We have chosen this normalization over the one proposed in [40] because it facilitates our later
construction of the corrections to the classical curve by simplifying the relation between the Lax connection
and the Noether currents.
−m
e2im(σ+qτ)
κ − x
√Υ2−m2q2
κ
0
0
q
.
s =
r 1 + q
1 − q
0
0
m
q
κ − x
e−2im(σ+qτ) √Υ2−m2q2 .
κ
(4.1)
(4.2)
(4.3)
The quasimomenta associated to this Lax connection, obtained from the logarithm of the
eigenvalues of the monodomy matrix, are
pˆ1A(x) = −pˆ2A(x) = pˇ1A
= −pˇ2A
1
x
1
x
=
2πxw0
κ(x − s) x + 1s ,
pˆ1S(x) = −pˆ2S(x) =
pˇ1S(x) = −pˇ2S(x) = − κ(x + s) x − s
energy we add extra cuts to the Riemann surface. These cuts are infinitesimally small and
appear as poles on the quasimomenta. Depending on the sheets of the Riemann surface
the infinitesimal cuts connect, they correspond to different kinds of excitations. The precise
relations between them are
AdS3 : (pˆ1A, pˆ2A), (pˇ2A, pˇ1A) ,
S3 : (pˆ1S, pˆ2S), (pˇ2S, pˇ1S) ,
Fermions : (pˆ1A, pˆ2S), (pˆ1S, pˆ2A), (pˇ2A, pˇ1S), (pˇ2S, pˇ1A) .
The explicit residues of these poles are
res
x=xˆnAX
res
x=xˇnAX
δpˆiA = −(δ1i − δ2i)αˆ(xˆnAX )NnAX ,
δpˇiA = +(δ1i − δ2i)αˇ(xˇnAX )NnAX ,
res
x=xˆSnX
res
x=xˇSnX
δpˆiS = +(δ1i − δ2i)αˆ(xˆnSX )NnSX ,
δpˇiS = −(δ1i − δ2i)αˇ(xˇnSX )NnSX , (4.7)
where X is either A or S depending on which sheet the cut ends. The functions αˆ(x) and
αˇ(x) are defined as
αˆ(x) =
x
2
κh(x − s) x + 1s ,
αˇ(x) =
x
2
κh(x + s) x − s
To fix the positions where we have to add these poles we have to use the relation between
the quasimomenta above and below the branch cuts of the Riemann surface Cij, where i
and j (comprising both A or S and 1 or 2) label the sheets the cut connects,
pi+(x) − pj−(x) = 2πn ,
x ∈ Cij .
This equation not only constraints the positions of the poles xinj, but also the behaviour of
the corrections to the quasimomenta on them
pi(xinj) − pj(xinj) = 2πn ,
(δpi)+(xinj) − (δpj)−(xinj) = 0 .
(4.8)
(4.9)
(4.10)
On top of that, the quasimomenta also present poles on the same points as the Lax
connection. From the residue of the Lax connection we infer that
1
1
s x=s
s x=s
res δpˆ1A =
res δpˆ2A =
1
1
s x=s
res δpˆ1S = −s x=re1s/s δpˇ1A = −s x=re1s/s
s x=s
res δpˆ2S = −s x=re1s/s δpˇ2A = −s x=re1s/s
δpˇ1S ,
δpˆ1A = −s x=r−es1/s
−s x=r−es1/s
δpˆ2A = −s x=r−es1/s
δpˆ1S =
δpˆ2S =
1
1
s x=−s
res δpˇ1A =
s x=−s
res δpˇ1S ,
s x=−s
res δpˇ2A =
s x=−s
res δpˇ2S .
1
1
These restrictions provides us with enough information to completely fix the corrections
to the quasimomenta. The details of the construction are lengthy, so we have relegated
them to appendix B. We eventually obtain the following expression for the correction to
the dispersion relation
w0δΔ = X
nκxˆnAA − qn − w0 NˆnAA +
n
nκ
xˇAA − qn NˇnAA +
n
(n + 2m)κ
xˇSS
n
− qn NˇnSS
+ nκxˆnSS − qn + 2K(0) NˆnSS +
− qn (NˇnAS + NˇnSA)
+ nκxˆnF − qn − K(0) − w0κ (NˆnAS + NˆnSA)
where we still have to implement the expressions for each pole. We can check that in
the q → 0 limit we recover the undeformed AdS3 × S3 ⊂ AdS5 × S5 expressions [29].
Note that all −2qn terms in the expressions from appendix B have been replaced by −qn
terms to facilitate a later comparison with the frequencies obtained through the quadratic
fluctuations. This substitution is legitimated by the level matching condition, expressed as
X n
n
X
all exc.
Nn = 0 .
It is also convenient to define the frequencies Ωin as
w0δΔ = X h(Ωˆ nAA − w0)NˆnAA + Ωˇ nNˇnAA + Ωˆ nSS + 2K(0) NˆnSS + Ωˇ nSSNˇnSS
n
+ (Ωˆ nF − K(0) − κw0)(NˆnAS + NˆnSA) + ΩˇnF (NˇnAS + NˇnSA)i .
The value of the poles is determined by equation (4.10), which can be solved as a series
in Υ−1 for general values of q. Nevertheless, when taking the q → 1 limit, those equations
heavily simplify and we can find exact solutions after an appropriate regularization the
poles with κ factors. The solutions for general q are collected in the appendix C. Here we
(4.11)
HJEP07(218)4
(4.12)
(4.13)
(4.14)
just write down the solutions for q = 1, which read
κxˆqA→A1 = 2
κxˆqS→S1 = 2
κxˆqA→S1 = 2
n + Υ
n
n
n
n + Υ
n + Υ
,
,
,
xˇqA→A1 =
xˇqS→S1 =
xˇqA→S1 =
κ
κ
κ
n
2(n − Υ)
,
,
ωnB = 2n + (n + w0) + (n − w0) + 4n = 8n ,
ωnF = 2 h2 n +
w0
2
+ 2 n − 2
w0 i = 8n .
The κ factors cancel after substituting into the oneloop correction (4.12), revealing that
HJEP07(218)4
the limit is well behaved despite the apparent singularities that appear. Plugging back into
the aforementioned equation and using the definitions (4.14) we get
Ωˆ qA→A1 = n + 2Υ ,
Ωˆ qS→S1 = n + 2Υ ,
Ωˆ qF→1 = n + 2Υ ,
Ωˇ qA→A1 = n − 2Υ ,
Ωˇ qS→S1 = n + 4m − 2Υ ,
Ωˇ qF→1 = n + 2m − 2Υ .
We end this section comparing the expressions of the characteristic frequencies obtained
through both methods and discuss their differences. Here we focus on the su(2) sector in the
pure NSNS limit. The comparison for general values of q is relegated to the appendix C,
but the arguments presented in the discussion below are still valid.
When we collate equations (3.11), (3.17) and (3.24) with (4.16) we observe that they
are equal up to some shifts. Such shifts fall into two categories, shifts of the mode number
and shifts of the frequencies, and can be understood as a change of reference frame [81]. As
our frequencies present the same shift structure and the shifts at q = 1 cancel each other
when summed, we confirm that both computations are in agreement. Therefore, we can
extract the oneloop correction to the dispersion relation using the frequencies from either
of the methods.
5
Computation of the oneloop correction
In this section we put together the characteristic frequencies from previous sections to
compute the oneloop shift to the dispersion relation in the su(2) sector. This correction
is given by the sum of the fluctuation frequencies
E1loop = E0 + δE ,
δE =
1
2w0 n∈Z
X (ωnB − ωnF ) ,
where ωnB and ωnF are the bosonic and fermionic contributions respectively.
Firstly, we consider the pure NSNS the limit in the su(2) sector. Using the frequencies
obtained from the quadratic fluctuations, these contributions are
(4.15)
(4.16)
(5.1)
(5.2)
The net contribution from the frequencies thus vanishes, resulting in the vanishing of the
oneloop correction in the pure NSNS limit
This vanishing remains valid out of the su(2) sector, i.e. if m1 6= −m2.
Let us focus now on the case of general mixed flux. Here
ωnAdS = qn2 + 2qnw0 + w02 + qn2 − 2qnw0 + w02
= q(n + qw0)2 + κ2w02 + q(n − qw0)2 + κ2w02 ,
ωnT = 4n ,
ωnF = 4n + 4qn2 − q2w02 + Υ2 ,
whereas the S3 contribution is more conveniently written after summing part of the series
in a square root, by analogy with the results we have obtained from the algebraic curve,
ωnS = Υ + pn2 − 2nqΥ + Υ2 + O Υ−2
+
−Υ + pn2 + 2nqΥ + Υ2 + O Υ−1
= p(n + qΥ)2 + κ2Υ2 + p(n − qΥ)2 + κ2Υ2 + O Υ−1 .
In order to perform the infinite sum on the mode number we use the method presented
in [54], which consists in replacing such sum by an integral weighed with a cotangent
function
2πi X ωn =
n∈Z
C
I dz π cot(πz)ωz ,
where the contour C encircles the real axis.
The frequencies related to the AdS space (5.4) and fermions (5.6) contain square roots
with complex branch points. The same applies to the leading order of the frequencies
related to S3 (5.7). Choosing our branch cuts from each branch point to infinity allow
us to deform the contour of the integral so it encircles them. As the branch points are of
order iΥ, we can consistently approximate the cotangent by one in the semiclassical limit,
reducing the contour integral to the usual integral of a square root. In our case, (5.8)
involves integrals of the form
I(a, b) = −
dzp(z − a)2 + b2 =
p(z + ia)2 − b2
=
1
Z iΛ
a+ib
Z Λ
b−ia
(5.9)
where Λ is a sharp cutoff which regularizes (5.9). The sum of these contributions gives
X
n∈Z
ωnS + ωnAdS + ωnT − ωn
F
= I(qΥ, κΥ) + I(−qΥ, κΥ) + I(qw0, κw0) + I(−qw0, κw0)
−4I 0, qΥ2 − q2w02
A direct inspection shows that the quadratic and linear contributions of the regulator
cancel but the logarithmic contribution does not. This fact reflects that higher orders in
the expansion in the mode number of the S3 frequencies contribute to the cancellation.
In order to check this statement we should expand the characteristic frequencies of the S3
modes in the regime of large mode number n
S
ω1,n = n + (1 − q)w0 +
S
ω2,n = n − (1 − q)w0 +
κ2[Υ2
− m2(1 + q2)]
where we have written down just the solutions with +n as leading contribution. Since
matching with the frequencies expanded in Υ is not direct, the labelling here is arbitrary.
Taking into account the relation
− m2(1 + q2)] = 4(Υ2
− q2w02) − 2κ2w02 ,
(5.12)
we can check that after replacing the first two contributions in (5.10) by the sum over
n of the frequencies (5.11), the logarithmic divergence now cancels. This proves that the
oneloop correction to the dispersion relation is finite for all values of the mixing parameter.
6
A comment about nonrigid strings
In this section we provide a plausibility argument for the vanishing of the oneloop
correction for the nonrigid case in the pure NSNS limit.
First we must summarize some results about the nonrigid spinning string. The
existence of an Uhlenbeck constant of motion allows us to reduce the equations of
motion (2.10), (2.11) and (2.12) into a firstorder differential equation. This equation is solved
in terms of Jacobi elliptic functions, giving
r12(σ) = c1 + c2 sn2 c3σ, ν ,
where ci and ν are constants that depend on ωi, vi and q whose explicit expressions can be
found in [64]. For our purposes it is enough to know that the elliptic parameter ν vanishes
the same as the one for nonrigid spinning strings in AdS5 × S5 [73, 74].
when q → 1. Another important feature of the solution (6.1) is that its functional form is
The oneloop corrections for spinning folded and pulsating strings in AdS5 × S5 were
computed in [56–58], where it was shown that the EulerLagrange equations of all the
fluctuations can be rewritten as the eigenvalue problem of a singlegap Lam´e operators
[∂x2 + 2ν¯2sn2(xν¯) + Ω2]fΛ(x) = ΛfΛ(x) .
Here ν¯ is related to the elliptic modulus of the Jacobi function involved on the classical
solution, x is a linear function of σ for spinning strings and Ω2 is a linear function of
the square of the characteristic frequency. The specific relations depend on which of the
(6.1)
(6.2)
fluctuations are we considering, but in all the cases ν¯ vanishes if the elliptic modulus of
the classical solution vanishes.
Since ν vanishes in the limit q = 1, all the Lam´e equations reduce to a wave equation
in such limit as long as the functional form of this eigenvalue problem remains unaltered
when we include the NSNS flux. Accordingly, the characteristic frequencies should become
a linear function of the mode number n and the oneloop correction would vanish in the
nonrigid case.
7
Conclusions
In this article we have derived the oneloop correction to the dispersion relation of rigid
closed spinning strings on the R
×S3 subsector of a type IIB string theory on AdS3 ×S3 ×T 4
background supported by a mixture of RR and NSNS fluxes. This correction has been
obtained through the computation of the characteristic frequencies via both background
field expansion and the finitegap equations without massless excitations. We have proved
that the correction remains finite for all values of the parameter that controls the mixing
of the two fluxes. In addition, all the frequencies can be computed analytically in the pure
NSNS limit, where they become linear and the oneloop correction vanishes. We have also
argued that this vanishing probably extends to nonrigid strings.
It is important to remark that our finitegap computation does not take into account
the massless fields because the background field expansion showed that their net
contribution vanishes. Including the massless contributions into the finitegap equations would need
a further extension of the procedure followed here. It would be desirable to generalize the
method presented in [61] to deal with these massless excitations in the case of mixed flux.
A natural generalization of our analysis would be the precise computation of the
oneloop correction for nonrigid strings. Even though the vanishing of this correction on the
in [56–58] for AdS5 × S5 string theory could be generalized to this end.
q → 1 limit seems plausible, an explicit check is needed. In principle, the procedure used
Semiclassical giant magnon solutions support this statement, as they can be obtained
as a particular regime of general spinning strings. These solutions were studied in [
43, 69
],
where it was shown that they display a linear dispersion relation.9 Furthermore, in [43] it
is argued that such dispersion relation holds at each perturbative order for giant magnons
understood as magnonic bound states, which in particular implies the vanishing of its
oneloop correction up to corrections in the coupling constant h. The construction of
the Smatrix for elementary magnonic excitations starting from symmetry considerations
strongly supports this fact [15]. Equation (5.3) and the discussion of section 6 suggests that
the vanishing of the oneloop correction in the pure NSNS limit is not a characteristic of
rigid spinning string solutions but it might be a feature of general spinning strings. Proving
this statement would shed light on the role of spinning semiclassical solutions in the mixed
flux scenario, and its q → 1 limit.
9Note that in [
69
] a more general magnonic dispersion relation has been derived. Nonetheless,
semiclassical solutions can be typically mapped via AdS/CFT duality in the Ji → ∞ limit, according to which the
dispersion relation therein indeed becomes linear.
Besides, it would be desirable to compare our results with the prediction from the
string Bethe ansatz for the dressing phase. The comparison is based on the realization
of rigid spinning string solutions in the su(2) sector from the point of view of the Bethe
equations, along the lines of Hern´andezL´opez construction [79]. In particular, such
realization involves a large quantity of su(2) Bethe roots lying in a single connected curve of the
complex plane. The construction shows that there are two possible sources of corrections:
quantum corrections to the classical integrable structure and wrapping corrections coming
from finite size effects. In the AdS5 × S5 background, the comparison allowed to extract
strongcoupling corrections to the dressing phase that later were proven to be in agreement
with the predictions derived from crossing relations. In contrast to the AdS5 × S5
comparison, in AdS3 × S3
× T 4 with pure RR flux both computations of the dressing phase did
not match. The presence of massless modes, with no analogy in AdS5 × S5, is believed to
underlie this disagreement as wrapping contributions are exponentially supressed by the
mass of the excitations involved.10 On the other hand, it has been recently shown [46, 76]
that in the pure NSNS limit wrapping corrections are absent and both the Smatrix and
the string Bethe equations present a remarkably simple structure [46, 76].11 Taking all
of this into account, we expect that some control over massless wrapping corrections in
the Hern´andezL´opez construction can be gained by means of the parameter q, helping to
elucidate the disagreement.
Finally, we want to point out that the spectrum of closed strings with zero winding
and zero momentum on the torus on a F1/NS5brane supported with RR moduli has been
studied in [47]. It is also shown there that the mixed flux background studied here can be
retrieved from the nearhorizon geometry of such scenario. Thus, our computation may
also be relevant to the pure NSNS theory at a generic point in its moduli space.
Acknowledgments
The authors want to thank Rafael Hern´andez, Alessandro Torrielli and Santiago Varona
for valuable comments on the manuscript. We also want to thank Ben Hoare for feedback
on this work and for pointing us the articles [56, 57]. J. M. N. also want to thanks the
Trinity College Dublin for its hospitality during the “Workshop on higherpoint correlation
functions and integrable AdS/CFT”.
A
Conventions
In this appendix we collect the conventions we used during the article, in particular those
concerning the fermionic Lagrangian (3.19).
10Perturbative analyses around the BMN vacuum of the dressing phase in the RR [39] and mixed flux [45]
regimes has also led to discrepancies with the allloop Smatrix predictions and the dispersion relation,
obtained both from the underlying symmetries of the system, when dealing with massless excitations.
11It is not obvious how to apply the pure NSNS limit to the fluxdeformed Bethe equations proposed
in [9], based on the centrally extended lightcone symmetry algebra. This is so because such algebra cannot
be centrally extended when q = 1.
HJEP07(218)4
Firstly, we fix our index notation. We use greek indices for the worldsheet coordinates,
lowercase latin indices for the tendimensional Minkowski flat spacetime, uppercase
undotted indices for target spacetime and uppercase dotted indices separate the 32 components
MajoranaWeyl spinors in 10 dimensions into two 16 components spinors.
The worldsheet coordinates, denoted as τ and σ, are raised and lowered with the flat
metric ηαβ =diag(−1, 1) and its associated LeviCivita symbol is defined so ǫτσ = 1. Flat
Minkowskian coordinates takes values in {0, . . . , 9}, being raised and lowered with the flat
metric ηab =diag(−1, 1, . . . , 1).
ΩcabEc, obtained from the former using
In order to relate target space coordinates and flat Minkowskian coordinates we
construct the vielbeins Ea = EAadXA and the spin connection differential 1forms Ωab =
For the AdS3 × S3 space the vielbeins are given by
E0 = z0 dt ,
E3 = −r1 dr2 + r2 dr1 ,
E1 = −z0 dz1 + z1 dz0 ,
E4 = r1 dϕ1 ,
E2 = z1 dφ ,
E5 = r2 dϕ2 ,
The spin connection differential oneforms are written down as
Ω01 = −z1 dt ,
Ω21 = z0 dφ ,
Ω43 = r2 dϕ1 ,
Ω53 = −r1 dϕ2 ;
the remaining components either could be obtained through Ωab = −Ωba, or are zero. Both
vielbeins and spin connection can be pulled back to the worldsheet using a solution of the
equations of motion as
eaα = EAa∂αXA ,
ωαab = ecαΩcab .
When the constant radii classical solution presented in section 2 is plugged, the nontrivial
pulled back vielbiens are
e0 = κ dτ ,
e4 = a1(ω1dτ + α1′dσ) ,
e5 = a2(ω2dτ + α2′dσ) ,
and the nontrivial pulledback spin connection differential oneforms are
ω43 = a2(ω1dτ + α1′dσ) ,
ω53 = −a1(ω2dτ + α2′dσ) .
Habc and Fabc refer the Minkowskian components of the NeveuSchwarzNeveuSchwarz
and RamondRamond threeform fluxes respectively, given by [15]
H/ a = 2q E/ a(Γ012 + Γ345) + (Γ012 + Γ345)E/ a ,
Fabc = 12κ(Γ012 + Γ345) ,
where the slash denotes contraction with the gamma matrices. Integrability and conformal
symmetry fix q2 + κ2 = 1 [40].
(A.1)
(A.2)
(A.3)
(A.4)
(A.5)
Computation of the corrections to the quasimomenta of the onecut
We treat the AdS3, the S3 and the fermionic contributions separately to simplify the
computation. This separation allow us to alleviate notation by dropping the sheet labels
both on the poles x and on the number of excitations/cuts Nn.
B.1
Contribution from AdS3 excitations
The classical algebraic curve presents a cut only on the sheets related to the sphere, making
the process of computing the AdS3 modes equivalent to the computation of fluctuations
around the BMN string solution. This computation has been already performed in [60], so
we just quote their result here (without imposing the level matching condition)
δΔ = 2 X
n
ˆ
Nn hκαˆ(xˆn) − 1 − w0
2qn
+ Nˇn
hκαˇ(xˇn)
xˇ2
n
2qn
− w0
.
(B.1)
Substituting the quasimomenta (4.4) on the cut condition (4.10) relates the functions αˆ
and αˇ with the mode number and the position of the poles
(B.2)
(B.3)
(B.4)
(B.5)
(B.6)
pˆ1A(xˆn) − pˆ2A(xˆn) = 2
pˇ2A(xˇn) − pˇ1A(xˇn) = 2
2πxˆnw0
κ(xˆn − s) xˆn + 1s
2πxˇnw0
1
κ(xˇn + s) xˇn − s
=
=
xˆn
xˇn
4πhαˆ(xˆn)w0 = 2πn ,
4πhαˇ(xˇn)w0 = 2πn .
Plugging them into the previous expression we obtain
δΔ =
1
w0 n
X
Nˆn (nκxˆn − 2nq − w0) + Nˇn
nκ
xˇn − 2nq
.
B.2
Contribution from S3 excitations
Adding infinitesimal cuts in the sheets related to S3 has to take into account the existence
of a branch cut on the classical algebraic curve. The presence of both cuts generates two
kinds of corrections to the quasimomenta, one coming from the poles associated to the
infinitesimal cuts and another coming from shifts of the branch points of the cut due to
the addition of these poles. Thus we have to split the ansatz for the corrections to the
quasimomenta in two contributions
where the factor K(x) dividing the second term comes from the fact that ∂x0 x − x0 ∝
1/√x − x0. Using both the analytic properties of the corrections to the quasimomenta on
the cut (4.10) and the inversion symmetry of the algebraic curve
√
δpˇ2S = f (x) +
δpˆiS(x) = δpˇiS
g(x)
K(x)
1
x
.
,
δpˇ1S(x) = f (x) − K(x)
,
g(x)
δpˆ1S(x) = f
1
− Kg x1x1 ,
δpˇ2S(x) = f (x) +
δpˆ2S(x) = f
1
x
g(x)
K(x)
,
+
g x1
K x1 .
The explicit form of these functions is obtained from the known pole structure of the
corrections and their asymptotic properties. Equation (4.7) entails that the combinations
δpˇ1S + δpˇ2S and δpˆ1S + δpˆ2S have no poles, hence we are free to choose f (x) = 0. On the other
hand, δpˇ2S − δpˇ1S has a simple pole with residue 2αˇ(xˇ)Nˇn at xˇ and δpˆ2S − δpˆ1S has a simple
pole with residue 2αˆ(xˆ)Nˆn at xˆ. Furthermore, δpˇiS (respectively δpˆiS) have poles at −s and
1s (respectively s and − s ) whose residues are correlated with those of the δpˇiA (respectively
1
δpˆiA) quasimomenta at the same points, see (4.11). As a consequence, we can write down
the following ansatz for the function g(x)
g(x) = a +
sδa1 +
x + s
δa2
sx − 1
+ X
n
Nn
ˇ αˇ(xˇn)K(xˇn)
− Nˆn αˆxˆ(xˆnn(1)K−(x1x/ˆxnˆ)n)
= a + −2κδa− + 2x(δa+ + qδa−)
1
κ(x + s) x − s
+ X
n
Nn
ˇ αˇ(xˇn)K(xˇn)
− Nˆn αˆxˆ(xnˆn(1)K−(x1x/ˆxnˆ)n)
where a and δai are unknown constants.
As we mentioned in section 4, our conventions for the definition of the Lax connection
allow us to relate it with the Noether currents of the system for large values of the spectral
parameter in a simple way. Therefore, the asymptotic behaviour of the quasimomenta can
be related with conserved global charges. In particular, for the excitations we are interested
in we have
xl→im∞ κhx δpˇ1S(x) = −Nˇ ,
xl→im∞ κhx δpˆ1S(x) = Nˆ ,
xl→im∞ κhx δpˇ2S(x) = Nˇ ,
xl→im∞ κhx δpˆ2S(x) = −Nˆ ,
with N = Pn Nn. These asymptotic properties are mapped to asymptotic properties of
g(x) at x → ∞ and x → 0. The behaviour at infinity fixes the constant a to
xl→im∞ g(x) = a = xl→im∞
(δpˇ2(x) − δpˇ1(x)) =
K(x)
2
mNˇ
h
.
Besides, the behaviour around zero provides us two conditions that fix the remaining two
unknown constants
we can fix the rest of the quasimomenta related to the sphere
g(x)
K(x)
= δpˆ2(1/x) − δpˆ1(1/x) ,
g(0)
K(0)
+
g′(0)
g(0) K′(0)
K(0) − K(0) K(0)
ˆ
N
x + O(x2) = − κh
x + O(x2) .
(B.7)
(B.8)
,
(B.9)
(B.10)
(B.11)
(B.12)
To simplify our results we write the residues αˆ and αˇ in terms of the poles using the
equation (4.10) and the classical values of the quasimomenta (4.5) and (4.6)
pˆ1S(xˆn) − pˆ2S(xˆn) = 2πn ⇒ 2
pˇ2S(xˇn) − pˇ1S(xˇn) = 2πn ⇒ 2
xˆn
xˇn
αˆ(xˆn)hK(1/xˆn) = n ,
αˇ(xˇn)hK(xˇn) = n + 2m .
The O(1) of (B.12) reads
g(0) = a + 2δa− − X
while the O(x) gives us
δa+ − qδa− = − X
n 4h
xˇn
n + 2m Nˇn + nxˆn −
2K(0)
κ
ˆ
Nn .
Now that we know the residues at −s and 1/s of the quasimomenta associated to the
sphere, we can compute the correction to, for example, the pˇ2A quasimomenta, as δΔ is
computed from its asymptotic behaviour. Using that K(−s) = K(1/s) = w0 we can write
δpˇ2A(x) = −2κδa− + 2x(δa+ + qδa−) ,
1
w0κ(x + s) x − s
and thus
δΔ
= xl→im∞ x δpˇ2A(x) =
2(δa+ + qδa−) ,
w0κ
n
Nˆnn
2h
+
Nˇn(n + 2m) !
2h
= 0 ⇒ δa− = X (Nˆn + Nˇn)n
4h
n
, (B.15)
δΔ = X
κ
n w0
xˇn
− 2qn Nˇn + nxˆn − 2qn −
ˆ
Nn .
(B.18)
2K(0)
κ
B.3
Contribution from fermionic excitations
As opposed to the previous cases, we have to distinguish between the fermionic
contributions N AS and N SA, both hatted and checked.
Nevertheless, equations (4.4), (4.5)
and (4.6) imply that the differences of classical quasimomenta are equal two by two
pˆ1A(x) − pˆ2S(x) = −pˆ2A(x) + pˆ1S(x) ,
pˇ2A(x) − pˇ1S(x) = −pˇ1A(x) + pˇ2S(x) ,
(B.19)
hence both kind of poles are equal in pairs xˆAS = xˆSA and xˇAS = xˇSA.
As we have to deal with the cut of the classical solution too, we can use an analogue
ansatz to the one we used for the S3 modes (B.5), but with different functions f (x) and
g(x) that reflect the pole structure of fermionic fluctuations (4.7). The ansa¨tze for such
functions are
g(x) = b +
1
κ(x + s) x − s
2(δa+ + qδa−)x − 2κδa− + X
1
κ(x + s) x − s
2(δb+ + qδb−)x − 2κδb− + X
n
n
αˇ(xˇn) nˇn − xˆn(1 − xxˆn) nˆn ,
αˆ(xˆn)
x − xˇn
K(xˇn)αˇ(xˇn) Nˇn − xˆn(1 − xxˆn)
x − xˇn
K(1/xˆn)αˆ(xˆn) Nˆn
(B.13)
(B.14)
(B.16)
(B.17)
(B.20)
(B.21)
where we have defined 2nˇ ≡ Nˇ AS
− Nˇ SA and 2Nˇ ≡ Nˇ AS + Nˇ SA, with similar expression
for the hatted ones. Again, the known asymptotic behaviour of the quasimomenta fixes
the unknown constants.
Let us start with the function f (x)
n
xˆ2
n
+ O
1
x2
f (x) = a + 2δa− − X αˇ(xˇn)nˇn
− X αˆ(xˆn)nˆn + −
+ 2δa− s − s
+ O
1
− X αˇ(xˇn)nˇn
xˇ2
n
n
These are four equations for three unknowns. We can check that the equations are
compatible and, in fact, solved by
− X αˆ(xˆn)nˆn x + O(x2) = − κh
+ O(x2) .
xˆn
2(δa+ + qδa−)
nˇx
αˇ(xˇn)nˇn +
αˆ(xˆn)nˆn ,
−2δa+ = X αˇ(xˇn)nˇn
xˇn − q + X αˆ(xˆn)nˆn
κ
xˆn
n
κ
xˆn − q .
Substituting back these expressions, f (x) can be rewritten in a more compact form
1
hκ(x + s) x − s
xnˇn +
nˆn
In addition, the function g(x) has the expansions
g(x) = b + O
= K(∞) κhx
+ O
g(x) = b + 2δb− − X αˇ(xˇn)K(xˇn)Nˇn
− X αˆ(xˆn)K(1/xˆn)Nˆn + −
1
x
= mNˇ + O
h
1
x
,
+2δb− s − s
= −
K(0)Nˇ x
κh
+ O(x2) .
n
− X αˇ(xˇn)K(xˇn)Nˇn
xˇ2
n
The computation of the explicit value of the unknown constants is simplified when the
residues are rewritten in terms of the poles via eq. (4.10)
pˇ2S(xˇn) − pˇ1A(xˇn) =
2πhαˇ(xˇn)K(xˇn) +
xˇn
2πwˇn
h
− m = 2πn ⇒
n
xˆn
− X αˆ(xˆn)K(1/xˆn)Nˆn x + O(x2)
a = 0 ,
2δa− = X
n
f (x) =
1
x
n
1
xˇ
n
xˇn
xˇn
x
ˇ
N
xˇn
xˆn
X
n
n
pˆ1A(xˆn) − pˆ2S(xˆn) = 2πn ⇒
(B.22)
1
(B.23)
(B.24)
(B.25)
(B.26)
2(δb+ + qδb−)
κ
#
xˇn
xˆ
αˇ(xˇn)K(xˇn) =
n + m − wˇn
αˆ(xˆn)K(1/xˆn) =
n − wˆn
h
h
(B.27)
write the unknown coefficients as
xˇnw0 xˆnw0
where we have defined wˇn ≡ κ(xˇn+s)(xˇn− 1s ) and wˆn ≡ κ(xˆn−s)(xˆn+ 1s ) . Therefore we can
b =
,
n
2h
κ
xˇn
−2δb− = X
wˇn − n Nˇn +
wˆn − n ˆ
Nn ,
δb+ =
κ " 2qhδb− + X wˇn − n − m Nˇn + X
n
K(0)
+ (wˆn − n)xˆn Nn ,
ˆ
#
Now that we have reconstructed the δpˇiS quasimomenta, we can focus on the antide
Sitter sheets. For example, we can construct δpˇ2A and extract δΔ from the comparison of
the residues of the former at −s and 1/s with the residues of δpˇ2A. As the steps involved
in the construction of δpˇ2A are similar to those in fixing f (x), we can write down directly
δpˇ2A =
1
hκ(x + s) x − s
−δΔ
2
+ X Nn − nˆn + X x(nˇn − Nˇn) !
ˆ
.
1 − xxˆn
n
res δpˇ2A + res δpˇ2A = res δpˇ2S + res
and using that K(−s) = K(1/s) = w0, we get
δΔ
− hκ
+ X
n
1
hκ s + 1s
X
n
+
s
1+ sxˆn
s − xˆn
1
+
1
s − xˆn
hnˆκn −s+Nˆ1ns +
nˆn +
s
2
s + xˇn
+
s(1 + sxˇn) hκ s + 1s
1
nˇn − Nˇn #
s
2
s + xˇn
+
1
s(1+ sxˇn)
nˇn +
2(δb+ + qδb−) ,
κw0
(B.28)
(B.29)
which reduces to
xˇn/κ − 2qn Nˇn + (nκxˆn − K(0) − 2qn − κw0) Nˆn .
(B.30)
C
Finite gap frequencies for general values of q
In this appendix we write down the frequencies computed using the finitegap equations
for general values of the mixing parameter q. The equations (4.10) for the placement of
the poles can be solved exactly for general q for the AdS3 and the fermionic fluctuations,
but not for the S3 fluctuations, which have to be expressed as a series on Υ−1.
The values for the poles involving first two kinds of excitations are
xˆnAA =
nq + w0 ± pn2 + 2nqw0 + w02 ,
xˇnAA = −nq + w0 ∓ pn2 + 2nqw0 + w02 =
nκ
nκ
xˆnAS = xˆnSA =
xˇnAS = xˇnSA =
nq + w0 ± pn2 + Υ2 + 2nqw0 ,
nκ
q(n + m) − w0 ± pΥ2 + (m + n)(m + n − 2qw0)
nκ
nq − w0 ± pn2 − 2nqw0 + w02
,
,
(C.1)
while for the sphere are
κxˆnSS =
κxˆnSS =
κ
xˇSS =
n
xˇSS =
n
−
= ±
nq + Υ + pn2 + 2nqΥ + Υ2
n
nq + Υ − pn2 + 2nqΥ + Υ2
= ±
κ2√n2 − 4m2
2Υ
+
(n − 2m)qκ2
2Υ2
+
+
m2qκ2
Υ2
q(2m + n) − Υ − p(2m + n)2 − 2q(2m + n)Υ + Υ2
(2m + n)
(2m + n)
q(2m + n) − Υ + p(2m + n)2 − 2q(2m + n)Υ + Υ2
2Υ
2Υ
+
(n + 2m)κ2 ± κ2pn(n + 4m)
κ2pn(n + 4m) (n2 + 4mn + 2m2)qκ2
−
m2qκ2
Υ2
2Υ2
Substituting into equation (4.12) and using the definition (4.14) we get
ΩˆnAA = nq + w0 ±
ΩˇnAA = nq − w0 ±
qn2 + 2nqw0 + w02 ,
qn2 − 2nqw0 + w02 ,
Ωˆ nF = nq − Υ ± pn2 + Υ2 + 2nqw0 ,
Ωˇ nF = (m + n)q − w0 ± pΥ2 + (m + n)(m + n − 2qw0) ,
and
Ωˆ nSS = nq − Υ + pn2 + 2nqΥ + Υ2 +
κ2n√n2 − 4m2
2Υ
m2nqκ2
Υ2
+
−
+
m2qκ2
Υ2
qκ2m2
Υ2
Ωˇ nSS = q(2m + n) − Υ + p(2m + n)2 − 2q(2m + n)Υ + Υ2 +
= −2Υ + 2(n + 2m)q −
−
(n + 2m)2κ2
qκ2(n + 2m)[(n + 2m)2 − 2m2]
Υ2
Ωˇ nSS = ±
= ±
2Υ
2Υ
κ2(n + 2m)pn(n + 4m)
qκ2(n + 2m)(n2 + 4mn + 2m2)
κ2(n + 2m)p(n + 2m)2 − 4m2
qκ2(n + 2m)[(n + 2m)2 − 2m2]
+
2Υ2
We immediately observe that Ωˆ nAA match frequencies ω1,n and ω3,n from equation (3.16) up
to a shift by a constant, while Ωˇ nAA match frequencies ω2,n and ω4,n from equation (3.16)
up to the opposite shift.
The comparison between fermionic frequencies is not so immediate, since it requires a
shift of the mode number n. In fact the combination Ωˆ nF−qw0 + w20 is identical to ω5,n and
ω7,n from the equation (3.23). A similar relation relates ΩˇnF with ω6,n and ω8,n with an
extra contribution from the winding.
The comparison of sphere frequencies is more involved since we do not have a closed
expression for general values of q. Instead we can compare them at the level of the
characteristic equation. From the equation (3.8) and equations (B.13) and (B.14) we infer
ωnS = −ω−Sn, these relations can be rewritten as Ωˆ S−n = ωnS and ΩˇS−n−2m = ωnS .
that ΩˆnS = −ωnS and Ωˇ nS−2m = −ωnS. Using the reality condition for the fluctuation fields,
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