Semiclassical strings in AdS(3) X S^2

0.16 MB·

by  Bogeun Gwak

PreprinttypesetinJHEPstyle-HYPERVERSION
AdS S2
Semiclassical strings in
3
×
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2
Bogeun Gwaka, Bum-Hoon Leeab, Kamal L. Panigrahic and Chanyong Parkd

n
a
J

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3 a
Department of Physics, Sogang University, Seoul 121-742, Korea

]
h b
Center for Quantum Spacetime (CQUeST), Sogang University, Seoul 121-742, Korea
t
-
p
c
e Department of Physics, Indian Institute of Technology Guwahati, Guwahati-781 039,
h India
[

2 d National Institute for Mathematical Sciences, 385-16 Doryong-dong, Yuseong-gu,
v
Daejeon 305-340, Korea
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9
7
2
.
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E-mail: [email protected], [email protected], [email protected],
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9 [email protected]
0
:
v
Xi Abstract:Inthispaper,weinvestigate thesemiclassicalstringsinAdS3 S2,inwhichthe
×
r string configuration of AdS3 is classified to three cases dependingon the parameters. Each
a of these has a different anomalous dimension proportional to logS, S1/3 and S, where S is
a angular momentum on AdS3. Further we generalize the dispersion relations for various
2
string configuration on AdS3 S .
×
Keywords: dispersion relation, spike.
Contents
1. Introduction 1
2
2. Semiclassical String configurations in AdS3 S 2
×
2
3. Classification of the string configurations on AdS3 S 4
×
2
3.1 Spike on AdS3 and a point-like string on S 6
2
3.2 Spike on AdS3 and circular string on S 8
2
3.3 Circular string on AdS3 and magnon on S 10
4. More general solution 10
5. Conclusions 12
1. Introduction
A remarkable development in the study of string theory of last decade is the celebrated
string theory-gauge theory duality [1], which relates the spectrum of semiclassical string
5
states on AdS5 S to the operator dimensions of = 4 supersymmetric Yang-Mills
× N
(SYM) theory in four dimensions[2, 3]. This state-operator matching has been achieved in
the planar limit, which simplifies considerably both sides of the duality. Recently another
exampleofgaugetheory-stringtheorydualityhasbeenproposedbasedontheworldvolume
3
dynamics of multiple M2-branes. This relates type IIA string theory in AdS4 CP with
×
1
= 6 Chern-Simons matter theory in three dimensions [4] .
N
In the study of the gauge/gravity duality, an interesting observation is that the = 4
N
SYMtheorycanbedescribedbytheintegrable spinchain model[12,13,14]. Itwasfurther
noticed that the string theory also has as integrable structure in the semiclassical limit.
This integrability nature of both the gauge theory and string theory side has been used
extensively to understand the duality better. In this connection, Hofman and Maldacena
(HM)[15]consideredaspeciallimitwheretheproblemofdeterminingthespectrumofboth
sides becomes rather simple. The spectrum consists of an elementary excitation known as
magnon which propagate with a conserved momentum p along the long spin chain. In the
dualformulation, themostimportantingredientis semiclassical stringsolutions, which can
be mapped to long trace operator with large energy and large angular momenta. For some
related work see for example [16]-[32].
2 3
Not so long back the giant magnon and spike solutions for the string on S or S
have been studied [33, 34, 35]. In addition, the solitonic string configuration in the AdS
1for related interesting work see [5, 6, 7, 8, 9, 10,11].
– 1 –
space, whose anomalous dimension corresponds to that of the twist two operator of SYM
theory or thecuspanomaly, was considered[36,37,38,39]. Morerecently thesemiclassical
3
string configuration on AdS3 S has also been investigated in a special parameter region
×
3
[31, 40]. However a general class of solutions for string moving in AdS3 S background
×
has been lacking. The aim of this paper is to classify such solutions with the hope that a
better understanding of these will enable us to investigate more complicated solutions of
the dual theory on the boundary.
2
In this paper, we consider the semiclassical strings moving in AdS3 S background
×
and investigate the relation among various conserved charges of the string configuration at
various parameter regions. Dependingon the parameter region, the macroscopic strings on
AdS3 can be classified to three cases, whose anomalous dimension is proportional to logS,
S1/3 or S, where S is a conserved angular momentum on AdS3. After that, we generalize
2
rotating string on AdS3 to the one rotating on AdS3 S and study the corresponding
×
dispersion relation or the anomalous dimension.
The rest of the paper is organized as follows. In the section 2, we write down the most
2
general equations describing the rotating string in AdS3 S background. In the section
×
3, first we classify various possible string configuration in AdS3. Depending on the region
of parameter there are three distinct solutions with different anomalous dimensions. We
further generalize them further by looking at the various shapes of the string. We write
down the relevant dispersion relations in all the cases. Section 4 is devoted to the study of
2
more general string configurations in AdS3 S and we find out the dispersion relations
×
among various conserved charges. In section 5, we present our conclusions.
2. Semiclassical String configurations in AdS S2
3
×
2
In this section we will study the general rotating string solution in AdS3 S . We start
×
2
by writing down the relevant metric for AdS3 S in a particular coordinate system,
×
1
2 2 2 2 2 2 2 2 2 2
ds = R [ cosh ρdt +dρ +sinh ρdφ +dθ +sin θdψ ]. (2.1)
4 −
The Polyakov action for the string moving in this background can be written as
1
S = d2σ√ dethhαβ∂ xµ∂ xνG . (2.2)
α β µν
4π −
Z
We choose the following parametrization for the semiclassical rotating string in this back-
ground
t = kτ +h1(y), ρ= ρ(y), φ= ωτ +h2(y),
θ = θ(y), ψ = ντ +g(y), (2.3)
where y = aτ +bσ. The relevant equations of motion reduce to
1 π0
h = (ak ),
′1 b2 a2 − cosh2ρ
−
– 2 –
1 π2
h = (aω ),
′2 b2 a2 − sinh2ρ
−
1 π4
g = (aν ), (2.4)
′ b2 a2 − sin2θ
−
where π0, π2 and π4 are integration constants. The above equations of motion have to be
supplemented by the Virasoro constraints, which are simply given by, in a particular form
2 2
a +b
0 = T +T T ,
ττ σσ τσ
− ab
0 = T +T 2T . (2.5)
ττ σσ τσ
−
The first Virasoro constraint gives rise to a relation among various parameters
νπ4 = kπ0 ωπ2. (2.6)
−
The second one becomes
1
2 2
0 = −(b2 a2)2(b k +2bkπ0)
−
2 2 2 2
1 (π π )sinh ρ+π
2 2 2 2 2 2 0 2
+ρ′ + (b2 a2)2 2bωπ2+b (ω −k )sinh ρ+ −cosh2ρsinh2ρ
− (cid:18) (cid:19)
2
1 π
2 2 2 2 4
+θ′ + (b2 a2)2 ν b sin θ+2bνπ4+ sin2θ .
− (cid:18) (cid:19)
2 2 2
= k +k +k , (2.7)
− 0 1 s
where in the last equality we have defined
1
2 2 2
k0 = (b2 a2)2(b k +2bkπ0),
−
2 2 2 2
1 (π π )sinh ρ+π
2 2 2 2 2 2 2 0 2
k1 = ρ′ + (b2 a2)2 2bωπ2+b (ω −k )sinh ρ+ −cosh2ρsinh2ρ ,
− (cid:18) (cid:19)
2
1 π
2 2 2 2 2 4
ks = θ′ + (b2 a2)2 ν b sin θ+2bνπ4+ sin2θ . (2.8)
− (cid:18) (cid:19)
Then, the differential equations for ρ and θ can be rewritten as
1
2 2 2 2 6 2 2 2
ρ′ = (b2 a2)coshρsinhρ b (ω −k )sinh ρ+ 2bωπ2+b (ω −k )
− h (cid:16)
2 2 2 4 2 2 2 2 2 2 2 2
+(b a )k1 sinh ρ+ 2bωπ2+(b a ) k1 +(π2 π0) sinh ρ+π2 ,
− − −
θ′2 = (b2 1a2)2sνin2(cid:17)b22θ −sin4(cid:16)θ+ (b2−b2aν22)2ks2 − 2bπν4 sin2θ−(cid:17)bπ2ν422 . i (2.9)
− (cid:20) (cid:18) (cid:19) (cid:21)
The equation for θ can be rewritten as
bν
2 2 2 2
θ = (sin θ sin θ)(sin θ sin θ ), (2.10)
′ (b2 a2)sinθ max − − min
− q
– 3 –
2 2 2 2
for ks(b −a ) > 2bνπ4 only. The condition sinθmax = 1 for the infinite size magnon or
spike gives a relation
bν +π4
k = . (2.11)
s b2 a2
−
Using this, sinθ becomes
min
π4
sinθ = , (2.12)
min
bν
where bν > π4.
3. Classification of the string configurations on AdS S2
3
×
From now on, we will consider the special case, k = ω > 0 only. An example for k = ω was
6
investigated in Ref. [31]. For κ= ω, the equations of the AdS part are reduced to
1
4 2 2
ρ = ∆sinh ρ+Γsinh ρ π ,
′ (b2 a2)coshρsinhρ − 2
− q
1 π2
φ = aω , (3.1)
′ (b2 a2) − sinh2ρ
− (cid:18) (cid:19)
with
2 2
Γ = ∆+π π ,
0 2
−
2 2 2 2
∆ = (a b ) k1 2bωπ2. (3.2)
− −
Since there exists a solution only when the inside of the square root in the first equation
of Eq. (3.1) is positive, we will investigate the range of ρ accordingly. Depending on the
parameters, the range of ρ can be classified as the follows:
I. ∆ > 0
In this case, there exists one boundary value ρ so that the range of ρ is given by
min
ρ ρ < ρ = and ρ is rewritten as
min max ′
≤ ∞
√∆ sinh2ρ+sinh2ρ0 2 2
ρ = sinh ρ sinh ρ , (3.3)
′ b2 a2 sinhρ coshρ − min
p
− q
where depending on the sign of Γ, on has the following choices
i) Γ > 0
Γ2+4π2∆+Γ
2 2
sinh ρ0 =
2∆
p
Γ2+4π2∆ Γ
2 2
sinh ρ = − , (3.4)
min
2∆
p
ii) Γ = 0
2
∆+π
2 2 0
sinh ρ0 = sinh ρmin = , (3.5)
√∆
p
iii) Γ <0
– 4 –
Γ 2+4π2∆ Γ
2 2
sinh ρ0 = | | −| |
2∆
p
Γ 2+4π2∆+ Γ
2 2
sinh ρ = | | | |. (3.6)
min
2∆
p
In addition, the string configuration can also beclassified into various cases by looking
at the shape of the string. For this, we define the slope of the string at a fixed τ as
∂ρ ρ
′
= . (3.7)
R ≡ ∂φ φ
(cid:12) (cid:12) (cid:12) ′(cid:12)
(cid:12) (cid:12) (cid:12) (cid:12)
Forexample,thestringslopeisinfinity,an(cid:12)dth(cid:12)est(cid:12)rin(cid:12)gconfigurationhasacuspatρ . For
(cid:12) (cid:12) (cid:12) (cid:12) min
more details, we introduce sinh2ρ = π2 satisfying φ = 0 in which the slope diverges.
c aω ′ R
case (i) If ρ < ρ , the string slope becomes 0 at ρ and a constant at ρ = .
c min min max
∞
case (ii) In the case of ρ = ρ , as previously mentioned, the slope at ρ becomes
c min min
infinity so that this string configuration has a cusp at ρ and a constant slope at ρ =
min max
.
∞
case (iii) If ρ > ρ but finite, the string configuration is same as case (i) except
c min
that it includes a point ρ where the sign of the slope is opposite.
c
case (iv) If ρ = ρ = , the slopebecomes zero at ρ andinfinity at ρ . This
c max min max
∞
case can be obtained when we consider a= 0.
II. ∆ = 0
In this case, the range of ρ is given by ρ ρ < . For Γ > 0 ρ becomes
min ′
≤ ∞
√Γ sinh2ρ sinh2ρ
min
ρ′ = b2 a2 sinhρ−coshρ , (3.8)
p
−
with
2
π
2 0
sinh ρ = . (3.9)
min 2 2
π π
0 2
−
Theclassification of thestringconfiguration is similar totheprevious case. Note that since
ρ becomes zero at ρ = , the string configuration is slightly different from the previous
′
∞
cases. For ρ < ρ , the slope vanishes at ρ and ρ = . For ρ = ρ , the
c min min max c min
R ∞
string slope becomes infinity at ρ and zero at ρ = . For ρ > ρ but finite,
min max c min
∞ R
becomes zero at both ρ and ρ = . For ρ = ρ , with a = 0, is zero at ρ
min max c miax min
∞ R
and infinity at ρ = .
max
∞
In the case of Γ 0, there is no string configuration since the inside of the square root
≤
in the first equation of Eq. (3.1) becomes negative and hence ρ becomes imaginary.
′
III. ∆ < 0
In this case, there exist two boundary values for Γ > 0 so that the range of ρ is given
by ρ ρ ρ . Then, ρ is
min max ′
≤ ≤
2 2 2 2
sinh ρ sinh ρ sinh ρ sinh ρ
∆ max min
− −
ρ = | | , (3.10)
′ bp2−a2q(cid:0) sinhρ (cid:1)co(cid:0)shρ (cid:1)
– 5 –
2 2
where sinh ρ and sinh ρ are
max min
Γ+ Γ2 4π2 ∆
2 2
sinh ρ = − | | ,
max
2∆
p | |
Γ Γ2 4π2 ∆
2 2
sinh ρ = − − | |. (3.11)
min
2∆
p | |
For ρ < ρ or ρ > ρ , the slope vanishes at ρ and ρ . Unlike the previous
c min c max min max
cases, the case of a = 0 corresponds to the case of ρ > ρ . For ρ = ρ , the string
c max c min
configuration has a cusp at ρ where becomes infinite and a zero slope at ρ . For
min max
R
ρ < ρ < ρ , the slopes at ρ and ρ vanish and there exist a point where the
min c max min max
sign of slope is changed. For ρ = ρ , the slope vanishes at ρ and becomes infinite at
c max min
ρ . Once again for Γ 0, there is no string configuration because ρ becomes imaginary.
max ′
≤
2
In the infinite size limit for the giant magnon or spike solutions for the string on S
with θ = π/2, the conserved charges are given by:
max
2T ρmax (b2ωcosh2ρ aπ0)
E = dρ − ,
(b2 a2)b ρ
(cid:12) − Zρmin ′ (cid:12)
(cid:12)(cid:12) 2T ρmax (b2ωsinh2ρ aπ2)(cid:12)(cid:12)
S = (cid:12) dρ − (cid:12),
(b2 a2)b ρ
(cid:12) − Zρmin ′ (cid:12)
(cid:12)(cid:12) 2T π/2 (b2νsin2θ aπ4) (cid:12)(cid:12)
J = (cid:12) dθ − , (cid:12)
(cid:12)(cid:12)(b2−a2)b Zθmin θ′ (cid:12)(cid:12)
(cid:12)(cid:12) 2 ρmax 1 π2 (cid:12)(cid:12)
∆φ = dρ (aω ) ,
(cid:12)(b2 a2) ρ − sinh2ρ (cid:12)
(cid:12) − Zρmin ′ (cid:12)
(cid:12)(cid:12) 2 π/2 1 π4 (cid:12)(cid:12)
∆ψ = (cid:12) dθ (aν ) , (cid:12) (3.12)
(cid:12)(cid:12)(b2−a2)Zθmin θ′ − sin2θ (cid:12)(cid:12)
(cid:12) (cid:12)
(cid:12) (cid:12)
where we consider only the positive quantities. For simplicity, we assume that the ends of
(cid:12) (cid:12)
2
the open string are located at ρ in the AdS space and at the same time, θ on S .
max max
2
3.1 Spike on AdS3 and a point-like string on S
2
At first, we consider the string configuration located at a point on S (θ = 0). For this,
′
we choose ν = π4 = 0, which makes the angular momentum and the angle difference in
the ψ directions to vanish. So, the string solution for ν = π4 = 0 corresponds to the string
2
extend in AdS but a static point particle on S . Once again we consider the situation case
by case like the previous section for different parameter regions.
I. ∆ > 0
2
1) We first consider the case with a = 0 and b ω = a(π0 π2). The interesting physical
6 6 −
quantities of this string configuration are
2 b2ω a(π0 π2) T ρmax sinhρ coshρ
E S = − − dρ ,
− b√∆ 2 2 2 2
(cid:0) (cid:1) Zρmin sinh ρ+sinh ρ0 sinh ρ sinh ρmin
−
q
(cid:0) (cid:1)(cid:0) (cid:1)
– 6 –
2T ρmax sinhρ coshρ b2ωsinh2ρ aπ2
S = dρ − ,
b√∆ 2 2 2 2
Zρmin sinh ρ+sinh ρ(cid:0)0 sinh ρ sinh(cid:1)ρmin
−
2 ρmax q(cid:0) coshρ a(cid:1)ω(cid:0)sinh2ρ π2 (cid:1)
∆φ = dρ − , (3.13)
√∆ 2 2 2 2
Zρmin sinhρ sinh ρ+s(cid:0)inh ρ0 sinh ρ(cid:1) sinh ρmin
−
q
where ρ = . In the case of(cid:0)sinhρ >> 1, t(cid:1)h(cid:0)e dispersion relation(cid:1)can be approxi-
max min
∞
mately written as
2
b ω a(π0 π2)
E S − − T∆φ. (3.14)
− ≈ abω
Note that the dominant contribution of the above integral comes from the large ρ region.
So the calculation of the integral gives rise to
2
2 b ω a(π0 π2) T
E S − − ρ ,
max
− ≈ b√∆
(cid:0) (cid:1)
bωT
S e2ρmax. (3.15)
≈ 4√∆
As a result, the dispersion relation for ρ = becomes
max
∞
b2ω a(π0 π2) T 4√∆
E S − − log S , (3.16)
− ≈ (cid:0) b√∆ (cid:1) bωT !
which is the generalization of the GKP string configuration [36].
2
2) Let us consider the case b ω = a(π0 π2), the dispersion relation is
−
E S = 0, (3.17)
−
which looks like that of the BPS vacuum state. In this case, S and ∆φ are given by
S ≈ 4b√ωT∆ e2ρmax +4 sinh2ρmin−sinh2ρ0− 2ba2πω2 ρmax ,
(cid:20) (cid:18) (cid:19) (cid:21)
2aω
∆φ ρ . (3.18)
max
≈ √∆
So S and E can be rewritten in terms of ∆φ as
E = S bωT e∆φ√∆/aω e2ρ0 e2ρmin + 2aπ2 ∆φ√∆ . (3.19)
≈ 4√∆ " − − b2ω 2aω #
(cid:18) (cid:19)
3) Now, consider the case a= 0. In this case, ∆φ has a finite value because the integral is
regular even at ρ and ρ = . On the contrary, E S and S diverge at ρ =
min max max
∞ − ∞
so that we can not represent E S and S in terms of ∆φ. However, we can compute the
−
dispersion relation for this string configuration, which is
bωT 4√∆
E S log S . (3.20)
− ≈ √∆ bωT !
– 7 –
Notice the logarithmic behavior of the anomalous dimension.
II. ∆ = 0
For this case, E S and S are given by
−
2
b ω a(π0 π2) T
E S − − eρmax,
− ≈ b√Γ
(cid:0) (cid:1)
bωT
S e3ρmax, (3.21)
≈ 12√Γ
so that the dispersion relation becomes
E S b2ω−a(π0−π2) T 12√Γ 1/3S1/3. (3.22)
− ≈ (cid:0) b√Γ (cid:1) bωT !
2
Like the previous case, if we put b ω a(π0 π2) = 0, the string configuration becomes
− −
BPS-like configuration. In this case, the dispersion relation becomes
2/3
bωT
E S 121/3 S1/3. (3.23)
− ≈ √Γ
(cid:18) (cid:19)
III. ∆ < 0
For this case, the conserved charges are given by
2
b ω a(π0 π2) T
E S = − − π,
− b ∆
(cid:0) (cid:1)
| |
2
T b ω
p 2 2
S = (sinh ρmax +sinh ρmin) aπ2 π. (3.24)
b ∆ 2 −
| | (cid:18) (cid:19)
From these, the dispersionprelation can be rewritten as
2
2 b ω a(π0 π2) ∆
E S = − − | |S. (3.25)
− b2ωΓ 2aπ2 ∆
(cid:0) − | (cid:1)|
For a= 0, the dispersion relation is reduced to a simple form
2 ∆
E S = | |S. (3.26)
− Γ
2
3.2 Spike on AdS3 and circular string on S
Here, we consider the following parameter region: ν = 0 and π4 = 0. As previously
6
mentioned, ν = 0 implies θ = 0. So the string configuration in this parameter region
′
2
describes a circular string on S , which is extended in the ψ-direction with an angular
momentum J at the fixed θ = θ . To describe the angular momentum J and the angle
c
difference ∆ψ, θ is not a good variable so that using
dθ θ
′
= , (3.27)
dρ ρ
′
– 8 –
we can change the integral with respect to θ in Eq. (3.12) to
2T ρmax aπ4
J = dρ ,
−(b2 a2)b ρ
(cid:12) − Zρmin ′ (cid:12)
(cid:12)(cid:12) 2 ρmax π4(cid:12)(cid:12)
∆ψ = (cid:12)−(b2 a2) dρρ sin(cid:12)2θ . (3.28)
(cid:12) − Zρmin ′ c(cid:12)
(cid:12) (cid:12)
(cid:12) (cid:12)
From these, we find (cid:12) (cid:12)
2
a sin θ
c
J = T∆ψ. (3.29)
b
Especially, since the angular momentum becomes zero for a = 0, the string solution is
2
reduced to a static circular string on S .
For ∆ > 0, the dispersion relation is slightly modified to
b2ω a(π0 π2+π4) T 4√∆
E S J − − log S . (3.30)
− − ≈ (cid:0) b√∆ (cid:1) bωT !
2
So the BPS-like configuration appears at b ω = a(π0 π2 +π4). The dispersion relation
−
2
for a = 0 can be represented in terms of the angle difference ∆ψ on S instead of ∆φ on
AdS
2
bωsin θ
c
E S = T∆ψ
− π4
bωT 4√∆
log S . (3.31)
≈ √∆ bωT !
For ∆ = 0, when a = 0 the modified dispersion relation is
6
E S J b2ω−a(π0−π2+π4) T 12√Γ 1/3S1/3, (3.32)
− − ≈ (cid:0) b√Γ (cid:1) bωT !
and at a = 0 it becomes
2/3
bωT
E S 121/3 S1/3 (3.33)
− ≈ √Γ
(cid:18) (cid:19)
In the case of ∆ < 0, the dispersion relation for a = 0 is modified to
6
2
2 b ω a(π0 π2+π4) ∆
E S J = − − | |S, (3.34)
− − b2ωΓ 2aπ2 ∆
(cid:0) − | | (cid:1)
and one for a= 0 becomes
2 ∆
E S = | |S. (3.35)
− Γ
2
As shown in the above results, at a= 0 the dispersion relations for a circular string on S
2
are same as ones for a point-like string on S .
– 9 –