Nambu-Goto Like Action for the AdS_5 x S^5 Superstrings in the Generalized Light-Cone Gauge

0.2 MB·English

by  Hiroshi Itoyama

January, 2008
OCU-PHYS 288
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2 Nambu-Goto Like Action

n
a for the AdS S5 Superstrings
J 5
×
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in the Generalized Light-Cone Gauge
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Hiroshi Itoyamaab∗, Takeshi Ootab† and Reiji Yoshiokaa‡
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v
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aDepartment of Mathematics and Physics, Graduate School of Science,
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4 Osaka City University
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. 3-3-138 Sugimoto, Sumiyoshi, Osaka 558-8585, JAPAN
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bOsaka City University Advanced Mathematical Institute (OCAMI)
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v 3-3-138 Sugimoto, Sumiyoshi, Osaka 558-8585, JAPAN
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X
r
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Abstract
We reinvestigate the κ-symmetry-fixed Green-Schwarz action in the AdS S5
5
×
backgroundinaversion ofthelight-cone gauge. Inthegeneralized light-cone gauge,
the action has been written in the phase space variables. We convert it into the
standard action written in terms of the fields and their derivatives. We obtain a
Nambu-Goto type action which has the correct flat-space limit.
∗ [email protected]
† [email protected]
‡ [email protected]
1
1 Introduction and Summary
It is a challenging problem to quantize the Green-Schwarz (GS) action [1, 2] in the
AdS S5 background [3]. The knowledge of the spectrum will tell us the strong coupling
5
×
dynamics of the large N gauge theory through the AdS/CFT correspondence. One of the
difficulties in covariant quantization of the GS action stems from the existence of the local
κ-symmetry, which halves the fermionic degrees of freedom [1, 2, 4]. One approach to this
problem is to abandon the covariance and fix the κ-symmetry non-covariantly. But after
the κ-symmetry fixing, the model is still a constrained system due to the world-sheet dif-
feomorphism. Various gauges [5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16] have been proposed
to fix these symmetries. Especially, the uniform light-cone gauge, a generalization of
the flat-space light-cone gauge [17] to the curved space background, has been extensively
investigated in [11, 12, 13, 14, 15, 16].
In our previous paper [18], we adopted the light-cone gauge and considered the Hamil-
tonian dynamics of the GS action by using the physical degrees of freedom. The action
is formulated in the first order formalism, i.e., is written in terms of the phase space
variables. This first-order formulation is suited for considering the problem of canonical
quantization.
Unfortunately, the reduced action in the generalized light-cone gauge still has an
involved form. Beforeconsidering thequantization problem ofthe action, we would like to
investigate quantum fluctuation in various limits. Extensive study around the plane-wave
region was done in [11]. But, in general, the first-order Lagrangian is not so convenient to
study the quantum spectrum in various limits, such as the BMN limit [19], the Hofman-
Maldacena limit [20], or Maldacena-Swanson limit [21]. They can be better investigated
by using the Lagrangian written in terms of the fields and their derivatives. Therefore, in
thispaper, wereformulatetheGSactiontothestandardforminthegeneralizedlight-cone
gauge.
After κ-symmetry fixing and taking the generalized light-cone gauge
X+ = κτ, P = √λω, (1.1)
−
with κ and ω are constants, we find the Nambu-Goto like action for the GS action in the
AdS S5 background. Its bosonic part has the following form:
5
×
1 G
S = d2ξ λdet( + )+√λκω +− . (1.2)
ij ij
2π − J G G
Z (cid:18)q −−(cid:19)
Here = G ∂ Xm∂ Xn is the induced metric, induced from the target space metric
ij mn i j
G
for the transverse spatial directions. (The indexes m,n run over the transverse directions:
2
m,n = 1,2,...,8). The additional term comes from the longitudinal directions. The
ij
J
’t Hooft coupling λ is related to the radius R of AdS and S5 as √λ = R2/α′. The full
5
Lagrangian which includes the fermions will be given by (3.45). This is a Nambu-Goto
type Lagrangian.
It is natural to appear the Nambu-Goto type action when the world-sheet diffeomor-
phism is fixed by certain gauge conditions other than the conformal gauge. Solving the
equations of motion for the world-sheet metric yields the Nambu-Goto type action. For
example, the Nambu-Goto type action for the GS model in AdS S5 in the static gauge
5
×
can be found in [8].
IncontracttoordinaryNambu-Gotoactions, wehavechosenthesignbeforethesquare
root term to be positive. This comes from the requirement that the action must have the
correct flat-space limit. Indeed, the Lagrangian (3.45) goes to the correct κ-symmetry
fixed light-cone gauge Lagrangian in the limit.
The Lagrangian (3.45) will serves as a starting point for developing the various limits
and investigating the quantum fluctuations.
This paper is organized as follow. In section 2, using the bosonic sigma model as an
example, we explain the procedure to obtain the standard Lagrangian in the generalized
light-cone gauge. In section 2.1, we start from the Lagrangian in the first-order formalism
and arrive at the standard one. In section 2.2, we derive it without going to the first-
order form. In section 3, we first briefly review our notation for the GS action. In section
3.1, the κ-symmetry fixing is done and the action in the AdS S5 background is given.
5
×
In section 3.2, the κ-symmetry fixed GS action in the generalized light-cone gauge is
obtained. This is our main result. In section 3.3, it is discussed that the action has the
correct flat-space limit. Some of our notations are summarized in Appendix.
2 The bosonic sigma model
The action for the bosonic sigma model in the D-dimensional curved target space is given
by
1
S = d2ξ , (2.1)
2π L
Z
where
1
= √λhijG ∂ Xm∂ Xn, (2.2)
mn i j
L −2
Here m,n = 0,1,...,D 1, ξ0 = τ, ξ1 = σ, ∂ = ∂/∂ξi, hij = √ ggij (i,j = 0,1).
i
− −
3
The conjugate momenta are given by
P = √λh0jG ∂ Xn. (2.3)
m mn j
−
The target space metric is assumed to have the following form
G dXmdXn = G dXadXb+G dXmdXn. (2.4)
mn ab mn
Here a,b = denote the longitudinal directions, m,n = 1,2,...,D 2 denote the
± −
transverse directions. We assume that ∂/∂X± is a Killing vector.
Let us decompose the Lagrangian into two pieces:
= + , (2.5)
1 2
L L L
1
= √λhijG ∂ Xa∂ Xb, (2.6)
1 ab i j
L −2
1
= √λhijG ∂ Xm∂ Xn. (2.7)
2 mn i j
L −2
The first part and the second part are related to the metric for the longitudinal
1 2
L L
directions and the metric for the transverse directions respectively.
Under the target space metric ansatz (2.4), the momenta P which is conjugate to
−
X− is given by
P = √λh0jG ∂ Xa. (2.8)
− −,a j
−
The generalized light-cone gauge is given by the following two conditions1:
X+ = κτ, P = √λω = const, (2.9)
−
which fixes the world-sheet diffeomorphism.
2.1 From the first order form to the standard Lagrangian
The reduced action in the generalized light-cone gauge is given by (we use the notation
of [18])
1
S = d2ξ P X˙m , (2.10)
red m LC
2π −H
Z
(cid:0) (cid:1)
1 Originally,thesecondconditionwasgivenby∂1(P−)=0. MostgeneralsolutionisP− =P−(σ). But
withoutlossofgenerality,wecansetP to aconstantbyredefiningthe world-sheetspacevariableσ and
−
conjugate momenta such that P (σ)dσ = P′dσ′ with P′ constant. Therefore, we adopt the condition
− − −
P =const as one of the gauge conditions.
−
4
where
= κP . (2.11)
LC +
H −
This is the first-order Lagrangian = (Xm,P ) written in terms of the transverse
m
L L
coordinates Xm and their conjugate momenta P .
m
Here P is a solution of
+
G++P2 +2√λωG+−P +C = 0, (2.12)
+ +
where
C = λω2G−− +λG ∂ Xm∂ Xn +KmnP P , (2.13)
mn 1 1 m n
1
Kmn := Gmn + G ∂ Xm∂ Xn. (2.14)
ω2 −− 1 1
Explicitly, P is given by
+
ε G+−
P = λ(G+−)2ω2 G++C √λω , (2.15)
+ G++ − − G++
p
where ε = 1.
±
Now let us convert this first-order Lagrangian into the standard form. The equations
of motion for P
m
∂P
X˙m +κ + = 0 (2.16)
∂P
m
yield the following relations
ε λω2(G+−)2 G++CX˙m = κKmnP . (2.17)
n
−
p
It is convenient to introduce and by
ij ij
J G
κ2 ω2
:= , := , = := 0, (2.18)
J00 G++ J11 G J01 J10
−−
:= G ∂ Xm∂Xn, i,j = 0,1. (2.19)
ij mn i
G
Let K be the inverse of Kmn:
mn
(Gmm′∂1Xm′)(Gnn′∂1Xn′)
K = G . (2.20)
mn mn
− +
11 11
J G
Then
ε
P = λω2(G+−)2 G++CK X˙n. (2.21)
m mn
κ −
By substituting this relation intop(2.13), we have
1
C = λω2G−− +λ + λω2(G+−)2 G++C K X˙mX˙n. (2.22)
G11 κ2 − mn
(cid:0) (cid:1)
5
Then we have
λω2G−− +λ +(λω2/κ2)(G+−)2K X˙mX˙n
11 mn
C = G . (2.23)
1+(1/κ2)G++K X˙mX˙n
mn
Note that
+det( )
K X˙mX˙n = J11G00 Gij . (2.24)
mn
+
11 11
J G
With some work, we find
λκ2
λω2(G+−)2 G++C = ( + )2. (2.25)
11 11
− −det( + ) J G
ij ij
J G
Assuming det( + ) < 0, we have
ij ij
J G
λ
λω2(G+−)2 G++C = ε′ κ( + ). (2.26)
11 11
− s−det( ij + ij) J G
J G
p
Here ε′ = sign(κ( + )).
11 11
J G
Let J be a matrix defined by
ij
J := + , (2.27)
ij ij ij
J G
and Jij be its inverse. We can see that
1 det( + )
K X˙n = G ( + )X˙n ∂ Xn = Jij Gij G J0j∂ Xn.
mn mn 11 11 01 1 mn j
+ J G −G +
11 11 11 11
J G J G
(cid:2) (cid:3) (2.28)
Now we finally have
P = εε′ λdet(J )G J0j∂ Xn. (2.29)
m ij mn j
− −
q
By substituting this expression into the first order form of the action, we get the reduced
Lagrangian in the generalized light-cone gauge.
Summary: The Lagrangian of the bosonic sigma model in the generalized light-cone
gauge is given by
1
S = d2ξ , (2.30)
red LC
2π L
Z
where
G
= εε′ λdet( + ) +√λκω +−. (2.31)
LC ij ij
L − − J G G
−−
q
Here
κ2 ω2
= , = , = = 0, = G ∂ Xm∂ Xn. (2.32)
J00 G++ J11 G J01 J10 Gij mn i j
−−
6
2.2 Rederivation of the reduced Lagrangian
In this subsection, we rederive the reduced Lagrangian (2.31) without bypassing the first-
order formalism.
Let us restart from the Lagrangian (2.2). Let us decompose it as follows
= + , (2.33)
1 2
L L L
1 1
= √λhijG ∂ Xa∂ Xb, = √λhijG ∂ Xm∂ Xn. (2.34)
1 ab i j 2 mn i j
L −2 L −2
The generalized light-cone gauge conditions are given by
X+ = κτ, P = √λh0jG ∂ Xa = √λω = const. (2.35)
− −,a j
−
We interpret the second condition as the following relation for X˙−:
h01 ω G
X˙− = ∂ X− κh00 +− . (2.36)
− h00 1 − G − G
(cid:18) (cid:19) −− (cid:18) −−(cid:19)
With some work, we have
= ′ +P X˙−, (2.37)
L1 L1 −
where
G 1 ω2
′ = √λκω +− + √λ
L1 G 2 h00G
(cid:18) −−(cid:19) −− (2.38)
1 κ2 1 G h01
√λh00 + √λ −−(∂ X−)2 +√λω ∂ X−.
− 2 G++ 2 h00 1 h00 1
(cid:18) (cid:19)
Note that P X˙− is a total τ-derivative term. So, we use ′ as the Lagrangian in the
− L1
generalized light-cone gauge.
In ′, the field X− appears only through the form of ∂ X−. The field ∂ X− plays the
L1 1 1
role of an auxiliary field. The equations of motion for ∂ X− gives
1
ωh01
∂ X− = . (2.39)
1
−G
−−
By substituting this solution into ′, we have
L1
G 1 κ2 1 ω2
′ = √λκω ++ √λh00 √λh11 . (2.40)
L1 G − 2 G++ − 2 G
(cid:18) −−(cid:19) −−
Let us introduce a world-sheet symmetric tensor by
ij
J
κ2 ω2
:= , := , = := 0. (2.41)
J00 G++ J11 G J01 J10
−−
7
The reduced action now has the form
′ = ′ +
L L1 L2
G 1 (2.42)
= √λκω +− √λhij( + ),
ij ij
G − 2 J G
(cid:18) −−(cid:19)
where
= G ∂ Xm∂ Xn. (2.43)
ij mn i j
G
Sincetheworld-sheet diffeomorphismisfixedbythelight-conegaugeconditions(2.35),
hij are determined by solving the equations of motion for hij:
hij = det(J )Jij, (2.44)
ij
± −
q
where J = + , and Jij is the inverse of J .
ij ij ij ij
J G
Then, we finally have the reduced Lagrangian in the generalized light-cone gauge
G
′ = λdet( + )+√λκω +− . (2.45)
ij ij
L ± − J G G
q (cid:18) −−(cid:19)
3 The GS action
Now let us consider the GS action in the AdS S5 background. The GS action in
5
×
the flat target space [1, 2] is generalized in the curved supergravity background in [22].
More explicit GS action in the AdS S5 background was constructed in [3]. (See also
5
×
[23, 24]). Originally, the Wess-Zumino term is written in the three-dimensional form. The
manifestly two-dimensional form of the Wess-Zumino term was presented in [25, 26, 27].
We write the GS action in the AdS S5 background as follows:
5
×
1
S = d2ξ , (3.1)
GS GS
2π L
Z
= 1√λhijη EaEb +√λǫij(Eα̺ Eβ Eα¯̺ Eβ¯). (3.2)
LGS −2 ab i j i αβ j − i α¯β¯ j
Here a,b = 0,1,...,9, α,β,α¯,β¯ = 1,2,...,16, hij = √ ggij (i,j = 0,1), ǫ01 = 1.
−
η = diag( 1,1,...,1). (3.3)
ab
−
The induced vielbein for the type IIB superspace EA is denoted by
i
EA = EA ∂ ZM = EA ∂ Xm +EA ∂ θα +EA ∂ θ¯α¯, A = (a,α,α¯). (3.4)
i M i m i α i α¯ i
8
We use a Majorana-Weyl representation for the Gamma matrices:
0 (γa)αβ
Γa = , (Γa)∗ = Γa, Γa,Γb = 2ηab1 , (3.5)
(γa)αβ 0 ! { } 32
a = 0,1,...,9, α,β = 1,2,...,16. We denote the n n identity matrix by 1 . For our
n
×
specific choice of the Gamma matrices, see appendix.
A 32-component Weyl spinor Θ with positive chirality has the following form in the
Majorana-Weyl representation:
θα
Θ = . (3.6)
0!
Above, we have used the 16-component notation for the Weyl spinors. A spinor with
upper index α represent a Weyl spinor with positive chirality.
The constant matrix ̺ in the Wess-Zumino term is given by
̺ 0
CΓ01234 = αβ . (3.7)
0 ̺αβ!
Here C is the charge conjugation matrix.
3.1 κ-symmetry fixing
Let us decompose each of the two 16-component Weyl spinors into two 8-component
SO(4) SO(4) spinors:
×
θ+α θ¯+α¯
θα = , θ¯α¯ = , (3.8)
θ−α˙! θ¯−α¯˙!
where α = 1,2,...,8, α˙ = 1˙,2˙,...,8˙, α¯ = ¯1,¯2,...,¯8 and α¯˙ = ¯1˙,¯2˙,...,¯8˙.
We first fix the κ-symmetry by setting θ−α˙ = θ¯−α¯˙ = 0. In the 32-component notation,
these conditions are equivalent to the condition Γ+Θ = 0.
To simplify expressions, we combine the remaining fermionic coordinates into Ψαˆ:
θ+α
(Ψαˆ) = , αˆ = ˆ1,ˆ2,...,1ˆ6. (3.9)
θ¯+α¯!
Let 2 be a 16 16 matrix
M ×
( 2)α ( 2)α
2 = M β M β¯ , (3.10)
M (M2)α¯β (M2)α¯β¯!
9
with elements constructed purely from the fermionic variables:
1 1
( 2)αβ = (θ+γab)α(θ¯+γab̺)β (θ+γa′b′)α(θ¯+γa′b′̺)β,
M 2 − 2
1 1
(M2)αβ¯ = −2(θ+γab)α(θ+γab̺)β¯+ 2(θ+γa′b′)α(θ+γa′b′̺)β¯,
(3.11)
1 1
( 2)α¯β = (θ¯+γab)α¯(θ¯+γab̺)β (θ¯+γa′b′)α¯(θ¯+γa′b′̺)β,
M 2 − 2
1 1
(M2)α¯β¯ = −2(θ¯+γab)α¯(θ+γab̺)β¯+ 2(θ¯+γa′b′)α¯(θ+γa′b′̺)β¯.
Here a,b = 1,2,3,4, a′,b′ = 5,6,7,8.
For later convenience, let us introduce the following matrices:
cosh 1 (K )α (K )α sinh (L )α (L )α
M− 16 = 11 β 12 β¯ , M = 11 β 12 β¯ . (3.12)
M2 (K21)α¯β (K22)α¯β¯! M (L21)α¯β (L22)α¯β¯!
The κ-symmetry fixed action in the AdS S5 can be written as [18]:
5
×
1 1
= √λhijG Xm Xn + √λǫijB Ψαˆ Ψβˆ. (3.13)
LGS −2 mnDi Dj 2 αˆβˆDi Dj
Here m,n = 0,1,...,9, αˆ,βˆ = ˆ1,ˆ2,...,1ˆ6. The target space metric G is the bosonic
mn
AdS S5 metric, chosen as follows:
5
×
ds2 = ds2 +ds2 = G dXmdXn = G dXadXb+G dXmdXn. (3.14)
AdS5 S5 mn ab mn
The AdS part metric is chosen as
5
1+(z2/4) 2 4
ds2 = dt2 +G (dza)2, (3.15)
AdS5 − 1 (z2/4) z
(cid:18) − (cid:19) a=1
X
and the S5 part metric is chosen as
1 (y2/4) 2 4
ds2 = − dϕ2 +G (dys)2, (3.16)
S5 1+(y2/4) y
(cid:18) (cid:19) s=1
X
where
1 1
G = , G = . (3.17)
z (1 (z2/4))2 y (1+(y2/4))2
−
Here
4 4
z2 = (za)2, y2 = (ys)2. (3.18)
a=1 s=1
X X
10