Coupling of Higher Spin Gauge Fields to a Scalar Field in $AdS_{d+1}$ and their Holographic Images in the $d$-Dimensional Sigma Model

0.22 MB·English

by  T. Leonhardt

hep-th/ February 2004
4
0
Coupling of Higher Spin Gauge Fields to a
0
2 Scalar Field in AdS and their Holographic
d+1
n
Images in the d-Dimensional Sigma Model
a
J

0
3

Thorsten Leonhardt, Ruben Manvelyan and Werner Ru¨hl
1 ∗
v
Department of Physics
0
4 Erwin Schro¨dinger Straße
2
University of Kaiserslautern, Postfach 3049
1
0 67653 Kaiserslautern, Germany
4
0 tleon,manvel,[email protected]
/
h
t
-
p
e Abstract
h
:
v
The three-point functions of two scalar fields σ and the higher spin field h(ℓ) of
i
X
HS(4)on the one side and of their proposed holographic imagesα and (ℓ) of the
r J
a minimal conformal O(N) sigma model of dimension three on the other side are
evaluated at leading perturbative order and compared in order to fix the coupling
constant of HS(4). This necessitates a careful analysis of the local current Ψ(ℓ)
to which h(ℓ) couples in HS(4) and which is bilinear in σ.
∗
On leave from Yerevan Physics Institute, e-mail: [email protected]
1 Introduction
In any known model the AdS/CFT correspondence is an unproven hypoth-
esis still. If such model is derived from string theory as the standard case of
AdS supergravity and SYM ( =4), supersymmetry permits geometric argu-
5 4
N
ments based on representation theory that support AdS/CFT correspondence
and these arguments look quite convincing indeed.
But in models of the type of higher spin gauge fields (HS(d+1)) there is no
supersymmetry a priori and the correspondence can be proved only by dynamical
calculations both in AdS and CFT cases. Since in these models perturbative
d+1 d
expansions with small coupling constants are mapped on each other, such calcu-
lations are technically feasible and the holographic mapping is order by order.We
shall start such calculation for HS(4) and the 3-dimensional conformal O(N)
sigma model now.
We concentrate on three-point function of two scalar and one higher spin field
AdS CFT
4 3
Scalar σ(z) α(x)
HSF h(ℓ)(z) (ℓ)(x)
J
whereα(x)isthe“auxiliary”or“Lagrangemultiplier” fieldand (ℓ)(x) analmost
J
conserved current, which is a traceless symmetric tensor. In the sigma model
case the coupling constant is O( 1 ). In the higher spin field theory the coupling
√N
constant for σσh(ℓ) interaction is g , so that we expect
ℓ
1
g = C(ℓ) . (1)
ℓ
√N
We determine C(ℓ) first in an ad hoc wave function normalization such that
α(x)α(0) = x2 −β (2)
CFT
h i
1β, 1(β +1)
hσ(z1)σ(z2)iAdS =(cid:0)(2ζ(cid:1))−βF 2β 2 µ+1 ; ζ−2 (3)
(cid:20) − (cid:21)
(z0)2 +z0)2 +(~z ~z )2 1
ζ = 1 2 1 − 2 , µ = d , (4)
2z0z0 2
1 2
so that (2) is obtained from (3) by a “simple” boundary limit
lim lim z0z0 −β σ(z )σ(z ) = α(~z )α(~z ) . (5)
z0 0z0 0 1 2 h 1 2 iAdS h 1 2 iCFT
1→ 2→
(cid:0) (cid:1)
The higher spin fields are assumed to be normalized in the same fashion. At the
end we renormalize the higher spin field such that C(ℓ) is replaced by one.
2
We shall treat two versions of the minimal O(N) sigma model. In the “free”
case we have as a scalar field
1
α (x) = φ (x) φ (x) , (6)
f i i
√2N
where φ (x),i = 1,2,...N is the O(N) vector and space-time scalar field normal-
i
ized so that
φ (x)φ (x) = x2 −δδ , δ = µ 1 (7)
i j CFT ij
h i −
and (2) follows from (7) and (6) with(cid:0) (cid:1)
β = 2(µ 1) = d 2 . (8)
f
− −
In the “interacting” sigma model we have an interaction
z1/2 dxφ (x) φ (x)α(x) (9)
i i
Z
and the interaction constant z is expanded
∞ z
z = k . (10)
Nk
k=1
X
The “free” theory is unstable and by renormalization flow approaches the sta-
ble “interacting” theory. The conformal scalar field σ(z) on AdS is massive
d+1
(tachyonic duetoconformalcoupling withtheAdSmetric) andhastwo boundary
values from the two roots of the dimension formula
1
∆ = µ µ2 +m2 2 , (11)
±
where for d = 3 (cid:0) (cid:1)
2 in the free case
m2 = − (12)
2+O(1) in the interacting case
(cid:26)− N
so that
β = 1 from (8)
∆(d = 3) = f (13)
β = 2+O(1) from (9) .
(cid:26) N
2 Currents coupled to (conformal)
higher spin fields in AdS
We assume that the interaction of a spin ℓ gauge field h(ℓ) and two scalar fields
σ(z) is local and mediated by a current Ψ(ℓ)
dz
Tr Ψ(ℓ)(z)h(ℓ)(z) . (14)
(z0)d+1
Z
(cid:8) (cid:9)
3
Ψ(ℓ) and h(ℓ) are symmetric tensors of rank ℓ. If we postulate that the covariant
divergence of Ψ(ℓ) is a trace term, the interaction is gauge invariant. Namely a
gauge transformation of h(ℓ), being of the form (classical)
h(ℓ) h(ℓ) + Λ(ℓ 1) , (15)
−
→ ∇
where Λ(ℓ 1) is a symmetric traceless tensor and Λ(ℓ 1) is symmetrized, leads
− −
∇
to the zero gauge variation of (14)
Tr Ψ(ℓ)Λ(ℓ 1) = 0 . (16)
−
∇
(cid:8) (cid:9)
This consideration is in agreement with the so called ”Fronsdal” theory [1] of
higher spin with double-traceless gauge fields and currents. Truncation of this
higher spin theory to the conformal higher spin theory can be observed if we
consider the corresponding double-traceless current and gauge field as a sum of
two traceless objects, namely
Ψ(ℓ) = J(ℓ) +g(2)ψ(ℓ 2) , (17)
−
where J(ℓ) and ψ(ℓ 2) are now the traceless tensors, g(2) is the D = d+1 dimen-
−
sional AdS metric and symmetrization is assumed. Following [2] we call ψ(ℓ 2)
−
the compensator field. It is easy to see that ψ(ℓ 2) plays the role of the traceless
−
trace of the double-traceless current Ψ(ℓ) and has to decouple in the conformal
limit of higher spin theory. In other words we will assume that at the d dimen-
sional boundary M = ∂AdS , Ψℓ behaves as a conformal tensor field. Now
d D
we will consider the general structure of conformal higher spin currents in the
AdS space constructed from the conformally coupled scalar field σ(z) with the
D
corresponding on-shell condition
D(D 2)
2σ(z) = σ(z) = − σ(z) . (18)
∇·∇ 4L2
The tachyonic mass here (we use in this section the mainly minus signature
of the AdS metric 1) arises as a result of conformal coupling of the conformal
1WewilluseAdSconformalflatmetric,curvatureandcovariantderivativescomutationrules
of the type
L2 1
ds2 =gµνdzµdzν = (z0)2ηµνdzµdzν, ηz0z0 =−1,√−g = (z0)d+1 ,
[ , ]Vρ =R ρVσ R σVρ , (19)
∇µ ∇ν λ µνσ λ − µνλ σ
1 1
R ρ = η δρ η δρ = g δρ g δρ ,
µνλ −(z0)2 µλ ν − νλ µ −L2 µλ ν − νλ µ
D 1 (cid:0) D 1 (cid:1) (cid:0) D(D 1)(cid:1)
R = − η = − g , R= − .
µν − (z0)2 µν − L2 µν − L2
4
scalar σ(z) with the AdS curvature S = dDz√ g1 gµν∂ σ∂ σ D 2 Rσ2 .
− 2 µ ν − 4(D−1)
−
For the investigation of the conservation and traceless(cid:16)ness conditions for gener(cid:17)al
R
spin ℓ symmetric conformal current J(ℓ) we contract it with the ℓ-fold tensor
µ1µ2...µℓ
product of a vector aµ and make the ansatz including a first curvature correction
in contrast to the free flat case [3]
ℓ
1
J(ℓ)(z;a) = Ap(a )ℓ−pσ(z)(a )pσ(z)
2 ∇ ∇
p=0
X
a2 ℓ−1
+ Bp(a )ℓ−p−1 µσ(z)(a )p−1 µσ(z) (20)
2 ∇ ∇ ∇ ∇
p=1
X
a2 ℓ−1 1
+2L2 Cp(a∇)ℓ−p−1σ(z)(a∇)p−1σ(z)+O(a4)+O(L4) ,
p=1
X
where A = A ,B = B ,C = C and A = 1. Now we try to define the
p ℓ p p ℓ p p ℓ p 0
− − −
set of unknown constants A ,B and C using the current conservation condition
p p p
∂
∂ J(ℓ)(z;a) = µ J(ℓ)(z;a) = 0 (21)
∇· a ∇ ∂aµ
and the tracelessness condition connected with the conformal nature of our scalar
field σ(z)
∂2
2 J(ℓ)(z;a) = J(ℓ)(z;a) = 0 . (22)
a
∂a ∂aµ
µ
Using the following basic relations
p(p 1)
[∇µ,(a∇)p]σ = 2L−2 aµ(a∇)p−1σ −a2(a∇)p−2∇µσ , (23)
p(p (cid:0)1) (cid:1)
[∇µ,(a∇)p]∇νσ = 2L−2 aµ(a∇)p−1∇νσ −a2(a∇)p−2∇µ∇νσ
p
+ g (a(cid:0) )pσ a a p 1 σ , (cid:1) (24)
L2 µν ∇ − ν ∇ − ∇µ
∂ (cid:0) (cid:0) (cid:1)(cid:1)
∂aµ (a∇)pσ = p(a∇)p−1∇µσ
p(p 1)(p 2)
+ −6L2 − aµ(a∇)p−2σ −a2(a∇)p−3∇µσ , (25)
∂ 1 1 (cid:0) (cid:1)
(a )pσ = pD(D 2)
∇· ∂a ∇ L2 4 −
(cid:20)
2 7 1
+p(p 1) D + p (a )p−1σ +O( ) , (26)
− 3 − 3 ∇ L4
(cid:18) (cid:19)(cid:21)
1 1
2 (a )pσ = p(p 1)D(D 2)
a ∇ L2 4 − −
(cid:20)
1 1
+ p(p 1)(p 2)(p+2D 5) (a )p−2σ +O( ) (27)
3 − − − ∇ L4
(cid:21)
5
we can derive recursion relations for A , B and C coming from conservation
p p p
condition (21)
pA +(ℓ p+1)A +2B +2B = 0, (28)
p p 1 p p 1
− − −
s (p)A +s (p,ℓ,D)A +s (ℓ p+1,ℓ,D)A
3 p+1 2 p 2 p 1
− −
+s (ℓ p+1)A +2C +2C = 0, (29)
3 p 2 p p 1
− − −
1 1 1 7
s (p,ℓ,D) = pD(D 2)+p(p 1)(D + ℓ+ p ), (30)
2
4 − − 2 6 − 3
1
s (p) = (p+1)p(p 1) . (31)
3
6 −
The relation (28) relates A and B recursively as in the flat case [3]. The next
p p
relation (29) arises from the 1 correction and relates recursively C and A
L2 p p
coefficients from our ansatz (20). From the other side the tracelessness condition
(22) gives us two further relations between these coefficients
p(ℓ p)
B = − A , (32)
p −(D +2ℓ 4) p
−
1
C = − [s (p+1,ℓ,D)A +s (ℓ p+1,ℓ,D)A ], (33)
p 2(D+2ℓ 4) t p+1 t − p−1
−
1 1
s (p,ℓ,D) = p(p 1)D(D 2)+ p(p 1)(p 2)(ℓ+2D 5) . (34)
t
4 − − 3 − − −
Again the relation (32) is the same as in the flat case and leads the Eq. (28) to
the recursion
A = s (p,ℓ,D)A , (35)
p 1 p 1
− −
(ℓ p+1)(2ℓ 2p+D 2)
s (p,ℓ,D) = − − − . (36)
1
p(D +2p 4)
−
From this we can obtain the same solution for the A coefficients [3] as in the flat
p
case
ℓ ℓ+D 4
A = ( 1)p p p+D2−−2 . (37)
p − (cid:0) (cid:1)ℓ(cid:0)+DD−24 (cid:1)
2−
For the important case D = 4 this formul(cid:0)a simp(cid:1)lifies to
ℓ 2
A = ( 1)p . (38)
p − p
(cid:18) (cid:19)
It means that if our ansatz (20) and our consideration for the 1 correction are
L2
right, the recursion relation for the A coefficients obtained by substituting the
p
C coefficients in (29) by those of the 1 tracelessness condition (33) must be
p L2
6
consistent with (35). Indeed using (33) and (35) we can rewrite the relation (29)
in the form
(29) = A s (p,ℓ,D)+A s (ℓ p+1,ℓ,D) = 0 (39)
p f p 1 f
− −
s (p,ℓ,D)
s (p,ℓ,D) = s (p,ℓ,D) t
f 2 − D+2ℓ 4
(cid:20) −
s (p+1,ℓ,D)
s (p+1,ℓ,D) s (p) t (40)
1 3
− − D +2ℓ 4
(cid:18) − (cid:19)(cid:21)
(ℓ+D 3)(2ℓ+D 2)p(D+2p 4)
= − − − .
4(D+2ℓ 4)
−
It is easy to see that the relation (39) coincides with (35) because
s (p,ℓ,D)
f = s (p,ℓ,d) . (41)
s (ℓ p+1,ℓ,D) 1
f
−
So we obtain a result that the structure of the conformal higher spin currents
constructed from the conformal coupled scalar field in the fixed AdS background
remains the same as in the free flat space case. We prove that our ansatz with
1 correction connected with the difference between the traces in flat and AdS
L2
case does not violate the conservation condition (recursion relation (35)) for the
coefficients A if they obey the tracelessness condition (33) for the currents. It
p
means that the traceless conserved higher spin current constructed from confor-
mal scalar field in AdS can be obtained from the flat space expression replacing
usual derivatives with covariant ones and adding corresponding curvature cor-
rections to the expression for the traces. For completeness we present in the
Appendix an explicit derivation of the conformal conserved current in the case
ℓ = 4,D = 4 in all orders of 1 .
L2
This phenomenon we can explain now in the following way: The conformal
group for D- dimensional flat(with SO(D 1,1) isometry) and AdS space (with
−
SO(D 1,2) isometry) is the same -SO(D,2) 2. So we can say that the con-
−
formal primaries or the traceless conserved currents are the same due to the 1
L2k
corrections. But these originate from the curvature corrections to the flat space
equation of motion and noncommutativeness of the covariant derivatives. Then
because all currents are traceless we get the cancellation of all 1 accompanying
L2k
terms coming from these two sources of deformation of the flat case relations in
the conservation condition (21).
Now we will fix the coefficients A from the CFT consideration. We assume
p
thatontheboundary∂AdS , Ψ(ℓ) behaves asa conformaltensor field(thetrace
d+1
is decoupled). Moreover this conformal tensor must be local bilinear in α(x) of
rank ℓ and of dimension
1
2β +ℓ+O( ) . (42)
N
2Note that this is about conformal group of AdS space-not boundary
7
For this purpose we evaluate the 3-point function
ℓ
1
α(x )α(x ) A a ∂ pα(x ) a ∂ ℓ pα(x ) , (43)
h 1 2 2 ph · i 3 h · i − 3 iCFT3
p=0
X
where a ∂ = ai∂ , i = 1,2,3.
i
h · i
From the propagator (2) for α(x) we obtain for (43)
ℓ (x2 x2 )−β
2ℓ A (β) (β) 13 23 a x p a x (ℓ p) (44)
p p ℓ−p x2px2(ℓ−p) h · 13i h · 23i −
p=0 13 23
X
+ trace terms ,
where we define the Pochhammer symbols (z) = Γ(z+n).
n Γ(z)
As a 3-point function of a conformal tensor is unique up to normalization
x2 x2 −β a ξ ℓ + trace terms , (45)
C 13 23 h · i
xi xi
ξi(cid:0)= 13 (cid:1) 2(cid:8)3 , (cid:9) (46)
x2 − x2
13 23
it follows
( 1)p ℓ
A = C − p . (47)
p 2ℓ(β) (β)
p (cid:0)ℓ(cid:1)p
−
This expression, for β = 1, is in agreement with the previous one (38) obtained
from AdS consideration, if we will normalize in (45) = 2ℓℓ!.
4
C
For β = 2 we have to change the constraints imposed in (20) to allow com-
patibility of (47) with (20). We propose to give up the condition of tracelessness
so that the coefficients in (20) are determined from (28), (29) and (47). This
implies a coupling of the first trace compensator field of h(ℓ) to Ψ(ℓ).
3 The 3-point function σσh(ℓ) in AdS to first
d+1
order
Based on the interaction (14) with the general form of ansatz (20) we calculate
the AdS 3-point function
g 1 σ(z )σ(z )h(ℓ) (b )ℓ . (48)
ℓ− h 1 2 µ1...µℓi µi
⊗
Wewilluse fromthissection thenotationofEuclidian AdSwiththeEuclidian
scalar product ... both in the boundary and in the bulk space.
h i
The bulk-to boundary propagator of a scalar field of dimension β is
w β
K (w,~x) = 0 (49)
β |~x=0 w2 +w~2
(cid:18) 0 (cid:19)
8
and of a tensor field which is traceless symmetric of dimension λ
wλ ℓ
K(ℓ)(w,~x) = 0− a, R(w)~b ℓ traces , (50)
λ |~x=0 (w2 +w~2)λ h i −
0 h i
where
w w
R (w) = δ 2 µ ν , (51)
µν µν − w2+w~2
0
d
R (w)R (w) = δ . (52)
µν νλ µλ
ν=0
X
This kernel R is connected with the Jacobian of the inversion
µν
z
z z = µ . (53)
µ → µ′ z2 +~z2
0
So that
∂w 1
µ′ = R (w), (54)
∂w w2 +w~2 µν
ν 0
R (w) = R (w ) . (55)
µν µν ′
To first order in the interaction we obtain (48) with all arguments z taken to
1,2,3
the boundary
ℓ ℓ
1 dw
A ( )n K (w,~x ) ℓ−nK (w,~x ) gµiνi(w)
2 n wd+1 ∇µi β 1 ∇µj β 2
n=0 Z 0 ⊗ ⊗ i=1
X (cid:0) (cid:1) Y
ℓ ℓ
K(ℓ)(w,~x ) b = α(~x )α(~x ) (ℓ) b , (56)
× λ 3 ν1...νℓ,σ1...σℓ σk h 1 2 Ji1...iℓi ik
k=1 k=1
Y Y
where the equality is based on AdS/CFT correspondence.
On this integral we apply a d-dimensional translation
w~ w~ +~x (57)
→
and inversion (53) to
w w , ~x x~ , i 1,2 . (58)
′ i ′i
→ → ∈ { }
This gives by contracting the R’s and by (52)
ℓ
dw
x213x223 −β (w )d′+1 ∇′µi n Kβ(w′−x~′13) ∇′µi ℓ−nKβ(w′ −x~′23)(w0′)λ+ℓ bµi .(59)
Z 0′ ⊗ ⊗ i=1
(cid:0) (cid:1) (cid:0) (cid:1) (cid:0) (cid:1) Y
9
Since b = 0 the can be applied to x~ , x~ respectively and it follows
0 ′ ′13 ′23
∇
x213x223 −β(−1)ℓh~b·∂~′13inh~b·∂~′23iℓ−nIβ,β,λ+ℓ x~′13, x~′23 , (60)
(cid:16) (cid:17)
(cid:0) (cid:1)
where3
dw
I (~x , ~x ) = K (w ~x )K (w ~x )w∆3
∆1,∆2,∆3 1 2 wd+1 ∆1 − 1 ∆2 − 2 0
Z 0
3
1 Γ(Σ ∆ )
= πµΓ(Σ µ) − i ~x ~x 2(Σ ∆3) (61)
1 2 − −
2 − Γ(∆ ) | − |
" i #
i=1
Y
and
1 1
µ = d, Σ = (∆ +∆ +∆ ). (62)
1 2 3
2 2
In our case
∆ = ∆ = β, ∆ = λ+ℓ,
1 2 3
Σ = β +δ +ℓ, Σ µ = β +ℓ 1, (δ as in (7)) (63)
− −
Σ ∆ = Σ ∆ = δ +ℓ, Σ ∆ = β ℓ δ .
1 2 3
− − − − −
As in (46) we introduce
~x ~xi
ξ~= x~ x~ = 13 23 (64)
′13 − ′23 x2 − x2
13 23
and use
~b ∂~ n ~b ∂~ ℓ n(ξ2) ∆ = ( 1)n2ℓ(∆) (ξ~2) ∆ ℓ ~b ξ~ ℓ +trace terms .
′13 ′23 − − ℓ − −
h · i h · i − h · i
n o(65)
This yields finally
ℓ
g 1 α(~x )α(~x ) (ℓ) b
ℓ− h 1 2 Ji1...iℓiAdS ik
k=1
Y
= AdS(x2 )δ β(x2 x2 ) δ ~b ξ~ ℓ +trace terms , (66)
NααJ 12 − 13 23 − h · i
Γ(δ +ℓ) 2nΓ(β +ℓ 1)Γ(β δ)1o ℓ
AdS = 2ℓπµ − − ( 1)nA . (67)
Nαα Γ(β) Γ(2δ+2ℓ) 2 − n
J
(cid:18) (cid:19) n=0
X
The last factor gives using (47) and omitting the normalization factor
C
ℓ
1 1 ℓ, β ℓ+1 (2β +ℓ 1)
( 1)nA = F − − − ; 1 = − ℓ . (68)
2 − n 2ℓ+1(β) 2 1 β 2ℓ+1(β) (β)
n=0 ℓ (cid:20) (cid:21) ℓ ℓ
X
3since ℓ is even ( 1)ℓ =+1
−
10