Static Gauge Potential from Noncritical Strings

31 Pages·0.23 MB·English

by  Enrique Alvarez

IFT-UAM/CSIC-00-32
hep-th/0011219
Static Gauge Potential from Noncritical Strings
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2 Enrique A´lvarez 1 and Juan Jos´e Manjar´ın 2

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6 Instituto de F´ısica Te´orica, C-XVI, Universidad Aut´onoma de Madrid
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E-28049-Madrid, Spain3
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1 Departamento de F´ısica Te´orica, C-XI, Universidad Aut´onoma de Madrid
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0 E-28049-Madrid, Spain
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The static gauge potential between heavy sources is computed from a recently introduced
r
a
non-critical bosonic string background. When the sources are located at the infinity of
the holographic coordinate, the linear dilaton behavior is recovered, which means that
the potential is exactly linear in the separation between the sources. When the sources
are moved towards the origin, a competing overconfining cubic branch appears, which is
however disfavored energetically.
1E-mail: [e-mail]
2E-mail: [e-mail]
3Unidad de Investigaci´on Asociada al Centro de F´ısica Miguel Catal´an (C.S.I.C.)
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1 Introduction
There has been recently, starting with the basic insight of Maldacena [10] several works
purporting to the computation of Wilson loops in different gauge theories.
The general setup is as follows [11]: there is a Poincar´e invariant background, of the
type:
ds2 = a(r)d~x2 +dr2 (1)
where d~x2 stands for the flat metric in d-dimensional Minkowski space (although we shall
mostly work in the Euclidean version), M , and r is the holographic coordinate.
d
Then an idealized rectangular static loop is posited in spacetime, by placing heavy
sources at x1 x = L/2, xi = 0 (i = 1).
≡ ± 6
The basic postulate on which the computation relies [14] consists in representing it
semiclassicallythroughtheminimalareasurfacespannedbytheworldsheetoftheconfining
string, assumed to have structure in the holographic dimension only.
To be specific, we identify world sheet and spacetime coordinates by choosing
x = σ
t = τ (2)
and compute the minimal area through the Nambu-Goto action
S d2σ√h (3)
NG
≡
Z
where the induced metric, h is defined through
ab
h ∂ xA∂ xBg (4)
ab a b AB
≡
(where we have represented world-sheet coordinates by σa (τ,σ) and spacetime coordi-
≡
nates by xA).
The imbedding of the two-dimensional surface into the external spacetime is defined
by the unique function
r = r¯(σ) (5)
1
determinedthroughEuler-Lagrange’sequationsfortheNambu-Gotoaction,complemented
with Dirichlet boundary conditions at the endpoints of the string,
r¯(0) = r¯(1) = Λ (6)
The resulting embedding then resembles half a topological cylinder, with its length along
the temporal direction, and bounded by r = Λ (cf. Figure), in such a way that the total
action is proportional to the span of the temporal variable, S T.
∼
t
Lambda
x r
Figure 1: Plot of the configuration
Finally, the potential between the heavy sources (a physical observable) is defined as
1
V(L) lim S (7)
T NG
≡ →∞T
where T is the time interval, and S is the on-shell value of the Nambu-Goto action.
NG
2
IntheoriginalAds/CFTcorrespondence, therearegoodreasonstochooseΛ = :there
∞
is a definite sense in which the CFT is defined on an extension of Penrose’s boundary, cf.
[16], and, besides, the mass of the sources is related to their location in the holographic
direction.
In other backgrounds this is not so clear, and one of the two main purposes of the
present work is to study the dependence of the physical potential on the value of Λ (the
location of the heavy sources in the holographic direction).
On the other hand, even in the first AdS S background there are extra dimensions
5 5
×
which unavoidably lead to undesirable excitations. It is clearly preferable in this context
to have a background with the physical number of dimensions only. Recently [3], a new
bosonic background has been introduced, which makes sense in particular, in d = 4 di-
mensions (plus one holographic coordinate). Its specific form is of the Poincar´e invariant
type,
ds2 = a(r)d~x2 +dr2 (8)
with warp factor given by:
21
a(r) = th( r) (9)
6α
r ′
and a dilaton background as well,
1 21 1
Φ = log th( r) (10)
2  6α 
r ′ cosh2( 21 r)
6α
′
 q 
This background interpolates between the well-known linear dilaton solution when r
→
and the confining background of [3] when r 0. Linear dilaton (which is exact to all
∞ ∼
orders in α) is just Minkowski space,
′
ds2 = d~x2 +dr2 (11)
with a = 1, and a dilaton field
21
Φ = r (12)
6α
r ′
3
The confining background, on the other hand, is defined by:
ds2 = rd~x2 +l2dr2 (13)
c
(where l is a length scale) and a dilaton given by
c
Φ = log r (14)
Were the matching between the two performed at a fixed value r = r , the length scale l
0 c
would be fixed to be
1 6α
l ′ (15)
c
≡ r 21
0r
The second main purpose of the present work is precisely to study Wilson loops in this
non-critical background.
What we will find is that when the scale Λ is big enough, the effects of the confining
background are not felt, which means that the potential is purely linear ([2]). As we lower
Λ we reach a critical value below which the effects of the confining background start to
appear, which implies for the potential the appearance of a competing, overconfining, cubic
branch, which is not energetically favored.
2 Wilson Loops in the Conformal Gauge
The first thing we will study is how to work in the usual conformal gauge in Polyakov’s
action,
S γ ∂ax ∂bx gAB√γd2σ (16)
P ab A B
≡
Z
namely γ δ .
ab ab
∼
One can read the purported equivalence between Nambu-Goto and Polyakov actions
through the equations of motion. In the former case, avoiding the factors 1/4πα in the
′
Poyakov’s case and 1/2πα in Nambu-Goto, these read
′
4
√h
hab[2g (Xα)∂ ∂ Xµ +2∂ Xµ∂ Xρ∂ g (Xα)+∂ Xµ∂ Xη∂ g (Xα)] = 0 (17)
µν a b a b ρ µν a b ν µη
2
while for the Polyakov action one gets
2g (Xα)∂ ∂aXµ +2∂ Xµ∂aXρ∂ g (Xα)+∂ Xµ∂aXη∂ g (Xα) = 0 (18)
µν a a ρ µν a ν µη
It is plain that the equations (17) and (18) are equal if the induced metric on the world-
sheet is proportional to Kronecker’s delta
√h
hab = δab (19)
2
The condition that the induced metric is proportional to the standard flat two dimen-
sional Euclidean metric defines the so called isothermal coordinate system. That such a
coordinate system can be locally defined is a standard result whose proof can be found,
for example in [15].
Precisely in this frame the non-linear sigma model action turns out to be the Dirichlet
functional, and its stationary solutions can be interpreted as the surfaces of minimal area.
Thismeansthatonlyintheisothermalcoordinatesystem wecanexpect ongeneralgrounds
an area law behavior when computing the Wilson loops.
We can check explicitely this result in the case of AdS S5, where the metric is given
5
×
by
U2 dU2
ds2 = α dx2 +R2 +R2dΩ2 (20)
′ R2 U2 5
(cid:18) (cid:19)
and the actions are (neglecting the contribution of the S )
5
1 U2 U2 R2
S = d2σ (x)2 + (t˙)2 + (U )2 (21)
P ′ ′
4π R2 R2 U2
ZM (cid:20) (cid:21)
5
1 U4
S = d2σ(t˙) (U )2 + (x)2 (22)
NG 2π ′ R4 ′
ZM r
The corresponding equations of motion are given by
2
∂ ∂σx+ ∂ U∂ x = 0 (23)
σ σ σ
U
U2∂ ∂τt = 0 (24)
τ
1 U3
∂ ∂σU ∂ U∂ U (∂ x∂ x+∂ t∂ t) = 0 (25)
σ − U σ σ − R4 σ σ τ τ
in the Polyakov’s case, and by
U4x
′
∂ = 0 (26)
σ
R4 (U )2 + U4(x)2
′ R4 ′
 q 
U4
∂ (U )2 + (x)2 = 0 (27)
τ ′ R4 ′
r
U 2U3x2
′ ′
∂ = (28)
σ
R4 (U )2 + U4(x)2 R4 (U )2 + U4(x)2
′ R4 ′ ′ R4 ′
 q  q
in the Nambu-Goto one.
That this two results are identical is a simple consequence of (19).
However, there isaninteresting novel thing concerning therˆoleplayed bytheparameter
σ over the world-sheet. When working withthe Nambu-Gotoaction, one was freeto choose
the static gauge, τ = t and σ = x ([11], [4],[14]).
Now we can no longer make this choice, because all gauge freedom has been used.
What happens in the conformal gauge is that the range of the world-sheet coordinate is
dinamically determined by the field equations:
R2 dy R2 π 1
R(σ) = ∞ = F , (29)
TU0 Z1 y4 −1 √2TU0 (cid:18)2 √2(cid:19)
p
6
It is useful for later purposes to represent it in terms of the minimal value of the
holographic coordinate, U , ( in case the boundary conditions are located at U = Λ,) that
0
is:
R2 U 1
R(σ) = F cos 1 0 , (30)
−
√2TU Λ √2
0 (cid:18) (cid:18) (cid:19) (cid:19)
60
50
40
30
20
10
0 0.2 0.4 0.6 0.8 1
Figure 2: R(σ)TΛ vs. U0
R2 Λ
q
3 Wilson Loops in the Four-Dimensional Background
Let us now turn to the novel four-dimensional background recently obtained in [4]. The
metric reads
ds2 = a(r)η dxµdxν +dr2 (31)
µν
where
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2/√d
√ c c
a(r) = − 1 2 tanh √ c c (r +c ) (32)
1 2 3
c −
(cid:20) 1 (cid:21)
(cid:0) (cid:1)
for c , c different from 0, which is the case of interest in the prersent context.
1 2
In this background, the induced metric over the world sheet is
ds2 = a(r)(x)2 +(r )2 dσ2 +a(r)(t˙)2dτ2 (33)
ind ′ ′
(cid:2) (cid:3)
The Polyakov action is
1
S = d2σ a(r) (x)2 +(t˙)2 +(r )2 (34)
p ′ ′
4πα
′ ZM
(cid:2) (cid:0) (cid:1) (cid:3)
The energy-momentum tensor takes the form
j = a(r) (x)2 (t˙)2 +(r )2 (35)
′ ′
−
(cid:0) (cid:1)
and the momentum
p = a(r)x (36)
′
they follow the relation j = p2 a(r )(t˙)2. We can define j = ˆja(r )(t˙)2 and p2 =
a(r0) − 0 0
pˆ2a(r )2(t˙)2. Conformal invariance implies ˆj = 0 and pˆ2 = 1.
0
Using now the specific value for the warp factor (32) one gets:
c1/√d Y y(√d 1)/2dy
σ = 1 − (37)
(t˙)(−c1c2)(1+√d)/(2√d)tanh(1−√d)/√d(v0) Z1 (y2 −1)1/2(1−tanh2(v0)y√d)
c1/√d Y y(√d 3)/2
x = 1 − (38)
(−c1c2)(1+√d)/(2√d)tanh(1+√d)/√d(v0) Z1 (y2 −1)1/2(1−tanh2(v0)y√d)
whereas the action is given by:
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( c1c2)(1−2√d)/(2√d)T Y y(√d+1)/2dy
S = − (39)
p 2πα′c11/√dtanh(1+√d)/√d(v0) Z1 (y2 −1)1/2(1−tanh2(v0)y√d)
where Y = tanh2/√d(v )/tanh2/√d(v ), and v = √ c c √dr , v = √ c c √dΛ.
Λ 0 0 − 1 2 2 0 Λ − 1 2 2
In Minkowski signature only the integral condition for L is real, while R(σ) and S are
P
complex. We shall work in Euclidean space from now on.
We have not succeeded in solving these integrals in general. When the sources are
located in the vicinity of the singularity, (that is when Λ << l , these three integrals
s
can be approximately evaluated by treating the product by as a small quantity, where
b = tanh(c r ). In this case, and working by simplicity in d = 4, we obtain
1 0
b1/2 (Y2 1)1/2(3b4Y4 +2b2Y2 10) 3b2 Y z1/2dz
R(σ) = − − + +1 (40)
t˙c 5Y(1 b2Y2) 5 (z 1)1/2
1 (cid:26) − (cid:18) (cid:19)Z1 − (cid:27)
1 3+19b2 100b4 Y dz
L = − +
c b3/2 3(40b4 36b2 +5) z1/2(z2 1)1/2
1 (cid:26) − Z1 −
(Y2 1)1/2[(43 100b4)b2Y4 +(9 60b2)Y3 (10 40b2)Y2 (9 60b2)]
+ − − − − − − − (41)
3Y3/2(1 b2Y2)(40b4 36b2 +5)
− − (cid:27)
and the potential will be
1 97b2 55 Y dz
V = − +
2παc2b3/2 3(36b2 40b4 +5) z1/2(z2 1)1/2
′ 1 (cid:26) − Z1 −
(Y2 1)1/2[(70 109b2)Y4 15Y3 (10 22b2)Y2 +15]
+ − − − − − (42)
Y3/2(1 b2Y2)(36b2 40b4 5)
− − − (cid:27)
these results are basically the same one obtains fromthe confining background of [3], which
we have worked out in detail in Appendix A.
A problem with this approximation is that it is not valid when Λ l . We can improve
s
∼
on it by building the hyperbolic tangent out of two linear pieces; one corresponding to the
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