Holographic repulsion and confinement in gauge theory

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by  Viqar Husain

Holographic repulsion and confinement in gauge theory
Viqar Husain1,2∗ and Dawood Kothawala2†
1 Perimeter Institute for Theoretical Physics, Waterloo, ON, Canada and
2 Department of Mathematics and Statistics, University of New Brunswick, Fredericton, NB, Canada E3B 5A3
(Dated: January 30, 2012)
Weshowthatforasymptoticallyanti-deSitterbackgroundswithnegativeenergy,suchastheAdS
soliton and regulated negative mass AdS-Schwarzshild metrics, the Wilson loop expectation value
in the AdS/CFT conjecture exhibits a Coulomb to confinement transition. We also show that the
quark-antiquark(qq¯)potentialcanbeinterpretedasaffinetimealongnullgeodesicsontheminimal
stringworldsheet,andthatitsintrinsiccurvatureprovidesasignatureoftransitiontoconfinement
phase. The result demonstrates a UV/IR relation in that the boundary separation of the qq¯pair
exhibitsaninverserelationshipwiththeradialdescentoftheworldsheetintothebulk. Ourresults
2 suggest a generic (holographic) relationship between confinement in gauge theory and repulsive
1 gravity,which in turn is connected with singularity avoidance in quantumgravity.
0
2 PACSnumbers: ...

n
a I. INTRODUCTION where γ is the boundary loop, s is a bulk world sheet
γ
J
withγ asitsboundary,andS s istheNambu-Goto
NG γ
7 { }
Among the intriguing insights to come from studies in string action. The r.h.s of this equation is typically ap-
2
quantum gravity are the so called gauge-gravity duali- proximated at the saddle point, giving

] ties. These provide a map between a gauge theory with-
h
outgravityinD dimensionsandatheorywithgravityin W =exp S s S (2)
-t (D+1)dimensions. Suchdualitiesnaturallyincorporate h γi "− NG{ γ(min)}− reg!#
p
the appealing notion of holography, a direct realization
e where s is the minimal worldsheet area with the
h ofwhichistheAdS/CFTconjecturegivenbyMaldacena γ(min)
loop γ as its boundary. This is obtained by extremizing
[ [1]. This conjecture proposes a duality between a spe-
cific type of string theory on a D+1 dimensional AdS the classicalstring action SNG. Sreg in this expressionis
1 the action of two rectangular worldsheets extending into
background, and a conformal field theory living on the
v the bulk that have the two timelike edges of the Wilson
timelike boundary of this spacetime.
2 loopastheirboundaries. Thissubtractioncorrespondsto
3 Inessence,the diffeomorphismspreservingthe asymp-
the energy of the two free quarks moving on the bound-
8 totic AdS structure of the background generate confor-
ary. (The suffix “reg” signifies the fact that in an AAdS
5 mal diffeomorphisms of the boundary, and it is this iso-
spacetime,thistermremovesthedivergencecomingfrom
1. morphismbetweenthesymmetrygroupofAdSin(D+1)
the first term.)
0 dimensions and conformal group of D dimensional flat
This approach to computing a Wilson loop provides
2 spacetimewhichmakesitpossibletopostulateanequiva-
anotherconnectionbetweenclassicalAAdSsolutionsofa
1 lencebetweenthecorrespondingtheoriesatthequantum
: gravitationaltheoryandaquantumexpectationvalue in
v level. SinceitsproposalbyMaldacenaanditssubsequent
gaugetheory. Ofparticularinterestistheconfiningphase
i quantitative realization [2–4], the AdS/CFT conjecture
X ofgaugetheory,inwhichthequark-antiquarkpotentialis
has been used to study a wide range of boundary quan-
linearlyproportionaltotheirseparation. Therearemany
r tum systems using only the geometric tools of general
a papers in the literature that focus on confinement from
relativity. More recently this has been extended to the
AdS/CFT. The ones most relevant to the present work
study of condensed matter systems using gravitational
are [6] where the negative mass case was studied, and
theories.
[7] where AdS-scalar field metrics were studied. A class
In this paper we study one specific application of this
of conditions on metric functions that give confinement
duality, namely the prescription for computing the ex-
werederivedin[8]andsomeexamplesappearedin[9]. In
pectationvalueoftheWilsonloopoftheboundarygauge
addition to these works, there are many others that fo-
theory via a gravitycalculation[5]. This gives the quark
cus onthis aspect of the AdS/CFT correspondence;(see
anti-quarkinteractionpotentialofthegaugetheory. The
eg. [10–12]). In particular [13, 14] provide examples of
precise correspondence is
supergravity geometries that give confinement, and pro-
vide instances of the connection between negative bulk
Wγ = Dsγexp( SNG sγ ) (1) energy and confinement.
h i − { }
Z The confinement phase in gauge theory, which occurs
at large coupling and low energy (the opposite end of
asymptoticfreedom)isalsointerestingfromanotherper-
∗Electronicaddress: [email protected] spective, known as the UV/IR duality. This aspect of
†Electronicaddress: [email protected] holography states that the high energy regime in the
2
bulk theory corresponds to the low energy regime in the In the conventionalstatic gauge, the worldsheet is de-
boundary theory, and vice-versa. This in turn suggests scribedbyasinglefunctionr(x), andthe inducedmetric
thatthelowenergyconfinementphaseofgaugetheoryis is
connected with the high energy quantum gravity regime
r2f(r) r2 ℓ2 ∂r 2
in the bulk. (This connection may arise for some semi- h dxµdxν = dt2+ + dx2
classical geometries that exhibit bulk repulsion [13, 14]. µν − ℓ2 ℓ2 r2g(r)(cid:18)∂x(cid:19) !
)
(4)
This last consequence of holography is our motivation
for studying confinement in gauge theory using geome- Therefore, the Nambu-Goto action becomes
tries that violate energy conditions, for it is widely be-
lieved that quantum gravity should provide a cure for r4f(r) f(r)
S =T dx + r′2 (5)
curvature singularities,but doing so is concomitantwith NG s ℓ4 g(r)
Z
violating energy conditions. In other words, singularity
where r′ = ∂r/∂x and the integral over t is trivial due
avoidance requires that quantum gravity be repulsive at
to t independence of the integrand. To find the minimal
short distances.
surface,wenotethattheintegranddoesnotdependonx
Our aim is to investigate bulk geometries that ex-
explicitly,andhencethe“Hamiltonian”correspondingto
hibittheCoulombtoconfinementphasetransitioninthe
x translations is a constant. This constant can be fixed
boundarygaugetheory. Wefocusspecificallyontheroles
byevaluatingtheHamiltonianattheminimalradius,r ,
playedbythegeometryparameters,negativebulkenergy, m
of the worldsheet, where r′ =0. This gives
andgeometricalpropertiesoftheminimalsurface. Inthe
next section we review the basic construction of [5]. In
r2 r4 fg
Sec. III we present evidence of a relationship between r′ = g (6)
confinementandrepulsion. Sec. IVgivesanewinterpre- ℓ2srm4 fm −
tationofthepotentialasthedifferencebetweenbulkand
wheref =f(r ). Withtherequirementthattheworld-
boundary affine time separation of the qq¯pair, and also m m
sheet ends on the boundary loop, we have the condition
gives a possible signature of confinement via the Ricci
x( ) = L/2. Also, from symmetry, the minimal sur-
scalar of the intrinsic world sheet geometry. In the last ∞ ±
face is symmetric about x, hence x(r ) = 0. Therefore
sectionwe conjecture a generic relationshipbetween sin- m
the last equation gives
gularity avoidance in quantum gravity and confinement
in gaugetheory,basedon the evidence that both require L/2 ∞ ℓ2 1
negative bulk energy. dx= dr (7)
Z0 Zrm r2 rrm44 ffmg −g
q
which gives
II. BULK GEOMETRY AND THE WILSON
LOOP 2ℓ2 ∞ dy 1 1
L(r )= (8)
We summarize here the basics of the calculation [5], m rm Z1 y2 g(y;rm) ffm(y(;rrmm))y4−1
focusing on the planar metric p q
where y = r/r , and f (r ) = f(1;r ). This equation
m m m m
r2 ℓ2 dr2 gives the quark-antiquark separation L as a function of
ds2 = f(r)dt2+dx2+dy2+dz2 + (3)
ℓ2"− # r2g(r) the minimum radial coordinate value of the world sheet.
The interaction potential V(r ) is obtained using the
m
wherethefunctionsf(r)andg(r)aresuchthatthespace- prescription
time is asymptotically AdS . (The non-planar case is
5
S =TV(L) (9)
studied in [15] where the AdS analog of the Hawking- NG
Page transition is described; a planar analog of this for withtheactioncomputedfortheminimalsurface. Using
the AdS soliton appears in [16].) r′(x) from Eqn. (6) gives
The procedure is to calculate the Nambu-Goto action
for a worldsheet which is bounded by a rectangular loop S ∞ dr r4f(r) f(r)
NG =2 + r′2 (10)
itno laietiimnetlhikeetplaxnpelaonnetfhoerbdoeufinnditaernye.ssW. Tehtaekseidtehsisoflotohpe T Zrm r′ s ℓ4 g(r)
−
rectangle are T and L along t and x respectively; x which after some simplification leads to
∈
[ L/2,+L/2]. Thisloophasthestandardinterpretation
o−fdescribinginteractionbetweenaquark-antiquarkpair S = 2Trm
NG
f (r )
inthelimitT . Asdescribedintheprevioussection, m m
theexpectatio→nv∞alueofthe Wilsonloop,inMaldacena’s ∞ y2f(y;rm)
dy
pacretisocnri,patniodn,isisdgoivmeinnabtyetdhebypatthheinmtiengirmalaolfsNurafmacbeui-nGothtoe ×Z1 g(y;rm) ffm(y(;rrmm))y4−1
classical limit. p q (11)
3
To obtain the interaction potential from this, we first quarkworldsheetsextendinthe bulk. InpureAdSbulk,
needtoregularizethisexpressionbysubtractingthecon- one can take Λ = 0, whereas for a black hole in AdS,
c
tribution coming from the edges of the loop along the t Λ = r , the horizon radius. We can now calculate the
c 0
direction, which has the interpretation of energy of free potential V(r ) by subtracting this contribution from
m
quarks. Since these 2 worldsheets lie in the t r plane, the expression for minimal worldsheet contribution.
−
the corresponding Nambu-Goto action is
1
∞ f ∞ f V(r ) = (S S ) (13)
Sreg =2T dr =2Trm dy (12) m T NG− reg
ZΛc sg ZΛc/rm sg
wherer =Λ issomecut-offradiusuptowhichthesefree- Putting everything together we have
c
∞ 1
V(r ) 2r f(y;r ) f(y;r ) y2 f(y;r )
m m m m m
= dy 1 dy
~cr0/α′ r0  Z sg(y;rm) sfm(rm) f(y;rm)y4 1 − − Z sg(y;rm) 
 1 fm(rm) − Λc/rm 
L(r ) 2r ∞ dy 1  1 q  
m 0
= (14)
ℓ2/r0 rm Z1 y2 g(y;rm) f(y;rm)y4 1
fm(rm) −
p q
where we have put in all the constants such as string to L andthe result is interpreted as a screenedCoulomb
length scale, ℓ =√α′, ~ and c. These are the equations potential.)
s
wefocusonintheremainderofthepaper. Inalltheplots Toinvestigatetheissuefurther,wenotethattheabove
below, we have plotted the dimensionless combinations expressionsfor V andL suggestthat the qualitative fea-
on the left, as a function of r /r (which is the only tures of V(L) depend only on f(r), whereas all depen-
m 0
combination on which the RHS will depend, since the dence on g(r) is separated out in smooth analytic form.
entire ℓ dependence has been moved to the LHS). Therefore,one may imagine that the curvature singular-
itycanberemovedbyregulatingthefunctiong(r),while
maintaining the asymptotic form of the metric, and at
the same time ensuring that one does recover a confine-
III. NEGATIVE ENERGY AND
ment phase. This turns out to be possible for a wide
CONFINEMENT
range of functions g.
However, a further restriction on the choice of func-
Since our main interest in this work is the Coulomb
tionsarisesuponnoticingtwothings. First,inEqs.(14),
to confinementtransitionandits relationto negativeen- the square root in the denominator depends on y4, and
ergyinthebulk,wefirstpresentevidenceforthis,before
second,that confinementis obtainedfor a negativemass
attempting a geometric understanding in the following
AdS-Schwarzschild solution. This means that this forth
section. We consider three cases that exhibit this, the
power appears to be important for the confinement re-
negativemassSchwarzschild-AdSmetric,aregulatedsin-
sult. Infactitispossibletocheckthisnumerically. Based
gularity free version of it, and the AdS soliton.
on these observations, we make the following ansatz to
Acuriouspointnotedin[6]isthatiff andgarechosen remove the singularity at r =0:
to correspond to the negative mass AdS-Schwarzschild
solution, ie. f(r)=1+(r0/r)4 =g(r), then V(L) shows f(r) = 1+ r04 (15)
a transition from Coulomb to confinement phase. The r4
resultisintriguingsincethe bulk metrichasanakedsin-
r4
gularity, and one would not have expected any sensible g(r) = 1+ 0 1 Q(r/λ) (16)
result, let alone a confining phase, to come from such a r4" − #
pathological bulk geometry.
wherethelengthλissomeregulatorscale. Thefiniteness
A relevant question is whether this result, which ap-
ofKretschmannscalaratr =0isensuredbyputting the
pears in Figs. 2, is connected to either the presence of
conditions Q(0) = 1 and Q′(0) = Q′′(0) = Q′′′(0) = 0
a naked singularity, the absence of a horizon, or to the
on the otherwise arbitrary function Q. Further, to
negative energy, or all three. (It is known that the AdS
maintain the AAdS form of the metric, we must have
black hole solution does not lead to a confinement phase
lim r−4Q(r/λ) = 0. The specific choice we shall make
– the V(L) curve becomes multiple valued with respect
r→∞
4
to exhibit the plots etc. is Fig. 2 gives the potentials V(L) for the negative mass
AdS-Schwarzschild geometry and the AdS soliton. It is
f(r) = 1+ r04 (17) apparent that the confining regions for large L corre-
r4 spond to negative energy repulsion in deep bulk region
r4 r4 (ie. smallrm). TheterminationoftheAdSsolitonlinein
g(r) = 1+ 0 1 exp (18) Fig. 2 may be interpreted as a correspondence with the
r4" − (cid:18)−λ4(cid:19)# breaking of the streched qq¯string; this does not happen
for the negative mass Schwarzschildcase because, unlike
The Kretschmann scalaris finite everywhere,in particu-
the AdS soliton, it has no energy lower bound. Another
lar at r =0 the behaviour is
interesting feature is the significant difference in the in-
40 r 4 2 tercepts of the confining potentials. An understanding
R Rabcd r=0 1+ 0 (19)
abcd −→ ℓ4 λ ofthis in terms ofthe minimal worldsheet geometryap-
(cid:20) (cid:16) (cid:17) (cid:21) pears in Sec. III. Lastly Fig. 3 gives the V(L) curve for
In addition to this regulated negative mass solution, the regulated negative mass case; this is identical to the
we will also consider the AdS soliton [17] with metric unregulated case for the range of 0.01 r 5 in units
m
≤ ≤
of r (which appear in Fig. 1).
0
r2 ℓ2 dr2
ds2 = dt2+dx2+f(r)dy2+dz2 + (20)
ℓ2"− # r2g(r)
with f =g =1 r4/r4. This solutionhas a negative en-
− 0
ergy,which unlike the negative mass Schwarzschildcase,
is bounded below. The worldsheet we consider has the
same boundary loop in the t x plane, but its induced
−
metric is now
r2 r2 ℓ2
ds2 = dt2+ + r′(x)2 dx2 (21)
−ℓ2 −ℓ2 r2g(r)
(cid:18) (cid:19)
Therefore the boundary potential for the soliton is ob-
tained by simply setting f =1 and g =1 r4/r4 in the
− 0
formulas (14).
Letusfirstnotethatforsufficientlylarger theworld
m
sheet does not probe too deep into the AdS bulk, and
so we expect that V(L) behaviour in this limit should
be that of the pure AdS case, i.e. Coulomb. Therefore
deviations from Coulomb behaviour should come from
world sheets that descend deeper into the bulk. This
suggests looking at the integrands in the formulas for L
andV forr 1andΛ 1. Expandingtheintegrands
m ≪ c ≪ FIG. 1: qq¯ potential V(rm) and their separation L(rm) for
in a Taylor series at r = 0 for the regulated negative
m the negative mass AdS-Schwarzschild geometry. This shows
mass Schwarzschildcase gives
aUV/IRrelation: largeV andLontheboundarycorrespond
r 2 2λ2r2 1 to small rm and vice versa.
V = 0 L 0 +O(r3 ,Λ3) (22)
(cid:16) ℓ (cid:17) − r04+λ4Λc m c Numerical Fit: It is instructive to attempt a numerical
fit to V(L) to have some idea about order of magnitude
Thisdemonstratesthepconfiningbehaviour,andgivesthe
of numericalfactors involved. We give this below for the
slope and intercept of the V(L) line; the expansion that
unregulatednegativemass Schwarzschildcase. The fit is
gives this result obviously cannot be done for the posi-
of the form
tive mass case due to the square roots in the integrands.
a
−1
We note that the regulator and cutoff lengths λ and Λc 2πVnum(L)=−a0− L +a1L (23)
determine the intercept.
Fig. 1 givesthe plots ofV(r )andL(r )for the neg- where the constants are
m m
ative mass AdS-Schwarzschild geometry. Since rm mea- ~c r 2
0
sures the descent of the minimal world sheet into the a 0.9015
0
≈ r ℓ
bulk, it is apparent from this figure that the boundary (cid:18) 0(cid:19)(cid:18) s(cid:19)
mscaaylesbeVinatnedrpLrehtaevdeaasnainnvinerssteanrecleatoifotnhsehiUpVw/itIhRrrmel.atTiohnis, a−1 ≈ 1.2786 ~c 2gY2MN
which states that highenergy in the bulk correspondsto ~qc (r0/ℓs)4
a 0.9547 (24)
low energy on the boundary. 1 ≈ r2 2g2 N
(cid:18) 0(cid:19) YM
p
5
L for which V(L ) = 0 is given by L = (a /2a ) +
c c c 0 1
1
(a /2a )2+(a /a ) 1.72201 2g2 N 2 ℓ2/r .
0 1 −1 1 ≈ YM s 0
p (cid:0) (cid:1)
IV. INTRINSIC GEOMETRY OF THE
MINIMAL WORLDSHEET
It is known that for f = g = 1, the parametric equa-
tionV(r )andL(r )leadexactlytotheCoulombphase
m m
V 1/L [5]. Of much interest is the question of what
∼
f and g lead to confinement, i.e. V L and what the
∼
transitionregionbetweenthephaseslookslike. Itispos-
sibletoalgebraicallyanalysetheexpressionsofV(r )and
m
L(r ) to lay downconditions under which a givenback-
m
ground will lead to Coulomb/confinement phase, and
then verify what specific solutions fall into this class.
This was done for example in [8]. However, the ques-
tion we wish to consider is whether there is a geometric
rather than algebraic way of characterizing this feature,
since that might give a clearer insight into the whole is-
sue. We show in this section that this is indeed possible
FIG. 2: qq¯potential V(L) from AdS, AdS soliton and nega- using the intrinsic curvature and null geodesics of the
tivemass AdS-Schwarzschildgeometries; thelattertwo show minimal worldsheet.
a Coulomb to confinement transition. The AdS soliton line
terminatesatrm=r0,whichcorrespondstoitsenergybound.
A. qq¯potential and null geodesics
WeshowthatV(r )hasarathersimpleinterpretation
m
in terms of the affine parameter λ along null geodesics
k lying in the minimal worldheet. Consider the affine
parametersuch that k ∂ = 1, i.e. normalizedto have
t
unit Killing energy wit·h respe−ct to the ∂ Killing vector
t
of the bulk geometry. We shall see that the potential is
just the bulk affine time between the boundary quarks.
Interestingly, the term that regulates the minimal world
sheet action has exactly the same interpretation.
Theinducedgeometryontheminimalandfree quarks
worldsheets can be expressed in the generic form
ds2 = N(n)2dt2+dn2
−
dn = Ω(r)dr (25)
andΩ(r)andN(n(r))areknownfunctionsofr. Thenull
geodesics k for metrics of this form satisfying ∇kk = 0
are given by
1 1
k=C ∂ ∂ (26)
N2 t± N n
(cid:18) (cid:19)
FIG. 3: qq¯ potential V(L) for the regulated negative mass where the constant C = k · ∂ is the Killing energy
t
−
AdS-Schwarzschild metric; it is identical to the unregulated of these geodesics. We fix the scaling freedom in the
case for therange rm plotted. choice of affine parameter by setting C = 1, and also
choose (without any loss of generality due to x x
→ −
symmetry) the + sign above corresponding to outgoing
wherewehaveusedthestandardAdS-CFTdictionaryto geodesics. We therefore have
substitute ℓ2 =ℓ2 2g2 N in terms of ℓ andthe gauge
s YM s N(n)dn=dλ (27)
theory parameters g and N. Note that the numeri-
YM
p
cal factor 1.2786 in the coefficient for a is very close Letusdenotetheinteracting(minimal)andfreequarks
−1
to the pure AdS case for which a = 0 = a , and (nu- worldsheets respectively by r (x) and r (x). This gives
0 1 I F
merical coefficient in a ) 1.4355. Also, the length the following formulas for N and Ω.
−1
≈
6
1. Interacting quark worldsheet: x= L/2tox=+L/2. (Infact,ifonerescalesthemet-
ric n−ear the boundary by (ℓ/r)2 and then take r
The minimal worldsheet is given by Eqn. (6) for → ∞
limit, L is exactly the affine distance since the bound-
which the induced metric is
ary metric becomes flat.) Therefore, if T and t
sheet bound
r2f(r ) ℓ4 denote the bulk and boundary affine times, then
ds2 = I I dt2+ dr2
I ℓ2 − (f(r )r4 f r4 )g(r ) I
(cid:18) I I − m m I (cid:19) V(L) Tsheet(tbound) (33)
(28) ≡
This particulargeometricinterpretationofthe qq¯poten-
From this we have tial might be relevant for interpreting the behaviour of
r the V(L) curve in terms of a causal connection between
I
N NI(n(rI))= f(rI) events in the bulk and boundary.
≡ ℓ
ℓ rp 2 f(r )
I I
Ω Ω (r )=
≡ I I rI (cid:18)rm(cid:19) sg(rI) × B. Geometrical signature of confinement
1
(29) We nowexplore anotherfeature ofthe minimal world-
(r /r )4f(r ) f
I m I − m sheetgeometrywhichappearstobeaplausiblesignature
p of the Coulomb to confinement phase. The quantity we
2. Free quarks worldsheet: The induced metric of the consider is the intrinsic Ricci scalar (2)R of the minimal
worldsheets corresponding to free quarks is
worldsheet calculated at the radius r . For metrics of
m
the form (25), we have
r2f ℓ2
ds2 = F dt2+ dr2 (30)
F − ℓ2 r2g F 2 d2N
F (2)R= (34)
−N dn2
which gives
For the minimal wordsheet given by (28), the value of
rF f(rF) (2)R at y = r/rm = 1 can be obtained easily by noting
N NF(n(rF))= that Ω−1 0 as y 1. This gives
≡ pℓ I → →
ℓ
Ω ≡ ΩF(rF)= r g(r ) (31) (2)Rm = (2)R|y=1
F F
1 g(1;χ) d d
= lnN y4f
p −ℓ2f(1;χ) dy I dy
We arenow readyto establishthe connectionbetween (cid:20) (cid:21)y=1(cid:20) (cid:21)y=1
(cid:0) (cid:1)
the qq¯ potential V and the affine parameter λ along (35)
the null geodesic k. Using the above expressions for
where χ=r /r and g(1;χ) g(y;χ) etc.
N ,Ω ,N and Ω , V may be written as m 0 y=1
I I F F ≡ |
Weexaminethisquantityforthethreebulkgeometries
B B studied in the last section, since as we have seen, each
exhibits a Coulomb to confinement transition and each
V = 2 dr Ω N 2 dr Ω N
I I I F F F
− hasnegativeenergy. Theformulasfor(2)Rareasfollows:
Z Z
M C
B B 1. Negative mass AdS-Schwarzschild
= 2 dλ 2 dλ (32)
I F
−
Z Z
M C 4 χ4 1
unregulated: (2)R = −
where ,M and C symbolically denote the values of the m −ℓ2 χ4+1
B (cid:18) (cid:19)
integration parameter at the boundary, minimal radius 4 χ4 1
rm, and the cut-off radius Λc respectively. We have regulated: (2)Rm = −ℓ2 χ4−+1 ×
also suppressed the explicit dependence of r.h.s on r (cid:18) (cid:19)
m
for brevity. exp (rm)4
1 − λ
This is a compact form of the expression for V, and − χ(cid:2)4+1 (cid:3)!
comes with a nice geometric interpretation: the qq¯ po-
(36)
tential is (twice) the difference of affine distances along
thenullgeodesicsoftheminimalworldsheet,propagating
2. AdS Soliton
from minimal point to the boundary, and those on the
freequarkworldsheetpropagatingfromcut-offradiusto
the boundary.
4 χ4 1
We note also that L is (proportionalto)the affine dis- (2)R = − (37)
tance travelled by a null geodesic in the boundary, from m −ℓ2 χ4
(cid:18) (cid:19)
7
3. Positive mass AdS-Schwarzschild bulk geometry(since the bulk metric depends only
on 1+(r /r)4). Algebraically, it is easy to trace
0
this result to the (lnN) term in Eq. (35).
4 χ4+1
(2)R = (38) These cases suggestthat a geometric signatureof con-
m −ℓ2 (cid:18)χ4−1(cid:19) finement is present in the intrinsic curvature (2)Rm of
the minimal worldsheet at the minimal point; the dis-
tinguishing feature is the increase in this quantity as r
m
1 decreases, and, in particular, the change in sign (from
{2RH2L
m
4 ve to +ve) at r = r . This in turn is reflection of the
0
−
fact that the negative (quasi-local) energy region begins
1 to be probed by the worldsheet.
In fact, there is a more direct way of understanding
rm the “repulsion” associated with the negative energy
1 2 3 4 5 r0 solutions. To see this, note that, for the negative mass
AdS black hole, g = (1/ℓ2)(r2+r4/r2) has absolute
− 00 0
-1 minimum at r = r0. Therefore, acceleration of a static
particle, which is given by ai = (1/2)(g/f)∂ ( g )δi,
r − 00 r
changes sign from +ve to ve as r becomes less than
-2 r . That is, an inward rad−ial force is needed to keep
0
a particle static for r < r , implying repulsive gravity.
0
In fact, the (g/f)(dN/dy) term in expression (35)
for (2)R is proportional to ar, which provides a
FIG. 4: Plot of (2)R at r = rm as a function of rm/r0. (a) direct conmnection between this repulsive behavior and
-vemassAdS(blue)withtheblackdotrepresentinginflection
intrinsic worldsheet curvature, thereby strengthening
pointofthecurve(b)-vemassregulatedAdS(dotted(thick)
our assertion that the latter captures the main feature
blue) (c) +ve mass AdS (dashed red) (d) AdS soliton (dark
red). (gravitational repulsion) responsible for transition to
confinement. This remains true also for the AdS soliton
The distinctive features present in these expressions (g00 = (r/ℓ)2;f = 1), in which case, although g00 has
and their graphs are the following. no min−ima, ar changes sign at r0 since g(r) changes
sign. The above argument also hints at an intriguing
(i) The first three of the above expressions suggest connectionbetweenquasi-localenergy,effectivepotential
that, for solutions which do give a Coulomb to governing point particle kinematics, and the signature
confinement transition, (2)R at minimal radius of these features in the static worldsheet configuration.
changes sign at r =r .
m 0
(ii) Asexpected,r r isthesameforallfourcases,
m 0
≫
ie. V. DISCUSSION
(2)R 4/ℓ2
m
rm−≫→r0 − We have studied the qq¯potential modelled by asymp-
totically AdS bulk geometries with negative mass, both
Furthermore, for the negative mass AdS
withandwithoutcurvaturesingularities. Wehaveshown
Schwarzschild case, (2)R goes from 4/ℓ2
for r r to 4/ℓ2 for r m r , whereas fo−r the thatV(L)exhibitsatransitionfromCoulombtoconfine-
m ≫ 0 m ≪ 0 ment phase, corresponding to the minimal string world
AdS solitonit divergesas r 0. This is however
m → sheet radius dipping from near the AdS boundary to-
of no relevance since for the soliton, r [r , ).
m ∈ 0 ∞ wards the r =0 region of the bulk. Therefore it appears
that the low energy confinement phase corresponds to
(iii) For the positive mass AdS Schwarzschild solution,
(2)R diverges on the horizon. This is surprising what would be considered the high energy sector of the
m
bulk in the context of AdS/CFT duality. The essential
because all bulk invariants are known to be finite
points highlighted by the result presented here are: (i)
on the horizon surface. The divergence is perhaps
confinement is a generic feature related to the presence
areflectionofthefactthatastaticworldsheetcon-
of an additional length scale (other than the AdS scale)
figurationclosetothehorizonmightrequireinfinite
in the bulk spacetime, which nevertheless does not cor-
stresses,justasaninfiniteaccelerationisneededto
respondtoacausallyinaccessibleregioninthebulkwith
hold a point particle near the horizon.
an event horizon. (ii) this result has nothing to do as
(iv) It is also curious that (2)R senses the pres- such with the presence of naked singularities; indeed, as
m
ence of the scale r even for the negative mass wehavefound the singularitycanbe regulatedsmoothly
0
Schwarzschildsolution,despitethefactthatsucha while retaining the seemingly important r4 behaviour of
scalelooksquiteirrelevantfromthepointofviewof the metric functions, without affecting the confinement
8
phase, and (iii) from the cases studied, it appears that with quantum gravity via holography, while preserving
thekeyfeaturethatgivesrisetoconfinementisbulkneg- the UV/IR aspect of the AdS/CFT correspondence.
ative energyrepulsion. The latter is alsoassociatedwith
singularity avoidance in quantum gravity; ie.
singularity avoidance in QG negative energy Acknowledgements We thank Jaume Gomis and Rob
←→
confinement in gauge theory. Myers for comments and discussion. This work was sup-
←→ ported in part by the Natural Science and Engineering
This plausible chain indicates that confinement in gauge ResearchCouncil of Canada. The work of D.K. was also
theory might be intimately and generically connected supported by an AARMS fellowship.
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