Aspects of large N gauge theory dynamics as seen by string theory
David J. Gross
Institute for Theoretical Physics, University of California, Santa Barbara, California 93106
Hirosi Ooguri
Institute for Theoretical Physics, University of California, Santa Barbara, California 93106;
Department of Physics, University of California, Berkeley, California 94720;
and Theory Group, Lawrence Berkeley National Laboratory, Berkeley, California 94720
~Received 29 May 1998; published 8 October 1998!
In this paper we explore some of the features of large N supersymmetric and nonsupersymmetric gauge
theories using Maldacena’s duality conjectures. We show that the resulting strong coupling behavior of the
gauge theories is consistent with our qualitative expectations of these theories. Some of these consistency
checks are highly nontrivial and give additional evidence for the validity of the proposed dualities.
@S0556-2821~98!00618-3#
PACS number~s!: 11.25.Hf, 11.15.Pg
I. INTRODUCTION
The newest, and perhaps most interesting, of the dualities
of string theory is that conjectured by Maldacena, which re-
lates the large N expansion of conformal field theory in d
dimensions to string theory in a AdS
d1 1
3 M spacetime
background @where AdS
d1 1
is (d1 1)-dimensional anti–de
Sitter space and M is a compact space#@1#. The dictionary
that relates these dual descriptions identifies the 1/N expan-
sion of the field theory to the perturbative expansion of the
string theory, and the strong coupling expansion of the field
theory to the
a
8
expansion of the string theory. This conjec-
ture offers the exciting possibility of using perturbative
string theory to explore the large N limit of field theory.
The simplest case of Maldacena’s conjecture is the duality
between large N supersymmetric, conformally invariant,
SU(N) gauge theory in four dimensions ~with coupling g
YM
2
)
and type IIB string theory expanded about an AdS
5
3 S
5
background. Here the string coupling, g
st
, is proportional to
g
YM
2
; N equals, in string theory, the magnitude of the five-
form flux on the five-sphere; and (g
YM
2
N)
1/4
is proportional
to the radius of curvature of the background AdS
5
space.
One can therefore hope to calculate gauge theory correlation
functions, for large N and large (l5 g
YM
2
N), in terms of
weak coupling string theory in the semiclassical
approximation—i.e. supergravity.
The precise relation between the gauge theory correlation
functions and the supergravity effective action has been
given by @2,3#, following earlier works @4#. In particular this
prescription determines the dimensions of operators in con-
formal field theory in terms of the masses of particle in the
string theory. This correspondence has been checked for the
duality between SU(N) gauge theory in four dimensions and
type IIB string theory expanded about an AdS
5
3 S
5
back-
ground, where it was shown that there is a precise correspon-
dence between the chiral fields of the conformal gauge
theory and the finite mass string states in the above limit,
including the complete infinite tower of massive Kaluza-
Klein states of ten-dimensional supergravity on the 5-sphere
@5,6#.
In @7,8#, it was shown that the strong coupling limit of the
large Wilson loop for large N can be evaluated using semi-
classical string theory, thereby obtaining the interaction en-
ergy between infinitely massive quarks and anti-quarks @ex-
ternal sources in the fundamental representation of SU(N)],
separated by distance R as
E
qq
¯
52
4
p
2
G
~
1/4
!
4
A
2l
R
, ~1!
a result that is completely consistent with our limited under-
standing of the gauge theory, wherein the 1/R behavior is
dictated by conformal invariance. The proportionality to
A
l
suggests that the Coulomb force is somewhat reduced from
the weakly coupled value of l. Similar calculations have
been performed for the monopole-monopole and monopole-
quark potential, yielding, as expected, S-dual expressions
@9#.
One can regard Maldacena’s duality as realizing the long
sought goal of finding the master field representation of large
N gauge theory correlation functions. What is most surpris-
ing from this point of view is that the master field lives in a
compactified ten-dimensional space-time, and corresponds to
supersymmetric type IIB string theory. That there should ex-
ist a string representation of the N5 4 conformally invariant
large N gauge theory is somewhat surprising, since the tra-
ditional arguments for such a representation have been for
confining theories, whereas here we have a string theory for
the Coulomb phase of the gauge theory. Thus, even though
the Wilson loop is given by the minimal area classical string
configuration spanning the loop, the fact that the loop can
meander into the extra dimensions and the nature of the ge-
ometry of AdS space lead to a 1/R potential in this case.
Although the duality between SU(N) gauge theory in four
dimensions and type IIB string theory expanded about an
AdS
5
3 S
5
background is of great academic interest, the most
exciting extension of Maldacena’s conjecture is to non-
supersymmetric gauge theories, especially to the physically
relevant case of four-dimensional, non-supersymmetric
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gauge theory—namely QCD
4
. As Witten has shown @10#,it
is reasonable to extend the conjecture to cases where super-
symmetry is broken, thereby obtaining properties of non-
supersymmetric gauge theories in the large N limit. For ex-
ample, one can easily extend the duality to discuss the finite
temperature behavior of the N5 4 gauge theory, by compac-
tifying the ~Euclidean! time direction of the background
space time of AdS on a circle of radius }1/temperature,in
which case supersymmetry is broken by the boundary con-
ditions on the circle. One can argue that, since supersymme-
try is broken, the fermions and the scalars acquire a mass
and, at least for large temperature decouple, thus yielding a
duality to high temperature QCD. Witten showed that in this
case one derives many of the expected features of high tem-
perature gauge theory; including a non-zero expectation
value of temporal ~Polyakov! loops, an area law for spatial
Wilson loops and a mass gap ~i.e. a magnetic mass!.
Finally, Witten has proposed a strategy to study ordinary
four dimensional QCD at zero temperature using string
theory @10#. This can be done by using Maldacena’s conjec-
ture to relate the large N limit of the SU(N)-type (2,0)
theory in R
6
to M theory on AdS
7
3 S
4
and dimensionally
reducing these to four dimensions. ~Throughout this paper,
we consider theories on Euclidean signature spaces, and
AdS
7
here means its Euclideanized version.! To do this and
to break the supersymmetry one sets the (2,0) theory on S
1
3 S
1
3 R
4
with supersymmetry breaking boundary condition
on the fermions around one of the S
1
’s.
An obvious candidate for its M theory dual would be
obtained by periodically identifying points on AdS
7
corre-
sponding to the periodicity’s of the S
1
3 S
1
and by imposing
the supersymmetry breaking boundary condition on the fer-
mions by hand. There is, however, another candidate which
obeys the same boundary condition. It is the anti–de Sitter
Schwarzschild solution constructed by Hawking and Page
~for AdS
4
case!@15#. The supersymmetry breaking boundary
condition is automatically imposed by the Schwarzschild ge-
ometry. It turned out that the classical action for the AdS
Schwarzschild solution is smaller than that of the vacuum
AdS
7
, and therefore is dominant in the large N limit @10#.
To make contact with four dimensional QCD we must
shrink the radii of the two circles to zero in a certain limit. In
this construction, the six-dimensional ~2,0! theory is re-
garded as a regularization of the four-dimensional QCD. The
ultraviolet cut-off scale is therefore set by the size of the
compact space S
1
3 S
1
.
Denote the radius of the supersymmetry preserving circle
by R
1
and that of the supersymmetry breaking one by R
2
.
The gauge coupling constant g
YM
of QCD
4
is given by the
ratio of the radii g
YM
2
5R
1
/R
2
. In the ’t Hooft limit, where
one keeps g
YM
2
N to be finite, the circle S
R
1
1
shrinks to zero as
one takes N→`. This corresponds to the IIA limit of M
theory as S
R
1
1
is the supersymmetry preserving circle. There-
fore one could have started with the theory on N D4 branes
in the IIA theory, wrapped around a circle with nonsuper-
symmetric boundary conditions, rather than the six-
dimensional theory. We will take this approach throughout
the paper. QCD is then regarded as the dimensional reduc-
tion of the five dimensional theory at high temperature, with
coupling g
YM
2
5g
5
2
T, where g
5
is the five dimensional cou-
pling and T the temperature ~inverse radius! of the circle.
Witten has argued that Wilson loops exhibit a confining
area law behavior in this geometry for large N and large
g
YM
2
N. However, as he points out, this does not establish that
QCD is a confining theory. The gauge theory so constructed
has an ultraviolet cutoff (}T) and the coupling g
YM
should
be thought of as the bare coupling at distances corresponding
to 1/T. The string tension will turn out, for large l5 g
YM
2
N
~as we shall show below!, to be proportional to lT
2
.To
construct four dimensional QCD we must take
T→ ` and l→
b
ln
S
T
L
QCD
D
, ~2!
where L
QCD
is the QCD mass scale. Presumably we would
find, were we able to calculate the small l behavior of the
tension, that the tension behaves as exp
@
22b/l
#
T
2
;L
QCD
2
.
This calculation is beyond our control at the moment. For
small l the background geometry develops singular behavior
and the supergravity approximation surely breaks down. To
deal with this continuum limit one would have to be able to
calculate the properties of string theory with background
Ramond-Ramond ~R-R! charge in a rather singular back-
ground.
Thus, for the time being, the Maldacena-Witten conjec-
ture only informs us about the behavior of large N QCD,
with a fixed ultraviolet cutoff in the strong coupling ~large l)
regime. The resulting physics should be compared best with
strong coupling lattice gauge theory, where the lattice spac-
ing a is analogous to 1/T, the radius of the fifth dimension.
What is remarkable here is that the short distance cutoff,
unlike in the case of lattice, does not destroy the rotational or
Lorentz symmetry of the theory. Indeed, at short distances
we see a higher dimensional theory with more symmetry,
indeed enough symmetry to render the theory finite. We are
using the six dimensional, ultraviolet finite, ~2,0! theory to
define the theory in the ultraviolet, yet its infrared behavior
should be qualitatively the same as QCD.
In this paper we shall explore some of the features of
large N supersymmetric and nonsupersymmetric gauge theo-
ries using the above duality conjectures. We shall show that
the resulting strong coupling behavior of the gauge theories
is, in all cases, consistent with our qualitative expectations of
these theories. Some of these consistency checks are highly
nontrivial and give additional evidence for the validity of the
proposed dualities.
First we shall explore, in the next section, the connected
correlation function of Wilson loops. This kind of calculation
can be used for many purposes among which are the evalu-
ation of the electric mass ~or screening length! of high tem-
perature QCD, the glue ball spectrum of confining gauge
theories and the demonstration that in the confining phase of
QCD monopoles are condensed. In particular we outline how
the glueball spectrum of this version of strong coupling QCD
could be calculated.
DAVID J. GROSS AND HIROSI OOGURI PHYSICAL REVIEW D 58 106002
106002-2
In Sec. III, we generalize the discussion of QCD to the
case where the
u
parameter is non-zero and argue that we
can demonstrate oblique confinement.
In Sec. IV we generalize the evaluation of Wilson loops
in the fundamental representation to higher representations.
Here we find that the string theory naturally produces the
behavior of higher representations that we would expect in a
confining theory—a result that depends critically on the mas-
ter field being described by fermionic strings.
In Sec. V we argue that one can also use the duality to
discuss heavy quark baryonic states and determine the effec-
tive energy of N fundamental representation quarks in a sin-
glet state for large N. The construction of the baryon is pos-
sible because of the Chern-Simons term in the action for
supergravity on AdS. The same arguments allow us to show
that the interaction energy between any finite number of
quarks is zero for the conformally invariant supersymmetric
four dimensional gauge theory and infinite for the confining
theory.
Finally, we conclude with a discussion of the possibility
of a large N phase transition. If such a phase transition exists
the power of the conjectured duality would be significantly
weaker.
While this paper was being typed, we learned of the work
@11# where a similar construction of baryons is given.
II. CONFINEMENT, MONOPOLE CONDENSATION AND
GLUEBALL
In this section, we first review the works @10,12–14#
where it was shown how confinement in strong coupling
QCD
p
can be seen in the dual description based on AdS
supergravity. In particular, they demonstrated the area law
behavior of the Wilson loop expectation value. We then dis-
cuss implications of this result and clarify an issue that was
raised in @12–14# on the apparent divergence of the electric
and magnetic masses. It turns out that this is related to the
computation of the mass gap suggested in @10#. We discuss
how one can compute glueball masses in this description.
According to Maldacena’s conjecture @1,16#, the
(p11)-dimensional maximally supersymmetric gauge
theory realized as the low energy dynamics of N Dp branes
(p<5) is dual to type II string theory on the near horizon
geometry of the Dp brane, as given by
l
s
2 2
ds
2
5
A
gN
u
72p
du
2
1
A
u
72p
gN
(
i50
p
dx
i
2
1
A
gNu
p23
dV
82p
2
, ~3!
where l
s
is the string length, dV
82 p
is the line element of
S
82 p
, and g is related to the Yang- Mills coupling constant.
We have neglected numerical factors that are not relevant to
the following discussion. For pÞ3, the dilaton
f
depends on
u and is given by
e
f
5 g
S
gN
u
72p
D
~
32p
!
/4
. ~4!
In particular, for p53, the near horizon geometry ~3! is AdS
and the dilaton ~4! is constant, corresponding to the fact that
the theory on D3 brane is conformal.
Witten @10# proposed to study non-supersymmetric QCD
p
by compactifying the supersymmetric theory in (p1 1) di-
mensions on a circle and break the supersymmetry by impos-
ing anti-periodic boundary conditions on the fermions. In the
dual type II theory this corresponds to considering the AdS
Schwarzschild geometry
l
s
2 2
ds
2
5
A
gN
u
72p
du
2
12u
0
72p
/u
72p
1
A
u
72p
gN
~
12u
0
72p
/u
72p
!
d
t
2
1
A
u
72p
gN
(
i51
p
dx
i
2
1
A
gNu
p23
dV
82p
2
, ~5!
with the dilaton
f
given by Eq. ~4!. We can regard
(
t
,x
1
,..,x
p
) as coordinates for the (p11)-dimensional
gauge theory. To make the horizon at u5 u
0
regular, the
coordinate
t
has to be periodically identified as
t
→
t
1 1/T
with T being related to u
0
by
u
0
5
~
gNT
2
!
1/
~
52 p
!
. ~6!
Since the circle in the
t
direction is contractible at u5 u
0
, the
boundary condition on the fermions around the circle is au-
tomatically anti-periodic, breaking the supersymmetry. For
large T, the (p11)-dimensional theory becomes effectively
p-dimensional, the fermions and scalars decouple, and the
theory should resemble QCD
p
in the infra-red.
If QCD
p
is confining, the vacuum expectation value of the
Wilson loop operator W(C) should exhibit area law behav-
ior. In @10,12,13# this was shown to be the case, for large
gN, by evaluating the classical action of string world sheet
bounded by a loop on R
p
located at u5 `. Because of the
u-dependent factor
A
u
72 p
/gN in front of (
i
dx
i
2
in the met-
ric ~5!, it is energetically favorable for the world sheet to
drop near the horizon u5 u
0
before spreading out in the R
p
direction. At the horizon, the u-dependent factor becomes
A
u
0
72 p
gN
5
~
gN
!
1/
~
52 p
!
T
~
72 p
!
/
~
52 p
!
, ~7!
where we used Eq. ~6!. Therefore the area dependent part of
the Wilson loop expectation value becomes
^
W
~
C
!
&
5 exp„2
~
gN
!
1/
~
52 p
!
T
~
72 p
!
/
~
52 p
!
A
~
C
!
…, ~8!
where A(C) is the area bounded by the loop C. Since the
QCD
p
coupling constant g
YM
is related to g by g
YM
2
5gT,
the string tension derived from the above formula is
~
tension
!
p
5
~
g
YM
2
N
!
1/
~
52 p
!
T
~
62 p
!
/
~
52 p
!
. ~9!
For p5 3,4, this agrees with the formulas derived in @12,13#.
~See Fig. 1.!
ASPECTS OF LARGE N GAUGE THEORY DYNAMICS AS . . .
PHYSICAL REVIEW D 58 106002
106002-3
In four dimensions, it is expected that confinement is as-
sociated with magnetic monopole condensation. It is interest-
ing to see that this in fact happens in this construction.
1
To
discuss QCD
4
, we start with the five-dimensional theory on
D4 branes. The magnetic monopole in five dimensions is a
string which is realized as a D2 brane ending on a D4 brane
@17,18#. The monopole in four dimensions is obtained by
wrapping the string around the compactifying S
1
.Itisnow
straightforward to compute the potential between a mono-
pole (m) and an anti-monopole (m
¯
). Consider a pair of m
and m
¯
traveling along the x
1
-axis in R
4
and separated in x
2
direction by distance L. In the large gN limit, the force be-
tween them is mediated by a D2 brane bounded by S
1
times
the trajectories of m and m
¯
, which are located at u5 `.
Away from the boundaries, the D2 brane can spread in the u
direction. In its classical configuration, u would be a function
of x
2
only because of the symmetry of the problem. If we use
(
t
,x
1
,x
2
) as the coordinates on the D2 brane, the induced
metric on the brane is then
G
t
,
t
5
A
u
3
gN
S
12
u
0
3
u
3
D
G
11
5
A
u
3
gN
,
G
22
5
A
gN
u
3
~
du/dx
2
!
2
12u
0
3
/u
3
1
A
u
3
gN
. ~10!
By taking into account the dilaton configuration ~4!, which in
this case is
e
f
5 g
S
u
3
gN
D
1/4
, ~11!
the D2 brane action per unit length in the x
1
direction be-
comes
E
mm
¯
5
E
0
L
d
t
dx
2
e
2
f
A
G
tt
G
11
G
22
5
1
gT
E
0
L
dx
2
A
S
du
dx
2
D
2
1
1
gN
~
u
3
2u
0
3
!
, ~12!
and it gives the potential energy for the m-m
¯
pair.
The next task would be to minimize this action. In fact,
essentially the same problem has already appeared in @12,13#
where the correlation function of temporal Wilson loops in
five dimensions was studied. There one considers a string,
rather than the D2 brane, wrapping in the
t
direction and
spreading in the x
2
direction. Because of Eqs. ~10! and ~11!,
we have
e
2
f
A
G
tt
G
11
G
22
5
1
g
A
G
tt
G
22
. ~13!
Therefore the classical action of the string is equal to g times
that of the D2 brane discussed in the above paragraph.
Therefore, we can borrow the result of @12,13# to discuss the
m-m
¯
correlation.
The new feature of this problem is a classical instability
of the D2-brane world volume. When the distance L between
m and m
¯
is less than a certain critical distance L
crit
, which is
equal to 1/T times some numerical factor, there is a D2 brane
configuration minimizing the action ~12! and connecting m
and m
¯
.IfLexceeds this critical distance, there is no con-
nected D2 brane configuration minimizing the action ~Fig.
2!. This happens because the G
tt
component of the induced
metric ~10! can be made arbitrary small by going near the
horizon u5 u
0
reflecting the fact that the compactification
circle along
t
is contractible in the AdS Schwarzschild ge-
ometry. Since the circle is contractible, the D2 brane can
split into two pieces each of which has a topology of a disk
and is bounded by the trajectory of m or m
¯
. Therefore, for
L. L
crit
, the potential between m and m
¯
becomes constant
and the force between them vanishes. This suggests that the
magnetic monopole is completely screened. In this construc-
tion, therefore, confinement is in fact accompanied by mono-
1
While this work was in progress, we received @14# where a re-
lated issue was discussed.
FIG. 1. The string drops to the horizon first before spreading in
the R
p
direction.
FIG. 2. For L.L
crit
, there is no volume-minimizing D2 brane
configuration connecting to the m-m
¯
pair.
DAVID J. GROSS AND HIROSI OOGURI PHYSICAL REVIEW D 58 106002
106002-4
pole condensation. If we view this system as finite tempera-
ture QCD
p1 1
, such a complete screening indicates that the
magnetic mass is infinite. This is somewhat puzzling and we
will address this issue later in this section.
A similar classical instability also shows up when one
studies correlation functions of Wilson loops. If one consid-
ers two Wilson loops in R
4
and repeats the above analysis to
compute their correlation function, one finds that, beyond a
certain critical distance determined by the size of the loops,
the correlation function vanishes identically. Once again, this
is because the loops are contractible and the string stretched
between the loops becomes classically unstable beyond the
critical distance ~Fig. 3!. This result is again somewhat puz-
zling since one would expect that the Wilson loops correla-
tion for large distance would be characterized by glueball
exchange. This result seems to indicate that the glueball
mass in QCD
p
is infinite. To address this issue, it is useful to
look into the nature of the classical instability and discuss
what happens at the critical distance and beyond.
The instability of minimal surfaces has been known for a
long time. It was Euler who showed that a minimal surface
bounded by a two concentric circle in R
3
is given by a
catenoid. Let us put the two circles of radius R
0
at z5
6 L/2. Euler’s catenoid is given by
A
x
2
1 y
2
5 R
min
cosh
S
z
R
min
D
, ~14!
where R
min
is the minimum radius of the catenoid, which is a
function of the distance L between the circles and the radius
R
0
determined by the relation
R
0
5 R
min
cosh
S
L
2R
min
D
. ~15!
When the two loops coincide (L5 0), obviously this formula
gives R
min
5 R
0
. As one increases L, the minimum radius
R
min
decreases. As shown in Fig. 4, however, there is a criti-
cal value of L
crit
5 1.325R
0
. For L. L
crit
, there is no solution
to Eq. ~15!. There the only minimal surface is a pair of disks
bounded by the two circles, called the Goldschmit discon-
tinuous solution. At L5 L
crit
, the catenoid becomes unstable.
A small perturbation would make the surface to pinch and
split into the two disks.
At L, L
˜
crit
5 1.056R
0
, the area of the catenoid is smaller
than that of the Goldschmit solution and therefore the
catenoid is absolutely stable. At L5 L
˜
crit
, the areas of the
two solutions coincide and, for L
˜
crit
, L, L
crit
, the catenoid
becomes more voluminous than the Goldschmit solution.
Therefore the transition from the catenoid to the Goldschmit
solution at L5 L
crit
is of the first order.
What does this mean for the Wilson loop correlation func-
tion? When the distance between the loops C
1
and C
2
is less
than the critical distance L, L
crit
, the main contribution to
the connected part of the correlation function
^
W(C
1
)W(C
2
)
&
comes from the classical string connecting
C
1
and C
2
.AtL5L
crit
, the string world sheet becomes un-
stable and starts to collapse. Before the surface becomes dis-
joint, however, the supergravity approximation breaks down
when the radius of the cylinder becomes of the order of the
string length l
s
. After that, quantum fluctuations of the sur-
face start to support the world sheet against the total col-
lapse, and the two disks would be connected by a thin tube of
a string scale l
s
. For large L, the thin tube is represented by
the supergraviton exchange between the two disks. Therefore
the correlation between the Wilson loops does not com-
pletely vanish, but are mediated by the supergraviton ex-
change between the disks ~Fig. 5!. This indicates that the
supergravitons in the AdS
p1 2
Schwarzschild blackhole ge-
ometry should be identified with the glueballs of QCD
p
.
Another way to obtain glueball masses would be to com-
pute correlation functions of local operators in QCD
p
and
look for particle poles. According to @2,3#, a two-point cor-
relation function of local operators in the
(p11)-dimensional supersymmetric gauge theory is ob-
tained by computing the Green’s function of the correspond-
ing supergraviton ~or its Kaluza-Klein cousin! on AdS
p1 2
.
Similarly acorrelation function in QCD
p
should be related to
a Green’s function on the AdS
p1 2
Schwarzschild geometry.
FIG. 3. For L. L
crit
, there is no area-minimizing string world
sheet connecting the two Wilson loops. The critical distance L
crit
is
determined by the size of the loop.
FIG. 4. For 0, L, 1.3525R
0
, the solid curve R
5 R
min
cosh(L/2R
min
) intersects twice with the dotted line R5 R
0
,
determining the minimum radius R
min
of the catenoid. For L
. L
crit
, there is no intersection, indicating that a catenoid solution
does not exist.
ASPECTS OF LARGE N GAUGE THEORY DYNAMICS AS . . .
PHYSICAL REVIEW D 58 106002
106002-5
