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JHEP02(2001)014

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 Received: December 7, 2000, Accepted: February 8, 2000
HYPER VERSION

Fractional D-branes and their gauge duals∗

Matteo Bertolini, Paolo Di Vecchia, Raffaele Marotta

NORDITA
Blegdamsvej 17, DK-2100, Copenhagen Ø, Denmark
E-mail: teobert@nordita.dk, divecchia@nbivms.nbi.dk,

JHEP02(2001)014
marotta@nbivms.nbi.dk

Marialuisa Fraua , Alberto Lerdab,a , Igor Pesandoa

a Dipartimento di Fisica Teorica, Università di Torino and
I.N.F.N., Sezione di Torino
Via P. Giuria 1, I-10125 Torino, Italy
b Dipartimento di Scienze e Tecnologie Avanzate

Università del Piemonte Orientale

I-15100 Alessandria, Italy
E-mail: frau@to.infn.it, lerda@to.infn.it, ipesando@to.infn.it

Abstract: We study the classical geometry associated to fractional D3-branes of

type-IIB string theory on R4 /Z2 which provide the gravitational dual for N = 2
super Yang-Mills theory in four dimensions. As one can expect from the lack of
conformal invariance on the gauge theory side, the gravitational background displays
a repulson-like singularity. It turns out however, that such singularity can be excised
by an enhançon mechanism. The complete knowledge of the classical supergravity
solution allows us to identify the coupling constant of the dual gauge theory in terms
of the string parameters and to find a logarithmic running that is governed precisely
by the β function of the N = 2 super Yang-Mills theory.

Keywords: D-branes, Brane Dynamics in Gauge Theories, AdS-CFT

Correspondance.

∗
Work partially supported by the European Commission RTN programme HPRN-CT-2000-
00131 and by MURST.
Contents
1. Introduction 1

2. The geometry of fractional D3-branes 2

3. The enhançon and the dual gauge theory 10

1. Introduction

JHEP02(2001)014
After the seminal paper of ’t Hooft on the large N expansion [1], many attempts to
obtain a string theory out of QCD have been tried. Recently, a remarkable progress
in this direction has been achieved with the Maldacena conjecture [2] which states
that N = 4 super Yang-Mills theory in four dimensions is equivalent to type-IIB
string theory compactified on AdS5 × S5 . This provides the first concrete example
of how a string theory can be extracted from a gauge theory. On the other hand,
however, it was expected that a string model should emerge from a gauge field
theory because of confinement, while the N = 4 super Yang-Mills is in the Coulomb
phase. Thus, a lot of effort has been recently devoted to extend the Maldacena
conjecture and find new correspondences between string theories and non-conformal
and less supersymmetric gauge theories. These attempts include the study of the
renormalization group flow under a relevant perturbation in N = 4 super Yang-
Mills [3]–[5], the study of fractional branes on conifold singularities [6]–[9], the study
of the so-called N = 1∗ theory [10], and the search for the geometry of the stable
non-BPS D-branes [11].
The common feature found in all these different examples is that the classical
geometric backgrounds have naked singularities of repulson type. In some of these
cases however, an interesting phenomenon was discovered [12]: a massive probe
moving in these backgrounds becomes tensionless before reaching the singularity.
The geometric locus where this occurs is called enhançon. When this happens the
supergravity approximation is not valid beyond the enhançon and one is forced to
consider stringy effects which should change the description and eventually excise
the singularity. This phenomenon has been analyzed also in refs. [13, 14] for different
configurations and in ref. [15] for fractional D-branes on K3 orbifolds. In ref. [8]
however, it has been shown that the repulson singularity of fractional branes on
conifolds can be removed already at the supergravity level by suitably deforming the
conifold, thus obtaining a consistent gravitational dual that explains many features
of the gauge theory. More recently, in ref. [16] it has been shown that instead the

1
resolution of the conifold singularity is not sufficient to regularize the gravitational
background and also in this case an enhançon mechanism seems to be necessary.
In this paper we study the classical geometry generated by fractional D3-branes
of type-IIB string theory on the orbifold R1,5 × R4 /Z2 . These are BPS configura-
tions that are constrained to be at the orbifold fixed hyperplane and preserve eight
supercharges. The dual gauge theory corresponding to a stack of M such fractional
D3-branes is pure N = 2 super Yang-Mills theory in four dimensions with gauge
group U(M). This is known to be non conformal and thus it is interesting to check
whether the dual classical geometry displays a singular behavior. Some features of
this solution indicating that this is indeed the case were already found in refs. [6, 17].
In this paper we give the complete solution, with all physical quantities expressed in
terms of the string parameters α0 and gs , and analyze its properties in some detail.

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The first step for finding the exact solution is done using the boundary state
formalism that allows to determine which supergravity fields are coupled to the
brane and also their asymptotic behavior at large distance. Using this formalism it
is possible to infer the complete world-volume action for the fractional D3-branes
and thus to obtain the non homogeneous equations that the supergravity fields must
satisfy. These equations can be explicitly solved and one can check that they describe
configurations which satisfy the no-force condition as implied by the BPS bound. As
expected the classical solution exhibit a naked singularity, which is in fact a repulson.
Following the analysis done in refs. [12]–[14], we see that the enhançon mechanism
works also in our case. Hence the region of validity of the supergravity approximation
does not include the singularity. The properties of our solution suggest a physical
picture where the enhançon geometry is that of a ring like in ref. [13, 14], instead of
an hypersphere as in ref. [12].
Exploiting the detailed knowledge of the solution of the fractional D3-brane and
its world-volume action, we are able to find the metric of the moduli space of the dual
gauge theory and determine from it the Yang-Mills coupling constant gYM in terms of
the string parameters. We find that gYM is logarithmically running with a β-function
that exactly matches the one of N = 2 super Yang-Mills theory in four dimensions.

2. The geometry of fractional D3-branes

The action for type-IIB supergravity in ten dimensions can be written (in the Einstein
frame) as1
Z Z
1 √ 1 h
SIIB = 2 d x − det GR −
10
dφ ∧ ∗ dφ + e−φ H(3) ∧ ∗ H(3) + (2.1)
2κ10 2
1 e i
∗ φe ∗e ∗e
+ e F(1) ∧ F(1) + e F(3) ∧ F(3) + F(5) ∧ F(5) + C(4) ∧ H(3) ∧ F(3)
2φ
2
1
Our conventions for curved indices and forms are the following: ε0...9 = +1, signa-
√
ture (−, +9 ), ω(n) = 1/n!ωµ1 ...µn dxµ1 ∧ · · · ∧ dxµn , and ∗ω(n) = − det G/(n! (10 − n)!) ×
εν1 ...ν10−n µ1 ...µn ω µ1 ...µn dxν1 ∧ · · · ∧ dxν10−n .

2
where

H(3) = dB(2) , F(1) = dC(0) , F(3) = dC(2) , F(5) = dC(4) (2.2)

are respectively the field strengths corresponding to the NS-NS 2-form potential, and
to the 0-form, the 2-form and the 4-form potentials of the R-R sector, and

Fe(3) = F(3) − C(0) ∧ H(3) , Fe(5) = F(5) − C(2) ∧ H(3) . (2.3)

Moreover, κ10 = 8 π 7/2 gs α02 where gs is the string coupling constant, and the self-
duality constraint ∗ Fe(5) = Fe(5) has to be implemented on shell. The Dp-branes with
p odd are solutions of the classical field equations that follow from the action (2.1),
which are charged under the R-R (p + 1)-form potentials and preserve sixteen super-

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charges. The D3-brane solution, in which only the metric and the 4-form potential
C(4) are turned on, is particularly important because of the AdS/CFT correspon-
dence [2].
Let us now consider type-IIB supergravity on the orbifold

IR1,5 × R4 /Z2 , (2.4)

where Z2 is the reflection parity that changes the sign to the four coordinates of R4 ,
which we take to be x6 , x7 , x8 and x9 . This is to be understood as the singular limit
of the corresponding ALE manifold. The bulk action for this theory is still given
√
by eq. (2.1), but with κ10 replaced by κorb = 2 κ10 = (2π)7/2 gs α02 . In this case,
besides the usual Dp-branes (bulk branes) which can freely move in the orbifolded
directions, there are also fractional Dp-branes [18] which are instead constrained to
stay at the orbifold fixed hyperplane x6 = x7 = x8 = x9 = 0. These fractional branes
are the most elementary configurations of the theory, preserve eight supercharges
and can be viewed as D(p + 2)-branes wrapped on the (supersymmetric) vanishing
2-cycle of the orbifold.
In this paper we will consider in detail the fractional D3-brane. From the super-
gravity point of view, this is a configuration in which the dilaton φ and the axion
C(0) are constant, while the metric, the 4-form C(4) and the two 2-forms B(2) and
C(2) are non-trivial. More precisely, the latter fields, whose presence is a distinctive
feature of the fractional branes, are

C(2) = c ω2 , B(2) = b ω2 , (2.5)

where ω2 is the 2-form dual to the vanishing 2-cycle of the orbifold, and c and b are
scalar fields living in R1,5 .
The fact that these are the non-trivial fields for a fractional D3-brane has a
natural interpretation from a string theory point of view. In fact, let us consider the

3
vacuum energy Z between two fractional D-branes which is given by the one-loop
open string amplitude
Z ∞    
ds 1 + (−1)F 1 + g −2πs(L0 −a)
Z= TrNS−R e , (2.6)
0 s 2 2
where (−1)F is the GSO parity, g is the orbifold Z2 parity, and the intercept is a = 1/2
in the NS sector and a = 0 in the R sector. By making the modular transformation
s → 1/s, one can translate the one-loop open string amplitude (2.6) into a tree-level
closed string exchange diagram and, after factorization, one can obtain the boundary
state |Bi associated to the fractional brane [19, 20] (for a review of the boundary
state formalism and its applications see, for example, [21]). The boundary state
represents the source for the closed strings emitted by the brane and in this case

JHEP02(2001)014
it has four different components which correspond to the (usual) NS-NS and R-R
untwisted sectors and to the NS-NS and R-R twisted sectors. By saturating the
boundary state |Bi with the massless closed string states of the various sectors, one
can determine which are the fields that couple to the fractional brane. In particular,
following the procedure found in [22] and reviewed in [21], one can find that in the
untwisted sectors the fractional D3-brane emits only the graviton hµν 2 and the 4-form
potential C(4) . The couplings of these fields with the boundary state are explicitly
given by [15]
T3 T3
hB|hi = − √ hαα V4 , hB|C(4) i = √ C0123 V4 , (2.7)
2 2 κorb
√ √
where Tp = π (2π α0 )(3−p) is the normalization of the boundary state, which is
related to the brane tension in units of the gravitational coupling constant [22], V4 is
the (infinite) world-volume of the D3-brane, and the index α labels the longitudinal
directions. By doing this same analysis in the twisted sectors, we find that the
boundary state of the fractional D3-brane emits a massless scalar eb from the NS-NS
sector and a 4-form potential A(4) from the R-R sector. Of course these fields exist
only at the orbifold fixed hyperplane x6 = x7 = x8 = x9 = 0, and thus their dynamics
develops in the remaining six-dimensional space. The couplings of these fields with
the boundary state turn out to be given by [15]
T3 1 e T3 1
hB|ebi = − √ b V4 , hB|A(4) i = √ A0123 V4 . (2.8)
2 κorb 2π 2 α0 2 κorb 2π 2 α0
The twisted fields eb and A(4) are related to the fields b and c of eq. (2.5). In fact,
the scalar eb represents the fluctuation part of b around the background value which
is characteristic of the Z2 orbifold [23, 20]
1
b= (4π 2 α0 ) + eb , (2.9)
2
2
We recall that the graviton field and the metric are related by Gµν = ηµν + 2κorb hµν .

4
while the potential A(4) is dual (in the six dimensional sense) to the scalar c. To write
down this duality relation in a correct way, let us observe that the field equation for
C(2) that follows from the action (2.1) is


d ∗ dC(2) = F(5) ∧ H(3) = d C(4) ∧ H(3) , (2.10)

so that we can write

∗
dC(2) − C(4) ∧ H(3) = dA(6) . (2.11)
The 6-form A(6) is the dual (in the ten dimensional sense) to the R-R 2-form C(2) .
Let us now write this relation in the case of the wrapped brane, i.e. using eq. (2.5)
and taking A(6) = A(4) ∧ ω2 . Then, one can easily find that

dA(4) = − ∗6 dc − C(4) ∧ d b , (2.12)

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where the Hodge dual ∗6 is taken in the six-dimensional space where the twisted
fields live.
From the explicit couplings (2.7) and (2.8), it is possible to infer the form of the
world-volume action of a fractional D3-brane, namely
Z  
T3 p 1 e
Sboundary = − √ d x − det Gαβ 1 + 2 0 b +
4
(2.13)
2 κorb 2π α
Z   Z
T3 1 e T3 1
+√ C(4) 1 + 2 0 b + √ A(4) .
2 κorb 2π α 2 κorb 2π 2 α0
Of course, the boundary state calculations only determine the linear terms of the
world-volume action, but the higher order terms can be found for example by im-
posing reparametrization invariance on the world-volume (first line of (2.13)) or by
considering the WZ part of the action of a D5-brane wrapped on the (vanishing) 2-
cycle in the presence of a non-trivial B(2) field (second line of (2.13)). The structure
of the boundary action Sboundary is confirmed also by explicit calculations of closed
string scattering amplitudes on a disk with appropriate boundary conditions [24].
As explained in ref. [22], the boundary state formalism allows also to compute
the asymptotic behavior at large distance of the various fields in the classical brane
solution. For example, in our fractional D3-brane we find that the metric is
   
Q Q
ds ' 1 − 4 ηαβ dx dx + 1 + 4 δij dxi dxj + · · ·
2 α β
(2.14)
2r 2r
p
where α, β = 0, . . . , 3; i, j = 4, . . . , 9; r = xi xj δij and
κorb T3
Q≡ √ = 4π gs α02 , (2.15)
2 2 π3
while the untwisted 4-form potential is
Q
C(4) ' − dx0 ∧ dx1 ∧ dx2 ∧ dx3 + · · · . (2.16)
r4

5
The asymptotic behavior of the twisted fields is instead given by
 
eb ' K log ρ + · · · , (2.17)
 ρ 
A(4) ' K log dx0 ∧ dx1 ∧ dx2 ∧ dx3 + · · · , (2.18)

p
where ρ = (x4 )2 + (x5 )2 ;  is a regulator and
κorb T3 1
K≡ √ 2 0
= 4π gs α0 . (2.19)
2 π 2π α
It is interesting to observe that while the untwisted fields depend on the radial
coordinate r of the entire six-dimensional transverse space, the twisted fields which
do not see the four orbifolded directions, depend only on the radial coordinate ρ of

JHEP02(2001)014
the remaining two-dimensional transverse space. This particular feature was also
found in ref. [25] in the case of the non-BPS D-branes in non-compact orbifolds,
while the logarithmic asymptotic behavior of the twisted fields was already pointed
out in refs. [6, 26].
In the following we look for an exact solution of the field equations of type-IIB
supergravity with the asymptotic behavior described above. We start by writing the
equations of motion for the dilaton φ and the axion C(0) , which are
1 φ e 1
d∗ dφ = e2φ dC(0) ∧ ∗ dC(0) + e F(3) ∧ ∗ Fe(3) − e−φ H(3) ∧ ∗ H(3) , (2.20)
2 2
and


d e2φ ∗ dC(0) = − eφ Fe(3) ∧ ∗ H(3) . (2.21)
As we discussed above, we are interested in a solution in which both the dilaton and
the axion are constant, and the two 2-form potentials are as in eq. (2.5). To obtain
this solution, it is convenient to introduce the combination3

G(3) = F(3) − τ H(3) with τ = C(0) + i e−φ . (2.22)

For constant dilaton and axion, eqs. (2.20) and (2.21) imply that

G 3 ∧ ∗ G3 = 0 (2.23)

which, using eq. (2.5), in turn implies that

dγ ∧ ∗6 dγ ∧ ω2 ∧ ω2 = 0 (2.24)

where we have defined the complex scalar

γ = c − τ eb (2.25)
3
Note that G(3) is not the Sl(2, R) invariant 3-form that is usually used in the supergravity
literature, but differs from the latter simply by a multiplicative factor.

6
and taken into account the anti-selfduality of ω2 . If dγ ∧ ∗6 dγ has components along
x4 and x5 , i.e. along the transverse directions orthogonal to the orbifold, then in
order to satisfy eq. (2.24) we must require that

∂z γ ∂z̄ γ = 0 where z = x4 + i x5 (2.26)

which clearly can be satisfied by taking γ to be, for instance, an analytic function
of z [6]. If we do this, the dilaton and the axion can be consistently taken to be
constant, and, without any loss of generality, we set them to zero. With this choice,
of course we have τ = i .
Let us now turn to the other field equations. To derive them, it is convenient
to first insert the Ansatz (2.5) into the original action (2.1) and then use the fact
that the integral of the product of two forms ω2 over the four-dimensional orbifolded

JHEP02(2001)014
space is a constant that we choose so that the various fields in the bulk action have
the canonical normalization, apart from the overall factor of 1/(2κ2orb ). Proceeding
in this way, we obtain the following action
Z Z
1 √ 1
0
SIIB = 2 d x − det G R −
10
Fe(5) ∧ ∗ Fe(5) −
2κorb 4
Z   
1 ∗6 i
− dγ̄ ∧ dγ − C(4) ∧ dγ ∧ dγ̄ , (2.27)
2 2 6

where the subindex 6 in the second line indicates that the integral is over the six-
dimensional space orthogonal to the orbifolded directions. At this point we can
0
write the field equations that follow from the total action S = SIIB + Sboundary . The
4
equation for the 4-form potential C(4) is
 
∗e i 2κorb T3
d F(5) + dγ ∧ dγ̄ ∧ Ω4 + √ Ω2 ∧ Ω4 = 0 , (2.28)
2 2
where we have defined

Ω4 = δ(x6 ) · · · δ(x9 ) dx6 ∧ · · · ∧ dx9 ,

Ω2 = δ(x4 ) δ(x5 ) dx4 ∧ dx5 , (2.29)

while the equation for the complex scalar γ is

   
∗6 e 2κorb T3 1
d dγ + i F(5) ∧ dγ ∧ Ω4 + i √ 2 α0
dx0 ∧ · · · ∧ dx3 ∧ Ω2 ∧ Ω4 = 0 . (2.30)
2 2π
Finally, the field equations for the metric are

eµν ≡ Rµν − 1 e
R (F(5) )µλ1 ...λ4 (Fe(5) )ν λ1 ...λ4 = Lµν , (2.31)
4 · 4!
4
Note that, as usual, only the linear part of boundary action gives a non-trivial contribution to
the field equations.

7
where
L L
Lαβ = − √ Gαβ , Lij = √ Gij , (2.32)
− det G − det G
with
 p 
1 κorb T3 p
L= − det G6 ∂γ · ∂γ̄ + √ − det Gαβ δ(x ) δ(x ) δ(x6 ) · · · δ(x9 ) .
4 5
8 2 2
(2.33)
We remark that in writing eq. (2.33) we have used the analyticity of γ, and have
denoted by G6 the metric in the six-dimensional space orthogonal to the orbifold.
We now solve eqs.(2.28), (2.30) and (2.31) by using a 3-brane-like Ansatz for the
untwisted fields, namely

ds2 = H −1/2 ηαβ dxα dxβ + H 1/2 δij dxi dxj , (2.34)

JHEP02(2001)014



Fe(5) = d H −1 dx0 ∧ · · · ∧ dx3 + ∗ d H −1 dx0 ∧ · · · ∧ dx3 , (2.35)

Inserting these expressions into eq. (2.30), we easily obtain

2κorb T3 1
δ ab ∂a ∂b γ + i √ 2 α0
δ(x4 ) δ(x5 ) = 0 , (2.36)
2 2π
where a, b = 4, 5, whose analytic solution is

γ = − i K log(z/) , (2.37)

where K is defined in eq. (2.19) and z = x4 + i x5 . Taking the real and imaginary
parts of γ, we get the twisted scalars
 5
−1 x
c = K tan , (2.38)
x4
 
eb = K log ρ . (2.39)

It is interesting to see that the asymptotic behavior of eb given in eq. (2.17) coincides
with the complete solution (2.39). Furthermore, using the duality relation (2.12)
and the Ansatz (2.34)–(2.35), we can obtain the classical profile of the twisted R-R
potential A(4) appearing in the boundary action of the fractional D3-brane. In fact,
we have
A(4) = K log(ρ/) dx0 ∧ dx1 ∧ dx2 ∧ dx3 . (2.40)
Again the asymptotic form (2.18) obtained from the boundary state coincides with
the full solution (2.40).
Let us now find the equation that determines the warp factor H. Inserting the
Ansatz (2.34)–(2.35) into eq. (2.28), we get
2κorb T3
δ ij ∂i ∂j H + |∂z γ|2 δ(x6 ) . . . δ(x9 ) + √ δ(x4 ) . . . δ(x9 ) = 0 . (2.41)
2

8
The last contribution is the standard source term that is present also for the usual
bulk D3-branes, while the second contribution is a peculiar feature of the fractional
D3-branes and represents the fact that, in this case, the non-trivial flux of G(3) is a
source for the untwisted fields [17, 27]. The final consistency check is to show that eq.
(2.41) follows also from the Einstein equation (2.31). This is indeed what happens;
in fact, using our Ansatz, it is possible to show that the left-hand side of eq. (2.31)
becomes
ij lk
eαβ = δ ∂i ∂j H ηαβ ,
R Reij = − δ ∂l ∂k H δij , (2.42)
4 H2 4H
while the right-hand side becomes
 
1 2κorb T3
Lαβ = − |∂z γ| δ(x ) · · · δ(x ) + √
2 6 9
δ(x ) · · · δ(x ) ηαβ ,
4 9
4 H2 2
 

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1 2κorb T3
Lij = |∂z γ| δ(x ) . . . δ(x ) + √
2 6 9 4 9
δ(x ) . . . δ(x ) δij . (2.43)
4H 2
Hence, also the Einstein eq. (2.31) implies eq. (2.41). Using standard techniques, it
is possible to find the explicit solution of this equation, and H reads
   
Q 2K 2 r4 ρ2
H = 1 + 4 + 4 log 2 2 −1+ 2 . (2.44)
r r  (r − ρ2 ) r − ρ2

This expression is clearly in agreement with the results (2.14) and (2.16) obtained
from the boundary state.
Having the explicit form of the solution, we can now analyze its properties. First
of all, we can check that it respects the no-force condition, as it should be because
of its BPS properties. To see this, we can substitute our classical solution into the
boundary action (2.13) and find
   ρ 
T3 V4 −1 K
Sboundary = − √ H 1 + 2 0 log − (H −1 − 1) ×
2 κorb 2π α 
    ρ 
K ρ K
× 1 + 2 0 log − 2 0 log
2π α  2π α 
T3 V4
= −√ . (2.45)
2 κorb
The fact that all position dependent terms exactly cancel leaving a constant result
is a check on the no-force condition to all orders; therefore, one can safely form a
stack of M fractional D3-branes by simply piling them on top of each other. In this
case, the solution for such a configuration has still the same form as before, but with
Q → M Q and K → M K.
On the other hand, a closer look at the behavior of the function H in eq. (2.44)
shows that the metric of the fractional D3-branes has a naked singularity. As we
discussed in the introduction, this fact is a feature that is shared also by other

9
configurations which are dual to non-conformal gauge theories [12, 7, 13, 14, 16, 11,
15], and indeed possess a naked singularity at some r = r0 . Actually, the structure
of such singularity is that of a repulson because in its vicinity the gravitational force,
which is related to the gradient of the temporal component of the metric tensor,
becomes repulsive. Thus, there exists a region of anti-gravity and a distance r =
re > r0 where the gravitational force vanishes. A study of the shape of G00 indicates
that the singularity is not a point in the transverse six dimensional space but rather a
two-dimensional surface. In fact, the repulson is located near x6 = x7 = x8 = x9 = 0
and extends along the non-orbifolded transverse directions x4 and x5 . Note that the
breaking of the spherical symmetry in the six-dimensional transverse space is not
surprising since the starting vacuum geometry, eq. (2.4), already breaks it. Moreover,
a simple numerical analysis reveals that the singularity does not cover the full x4 , x5

JHEP02(2001)014
plane; in fact the temporal component of the metric tensor ceases to be singular at
some value ρ0 and smoothly goes to zero for bigger values of ρ, signaling the possible
appearance of an horizon. Clearly, a more detailed analytical study of the classical
geometry produced by a fractional D3-brane and of its singularity is needed. Here
we have just mentioned the most relevant features which are useful and sufficient for
the discussion of the following section.

3. The enhançon and the dual gauge theory

The discussion of the previous section and the presence in our solution of a geometri-
cal locus where the gravitational force vanishes, clearly indicates the possibility that
an enhançon mechanism may take place [12].5 This would excise the repulson from
the metric, thus yielding a singularity-free solution.
In order to see whether this really happens, we apply the same methods of
ref. [12] and study the (slow) motion of a probe brane in the geometry generated by
M fractional D3-branes (for a review on this approach see, for example, [28]). This
can be done by inserting the classical solution into the world-volume action (2.13)
and expanding it in powers of the velocity of the probe D3-brane in the two transverse
directions x4 and x5 . Doing this, we find that the position dependent terms cancel
exactly because of supersymmetry, as we have already seen in the previous section,
while the terms quadratic in the velocity of the probe survive and allow to define a
non-trivial two-dimensional metric on moduli space. More precisely, from the DBI
part of boundary action, we get
Z  a 2 !
1 T3 ∂x eb
√ d4 x 1+ 2 0 , (3.1)
2 2κorb ∂x0 2π α
5
Notice that in [12], the enhançon locus coincides with the locus where the gravitational force
vanishes, but this may not be necessarily the case for more general configurations. The ”physical”
enhançon occurs, by definition, where the probe brane becomes tensionless.

10
where the index a takes values 4 and 5. We now express the various constants
of (3.1) in terms of the string parameters, use eq. (2.39) and identify the coordinates
xa = 2πα0Φa with the Higgs fields, so that the kinetic term for the scalars can be
rewritten as follows Z
1
− d4 x∂µ Φa ∂ µ Φb gab , (3.2)
2
where the metric in moduli space is
  ρ 
1 MK
gab = 1 + 2 0 log δab . (3.3)
8πgs 2π α 
It is easy to see that this metric vanishes when ρ reaches the following value
ρe
= e−π/(2M gs ) . (3.4)


JHEP02(2001)014
This means that at ρ = ρe the probe fractional brane becomes tensionless; thus for
ρ < ρe the supergravity solution looses its meaning because there the probe gets a
negative tension. This fact can be interpreted also as the signal that new degrees of
freedom are becoming massless below ρe and that they have to be suitably taken into
account with a fully stringy description. A phenomenon similar to the one originally
discussed in [12] is at work here: the true microscopic configuration is not given by
a stack of coincident branes but rather by an hypersurface (defined by eq. (3.4))
on which the branes are smeared. However, differently from [12], our enhançon is
not an (hyper)sphere in the transverse space, but rather a ring depending on the
two coordinates, x4 and x5 through ρ. Another important point to notice is that at
the enhançon, the fluctuation part eb of the twisted scalar√field b exactly cancels its
background value which, in unit of the string length 2π α0 , is 1/2 (see eqs. (2.9)
and (2.39)).
A more detailed characterization of the structure of the regularized classical
solution deserves further study. Nevertheless, what we have found here is already
enough to get interesting information about the dual field theory. We recall that in
the case of M fractional D3-branes, the world-volume gauge theory is pure N = 2
super Yang-Mills in four dimensions with gauge group U(M). This can be simply
understood by analyzing the massless spectrum of the open strings attached to the
fractional D3-branes. Notice that no hypermultiplets are present since the corre-
sponding moduli would be related to displacements of the branes from the orbifold
fixed point, which are not possible for fractional branes.6
Remembering that the Higgs fields Φa are the two scalars of the N = 2 vector
multiplet, from the action (3.2) we can read that the Yang-Mills coupling constant
is given by
 0
−1
2 0 2 M (gYM )2
gYM (µ) = (gYM ) 1 + log µ , (3.5)
4π 2
6
This is to be contrasted with the case of N bulk branes where the gauge group is U(N ) × U(N )
and one expects also hypermultiplets to be present [29].

11
where
ρ
0
(gYM )2 ≡ gYM
2
(µ = 1) = 8πgs , µ≡ . (3.6)

Eq. (3.5) defines the running of the YM coupling constant with the variation of the
scale µ. Notice that from this point of view, the enhançon locus (3.4) defines the
scale at which gYM diverges. Remembering that ρ → ∞ corresponds to the ultraviolet
limit in the dual field theory, we see that the coupling constant (3.5) describes an
asymptotically free gauge theory! Finally, by computing the β-function, we find
∂ g 3 (µ)
β≡µ gYM (µ) = − YM 2 M (3.7)
∂µ 8π
which is precisely the β-function of the pure N = 2 super Yang-Mills theory (modulo
instanton corrections). It would be interesting to investigate the relation between

JHEP02(2001)014
our results and those of ref. [30] where the N = 2 gauge theories are obtained from
M5-branes of 11-dimensional supergravity wrapped on Riemann surfaces.
It is also worth pointing out that the R-R twisted scalar c of eq. (2.38) is directly
related to the θ-angle of the YM theory. In fact, by introducing a gauge field F in the
world-volume action of the probe D3-brane and expanding the WZ part in powers
of F , we can read from the coefficient in front of the Tr(F ∧ F ) term that

c = 2π α0 gs θY M . (3.8)

As a consequence, the complex scalar γ̄ = c + i b of the supergravity solution can be

nicely written as √
γ̄ = (2π α0 )2 gs τ , (3.9)
where τ is the standard combination of the YM coupling constant and θ-angle
θY M 4π
τ= +i 2 . (3.10)
2π gY M
We conclude by observing that it is straightforward to extend our analysis to the
case in which there are N bulk D3-branes besides the M fractional ones considered
so far. The only change in the classical solution corresponding to this configuration
occurs in the function H of eq. (2.44) in which the parameter Q must be replaced
by (2N + M) Q, while K must be replaced by M K as before. Furthermore, the
enhançon locus gets changed to
ρe
= e−π(1+2n)/(2M gs ) , (3.11)

where n = N/M.
Our results show that is possible to obtain precise non-perturbative information
on a gauge theory using the dual classical geometry provided by D-branes, even
in cases different from those of the AdS/CFT correspondence. This fact hints the
possibility that the gauge/gravity duality has an even deeper meaning than expected.

12
Acknowledgments

We would like to thank M. Billó, P. Frè, L. Gallot, A. Liccardo, R. Musto, M. Petrini

and R. Russo for very useful discussions. M.B. acknowledges support from INFN.

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