DFTT 20/2001

NORDITA-2001/15 HE

= 2 GAUGE THEORIES ON SYSTEMS

OF FRACTIONAL D3/D7 BRANES
^{1}^{1}1Work partially supported by the European Commission RTN
programme HPRN-CT-2000-00131 and by MURST.

M. Bertolini , P. Di Vecchia , M. Frau , A. Lerda , R. Marotta

NORDITA, Blegdamsvej 17, DK-2100 Copenhagen Ø, Denmark

Dipartimento di Fisica Teorica, Università di Torino

and I.N.F.N., Sezione di Torino, Via P. Giuria 1, I-10125 Torino, Italy

Dipartimento di Scienze e Tecnologie Avanzate

Università del Piemonte Orientale, I-15100 Alessandria, Italy

Dipartimento di Scienze Fisiche, Università di Napoli

Complesso Universitario Monte S. Angelo, Via Cintia, I-80126 Napoli, Italy

We study a bound state of fractional D3/D7-branes in the ten-dimensional space supersymmetric gauge theory with fundamental matter. using the boundary state formalism. We construct the boundary actions for this system and show that higher order terms in the twisted fields are needed in order to satisfy the zero-force condition. We then find the classical background associated to the bound state and show that the gauge theory living on a probe fractional D3-brane correctly reproduces the perturbative behavior of a four-dimensional

###### Contents

## 1 Introduction

During the last few years, it has become more and more evident that the low-energy properties of D-branes can be studied in two complementary ways: one based on the fact that a D-brane is a classical solution of the string effective theory (supergravity) charged under a RR potential; the other based on the fact that a D-brane supports a gauge field theory on its world-volume. The realization of this two-fold interpretation, which in the literature goes under the name of gauge/gravity correspondence, has been one of the most significant results of the recent research in string theory. In fact, in view of this correspondence, one can exploit the classical geometrical properties of D-branes to study gauge theories or, vice-versa, use the quantum properties of the gauge theory defined on a D-brane to study the dynamics of non-perturbative extended objects.

In the case of the D3-branes of the type IIB string theory in flat space, it was possible to carry this correspondence much further by taking the so-called near-horizon limit, and observing that in this limit the gravity degrees of freedom (closed strings) propagating in the entire ten-dimensional space decouple from the gauge degrees of freedom (open strings) living on the four-dimensional world-volume of the D3-brane. This decoupling led Maldacena [1] to conjecture an exact duality between the super Yang-Mills theory in four dimensions, which is the conformal field theory living on the D3-brane world-volume, and type IIB string theory on , which is the space-time geometry in the near horizon limit. This remarkable conjecture, which has been confirmed by all subsequent studies, has opened the way to the use of brane dynamics in the analysis of the strong coupling regime of four-dimensional gauge theories.

More recently, a lot of efforts have been made to find possible extensions of the Maldacena duality to less supersymmetric and non-conformal gauge theories with more realistic properties such as asymptotic freedom and a running coupling constant. The simplest theories with these features are those with supersymmetry that can be obtained, for instance, by studying fractional branes on orbifolds [2, 3, 4, 5]. In particular, the world-volume theory defined on a stack of fractional D3-branes in is a pure super Yang-Mills theory in four dimensions [6], and thus this is a very natural system to consider in order to study possible non-conformal extensions of the Maldacena duality. However, as shown in Ref.s [7, 8] (and for more general orbifolds in Ref. [9]) the supergravity solutions corresponding to these fractional D3-branes possess naked singularities of repulson type [10]. The appearance of naked singularities is a quite general feature of the supergravity solutions that are dual to non-conformal gauge theories, but in the orbifold case, it seems that there exists a general mechanism to resolve them. Indeed, by analyzing the action of a probe D3-brane in the singular background, one can see that the probe becomes tensionless before reaching the repulson singularity on a hypersurface called enhançon [11] (similar results hold also in the case of fractional branes on compact orbifolds, see Ref. [12]). This fact suggests that at the enhançon the classical solution cannot be trusted anymore because additional light degrees of freedom come into play and the supergravity approximation is no longer correct. Therefore, the presence of the enhançon allows to consistently excise the singularity region and obtain a well-behaved solution [13], but at the same time it does not allow to easily take the decoupling limit anymore (for a discussion on the physics of the enhançon for these and more complicated systems see Ref.s [14, 15]).

Despite these problems, the classical solution describing fractional
D3-branes on orbifolds has been successfully
used to study the perturbative
dynamics of supersymmetric gauge theories and obtain
their correct perturbative
moduli space [7, 8, 9].
Among other things, this analysis has also shown
that the enhançon corresponds, in the gauge theory,
to the scale where the gauge
coupling constant diverges (the analogue of in QCD).
These results, that
seem to be in contrast
with a duality interpretation à la Maldacena where the supergravity
solution gives a good description of the gauge theory for large ’t
Hooft coupling, can instead be easily understood if we regard the
classical supergravity solution as an effective way of summing over
all open string loops, as explained for example in Ref. [16].
From this point of view, in fact, one does not take the near-horizon limit
(i.e. , where is the distance from the
source branes), but rather expands the classical solution
around
where the metric is almost flat and the
supergravity approximation is valid. This expansion corresponds
to summing closed string diagrams at tree level, but, because of
the open/closed string duality, it is also equivalent to summing
over open string loops. Therefore, expanding the supergravity
solution around is equivalent to perform an expansion
for small ’t Hooft coupling.
In view of these considerations, it is then not surprising that the
supergravity solution of Ref.s [7, 8, 9]
encodes the perturbative properties of the
gauge theory living on the world-volume of a fractional D3-brane,
but at the same time it is also natural to expect that this approach
does not include
the non-perturbative instanton corrections to the moduli space.
In conclusion we can say that the previous results are not a
consequence of a Maldacena-like duality, but rather they follow
directly from the gauge/gravity correspondence or, equivalently,
from the open/closed
string duality ^{1}^{1}1Recent papers stressing the importance of the
open/closed string duality in our context
are in Ref.s [17, 18, 19]..
On the other hand, the presence of the enhançon, which excises
the region of space-time corresponding to scales where non-perturbative
effects become relevant in the gauge theory and which is reminiscent
of the curve separating strong from weak coupling in super
Yang-Mills theory [20], is consistent with the above picture.
To incorporate in this scenario also the non-perturbative instanton
effects, presumably one must include D-instanton corrections already
at the string level [21, 22], as done in Ref. [23]
for the super Yang-Mills theory, or start directly from
M-theory [24]. M-theory is also the starting
point of a very interesting alternative approach pursued recently in
Ref.s [25, 26] where the non-perturbative properties
of supersymmetric gauge theories have been obtained
by taking the near-horizon limit. More recently, the instanton
corrections for systems of fractional D-branes have been discussed
in Ref. [27],
while alternative approaches to investigate
supersymmetric gauge theories, at the perturbative level,
have been carried out for instance in Ref.s [28, 29, 30, 31].

In a recent paper [32], the approach of Ref.s [7, 8] has been extended to the case of a system of fractional D3/D7-branes on the orbifold and the perturbative properties of the supersymmetric gauge theory living on the D3-brane world-volume have again been recovered from supergravity. In this case the gauge theory includes also hypermultiplets in the fundamental representation, associated to the open strings stretched between the D3 and the D7-branes.

In this paper, motivated by the considerations discussed above and using essentially the information provided by the boundary state formalism, we discuss further properties of the fractional D7-branes and of the bound states of fractional D3/D7-branes in the orbifold background. In particular we construct the boundary action for a fractional D7-brane and find that in order to satisfy the no-force condition required for a supersymmetric system, terms of higher order in the twisted fields, which are not accounted by the boundary state, must be included. We then solve explicitly the supergravity field equations for the D3/D7 system and study the properties of the dual four-dimensional gauge theory, finding agreement with the analysis of Ref. [32]. An interesting feature of this system is that, unlike the case discussed in Ref.s [7, 8, 9], the twisted fields receive contribution from diagrams with an arbitrary number of open string loops or, equivalently, from closed string tree diagrams containing an arbitrary number of boundaries. However, as expected from the non-renormalization theorems, we find that in the gauge theory the twisted fields appear always in special combinations in which only the one-loop perturbative contribution is non-trivial.

The paper is organized as follows. In Sect. 2 we write the field equations of type IIB supergravity on the orbifold coupled to fractional branes. In Sect. 3 we use the boundary state to study the properties of the fractional D7-branes determining their couplings with the bulk fields and the first order approximation of their classical solution. In Sect. 4 we extend the previous analysis to a bound state of fractional D3/D7-branes finding its complete classical solution. Using the no-force argument, we are also able to fix the complete form of the D7 brane boundary action. Finally in Sect. 5, by means of the probe analysis, we derive the perturbative behavior of the corresponding gauge theory and discuss its properties. A few technical results on the boundary state construction are reviewed in Appendix A.

## 2 Field equations for fractional D-branes

We consider type IIB supergravity in ten dimensions on the orbifold

(2.1) |

where is the reflection parity along
, , and . Its action (in the Einstein frame) can be
written as ^{2}^{2}2Our conventions for curved indices and forms are the
following:
; signature ;
; ; ;
; , and

(2.2) |

where

(2.3) |

are, respectively, the field strengths of the NS-NS
2-form and the 0-, 2- and 4-form
potentials of the R-R sector,
and ^{3}^{3}3Notice that we use definitions where
with respect to Ref. [7].

(2.4) |

Moreover, where is the string coupling constant, and the self-duality constraint has to be implemented on shell.

We are interested in obtaining classical solutions of the field equations descending from the action (2.2) that describe fractional D branes characterized by the presence of “twisted” scalar fields and defined as

(2.5) |

where is the anti-self dual 2-form associated to the vanishing 2-cycle of the orbifold ALE space. In our normalizations [7], this form satisfies

(2.6) |

Inserting eq.(2.5) in the action (2.2), we easily get

(2.7) |

where the index refers to the six-dimensional space orthogonal to the orbifold directions.

The field equations for a fractional D-brane are obtained by varying the total action

(2.8) |

where describes the coupling of the bulk supergravity fields with the brane; we recall however, that only the linear part of the boundary action is relevant to yield the source terms for the inhomogeneous field equations [16]. For the moment we do not specify the form of which instead will be discussed in the following sections for the specific cases of the fractional D3 and D7 branes. Defining

(2.9) |

the field equation for the dilaton is

(2.10) | |||

and the one for the axion is

(2.11) |

The field equations for the two twisted fields and are respectively

(2.12) |

and

(2.13) |

Finally, the field equation for the untwisted -form is

(2.14) |

and the one for the metric tensor is

(2.15) |

where is the stress energy tensor of the scalars and whose explicit expression is not really needed in the following.

To analyze these equations it is convenient to introduce the following complex
quantities ^{4}^{4}4The relation
between and the usual complex -form of type IIB supergravity
is .

(2.16) |

In fact, with simple manipulations the four equations (2.10)–(2.13) can be combined into two complex differential equations for and , which are

(2.17) |

and

(2.18) |

In the following we are going to solve these equations for bound states made of fractional D3 and D7 branes. In particular we will consider configurations in which the D7 branes extend in the directions (i.e. entirely along the orbifold) and the D3 branes in the directions (i.e. transversely to the orbifold). With this arrangement, the twisted fields and , which are stuck at the orbifold fixed point, are functions only of the transverse coordinates and . Moreover, since the D3 branes emit neither the dilaton nor the axion , these two fields are produced only by the D7 branes and thus they too are functions only of the transverse coordinates and . For the remaining fields, the metric and the self-dual field strength , we take the standard Ansatz for a D7/D3 system [33], namely

(2.19) | |||||

(2.20) |

where the warp factor is a function of all coordinates that are transverse to the D3 brane ().

A drastic simplification occurs by analyzing the supersymmetry transformation
rules of the gravitinos and dilatinos and asking for
the solution to be supersymmetric.
Plugging the Ansatz (2.19) and (2.20) in the variations of the
gravitinos and dilatinos, one can show [32]
that the existence of a Killing spinor implies that both
and are holomorphic functions of
, i.e. ^{5}^{5}5The case of
constant was discussed in Ref.s [34, 35]
for the case of singular spaces
and in Ref. [36] for the blown-up case,
while that for vanishing
G-flux was discussed in Ref. [37]. This kind of type
IIB supersymmetric solutions
are dual to those discussed in Ref. [38].

(2.21) |

The analyticity of and in turn implies that is analytic too (see eq.(2.16)); thus the equations for and drastically simplify and reduce to

(2.22) |

and

(2.23) |

Finally, we can see that the field equations for the metric and are satisfied provided that be a solution of the following equation

(2.24) |

where denotes the six-dimensional volume form .

As anticipated, we want to find the solution of these field equations for bound states of fractional D7/D3 branes. However, before doing this, we present a discussion of the pure D7 branes on orbifold from a string theory point of view, using the boundary state formalism.

## 3 The fractional D7-branes

In this section we will analyze in some detail the D7 branes of type IIB in the orbifold background (2.1). In order to realize a BPS brane of type IIB, the number of orbifold directions along its world-volume must be even [39]. Thus, for a 7 brane we have only two possibilities: and . Here we discuss only the case , i.e. when the brane extends throughout the entire orbifold, since this is the relevant case to yield four-dimensional gauge theories in the presence of fractional D3 branes.

The fact that D7 branes with extend entirely along the orbifold has some peculiar consequences that we would like to emphasize. From a string theory point of view, these branes are sources not only of those fields that are typical of a D7 brane (i.e. the metric , the dilaton and the 8-form R-R potential ), but also of twisted fields, which, specifically, comprise a scalar from the twisted NS-NS sector and a 4-form potential from the twisted R-R sector. It is interesting to observe that these twisted fields are the same as those emitted by the fractional D3 branes studied for example in Ref.s [7, 8]. Moreover, the charge of these D7 branes under is a half of that carried by the D7 branes in flat space. Thus, it is natural to regard these configurations as fractional branes, despite the fact that, contrarily to what happens for the fractional branes of lower dimension, they cannot be interpreted as wrapped branes. Finally, the charge of these D7 branes under the twisted 4-form potential is a quarter of that carried by the fractional D3 branes.

All these features can be clearly seen by computing the vacuum energy of the open strings stretched between two such D7 branes that is given by

(3.1) |

where is the GSO projection, is the orbifold parity, and in the NS sector and in the R sector. When one takes the inside the bracket, one gets half of the contribution of the open strings stretched between two D7 branes in flat space, whereas when one takes the inside the bracket one obtains 1/16 times the contribution of the twisted sectors of the fractional D3 branes of Ref.s [7, 8] (see Appendix A for more details). After performing the modular transformation and factorizing the resulting expression in the closed string channel, one can derive the boundary state associated to the D7 brane along the orbifold (for a review of the boundary state formalism and its applications see, for example, Ref. [40]; for an analysis of the boundary state in orbifold theories see, for example, Ref.s [41, 39, 42]). The explicit expression of and a discussion of its properties can be found in appendix A. Here we just mention that this boundary state contains both an untwisted and a twisted part:

(3.2) |

The untwisted part is the same as that of the
D7 branes in flat space but with a normalization differing by a
factor of ; the twisted part is,
instead, similar to that of the fractional D3 branes
but with a normalization differing
by a factor of 1/4. By saturating the boundary state with
the massless closed string states of the various sectors, one can determine
which are the
fields that couple to the fractional D7 brane. In particular, following
the procedure found in Ref. [43] and reviewed in Ref. [40],
one can find that in the untwisted sectors the D7 brane
emits the graviton ^{6}^{6}6We recall that the graviton
field and the metric are related by
.,
the dilaton and the 8-form potential .
The couplings of these fields
with the boundary state are explicitly given by [12]

(3.3) |

where , appearing in the normalization of the boundary state, is related to the brane tension in units of the gravitational coupling constant [43, 44], is the (infinite) world-volume of the D7 brane, and the index labels its eight longitudinal directions. Notice that there is no coupling of the boundary state with the untwisted 4-form , in agreement with the observation [32] that the D7 branes do not carry charge under .

By doing this same analysis in the twisted sectors, we find that, as advertised before, the boundary state emits a massless scalar from the NS-NS sector and a -form potential from the R-R sector. These fields exist only at the orbifold fixed hyperplane , and their couplings with the boundary state turn out to be given by [12]

(3.4) |

where is the (infinite) world-volume of the 7 brane that lies outside the orbifold.

From the explicit couplings (3.3) and (3.4), it is possible to infer the form of the world-volume action of a fractional D7 brane. Of course, the boundary state approach allows to obtain only the terms of the world-volume action that are linear in the bulk fields. However, terms of higher order can be determined with other methods. For example, from our previous considerations, it is natural to write for the untwisted fields the same world-volume action of the D7 branes in flat space but with an extra overall factor of as dictated by the boundary state. Therefore, we write (in the Einstein frame)

(3.5) |

where and is the induced metric. It is easy to check that this action correctly accounts for the couplings (3.3). Furthermore, recalling that , we can see that the charge of our D7 brane is a half of that carried by the D7 branes in flat space.

For the twisted fields, instead, things are slightly more complicated. Using the couplings (3.4) and defining , one can write

(3.6) |

where in the first term the four-dimensional induced metric has been inserted to enforce reparametrization invariance on the world-volume, and the ellipses stand for terms of higher order which are not accounted by the boundary state approach but which, in principle, can be present. In the following section we will show that such higher order terms are indeed present in the complete world-volume action of the fractional D7 branes.

As explained in Ref. [43], the boundary state formalism allows also to compute the asymptotic behavior of the various fields in the classical brane solution. Applying this technique to the case of a stack of N coincident fractional D7 branes, we find that, to leading order in , the metric is

(3.7) |

where and is a regulator, while the dilaton is

(3.8) |

the 8-form R-R potential is

(3.9) |

and the 4-form potential vanishes at linear order. The asymptotic behavior of the twisted fields is instead given by

(3.10) | |||||

(3.11) |

If we insert these expressions into the world-volume action

(3.12) |

and consistently retain only terms of first order in , we obtain a constant result, thus verifying the no-force condition at first order.

To extend this analysis to all orders, one needs to solve the complete field equations which we have derived in the previous section. However, to do this it is first necessary to establish the correct relation between the fields that the string description suggests and those appearing in the supergravity field equations. In particular we have to find how , and are related to , and . It turns out that represents the fluctuation of the scalar of eq.(2.5) around the background value of the orbifold [47], i.e.

(3.13) |

The 4-form potential is instead the Hodge dual (in the six dimensional sense) to the twisted scalar , while the 8-form potential is the dual (in the ten dimensional sense) to the axion . These duality relations, which can be obtained from eq.s (2.11) and (2.13) remembering the analyticity of and and the absence of source terms, are

(3.14) | |||||

(3.15) |

The absence of source terms is due to the fact that the fields and are not coupled to the boundary state of a D7-brane (see eqs. (3.3) and (3.4)).

Using the asymptotic expressions for the various fields in these relations, one can easily find that

(3.16) | |||||

(3.17) |

where . In the next section we are going to determine the higher order terms and find the complete classical solution with the asymptotic behavior described above. In particular we will discover that the untwisted 4-form and the metric along the world-volume directions of the fractional D7 brane will develop a non trivial profile at higher order.

## 4 The fractional D7/D3 bound state

Since the world volume of a 7-brane is eight-dimensional, a stack of fractional D7 branes is not immediately useful to yield information on gauge theories in four dimensions. To obtain a classical solution capable of describing a four-dimensional field theory we must include also some D3-branes, and hence it is natural to study a bound state made of fractional D7-branes and fractional D3-branes. As shown in Ref. [32], this is a BPS configuration which preserves eight of the sixteen supersymmetries of the type IIB theory on the orbifold (2.1). Our task is then to solve the field equations derived in Section 2 and specify the appropriate boundary action for this configuration. For the D3 brane components, we can simply take times the action introduced in Ref. [7], and thus we can write

where and are defined after eq.(3.5) and before eq.(3.6). For the D7 brane components, instead, we can take as boundary action times the sum of (3.5) and (3.6). As mentioned in the previous section, the twisted part (3.6) may be non-complete, but it is certainly correct at linear order and thus yields the right source terms in the various field equations. Using these ingredients and the Ansatz (2.19) and (2.20), one can show that eqs.(2.22) and (2.23) become

(4.2) | |||||

(4.3) |

The holomorphic solutions to these equations can be immediately found and are (see also Ref. [32])

(4.4) |

and

(4.5) |

where we have chosen the integration constants to enforce the appropriate background values. Written in terms of the real supergravity fields, eqs.(4.4) and (4.5) become

(4.6) | |||||

(4.7) | |||||

(4.9) |

Notice that for this solution reduces to the one of pure fractional D3-branes discussed in Ref.s [7, 8]. It is interesting to observe, on the other hand, that putting and expanding in powers of , we recover at first order the solution (3.8), (3.16) and (3.17) obtained from the boundary state approach. In this respect, we observe that the axion does not receive corrections to higher orders while the twisted fields and acquire an infinite tail of logarithmic terms. This is to be contrasted with the solution of the pure fractional D3 branes [7, 8] where the twisted scalars had, instead, only terms at first order. Thus, if one wants to determine the classical profile of the twisted scalars using the boundary state formalism in the presence of fractional D7 branes, it is not sufficient to consider contributions with just one boundary, but it is necessary to sum over all contributions with an arbitrary number of boundaries as explained in Ref. [16], which, due to the open/closed string duality, is equivalent to sum over an arbitrary number of open-string loops.

Finally, if we insert the above expressions for and into eq.(2.24), we obtain the following equation for the warp factor :

In general, it is not possible to find an explicit solution of this equation in terms of elementary functions. For this equation was solved exactly in Ref. [7], whereas for , i.e. when the last term vanishes, this equation becomes of the same form that was considered in Ref. [33]. It is also interesting to observe that eq.(4) remains non-trivial even for . This fact means that for a system made of only D7 branes on orbifold both the longitudinal metric and the 4-form are not trivial, contrarily to what happens for D7 branes in flat space. However, it should be realized that these fields start developing only at the second order in the string coupling constant, as is clear from the structure of eq.(4) for .

Using the full solution (4.6)-(4.9) in the duality relations (3.14) and (3.15), and recalling, apart irrelevant additional terms, that

(4.11) |

we can easily obtain the complete expressions for the 8-form and for the twisted 4-form which are more natural from a stringy perspective. After some algebra, we find

(4.12) |

and

(4.13) |

Having the complete solution, we can verify the no-force condition and check the structure of the world-volume action of the bound state. If we substitute our solution into the D3-brane component (4) of the boundary action, we find, as expected, that all terms depending on the transverse coordinates cancel, leaving a constant result. Doing the same thing for the D7-brane components (3.5) and (3.6), we see that in the twisted part not all terms cancel, thus indicating the presence of a non-zero force. However, this result is unacceptable in view of the BPS properties of our solution. This problem is easily overcome if we recall that the boundary action (3.6) is actually justified only at the linear level, and thus may be non-complete. To construct the full boundary action we can start from the standard expansion of the WZ part of the action for a D7 brane, namely

(4.14) |

where the ellipses stand for curvature terms. We now decompose the forms , and into untwisted components (denoted by ) and into twisted components along the 2-form (denoted by ). For the case under consideration, the relevant expressions are

(4.15) | |||||

where and are numerical coefficients which will be determined later. If we substitute eq.(4.15) into the action (4.14) and recall that with given by eq.(3.13), after some simple manipulations we get

(4.16) | |||||

Notice that in writing the last expression we have used the fact that the (understood) curvature contribution exactly cancels the term linear in . This fact, shown in Ref. [32], is consistent with the boundary state of a fractional D7 brane which indeed does not couple to . Instead, it couples to the twisted 4-form , and matching the corresponding charge with the boundary state result (see eq.(3.6)) fixes

(4.17) |

If we substitute the classical solution (4.12)-(4.13) in eq.(4.16) and require no force, we can see that to cancel the contribution of we must add to the boundary action the expected DBI term

(4.18) |

while to cancel the contribution of we must add a term like

(4.19) |

and fix . In this way the no-force condition is fully satisfied, as it should be. We thus conclude that the world-volume action of a fractional D7 brane consists of an untwisted part given by eq.(3.5) and a twisted part given by

(4.20) | |||||

It would be interesting to confirm the structure of this boundary action with geometrical considerations and also with explicit calculations of closed string scattering amplitudes on a disk with boundary conditions appropriate for the fractional D7 brane, similarly to what has been done in Ref. [45] for the fractional D3 branes.

## 5 The probe action and the gauge theory

The supergravity solution found in the previous section can provide non-trivial information on its dual four-dimensional gauge theory. To see this we use the probe technique (for a review see Ref. [46]) and consider a probe fractional D3-brane carrying a gauge field and slowly moving in the supergravity background produced by D3 and D7 fractional branes. We then fix the static gauge and study the world-volume action of the probe, regarding the transverse coordinates as Higgs fields , and expanding up to quadratic terms in derivatives. From a gauge theory point of view, the resulting action describes the Coulomb phase of a gauge theory in which the symmetry breaking corresponds to taking one of the D3 branes (the probe) away from the others at a distance related to the energy scale where the theory is defined.

Applying this technique to our case, we find that the action of a probe fractional D3-brane can be written as

(5.1) |

where is given by eq.(4) with and

(5.2) |

where . Inserting in the solution for the closed string fields obtained in the previous section (i.e. eq.s (4.4)-(4.5) and the Ansatz (2.19)-(2.20)), we see that becomes independent of the distance between the probe and the source branes that yield the classical solution. This is in agreement with the fact that there is no interaction between a fractional D3-brane and a system of fractional D3/D7-branes.

Considering now eq.(5.2), we see that the dependence on the function drops out in this case too, while the kinetic terms for the gauge field strength and the scalar fields have the same coefficient, in agreement with the fact that the gauge theory living on the brane has supersymmetry. Indeed one gets

(5.3) |

where

(5.5) |

are the effective Yang-Mills gauge coupling and -angle, respectively. The renormalization group scale is defined by , while is the bare coupling, i.e. the value of the gauge coupling at the ultraviolet cutoff .

Eq.(5) clearly shows that is the running coupling constant of an supersymmetric gauge theory with gauge group and hypermultiplets in the fundamental representation. This is precisely the field theory living on the system of D3-branes and D7-branes, where the gauge vector multiplet corresponds to open strings stretched between two fractional D3-branes, while the hypermultiplets correspond to strings stretched between the D3 and the D7-branes. The reason why the hypermultiplet kinetic term is absent in is just because the probe is a D3-brane only, and therefore there are no 3-7 strings that can give rise to massless fields on the probe world-volume. Of course, this theory is ultraviolet free only for

From eq.(4) one sees that on the geometric locus defined by

(5.6) |

the D3-brane probe becomes tensionless, thus indicating the presence of an enhançon. At distances smaller than the probe has negative tension, while at the enhançon extra light degrees of freedom come into play [8]. This means that the supergravity approximation leading to the solution described in section 4 is not valid anymore, and that the region of space-time is excised. Notice that the vanishing of the tension of the probe at the enhançon is consistent with the fact that fractional branes are tension-full because of the presence of a non-vanishing flux, [47]. Indeed, by using eq.(5.6), one can write as follows

(5.7) |

and see that the quantity , which is proportional to the probe tension, vanishes at the enhançon, since there the fluctuation of the field cancels precisely its background value.