NORDITA-2002-78 HE

Non conformal gauge theories from D branes
^{1}^{1}1Work partially supported by the European
Commission RTN programme HPRN-CT-2000-00131.

P. Di Vecchia

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

We use fractional and wrapped branes to describe perturbative and nonperturbative properties of the gauge theories living on their worldvolume.

## 1 Introduction

One of the most important ideas developed in recent years has been the one that goes under the name of gauge-gravity correspondence. According to it one can either use the low-energy dynamics of branes to study the properties of the gauge theory living on them or, if one knows the properties of the gauge theory living on a brane, one can deduce its low-energy dynamics. This idea is also at the basis of the Maldacena conjecture that, by using it, has established a complete equivalence between a gauge theory ( super Yang-Mills ) and a superstring (supergravity) theory (type IIB string theory compactified on ). In this paper we want to use the gauge-gravity correspondence for studying the properties of less supersymmetric and non-conformal gauge theories. We will not try to establish an exact duality between these gauge theories and some superstring theory as in the case of the Maldacena conjecture, but we will use classical supergravity solutions corresponding to fractional and wrapped branes having supersymmetric non-conformal gauge theories living on them in order to study their perturbative and nonperturbative properties. In particular we will use the expression of the gauge coupling constant and of the angle in terms of the supergravity fields in order to compute them when a consistent classical solution is found.

For both wrapped D5 and fractional D3 branes of the orbifold the gauge coupling constant is given by:

(1.1) |

In the case of wrapped branes that we will consider in this paper we have to put , while for fractional D3 branes, that for the sake of simplicity we take those of the orbifold having only one vanishing two cycle, we get:

(1.2) |

Finally the angle both in the case of fractional D3 branes and wrapped D5 branes is given by:

(1.3) |

The paper is organized as follows. In the next section we will consider the case of fractional branes, while in section 3 we will use wrapped branes for studying the properties of the gauge theory living on them.

## 2 Fractional branes

In this section we will consider fractional D3 and D7 branes of the
orbifolds and in order to study the
properties of respectively and
supersymmetric gauge theories. The orbifold group acts on the
directions transverse to the worldvolume of the
D3 brane where the gauge theory lives. In particular in the case of
the first orbifold the nontrivial generator of acts
as ^{1}^{1}1We denote ,
and
while in the case of the second orbifold the three nontrivial
generators act as follows on the transverse coordinates:

(2.1) | |||||

They are both non compact orbifolds with respectively one and three
fixed points at the origin corresponding to the point
and to the three points , and . Each fixed point corresponds to a vanishing -cycle. Fractional
Dp branes are D(p+2) branes wrapped on the vanishing two-cycle and
therefore are, unlike bulk branes, stuck at the orbifold fixed point.
By considering fractional D3 and () fractional D7 branes of the
two previous orbifolds we are able to study
() super QCD with hypermultiplets.
In order to do that we need to determine the classical solution
corresponding to the previous brane configuration. For the case of the
orbifold the complete classical solution has been found in
Ref. [1] ^{2}^{2}2See also
Refs. [2, 3, 4, 5] and Ref. [6] for a review on
fractional branes..
In the following we
write it explicitly for a system of D3 fractional
branes with worldvolume along the directions and and D7 fractional branes containing the D3 branes and having
the remaining four worldvolume directions along the orbifolded ones. The
metric, the -form field strenght, the axion and the dilaton are
given by ^{3}^{3}3We
denote with and the four directions corresponding to
the worldvolume of the fractional D3 brane, with and those
along the four orbifolded directions and and
with and the directions and that are transverse to
both the D3 and the D7 branes.:

(2.2) | |||||

(2.3) |

(2.4) |

where the warp factor is a function of all coordinates that are transverse to the D3 brane (). The twisted fields are instead given by , where is the volume form corresponding to the vanishing -cycle and

(2.5) |

It can be seen that the previous solution has a naked singularity of the repulson type at short distances. But, on the other hand, if we probe it with a brane probe approaching the stack of branes corresponding to the classical solution from infinity, it can also be seen that the tension of the probe vanishes at a certain distance from the stack of branes that is larger than that of the naked singularity. The point where the probe brane becomes tensionless is called in the literature enhançon [7] and at this point the classical solution cannot be used anymore to describe the stack of fractional branes.

Inserting in eq.s (1.2) and (1.3) the classical solution we get the gauge coupling constant and the angle [1] :

(2.6) |

Actually in the case of an supersymmetric theory one gets in the gauge multiplet also a complex scalar field . This means that, when we derive the Yang-Mills action from the Born-Infeld action we also get a contribution from the kinetic terms of the brane coordinates and that are transverse to the brane and transverse to the orbifolded ones. This implies that the complex scalar field of the gauge supermultiplet is related to the coordinate of supergravity through the following gauge-gravity relation . This is a relation between a quantity of the gauge theory living on the fractional D3 branes and the coordinate of supergravity. This identification allows one to obtain the gauge theory anomalies from the supergravity background. In fact, since we know how the anomalous scale and transformations act on , from the previous gauge-gravity relation we can deduce how they act on , namely

(2.7) |

Those transformations do not leave unchanged the supergravity background in eq.s (2.5) and, as a consequence, they generate the anomalies of the gauge theory living on the fractional D3 branes. Acting with those transformations on eq.s (2.6) we get:

(2.8) |

The first equation implies that the -function of super QCD with hypermultiplets is given by:

(2.9) |

while the second one reproduces the chiral anomaly [8, 9]. In particular, if we choose , then is shifted by a multiple of . Since is periodic of , this means that the subgroup is not anomalous in perfect agreement with gauge theory results.

Using eq.s (2.6) it is easy to compute the combination:

(2.10) |

where is called in the literature the enhançon radius and corresponds in the gauge theory to the dimensional scale generated by dimensional transmutation. Eq. (2.10) reproduces the perturbative moduli space of super QCD, but not the instanton corrections. This corresponds to the fact that the classical solution is reliable for large distances in supergravity corresponding to short distances in the gauge theory, while it cannot be used below the enhançon radius where nonperturbative physics is expected to show up. This means that in order to study nonperturbative effects in the gauge theory we need to find a classical solution free from enhançons and naked singularities. This will be done in the next section. Before doing that let us first extend the previous results to super QCD that can be obtained as a particular case of the general one studied in Ref. [10]. In this case only the asymptotic behaviour for large distances of the classical solution has been explicitly obtained and this is sufficient for computing the gauge coupling constant and the angle of super QCD. As explained in Ref. [10], together with fractional D3 branes of the same type, one must also consider two kinds of fractional D7 branes in order to avoid gauge anomalies and one gets the following expressions for the gauge coupling constant and the angle ( ) [9, 11, 10]:

(2.11) |

As explained in Ref. [11] the anomalous scale and transformations act on as . This implies that the gauge parameters are transformed as follows:

(2.12) |

that reproduce the anomalies of super QCD. The differences between the anomalies in the (eq.(2.8)) and (eq.(2.12)) super QCD can be easily understood in terms of the different structure of the two orbifold considered. If we consider the two gauge coupling constants there is a factor between the contributions coming from the pure gauge part, while the contribution of the matter is the same. The factor is a consequence of the fact that the orbifold has three sectors, while the factor follows from an additional factor in the orbifold projection for the orbifold with respect to the orbifold . This explains the factor in the gauge field contribution to the -function. The matter part is the same because in the orbifold we have only one kind of fractional branes, while in the other orbifold, in order to cancel the gauge anomaly [10], we need two kinds of fractional branes. This factor cancels the factor coming from the orbifold projection. Similar considerations can also be used to relate the two chiral anomalies.

In conclusion, by using the fractional branes we have reproduced the one-loop perturbative behaviour of both and super QCD, but, because of the enhançon and naked singularities we are not able to enter the nonperturbative region in the gauge theory corresponding to short distances in supergravity. In order to do this we must find a classical solution free of singularities. That is why in the next section we turn to wrapped branes.

## 3 Running coupling constant from wrapped branes

In this section we turn to the case of wrapped branes and in particular we will focus on a D5 brane wrapped on whose corresponding solution, found in Ref. [12] in four dimensions, was riinterpreted as a ten dimensional one corresponding to a wrapped D5 brane and used in Ref. [13] for describing super Yang-Mills. A more detailed and pedagogical derivation of the classical solution is presented in Ref. [14] where the classical solution was used for determining the running coupling constant of super Yang-Mills as a function of the renormalization group scale . In particular, inserting the classical solution in eq.(1.1), one can determine how the gauge coupling constant depends on the distance from the branes. One gets:

(3.1) |

But in order to determine the behaviour of the gauge coupling constant as a function of the renormalization scale one must also give a relation between and . This was obtained in Ref. [14] by connecting a certain function of , called in Ref. [14] , to the gaugino condensate following the suggestion of Ref. [15]. The result was:

(3.2) |

The running coupling constant is determined once we fix the
function that depends on the two-cycle on which we wrap the
brane. On the other hand it is important to stress that the gauge coupling
constant depends on the renormalization scheme chosen and
therefore two different choices of the two-cycle can be interpreted
to correspond to two different renormalization schemes.
In Ref. [14] the brane was wrapped on the
spanned by the coordinates and
having chosen the other coordinates and at
constant values ^{4}^{4}4We use the notation of Ref. [14]..
This choice gave the following result:

(3.3) |

where is the elliptic integral and behaves as for large
values of . In Ref. [14], by considering only the leading
asymptotic behaviour of eq. (3.3) and
by combining it with eq.(3.2), it was derived that the
-function of super Yang-Mills was exactly the
NSVZ -function [16] plus non perturbative corrections due to
fractional instantons ^{5}^{5}5An extension to the noncommutative
case was done in Ref. [17].. This result was questioned in
Ref. [18]
where it was shown that, if one also includes the first non leading
logarithmic correction, one gets an extra contribution to the
-function that modifies the one derived in Ref. [16]
already at two-loop level. Then, in order to recover the correct
two-loop behaviour, it was suggested in Ref. [18] to add in
eq.(3.2) an extra function of the coupling
constant that can be fixed by requiring agreement with the correct
two-loop result. Of course it turns out that must be
singular at as the transformation that is needed in
going from the holomorphic to the wilsonian
-function [19]. But, if we are prepared to recover the
correct two-loop behaviour by simply changing the renormalization
scheme, in order to
obtain the NSVZ -function one could change immediately the scheme of
renormalization by trading the elliptic integral with just its asymptotic
behaviour: as was
done in practice in Ref. [14]. This way of thinking eliminates a problem
that seems to appear if we perform a gauge transformation on the
non-abelian gauge field of gauged supergravity. In fact, if one
performs a gauge transformation in such a way that the gauge field
is vanishing in the deep infrared (), one gets a function
that is different from the one in eq.(3.1).
One gets [20]

(3.4) |

that, when put in eq.(3.1), gives a Landau-pole singularity at unlike the function in eq.(3.3) that gave a smooth behaviour at . This is, however, not a problem if one also interprets the gauge transformation in supergravity as a change of renormalization scheme in the gauge theory.

A natural and elegant way to get directly the SNVZ -function without having to change the renormalization scheme as was implicitly done in Ref. [14], is presented in Ref. [21] and is based on the proposal of choosing the same cycle used in Ref. [14] if one uses the solution after having performed the previous discussed gauge transformation or equivalently use the original solution and integrate on any of the two following cycles: or . In both cases one gets precisely the expression in eq.(3.4) [20, 21]. This means that the definition of the two-cycle depends on which gauge we use for the gauge field of gauged supergravity and if one takes into account these changes one gets always the same result for the gauge coupling constant of the gauge theory living on the wrapped brane.

In conclusion if one follows the proposal of Ref. [21] the two equations that determine the running gauge coupling constant of super Yang-Mills as a function of the renormalization scale are the following:

(3.5) |

It is easy to check that they imply the NSVZ -function plus corrections due to fractional instantons. In fact from the previous two equations after some simple calculation one gets:

(3.6) |

This equation is exact and should be used together with the first equation in (3.5) in order to get the -function as a function of . It does not seem possible, however, to trade with in an analytic way. It can be done in the ultraviolet where, from the first equation in (3.5), it can be seen that can be approximated with

(3.7) |

that is equal to the NSVZ -function plus nonperturbative corrections due to fractional instantons.

Acknowledgement We would like to thank M. Bertolini, E. Imeroni, P. Merlatti, R. Marotta, P. Olesen, F. Pezzella and F. Sannino for useful discussions and specially A. Lerda for many exchanges of views on the subject of this talk.

## References

- [1] M. Bertolini, P. Di Vecchia, M. Frau, A. Lerda and R. Marotta, Nucl. Phys. B621 (2002) 157, hep-th/0107057.
- [2] M. Bertolini, P. Di Vecchia, M. Frau, A. Lerda, R. Marotta and I. Pesando, JHEP 02 (2001) 014, hep-th/0011077.
- [3] J. Polchinski, Int. J. Mod. Phys. A16 (2001) 707, hep-th/0011193.
- [4] M. Graña and J. Polchinski, Phys. Rev D65 (2002) 126005, hep-th/0106014.
- [5] M. Billò, L. Gallot and A. Liccardo, Nucl. Phys. B614 (2001) 254, hep-th/0105258.
- [6] M. Bertolini, P. Di Vecchia and R. Marotta, hep-th/0112195.
- [7] C.V. Johnson, A.W. Peet and J. Polchinski, Phys. Rev. D61 (2000) 086001, hep-th/9911161.
- [8] I.R. Klebanov, P. Ouyang and E. Witten, hep-th/0202056.
- [9] M. Bertolini, P. Di Vecchia, M. Frau, A. Lerda and R. Marotta, Phys. Lett. B540 (2002) 104, hep-th/0202195.
- [10] R. Marotta, F. Nicodemi, R. Pettorino, F. Pezzella and F. Sannino, JHEP 0209 (2002) 010, hep-th/0208153.
- [11] M. Bertolini, P. Di Vecchia, G. Ferretti and R. Marotta, Nucl. Phys. 360 (2002) 222, hep-th/0112187.
- [12] A. H. Chamseddine and M. S. Volkov, Phys. Rev. Lett. 79 (1997) 3343, hep-th/9707176; Phys. Rev. D57 (1998) 6242, hep-th/9711181.
- [13] J. Maldacena and C. Nuñez, Phys. Rev. Lett. 86 (2001) 588, hep-th/0008001.
- [14] P. Di Vecchia, A. Lerda and P. Merlatti, Nucl. Phys. B646 (2002) 43, hep-th/0205204.
- [15] R. Apreda, F. Bigazzi, A. L. Cotrone, M. Petrini and A. Zaffaroni, hep-th/0112236.
- [16] V. Novikov, M. Shifman, A. Vainstein and V. Zakharov, Nucl. Phys. B229 (1983) 381.
- [17] T. Mateos, J.M. Pons and P. Talavera, hep-th/0209150
- [18] P. Olesen and F. Sannino, hep-th/0207039.
- [19] M. A. Shifman and A. I. Vainshtein, Nucl. Phys. B277 (1986) 456 [Sov.Phys. JETP 64 (1986) 428].
- [20] P. Olesen, private communication.
- [21] M. Bertolini and P. Merlatti, hep-th/0211142.