# Beyond

UWO-TH-08/15

Beyond

Alex Buchel, Robert C. Myers and Aninda Sinha |

Perimeter Institute for Theoretical Physics |

Waterloo, Ontario N2L 2Y5, Canada |

Department of Applied Mathematics |

University of Western Ontario |

London, Ontario N6A 5B7, Canada |

###### Contents

- 1 Introduction
- 2 Effective description of conformal gauge theory/string theory duality
- 3 in superconformal gauge theories
- 4 The strongly coupled quark-gluon plasma?
- 5 Discussion
- A Comments on Field Redefinitions
- B String theory origin of

## 1 Introduction

Over the past decade, the AdS/CFT correspondence [1, 2] has been developed to provide a powerful tool to investigate the thermal and hydrodynamic properties for certain strongly coupled gauge theories [3]. At the same time, recent experimental results from the Relativistic Heavy Ion Collider (RHIC) have revealed a new phase of nuclear matter, known as the strongly coupled quark-gluon plasma (sQGP) [4]. Recently, there has been great interest in possible connections between these two advances, in particular, using the AdS/CFT to gain theoretical insight into the sQGP [5]. The primary motivation for this possible connection is the observation that a wide variety of holographic theories exhibit an exceptionally low ratio of shear viscosity to entropy density while the RHIC data seems to indicate that this ratio is unusually small for the sQGP and even seems yield roughly [6].

Motivated by the results from the AdS/CFT correspondence, Kovtun, Son and Starinets (KSS) proposed a now celebrated bound for the viscosity-to-entropy-density ratio [7]. That is, for all fluids in nature the ratio is bounded from below:

(1.1) |

This bound certainly appears to be satisfied by all common
substances observed in nature [8]. Using the AdS/CFT
correspondence, the bound has been shown to be saturated in all
gauge theories in the planar limit and at infinite ’t Hooft coupling
(with various gauge groups, matter content, with or without chemical
potentials for conserved charges, with non-commutative
spatial directions, in external background fields) that allow for a
dual supergravity description
[8, 9, 10, 11, 12, 13, 14, 15, 16, 17]. The bound is
not saturated but it is still satisfied in all
four-dimensional^{1}^{1}1Preliminary analysis indicates that the
bound is satisfied under the same conditions in three-dimensional
conformal gauge theories [18]. conformal gauge theories with
equal and central charges, again allowing for a string
theory dual and in the planar limit and with large but finite ’t
Hooft coupling [19, 20].

One may ask if the KSS bound (1.1) is indeed of fundamental importance to nature? However, the answer appears to be “no”. It was pointed out by [21] that the bound is violated in a nonrelativistic gas with increasing number of species and by [22, 23, 24], that it can be violated in effective theories of higher derivative gravity. Of course, the true question is whether or not the violation occurs in quantum field theories that allow for a consistent ultraviolet completion [25]. In fact, Kats and Petrov [24] proposed an explicit example of a gauge theory/string theory duality where a violation of the KSS bound occurs in a controllable setting — see also [26]. However, one may easily draw into question the veracity of this claim.

In particular, the calculations in [24] were presented in terms of an effective five-dimensional gravity theory. However, the proposed duality is between a gauge theory and a ten-dimensional string theory. Thus, it seems the gravity calculations should be performed within the full ten-dimensional string theory background constructed to required order in . Alternatively, beginning with the ten-dimensional background, one could carefully perform the Kaluza-Klein reduction but this would require keeping track of all of the fields and their interactions in the effective five-dimensional theory. For instance, the reduction may produce scalar fields which it seems are likely to effect the calculations at the order to which they must be performed to detect the potential violation of the viscosity bound.

Our primary motivation for the present work was to examine in detail the claimed violation of the viscosity bound (1.1) in [24]. In fact, we are able to sharpen the arguments in terms of an effective five-dimensional gravity dual and confirm that the KSS bound will be violated as long as the central charges of the conformal gauge theory satisfy a number of conditions: and are required to guarantee the reliability of the low energy effective action and then the inequality

(1.2) |

An outline of the paper is as follows: In section 2, we examine in detail when an effective five-dimensional gravity dual yields a reliable description of the superconformal gauge theory. In section 3 we compute in variety four-dimensional superconformal gauge theories. This produces new examples where we can reliably state that the KSS bound is violated. Given these observations, we consider the comparison of results from AdS/CFT calculations to the sQGP in section 4. Finally, we provide a concluding discussion in section 5. Appendix A elaborates on the discussion of field redefinitions in the presence of other bulk fields, while appendix B provides an explicit realization of our effective AdS/CFT duality in a stringy context where ten-dimensional supergravity plus probe branes is a reliable approximation.

## 2 Effective description of conformal gauge theory/string theory duality

According to the AdS/CFT correspondence [2], any four-dimensional superconformal gauge theory will have a dual description in terms of quantum gravity with a negative cosmological constant in five dimensions. Now for particular cases where it is sensible to consider the conformal gauge theory with large- and strong coupling, our intuition is that the dual description is well approximated by Einstein gravity in a five-dimensional AdS spacetime. In this framework, higher curvature (or more broadly higher derivative) interactions are expected to arise on general grounds, e.g., as quantum or stringy corrections to the classical action. Hence a more refined description will be given by an effective action where the cosmological constant and Einstein terms are supplemented by such higher curvature corrections. Here we consider when such an effective action approach yields a reliable description of the superconformal gauge theory.

A key assumption in our discussion will be that:

The effective five-dimensional gravity theory is described by a
sensible derivative expansion. That is, we expect that the higher
curvature terms are systematically suppressed by powers of the
Planck length, .

Hence we can expect the effective gravity action in five
dimensions to leading order to take the form

(2.1) |

where the scale will correspond to the AdS curvature scale, at leading order, and we assume that . We have parameterized the curvature squared couplings with the AdS curvature scale, as is convenient for explicit calculations, but we expect that the dimensionless couplings in accord with our assumption of a sensible derivative expansion. Further, compared to these interactions, the six- and higher derivative terms, which have been left implicit, are suppressed by further powers of . For example, an interaction of the form would have .

At this point, we note that we can simplify the form of the action by making a field redefinition with [22, 24]

(2.2) |

which then simplifies the action to

(2.3) |

The implicit terms implied by the ellipsis all contain six or more
derivatives suppressed by at least , as described
above. Hence, the field redefinition (2.2) has succeeded in
eliminating the and terms.^{2}^{2}2Of course,
the coefficients of these two interactions could be tuned to any
convenient values. For example, this would allow us to assemble the
curvature-squared terms to be the square of Weyl-curvature or the
Gauss-Bonnet term [22], either of which may be advantageous
for certain calculations. This makes clear that, at this order, the
gravity action contains two and only two dimensionless small
parameters: and .

We return to this point after making a number of observations: first, given the effective action (2.3), we might consider making a further field redefinition of the form

(2.4) |

which would modify the action by adding terms of the form

Hence, given the first term above, it would seem that we can use these field redefinitions to remove the term in (2.3). Note that the latter would require that and hence the six-derivative terms, appearing in the second line of (2), would only be suppressed by this same factor . However, our assumption is that the derivative expansion organizes the effective action so that any such term is suppressed by a factor of . Hence if we wish to maintain this structure, then we must require that and so this field redefinition could only make higher order corrections to .

Next, we observe that with the original field redefinitions (2.2), Newton’s constant (i.e., the coefficient of the Einstein term) has been kept fixed but the curvature scale has to be redefined as

(2.6) |

In principle, the coupling was also corrected with . However, we do not specify the latter in detail, as it actually requires specifying the field redefinition (2.2) more precisely, i.e., to order . But these expressions do illustrate the point that in general the parameters in this effective action (2.3) may be complicated functions of the microscopic parameters of the quantum gravity theory. For example, in a string or M-theory framework, they would arise upon the Kaluza-Klein compactification of the higher dimensional geometry and these low energy parameters would depend on all of the details for the compactification. In general, we would also expect that these parameters also receive quantum ‘corrections’, which might in turn include both perturbative and nonperturbative contributions.

Note, however, that the dual theory is assumed to be dual to a four-dimensional conformal field theory. Hence at any order in the derivative expansion, the gravity theory admits a five-dimensional anti-de Sitter vacuum, although the precise characteristics, i.e., curvature, of the latter may change as we increase the accuracy of our calculations. Given the action (2.3), the curvature of the AdS space is:

(2.7) |

Again, this curvature is dependent on the microscopic details of the quantum gravity theory.

The key observation, which we review here, is that the two dimensionless parameters identified above are simply related to parameters characterizing the dual CFT. First, we recall that the conformal anomaly of a four-dimensional CFT can be identified by putting the theory in a curved spacetime and observing [27]

(2.8) |

Here and are the two central charges of the CFT and and correspond to the four-dimensional Euler density and the square of the Weyl curvature, respectively. Explicitly,

(2.9) |

Holographic techniques allow precisely the same expression to be calculated with the result [28, 29, 30],

(2.10) |

Hence comparing (2.8) and (2.10), we arrive at

(2.11) |

In these expressions, we have used our assumption of a sensible derivative expansion, which dictates that .

One conclusion then is that if we require the quantum gravity theory is described by a low energy action with sensible derivative expansion, we are restricted to consider CFT’s for which

(2.12) |

Further, the effective action is expected to contain further higher curvature terms and the dimensionless coefficients appearing in these interactions would be related to new parameters characterizing the CFT — for example, see [31]. Our assumption of a sensible derivative expansion then restricts the size of these parameters, i.e., the CFT’s of interest should have these parameters being proportional to inverse powers of the central charge .

Above, we observed that the AdS/CFT correspondence dictates the values of the leading parameters in the effective gravity action terms of the central charges of the dual CFT according to (2.11). Hence if the central charges of the CFT are known (and the inequalities (2.12) are satisfied), we can be confident of the precise form of this effective action (2.3) to leading order, even if we do not understand the microscopic details underlying the quantum gravity theory. Then if we are careful to respect the limitations of the derivative expansion, we can work reliably with the gravity action (2.3) to determine the properties of the CFT using the standard AdS/CFT correspondence. Note that only the dimensionless ratios in (2.11), but not the Planck length , appear in any physical results for the CFT. Of course, this is in accord with the fact that for a supersymmetric CFT, supersymmetry combines with diffeomorphism and conformal invariance to completely dictate the form of the two- and three-point correlators of the stress-energy tensor in terms of these two central charges, and [33]. Hence while we can reproduce these correlators with the dual gravity action (2.3), the latter also allows us to calculate more interesting properties, such as thermal transport coefficients of the CFT. One interesting example is the shear viscosity [22, 24]

(2.13) |

where we have introduced

(2.14) |

Hence the sign of in the CFT or of the term in effective gravity action determines whether or not the viscosity bound (1.1) is respected or violated at this order. In particular, the bound is violated if .

Of course, according to the standard AdS/CFT dictionary, the metric is dual to the stress-energy tensor of the CFT and so with the gravity action (2.3), we are restricted to study the properties of this one operator. In general, we should expect the full CFT will have a spectrum of interesting operators, possibly including a variety of relevant, irrelevant and marginal operators. The latter would then be dual to other fields which may also play an interesting role in the gravity theory. Hence our preceding conclusions may seem somewhat naive since we have restricted the discussion to the pure gravity sector of the theory. Therefore we must show that such operators do not effect our conclusions.

As an example, consider the case where the gravity theory that contains a number of scalars . As above, we assume that the effective gravity theory is described by a sensible derivative expansion. In principle, a large number of four-derivative terms could appear in the effective action but as described in appendix A, field redefinitions can be used to greatly simplify the action. The final action can be written as

While more details are provided in the appendix, combines the remaining four-derivative interactions which explicitly contain derivatives of the scalar fields. An important point is that all of these interactions contain at least two factors of scalar derivatives. Then, since we are treating these terms perturbatively within the derivative expansion and the scalars will be constant in the leading solutions of interest, they remain constant at the next order. Hence we may ignore these terms for the remainder of the discussion.

In describing the rest of the terms in (2), we should begin by saying that we have adopted the convenient (supergravity) convention where the scalar fields are dimensionless. Below, we argue that the scalars vanish in the AdS vacuum and so we may assume that all of the expressions in the action are nonsingular at . Hence we can express each of the coefficient functions in terms of a Taylor series:

(2.16) | |||||

(2.17) | |||||

(2.18) |

Now in keeping with our assumption of the derivative expansion
above, a second key assumption here is that:

All of the coupling coefficients in each of (2.16),
(2.17) and (2.18) above are of the same order (with
the exception of ).

That is, all of the couplings in
(2.16) and in (2.17) may be of order one
(or higher order in ), with the exception of –
which we address below. Similarly, and all of the
subsequent couplings in (2.18) are assumed to be
of order (or higher). Of course, each of these couplings
may in general be a complicated function of and so here
we are demanding that and do not have
order one (or order ) contributions. Within this framework,
the corresponding scalar masses are of the order of the AdS
curvature scale, i.e., . Hence each of the dual
scalar operators has a conformal dimension of order one.
These operators may be relevant, irrelevant or marginal. An exactly
marginal operator is an exceptional case, which will receive
detailed consideration below. As before, the ellipsis in (2)
corresponds to six- and higher derivative terms which implicitly are
suppressed at least by couplings of order , as in the
previous discussion.

The dual theory is a conformal field theory, which again implies that at any order in the derivative expansion, the gravity theory (2) admits an AdS vacuum. Further, in the conformal vacuum, the expectation value of any of the operators must vanish, i.e.,

(2.19) |

This property is reflected in the gravity theory with the vanishing of the dual scalar fields in the AdS vacuum. A possible exception to this conclusion arises with an exactly marginal operator. In principle, the corresponding massless scalar in the dual gravity theory can take on any constant value. However, we will define this expectation value of the scalar field to be zero for the vacuum that we are studying here.

Let us now turn to the special case of the exceptional couplings . For the AdS space to be a solution with at leading order in the derivative expansion, i.e., dropping the curvature-squared and higher order terms, it must be true that at this order. However, when curvature-squared term is included, the scalar equations of motion yield

(2.20) |

assuming an AdS background with vanishing scalars. Hence we find that consistency demands that the scalar potential contains linear couplings of order :

(2.21) |

An alternate interpretation would be that if we set then the
AdS solution is stable to leading order but in general the
appearance of the curvature-squared term will then cause the
scalars to acquire an expectation value of order in the
vacuum. We are simply redefining the scalars to absorb this constant
shift in our approach.^{3}^{3}3 An alternate approach would be to
use the freedom of field redefinitions so that the square of the
Weyl tensor, rather than of the Riemann tensor, appears in the
effective action (2). Then because the AdS vacuum has
vanishing Weyl curvature, the scalar equations of motion would be
unaffected by the curvature-squared term and would remain zero
at this order. Note that in this approach, the AdS curvature would
also match precisely the scale appearing in the action. However,
the additional and interactions, appearing in
the Weyl-curvature squared, would modify the holographic anomaly
(2.10) in precisely such a way to reproduce the same
expressions as in (2.11).

Although the scalars vanish in the AdS vacuum, one can expect that the curvature-squared term will source the various scalar fields in more general backgrounds. However, one would still have that in such a background. We must consider two particular examples in our discussion. The first relevant example would be a black hole background and this effect implies that in a thermal bath the dual operators acquire expectation values . The second relevant case comes from the holographic calculation of the conformal anomaly (2.10). While the precise background is typically not specified in these calculations, implicitly, one must be working with more general backgrounds where, in particular, the Weyl curvature is nonvanishing. Hence we must argue that even though the scalars may have nontrivial profile at order , this will not affect the holographic calculations of the conformal anomaly or the thermal behaviour of the CFT. With a careful consideration below, we will show that the nontrivial scalars can only modify the results at order . Our general argument was originally formulated in a slightly different context in [11].

As an explicit example, let us consider the calculation of shear viscosity [34, 35, 3]. A key step would be calculating the effective quadratic action for the various graviton fluctuations, i.e., the shear, sound and transverse modes, in the black hole background. The nontrivial scalars can effect these modes in two ways. First they explicitly appear in the action. However, contributions of terms quadratic or higher powers in would be suppressed by or higher powers. There are two possible sets of linear terms in and in but, as discussed above, the couplings for both of these are already order and so they only contribute with an overall factor of . Secondly, the nontrivial scalars will modify the background geometry through Einstein’s equations, but similar reasoning shows that the modifications of the metric would again be order . Hence, even though the scalars themselves appear at order , their effect is only felt by the graviton modes at order . Hence the calculation of can be reliably made at order while ignoring all of the scalar fields, i.e., with the effective gravity action (2.3). The same general argument applies to calculations of other thermal properties from the black hole background or of the holographic conformal anomaly.

Next, we make a few comments on the extension of our discussion to include vectors in the gravity theory — see also appendix A. If we consider some number of Abelian gauge fields, the vectors are dual to conserved currents and the corresponding gauge symmetries are identified with global symmetries in the CFT [2]. A complete discussion of the contributions of these gauge fields to the four-derivative gravitational action would be quite lengthy and equally tedious and so we only remark on salient points. First, we restrict the discussion to having only constant gauge fields at leading order in the background. That is, we are only considering the case of vanishing chemical potentials. Next, it is relatively easy to show that the majority (i.e., all but one) of the new four-derivative interactions are at least quadratic in the field strengths of these gauge fields. Hence an argument similar to that below (2) applies here as well, with the conclusion that these terms are irrelevant at this order, as long as we are considering backgrounds where the vectors are constant. However, given a set of gauge fields , there is one four-derivative coupling which cannot be dismissed by this argument, namely,

(2.22) |

In keeping with the derivative expansion, these terms, which are
linear in the gauge fields, are characterized by a set of
dimensionless constants . Note that we require
that under local gauge transformations, only produces a surface
term and so even if the gravity theory contains scalars, we cannot
replace the constants by general functions .
An interaction of this form plays an interesting role in describing
the anomaly for the current in supersymmetric CFT’s
[36, 37]. In fact, in the context of gauged
supergravity, supersymmetry connects this interaction (2.22)
to an term [38, 39]. Since this
interaction is linear in the gauge potential, it will induce a
nontrivial profile in a background where is
nonvanishing. However, this combination of curvatures vanishes both
for the AdS vacuum and an AdS black hole background and so
no profile is induced for these backgrounds. Of course, this result
is in keeping with the intuition that a finite charge density is not
induced by introducing a finite temperature alone. We should also
consider the nontrivial backgrounds implicit in calculating the
holographic conformal anomaly (2.10). In general, we expect
that a nontrivial profile can be induced for the gauge potentials in
this case but we would still only find that in such a background. Hence following arguments analogous
to those presented to dismiss the effect of nontrivial scalar
profiles, we would again find that the nontrivial gauge potentials
can only modify the anomaly calculation at order .
Therefore the calculations of both the thermal properties and of the
conformal anomaly would remain unaffected by the appearance of
additional vector fields. Hence our conclusion once again is that
these calculations can be reliably made at order with
the effective gravity action (2.3), while ignoring any matter
fields in the gravitational theory.^{4}^{4}4Of course, the
interesting question of corrections in the presence of a chemical
potential would require a detailed analysis of the higher order
gauge field interactions.

In closing this section, we return to the special case of an exactly marginal (scalar) operator. As already mentioned above, the dual scalar field is precisely massless and so it can in principle be set to any arbitrary value in the AdS vacuum. This property implies special relations between the couplings for in the effective action (2), i.e.,

(2.23) |

where corresponds to in AdS, as in (2.20). Unfortunately, these relations make the discussion somewhat more complicated than necessary. So instead, we make a field redefinition such that the effective action (2) takes the form

(2.24) |

where is the Weyl curvature in five dimensions. Since the Weyl curvature vanishes in the AdS vacuum, the only restriction is that the scalar potential is completely independent of . Note, however, that in general remains a function of without any restrictions. Hence, even though the couplings in are naturally of order as in (2.18), this suppression could be overcome if the massless scalar has a very large expectation value, i.e., for large . More generally, since can become arbitrarily large, it can produce effective coupling coefficients for the higher derivative terms which are not suppressed as we initially assumed. Hence, our assumption of a sensible derivative expansion will implicitly restrict us to a limited region of the parameter space.

Of course, it is generally expected that a CFT with exactly marginal operators will be an exceptional case and in the absence of exactly marginal operators, these considerations are not required. However, this issue naturally arises in many string realizations of the AdS/CFT correspondence where the dilaton, i.e., the string coupling, is dual to an exactly marginal operator.

In particular, this is the case for the string theory construction which Kats and Petrov [24] suggested produces a violation the KSS viscosity bound (1.1). In this context, the gravitational theory is Type IIb string theory on a AdS background, which can be viewed as the decoupling limit of D3-branes overlapping with a coincident collection of four D7-branes and an O7-brane [40]. The dual CFT is four-dimensional super-Yang-Mills coupled to 4 hypermultiplets in the fundamental representation and 1 hypermultiplet in the antisymmetric representation. The central charges for this gauge theory are [42]:

(2.25) |

Now for large (but finite) , the central charges satisfy both of the inequalities in (2.12) and so it seems that we can confidently apply the results calculated from the five-dimensional effective action (2.3) with the gravitational couplings fixed by (2.11). Further, as noted by [24], and so the shear viscosity (2.13) is

(2.26) |

which violates KSS bound (1.1).

However, before accepting this result, we must first consider that in this construction, the string coupling remains a free parameter. That is, this corresponds to the case of an exactly marginal operator which is dual to the dilaton. As usual then, the results for the CFT can be considered in a double expansion in both inverse powers of and of the ’t Hooft coupling . Alternatively, we can think that the corrections to effective gravity action are governed by two independent scales: the Planck length and the string length . Hence, we must make sure that the higher curvature corrections beyond those explicitly shown in (2.3) are sufficiently suppressed according to the assumed derivative expansion. As we discuss in section 4, there will be no curvature cubed interaction. There is a universal term quartic in curvatures which appears in any (closed) superstring theory [41]. It is known that this term corrects the ratio of viscosity-to-entropy-density at [10, 43, 19]. Hence as noted in [24], in order for the correction in (2.26) to dominate, we must have

(2.27) |

The full correction to from the term also contains a contribution at , as well as various nonperturbative corrections [11, 20]. While the latter play no role in the present discussion, formally requiring the first correction to be subdominant yields . While the previous interaction can be associated with the closed string sector, one might also ask if the calculations could be significantly effected by interactions induced by the branes. As explained in appendix B, such higher curvature terms will be subdominant in the derivative expansion. In particular, a D7-brane induced term would be accompanied by an additional suppression factor of . In the present case with and , such an interaction would only contribute corrections at order . The final conclusion is that the Kats and Petrov result (2.26) calculated with a five-dimensional effective action (2.3) is reliable within in a certain parameter regime (2.27) and that we have at least limited violations of the KSS bound (1.1) in string theory.

In the above string theory example, the curvature-squared term can be associated with the world-volume action of the D7-branes [42, 37]. In appendix B, we have added a discussion which provides a schematic understanding of the origin of this term.

Note the requirement (2.27) is compatible with conventional restrictions implicit in considering the classical gravity limit of the AdS/CFT correspondence in string theory. That is, we have from requiring to minimize stringy contributions in the derivative expansion and to minimizes string loop contributions. While the derivative expansion, and hence , is central to the present effective action approach, there is no need to give a separate account of loop contributions. That is, using the five-dimensional effective action (2.3) did not require a detailed understanding of the underlying microscopic origin of each of the couplings in the full quantum gravity theory. Rather we advocated that if the CFT central charges were given, we could use the AdS/CFT dictionary to fix the gravitational couplings according to (2.11). Of course, consistency also required that these central charges satisfy the inequalities given in (2.12). With this approach, there is no reason that we could not consider the above or other string theory constructions where the string coupling is strong, i.e., , which implies that or . In particular, we can apply this approach to evaluate the thermal behaviour of CFT’s holographically described by the F-theory constructions of [42]. The case considered by Kats and Petrov corresponds to one of these constructions and in fact, it is the only case with a marginal coupling. In the remaining cases, there are no marginal couplings and the string coupling is pinned at . Further, as discussed in the following section, with large (but finite) , the central charges again satisfy the inequalities in (2.12) and so the shear viscosity (2.13) yields new violations of the KSS bound (1.1) since in each of these cases.

## 3 in superconformal gauge theories

From the results of the previous section, we can conclude that if the central charges of a four-dimensional superconformal gauge theory satisfy the two inequalities in (2.12), then we can reliably describe the theory with a gravity dual with a five-dimensional effective action (2.3) in which the gravitational couplings fixed by (2.11). Further, the shear viscosity is given by (2.13) and the superconformal theory will violate the KSS bound (1.1) provided that

(3.1) |

Hence in this section, we explore the central charges of superconformal gauge theories based on simple Lie groups with various matter fields. We only consider the gauge group to be a classical Lie group since we wish to take a large- limit so that the first inequality in (2.12) will be satisfied. We discuss two sets of theories, first those in which the gauge coupling is an exactly marginal operator and secondly models defined as isolated SCFTs. The gauge theories under consideration have either or supersymmetry in four dimensions; some of them have a known string theory dual while others do not (at this stage). Quite surprisingly, we find that in all of these models , which would seem to indicate a violation of the KSS bound (1.1). However, generically, as and so the second inequality in (2.12) is not satisfied. Therefore those theories do not have a gravity dual with a controllable derivative expansion, which is required for (2.13) to be valid. A similar analysis was carried out by Yuji Tachikawa and Brian Wecht [44]. Recently, [45] presented a complementary analysis of super-QCD with various relevant superpotentials. Related calculations also appear in [46].

A superconformal gauge theory has an anomaly free global
symmetry. The central charges, and , are relatively easy to
determine as they are related to gravitational anomalies in this
global symmetry. Consider a superconformal gauge theory with a gauge
group and matter multiplets in representations . Let
denote the -charges of the matter chiral
multiplets^{5}^{5}5We use susy representations to
describe theories with extended supersymmetry as well. in the
representation . It was found in [47] that

(3.2) |

Thus computation of reduces to the identification of the anomaly-free symmetry of the gauge theory at a superconformal fixed point. Our approach to this question depends whether the gauge coupling is marginal or the theory is at an isolated fixed point.

### 3.1 Superconformal gauge theories with exactly marginal gauge coupling

Let us begin with the identification of the anomaly-free symmetry for the case where the gauge coupling is exactly marginal. Resolving this question is straightforward in this case as it can be shown a symmetry with classical assignment of the -charges is anomaly free, given the vanishing of the one-loop perturbative -function. Consider classical assignment of -charges, i.e., all matter superfields have and a vector superfield has . The superconformal algebra then implies that anomalous dimensions of chiral superfields () must vanish. That is, the vanishing of the NSVZ exact perturbative -function, which is equivalent for zero anomalous dimensions to vanishing of one-loop perturbative -function,

(3.3) |

guarantees that the classical -charge assignment is in fact anomaly-free:

(3.4) |

In these expressions, and are group indices of the adjoint representation and a representation in , e.g., see [48] for explicit values. We only consider non-chiral theories in the following.

#### 3.1.1

Since , to satisfy the vanishing of -function as , we can consider (besides adjoint) only fundamental, symmetric and antisymmetric representations for the — any other representation has an index growing at least as as .

Suppose we have in the adjoint representation,
flavors^{6}^{6}6Recall that for a chiral representation, one
flavor is the sum of two conjugate representations. in the
fundamental representation, flavors in the symmetric
representation and flavors in the anti-symmetric
representation. Then, the vanishing of the NSVZ -function
implies

(3.5) |

which we can rearrange to yield

(3.6) |

Using this result, we can rewrite in (3.2) as

(3.7) |

Since , requiring that as necessitates

(3.8) |

which along with further implies that

(3.9) |

It is easy now to enumerate all the models with and as , as shown in the following table.

(a) | (3,0,0,0) | 0 | 0 |
---|---|---|---|

(b) | (2,1,0,1) | ||

(c) | (1,2,0,2) | ||

(d) | (1,1,1,0) | ||

(e) | (0,3,0,3) | ||

(f) | (0,2,1,1) |

Notice that model has a matter content corresponding to susy (and as a result ). Similarly, models and have a matter content corresponding to susy. In principle, the five models through are described by a gravity dual with the effective action (2.3). Further, we note that for each of these models and so they would seem to give violations of the KSS bound (1.1). However, the gauge coupling is marginal in all of these models and so we would have to make sure there is a regime in which gives the dominant correction to the ratio of the shear-viscosity-to-entropy-density (2.13). As discussed at the end of section 2, if we imagine that the gravity dual comes from a string theory construction, this should be possible for models with if the inequality (2.27) is satisfied. However, for model with , we should note that the interactions are also expected to contribute (positive) corrections at . The latter would always dominate since is also required for a sensible derivative expansion. However, the four superconformal gauge theories potentially have string theory duals which would produce violations of the KSS bound.

#### 3.1.2 and

The analysis proceeds precisely as before. As well as in the adjoint (or antisymmetric) representation, we consider in the vector representation and in the symmetric representation. For general , subject to vanishing -function, we find as

(3.10) |

which is supplemented with the condition

(3.11) |

This suggests that while , the only model with a controllable gravitational dual is the one with and the susy matter content since when , the condition also requires that .

#### 3.1.3

The analysis is the same as before. Besides in the adjoint (symmetric) representation, we consider in the fundamental representation and in the antisymmetric representation.

It is straightforward to establish that always as and to enumerate all the models with :

(a) | (3,0,0) | 0 | 0 |
---|---|---|---|

(b) | (2,1,4) | ||

(c) | (1,2,8) | ||

(d) | (0,3,12) |

Notice that model has a matter content corresponding to susy (and as a result ). Model is that originally identified by Kats and Petrov [24] and has a matter content corresponding to susy. Models and provide interesting new candidates for a controllable gravity dual which again yield violations of the KSS bound (1.1).

### 3.2 Isolated superconformal fixed points

There are several ways to engineer an isolated superconformal fixed
point. In a purely field theoretical construction, we can define an
asymptotically free gauge theory in the UV, which flows to a
strongly coupled interactive conformal fixed point in the IR
[49, 50]. These models have supersymmetry.
Alternatively, one can engineer isolated superconformal fixed points
arising from the large number of D3-branes at singularities in
F-theory [51, 52, 53, 54, 55]. The latter have
supersymmetry. All these theories have non-classical assignment of
-charges of the anomaly-free global symmetry for matter
fields, which implies anomalous dimensions of chiral
superfields — of course, this is simply a reflection of the strong
coupling at the isolated superconformal fixed point. Unlike the
examples of superconformal fixed points with exactly marginal
coupling discussed above, in the models which we review here with
supersymmetry, is always positive but it is not
suppressed by inverse powers of in the large limit.
Therefore these theories will not have a controllable gravity dual
and cannot be proven with the approach considered here to give
counterexamples to the KSS bound.^{7}^{7}7Besides models listed
below, we have considered Kutasov-Schwimmer model [56] and
there again we find but . On the other
hand, the models of [42] engineered directly in string theory
can violate the KSS bound (1.1).

#### 3.2.1 Conformal window for gauge theory

Consider gauge theory with flavors in the fundamental representation. As shown in [49], for the theory flows to a nontrivial superconformal fixed point in the IR. The matter fields global anomaly-free charge assignment is as follows [49]:

(3.12) |

which from (3.2) implies

(3.13) |

The work of [45] expands on these results by adding adjoint matter fields and studying the effect of various superpotential terms. In these theories, they again find that is always positive but also order one in the limit of large .

#### 3.2.2 Conformal window for gauge theory

#### 3.2.3 Conformal window for gauge theory

#### 3.2.4 superconformal fixed points from F-theory

Models constructed as D3-branes probing an F-theory singularity generated by a collection of coincident 7-branes and resulting in a constant dilaton were classified in [42]. Classifying the F-theory singularity with the symmetry group , one finds [42]

2 | 3 | 4 | 6 | 8 | 9 | 10 |

We emphasize that the F-theory analysis fully accounts for the back-reaction of the 7-branes, which generate a deficit angle in the internal geometry. Central charges of the dual four-dimensional superconformal gauge theories were also computed [42] and as , one has

(3.18) |

Notice then that with large but finite , each of the models
tabulated above yields and . Hence they all
have a controllable gravity dual and (2.13) yields a violation
of the KSS bound (1.1). The string coupling remains arbitrary
in the model and so this actually corresponds to a
superconformal gauge theory with an exactly marginal operator. Of
course, this is precisely the case that was examined by Kats and
Petrov [24].^{8}^{8}8In the present F-theory description, the
O7-plane is resolved as a combination of a (-1,-1) and a (1,-3)
7-brane.

## 4 The strongly coupled quark-gluon plasma?

Our analysis in section 2 demonstrates that the thermal properties of a large class of conformal gauge theories can be derived from a simple holographic framework. Of course, one is tempted to consider how these results might be applied to understand the strongly coupled quark-gluon plasma, which is currently under study with experiments at RHIC and soon at the LHC. In this direction, we would like to generalize a phenomenological approach originally advocated in [20]. The essential first step is to assume that the QCD plasma is described by an effective conformal field theory. Given this assumption, this effective CFT will have a holographic dual according to the AdS/CFT correspondence and if nature is gracious, the dual theory may be one for which we calculate. That is, the holographic dual may be approximated by the five-dimensional Einstein gravity coupled to a negative cosmological constant, with controllable higher curvature corrections.

In this case, we can ask if the sQGP is described by an effective CFT within the class of theories whose dual is governed the low energy action (2.3). We can then treat the parameters characterizing the CFT, i.e., the central charges and , or equivalently the dual gravitational parameters and , as phenomenological. That is, we can calculate the properties of the gauge theory plasma from the gravity dual and then compare the results to experimental observations of QCD to fix the effective parameters. One interesting property for such a comparison would be , as given in (2.13). As discussed in [20], if we denote the energy density of the conformal plasma and of the corresponding free theory as and , then the ratio provides another interesting quantity for comparison, as the ratio can also be determined by lattice QCD calculations. Hence our next step is to determine holographically with the effective action (2.3).

Working to first order in or , the equilibrium state of CFT plasma is encoded in the AdS-Schwarzschild background geometry [24]

(4.1) |

where

(4.2) |

The horizon appears at

(4.3) |

and the plasma temperature corresponds to the Hawking temperature which is given by

(4.4) |

Next we evaluate the black hole entropy for the solution (4.1) following the standard approach of [57] for gravity actions with higher curvature corrections. The general expression takes the form

(4.5) |

Of course, in the present case with a planar horizon, the horizon area diverges and so we calculate the entropy density. Dividing by the coordinate volume, the final result can be expressed as

(4.6) |

Note here that since the curvature above is multiplied by , we can evaluate it on the leading order solution, i.e., . To express this result in terms of CFT parameters, we use the relations (2.11) as well as our expressions above for the horizon radius (4.3) and temperature (4.4). Combining all of these, we arrive at the final result

(4.7) |

We would like to compare this result for the entropy density which is implicitly calculated for strong coupling to the entropy density of the free field limit. To produce a quantitative result, it turns out that we must assume that the underlying CFT is supersymmetric. We begin by noting that the central charges may be written as [27]:

(4.8) | |||||

where , and denote the number of vectors, (chiral) fermions and scalars, respectively. While these expressions assume that these are all massless free fields, the results are protected in a supersymmetric theory and so also apply at finite coupling in that case. In a supersymmetric theory, we have an equal number of bosonic and fermionic degrees of freedom, which we denote as , and therefore the entropy density is naturally proportional to . Hence we find the linear combination