A Appendix : Three forms and B_{2}

Thermodynamics of large N gauge theory from top down holography

Abstract:

By considering fluxes on branes and explicitly computing their back-reaction on the geometry with and without a black hole, we show how the UV divergence of Klebanov-Strassler model can be regulated. Using the form of the metric and fluxes in the extremal and non-extremal limit, we compute the on-shell gravity action including the localized sources up to linear order in perturbation parameter where and are units of and charges in the dual gauge theory. Using the gravitational description, we show how the gauge theory undergoes a first-order Hawking-Page like phase transition and compute the critical temperature . Finally, we obtain the equation of state for the gauge theory by computing thermodynamic state functions of the black hole and exhibit how black holes in deformed cone geometry can lead to results that are qualitatively similar to lattice QCD simulations.

1 Introduction

Asymptotic freedom guarantees that at high temperatures, nuclear matter is best described as a weakly interacting gas of quarks and gluons. The weak nature of the coupling allows a perturbative description of QCD where observables in principal can be computed with increasing accuracy as temperature is increased. However at low temperatures, nuclear matter is color neutral, indicating color degrees of freedom are strongly coupled and confined inside hadrons. The strong coupling regime of QCD is analyzed by either studying the theory on the lattice or resorting to effective field theories- both of which have their success and limitations.

On the other hand, gauge theories naturally arise from excitations of open strings ending on branes [1] while gravitons can be described by excitations of closed strings. By studying the interaction between open and closed strings, one can relate the Hilbert space of the gauge theory with that of gravity. The best studied example is the AdS/CFT correspondence [2]: Here the gauge theory is four dimensional maximally SUSY conformal field theory and the gravitons describe a ten dimensional space i.e. five dimensional anti-deSitter space times a compact five sphere. When the t’Hooft coupling for the gauge theory is large, Hilbert space of the gauge theory is conjectured to be contained in the Hilbert space of gravitons described by classical action of geometry. Thus a strongly coupled quantum gauge theory gets a classical description in terms of weakly coupled gravity. Expectation values in gauge theory which are otherwise extremely difficult to compute due to strong coupling, can easily be computed using the dual holographic description [3].

Since QCD is a gauge theory, the obvious question becomes is there a holographic description for a QCD like theory? Any model that attempts to mimic QCD should feature it’s two key attributes: deconfinement at high temperatures and confinement at low temperatures. Thus a holographic description of QCD, if it exists, should incorporate both the deconfined and confined phase while allowing us to study the thermodynamics near the critical region where phase transition occurs. Since QCD coupling is large near the critical temperature , the corresponding ’tHooft coupling is also large and we expect the holographic description to be most accurate near . While for , QCD coupling is small rendering pQCD techniques to be most reliable and we do not require a holographic description. In addition, lattice QCD simulations suggest that the conformal anomaly is largest near 1. Thus we should look for a holographic description near where the gauge coupling is large and the theory is highly non-conformal.

The AdS/CFT correspondence only considers conformal field theories, where there is no phase transition and no critical temperature . Black holes in , describing a thermal CFT can mimic large regime of QCD. But as already mentioned, at large we can simply use perturbative QCD to study the thermodynamics quite accurately. The regime near where pQCD breaks down is where holographic techniques can be most valuable. But since black holes in AdS space cannot describe a non-conformal theory, we must find a generalization of AdS/CFT correspondence to describe thermal gauge theories that undergo phase transitions.

Obtaining a geometric description of a gauge theory that resembles thermal QCD near phase transition is a formidable task. Before attempting to find a holographic map between QCD and gravity, we must first explore the general scope of gauge/gravity correspondence and understand how gauge theories with non-trivial Renormalization Group (RG) flows arise in string theory. In principal excitations of D branes placed in various geometries give rise to gauge theories. The fluxes and scalar fields sourced by the branes back-react and warp the geometry. This warped geometry is referred as the ‘dual geometry’. The fluxes and dilaton field in the dual geometry are used to obtain the RG flow of the gauge theory. While all the thermodynamic state functions of the gauge theory can be obtained by identifying the partition function of the dual geometry with that of the gauge theory.

Since QCD is non supersymmetric and non-conformal, the first objective is to find gauge theories with RG flows arising from D brane configurations with minimal SUSY. A great deal of progress has been made in that direction: In [20, 19] RG flows that connected conformal fixed points at IR and UV was incorporated, and [21] connected the UV conformal fixed point to a confining theory. But the model with QCD like RG flow and minimal supersymmetry is the Klebanov-Strassler (KS) model [22] which was obtained by considering IR modifications to Klebanov-Tseytlin (KT) model [23] (with an extension [24] to incorporate fundamental matter by considering D7 branes). However, at the highest energies the gauge theory is nothing like QCD: the effective degrees of freedom diverges and the gauge theory is best described in terms of bifundamental fields. Only at the lowest energies the gauge theory resembles SUSY QCD which confines. In the limit when effective brane charge is large, the gauge theory with large ’tHooft coupling has an equivalent description in terms of warped deformed cone. We can learn about this gauge theory which is very different from QCD in the UV, using the dual description .

In a series of papers [30]-[37], we proposed the general procedure to modify the UV dynamics of KS theory. In this paper we demonstrate how UV modifications are realized with an exact calculation of fluxes and metric as follows: We first consider world volume fluxes on D7 branes embedded in KS geometry with or without a black hole. These fluxes induce anti-D3 and anti-D5 charges such that the total effective charge vanishes, while the effective charge no longer diverges in the far UV. This way, the UV divergence of KS theory is removed. The resulting dual geometry takes the form of a warped deformed cone at small radial distances and far away from the tip of the cone. Thus, we have confinement at IR, dual to deformed cone at small radial distance and conformal gauge theory at UV, dual to AdS space.

It is worth mentioning the great deal of effort given in computing the black hole geometry in the presence of flux and scalar fields. For example in [25, 26, 27, 28] the cascading picture of the original KS model was extended to incorporate black-hole without any fundamental matter, while fundamental matter was accounted for in [29]. However, since these black holes are obtained in KS geometry, the dual gauge theories are UV divergent and quite distinct from QCD.

Additionally, most of the attempts are based on obtaining an effective lower dimensional action from KK reducing ten dimensional supergravity action. Dimensional reduction of a generic ten dimensional action can be quite challenging specially when there are non-trivial fluxes and scalar fields. It is also highly non-trivial to obtain a consistent truncation. Furthermore, it is not clear how RG flow of the dual gauge theory can be obtained since the fields in the effective action are not the dilaton or the flux in the original ten dimensional action.

In our approach, we directly work with the ten dimensional geometry, avoiding the difficulty of KK reduction. With the UV divergences of KS theory removed, we study the thermodynamics of the gauge theory by directly identifying the gauge theory partition function with that of the ten dimensional geometry. The thermal gauge theory arising from the brane excitations has a rich phase structure and as temperature is altered, we expect phase transitions. In this work, we make progress in that direction and obtain ten dimensional geometry (with or without a black hole) that arises from low energy limit of type IIB superstring theory including localized sources.

The role of localized sources is crucial in our analysis, since they allow us to modify the geometry at large radial distances. They also give rise to a radial scale and consequently an energy scale : Warped geometry in small region i.e. correspond to IR modes of the gauge theory while inclusion of large region i.e. correspond to including UV modes of the gauge theory. For any given temperature of the dual gauge theory, there are two geometries extremal (without black hole) and non-extremal (with black hole) but the geometry with lower on-shell action is preferred. At a critical temperature , both geometries are equally likely and we have a phase transition. We evaluate critical temperature using a perturbative analysis and depends on the boundary conditions as well as the scale . Thus the localized sources directly influence the thermodynamics of the gauge theory.

An alternative approach to construct gravitational description of gauge theories is to start with non-critical string theory and consider the resultant dual geometry [38]. For a QCD like gauge theory that confines in the IR and becomes free in the UV, one can obtain a five dimensional dual geometry [39]. The gravity action includes dilaton field and an effective potential for the dilaton. However, the geometry has large curvature and higher order terms in need to be included, which will modify the classical gravity action. By considering part of the higher order terms, one can find an effective dilaton potential which in turn can reproduce the QCD beta function. In a bottom up scenario, this effective potential is tuned to fit lattice QCD results for the conformal anomaly and the Polyakov loop. Since the potential is not derived directly from an underlying brane configuration, there is no guarantee that the geometry is in fact a holographic image of a gauge theory.

On the other hand, in our top down approach we proceed by first analyzing brane excitations in conifold geometries where the field theory has global and local symmetries common to that of QCD. At low energies, the excitations give rise to a four dimensional gauge theory which decouples from gravity and can be described holographically by the low energy limit of critical superstring theory i.e supergravity in ten dimensions. As we study a gauge theory arising from strings ending on branes, we know the field content of the theory and in some cases, the exact superpotential at zero temperature. Hence our top down approach is distinct from the bottom up models where a precise knowledge of the gauge theory is lacking or the gravity action is incomplete. To make meaningful quantitative comparisons with QCD, one must identify the gauge theory for which the dual gravity is being constructed. While in bottom up models this identification is not clear, in our top down approach, it is automatic. Thus the phenomenology that results from this gravity description can be directly compared to that of QCD as the gauge theory resembles large N QCD.

The paper is organized as follows: In section 2.1, we obtain exact values for type IIB fluxes in warped ten dimensional geometry in the presence of branes both in extremal (no black hole) and non-extremal (black hole) limit. The metric and fluxes are evaluated as a Taylor series in perturbative parameter where and are number of and branes in the dual gauge theory. For the metric, terms up to linear order are evaluated while the fluxes are obtained at zeroth order. In section 2.2 we propose a brane configuration that can source such fluxes and demonstrate how UV divergence of KS theory can be removed. Using the metric and flux, in section 3.1 the on-shell gravity action is exactly evaluated up to linear order and Hawking-Page like transition is analyzed. In section 3.2, the effect of localized source is incorporated and thermodynamic state functions are obtained. Finally in section 3.3 connections to QCD are established by considering small black holes in deformed cone geometry.

2 Gauge/String duality: From branes to geometry

As already mentioned in the introduction, holographic map between gauge theory and gravity can be constructed by studying excitations of branes placed in certain geometries. The gauge theory arises from open strings ending on the branes while the interactions between open and closed strings leave a holographic imprint of the gauge theory on the geometry. This imprint is captured by the warped dual geometry. At the lowest energies, open and closed string sector decouples and we are left with a gauge theory living in flat four dimensional space which can be described by the dual geometry.

The dual geometry has a classical action with fluxes and localized sources. The classical action is enough to describe the geometry, since the curvature will be small everywhere which in turn can be guaranteed by considering large . In the following section, we analyze this classical action and then in section 2.2, we will describe the brane configuration that can give rise to such geometry.

2.1 Geometry: Fluxes and localized sources in type IIB theory

Consider the type IIB action including number of coincident Dp branes in string frame2:

(1)

where , and , is the metric in string frame and . Here the action for a brane, upto quadratic order in flux is given by3

(2)

Here , is the pull back metric, , is NS-NS two form and is the RR flux. Also, is raised or lowered with the pullback metric in string frame.

The action (2.1) is complex due to the topological term . By taking the real part of the action and minimizing it, we can obtain the real valued fluxes , the metric and the scalar fields . We will consider the following real action

(3)

We can simplify equations resulting from variation of the above action by absorbing the scalar field in the definiton of the metric. This is done by going to the Einstein frame defined through , where the action (3) takes the following form

(4)
(5)

where , and is raised or lowered with the pullback metric in Einstein frame. The background warped metric takes the following familiar form

(6)

where the internal unwarped metric is given by with

(7)

Here is the metric of the base of deformed cone while is the perturbation due to the presence of fluxes and localized sources. Also is a constant, are one forms given by

(8)

and denote the internal ‘cone’ direction while run over Minkowski directions. The warp factor are functions of the cone coordinate . Observe that with a change of coordinates

(9)

for large , the metric becomes

(10)

which is the metric of regular cone with base . Thus only for small radial coordinate , the internal metric is a deformed cone while at large , we really have a regular cone with topology of .

We will now consider , that is we embed D7 branes in the large region where is the more convenient radial coordinate. Adding these sources in the large region means that we are only modifying the UV of the dual gauge theory and we expect that the IR of gauge theory remain mostly unaltered. The effect of this brane embedding for the gauge theory will be discussed in detail in section 2.2.

The D7 branes fill up Minkowski space , stretching along direction and filling up inside the . We consider two branches:

Branch I with parametrization . The brane fills up 4D Minkowski space and stretches along the direction, filling up an inside . It is a point inside and we pick a profile such that and is only a function of the . The DBI part of the world volume action for this branch takes the form

where , we have denoted world volume flux on branch I with and is the Hodge star with respect to the pullback metric of the branch. In obtaining the above action from (5), we have used the definition of tension of Dp brane, . Now observe that Branch I of D7 brane is a point on an with volume form

(12)

Thus the Chern-Simons action for Branch I can be written as

where is the pullback of the RR form to the world volume 4

(14)

Branch II with parametrization . Again the brane fills up 4D Minkowski space and stretches along the direction, fills up an inside but now is a point inside . Again we pick a profile such that and is only a function of the . The DBI part of the world volume action for this branch takes the form

where .

Branch
I - -
II - -

Observing that Branch II of D7 brane is a point on an with volume form

(16)

the Chern-Simons action for Branch II can be written as

where

(18)

Note that has no legs in the direction and thus the last term in both (2.1) and (2.1) do not contribute. Now varying the action (5) with respect to , we get the following equation

(19)

where or .

Note that , and where is some seven form. Then the action (where is given by (4)) to be stationary under variation and gives

where . In deriving the above equation from (4) and (5), we used the relation and .

Now observe that , where is some five form. Then for the action to be stationary under variation of gives the Bianchi identity

(21)

Now Lorentz invariance and self-duality fixes the five form to be

(22)

where is a scalar function. Variation of the action with respect to the space time metric gives the Einstein equations

(23)

where is the Ricci tensor for the metric and is defined through

(24)

We want to solve the flux equations (19),(2.1), (21), the Einstein equations (2.1) and simultaneously find the embedding that minimizes the action. But before we do so, observe that is closed form, i.e. . Then taking derivative of the first equation in (2.1) gives

(25)

which has the solution

(26)

On the other hand, the four form is given by

(27)

The scalar function can be obtained from (21), which is the Bianchi identity while the warp factor can be obtained from the Einstein equations. We now proceed as follows: The Ricci tensor in the Minkowski direction takes the following simple form

(28)

where and is the Christoffel symbol. Now using the ansatz (6) for the metric, (28) can be written as

(29)

where the Laplacian is defined as

(30)

The set of equations can be simplified by taking the trace of the first equation in (2.1) and using (2.1). Doing this we get

(31)

On the other hand using (2.1) in (2.1), one gets

(32)

which in turn would immediately imply

(33)

Now let’s look at the Bianchi identity. Using (22) in (21) gives

where is the Hodge star for the metric and

(35)

Now taking the trace of the first equation in (2.1), subtracting it from (2.1), we get

(36)

where . Now, we can ignore the localized term

(37)

which is in fact second or higher order in our perturbation, as we shall see in what follows. Solving (31), (33) and (2.1) together will give the scalar functions and . We can solve the system perturbatively, order by order in our perturbative parameter

(38)

where is a unit less constant defined through , being a 3-form and is the number of D7 branes.

Now writing , we get the following equation from , by varying with respect to

(39)

which leads to

(40)

Using this normalization of the axion field, we get following scaling of the dilaton field, using F-theory

(41)

where is some function describing the running of dilaton. We solve (31), (33) and (2.1) by only considering terms up to i.e. linear order in our perturbation. In the large region, by the switching to coordinate, we find following scaling of the solution with our perturbative parameter

(42)

It is instructive to note that at zeroth order in our perturbation, the above solution gives an AdS warp factor and a Schwarzchild black hole with horizon radius . When perturbation is included, the true horizon surface defined through the relation , would have radial location given by

(43)

Using (2.1) and the form of as given in (27), equation (19) can be solved with

(44)

where is the Hodge star for the metric .

Combining (26) and (2.1), we see that is self dual (or anti-self dual) while is closed at zeroth order in our perturbation. Thus it takes the following form

(45)

where are constants. The factor makes the flux self dual while the function is exactly chosen for closure. One can readily check that using as the internal metric, indeed the above flux satisfies closure and self duality.

The form in (2.1) also makes it clear that localized term that we ignored to obtain (2.1), are

(46)

Then viewing (2.1) as an equation for gives that the localized terms are of second order in our perturbation- which justifies ignoring them. On the other hand, solving the second equation in (2.1), one obtains that