^{3}.

The holographic renormalization of a charged black brane with or without a dilaton field, whose dual field theory describes a dense medium at finite temperature, is investigated in this paper. In a dense medium, two different thermodynamic descriptions are possible due to an additional conserved charge. These two different thermodynamic ensembles are classified by the asymptotic boundary condition of the bulk gauge field. It is also shown that in the holographic renormalization regularity of all bulk fields can reproduce consistent thermodynamic quantities and that the Bekenstein-Hawking entropy is nothing but the renormalized thermal entropy of the dual field theory. Furthermore, we find that the Reissner-Nordström AdS black brane is dual to a theory with conformal matter as expected, whereas a charged black brane with a nontrivial dilaton profile is mapped to a theory with nonconformal matter although its leading asymptotic geometry still remains as AdS space.

The AdS/CFT correspondence [

In general, the partition function of a quantum field theory suffers from the UV divergence which should be renormalized by adding appropriate counter terms. This is also true in the dual gravity where the corresponding IR divergence is caused due to infinite volume of the background geometry. In [

In holography, a conformal field theory matches to asymptotic AdS geometry. Introducing matter may break the conformal symmetry depending on the properties of matter. In the strong coupling regime, it is hard to investigate such matter effects through the traditional quantum field theory method. Although many things still remain to be clarified, recent numerous works based on the AdS/CFT correspondence provided some clues for understanding a strongly interacting system. In the holographic set-up, matter of the dual CFT can be realized as a field in AdS space. In particular, a massless vector field in AdS is mapped to quark or hadronic number operator in the deconfining or confining phase, respectively [

The holographic renormalization of the

The rest of paper is organized as follows. In Section

Let us first take into account a 5-dimensional RNAdS black brane. A charged black brane solution usually has two hairs, charge, and mass, so that its thermodynamics can be represented as two different thermodynamic systems depending on the choice of the fundamental thermodynamic variables [

For the direct comparison with the black brane thermodynamics, we regard an Euclidean Einstein-Maxwell action with a negative cosmological constant. To obtain the Euclidean action from the Lorentzian one in (

The requirement of the regular metric at the horizon generally leads to specific Euclidean time periodicity

Physically, it is quite natural to assume that all fields are regular in the entire space. If not so, the theory becomes singular. In the Einstein-Maxwell theory, there exist two bulk fields, metric and vector fields. At the horizon, the absence of the conical singularity in the metric determines the time periodicity. The Hawking temperature is represented as the inverse of the time periodicity:

In general, the gravity action without additional boundary terms is not well defined because its variation becomes problematic at the boundary. Therefore, the following two boundary terms should be added:

The useful and crucial relation in the AdS/CFT correspondence is that the on-shell gravity action is proportional to the free energy (or the generating functional) of the boundary (or dual) field theory:

The correct counter terms should satisfy the following prescriptions.

The on-shell action should be finite.

The variation of the on-shell action with respect to the leading boundary value of all bulk fields should also be finite.

As mentioned before, the first prescription corresponds to finiteness of the free energy. The second implies the physical quantities of the dual theory should be also finite. In order to understand this, it should be noticed that, in general, the bulk fields denoted by

In the charged black brane geometry, the regularity conditions at the horizon in (

Let us first impose the Dirichlet boundary condition on the gauge field, which fixes the chemical potential corresponding to the boundary value of the gauge field. Then, the dual theory is described by the grand potential of a grand canonical ensemble. In this case, since the variation of the renormalized action with respect to the gauge field is well defined, an additional boundary term is not needed. When the Neumann boundary condition instead of the Dirichlet boundary condition is imposed, this is not true anymore as will be shown in the next section. The resulting renormalized action with the Dirichlet boundary condition becomes

From the renormalized action, one can easily read the thermodynamic quantities by evaluating the boundary stress tensor. At the UV fixed point (

Note that the chemical potential

Before concluding this section, it is worth to note that the trace of the boundary stress tensor vanishes. Since the second terms in (

As shown in [

One can also take into account another deformed charged black brane which still has an asymptotic AdS geometry. One of the known examples is the deformed RNAdS black brane with two different Liouville scalar potentials, where the scalar field was identified with a dilaton because it controls the physical coupling [

With the Lorentzian signature, the action describing the deformed charged black brane is given by

The absence of a conical singularity at the horizon gives rise to Hawking temperature which is associated with the horizon

With the Euclidean signature, the action for the grand canonical ensemble is

For

In order to describe the canonical ensemble instead of the grand canonical ensemble, we should impose the Neumann boundary condition on the vector field which requires one more boundary term

We have studied the holographic renormalization of the charged black branes whose asymptotic geometry is given by an AdS space. In the general renormalization scheme of the field theory, finding correct counter terms is very important because they cancel the UV divergences and at same time provide some finite contributions. Therefore, the renormalized physical quantities usually depend on the finite contributions of the counter terms. This is also true in the holographic renormalization. Following the AdS/CFT correspondence and using the holographic renormalization, in this paper, we have investigated the physical quantities of the dual field theory with matter. As mentioned before, for obtaining the correct physical results one should take into account the correct counter terms. In the charged black brane with an asymptotic AdS geometry, the bulk gauge field dual to matter does not require new divergent terms in the on-shell gravity action so that one need to introduce more counter terms except one used in the Schwarzschild (neutral) AdS black brane. Although there are no additional counter terms, the change of the metric caused by the bulk gauge field leads to a new finite contribution to the on-shell gravity action which depends on the physical quantities of matter, the chemical potential, or number density.

In the RNAdS black brane case, the boundary stress tensor generated by the on-shell gravity action becomes traceless. As expected usually, the dual matter does not break the conformal symmetry of the asymptotic AdS geometry. Furthermore, the expectation value of the number operator shows that the Fermi surface energy at zero temperature is proportional to

In the RNAdS black brane, there is no nontrivial dilaton field associated with the running coupling constant of the dual field theory. We further regarded the charged black brane in the Einstein-Maxwell-dilaton theory. The asymptotic geometry of it is also given by an AdS space, which is the leading part of the metric, and the next to leading part gives rise to the nontrivial finite contribution to both the on-shell gravity action and the boundary stress tensor. Unlike the RNAdS black brane, the boundary stress tensor of the charged dilatonic black brane is not traceless which implies that the dual matter breaks the conformal symmetry although the leading part of the asymptotic metric is not changed.

Finally, we also showed that the holographic renormalization results can be reinterpreted as two different ways depending on the asymptotic boundary condition of the gauge field. These results are exactly matched with the thermodynamic laws of the charged black brane geometries that we considered. All above results are derived at the UV fixed point. So it is interesting to investigate the renormalizaiton group flow of them.

Here, we will explain the thermodynamics of the RNAdS black brane in detail for the comparison with the holographic renormalization results. The action for the RNAdS black brane with a Lorentzian signature is given by

Let us first take into account the case with the Neumann boundary condition. At the asymptotic boundary, the Neumann boundary condition for the gauge field is given by

In the canonical ensemble, the most important thermodynamic function is the free energy

Now, let us consider the Dirichlet boundary condition instead of the Neumann boundary condition, which is given at the asymptotic boundary by

The author declares that there is no conflict of interests regarding the publication of this paper.

The author would like to thank Yun Soo Myung and Mu-in Park for valuable discussion. This work has been supported by the WCU Grant no. R32-10130 and the Research Fund no. 1-2008-2935-001-2 by Ewha Womans University. The author was also supported by Basic Science Research Program through the National Research Foundation of Korea (NRF) funded by the Ministry of Education (Grant no. NRF-2013R1A1A2A10057490).

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