We review a construction of holographic geometries dual to SYM theory on a Friedmann-Robertson-Walker background and in the presence or absence of a gluon condensate and instanton density. We find the most general solution with arbitrary scale factor and show that it is diffeomorphic to topological black holes. We introduce a time-dependent boundary cosmological constant and show energy-momentum conservation in this background. For constant , the deconfinement properties of the temporal Wilson loop are analysed. In most cases the Wilson loop confines throughout cosmological evolution. However, there is an exceptional case which shows a transition from deconfinement at early times to confinement at late times. We classify the presence or absence of horizons, with important implications for the Wilson loop.
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Holographic Cosmological Deconfinement]Holographic Cosmological Backgrounds, Wilson Loop (De)confinement and Dilaton Singularities
J. Erdmenger]Johanna Erdmenger111E-mail: firstname.lastname@example.org K. Ghoroku]Kazuo Ghoroku222E-mail: email@example.com R. Meyer]René Meyer333Corresponding author and conference speaker, E-mail: firstname.lastname@example.org, Phone: +30 2810 39 4233, Fax: +30 2810 39 4274. The work of RM was in part supported by the European grants FP7-REGPOT-2008-1: CreteHEPCosmo-228644 and PERG07-GA-2010-268246, as well as the EU Program “Thalis” ESF/NSRF 2007-2013. RM gratefully acknowledges the support and hospitality of the Korean Institute for Advanced Study (KIAS), of the Center for Quantum Spacetime at Sogang University (Seoul) and of the Isaac Newton Institute for Mathematical Sciences during the final stages of this work. IP wishes to thank the University of Crete for the hospitality, where part of this work was carried out. I. Papadimitriou]Ioannis Papadimitriou444E-mail: Ioannis.Papadimitriou@csic.es CCTP-2012-04, IFT-UAM/CSIC-12-26, KIAS-P12021
1 Background and Holographic Stress Energy Tensor
In AdS/CFT, the holographically defined theory is only sensitive to the conformal class of the boundary metric, not the metric itself.555In the presence of a conformal anomaly observables depend on the chosen representative of the conformal class, but transform in a well-defined way, governed by the anomaly, into each other . It is hence possible to put any holographically defined flat space field theory on or via an appropriate boundary conformal transformation, amounting to particular bulk diffeomorphisms. The same holds for any metric of Friedmann-Robertson-Walker (FRW) type,
It is hence conceivable that holographic duals for field theories living on an FRW universe can be constructed in such a way. Such a construction from 5D Einstein-Hilbert theory has been put forward in , describing a universal sector of any holographic theory with a stress-energy tensor.666For more details on the effective holographic approach to model building, cf. . In the following we review such a construction  based on the background of . This supersymmetric background describes SYM theory in a vacuum of constant (anti) self-dual instanton density, and is known to lead to Wilson loop confinement. We will first review the derivation of the cosmological background presented in , and then show its diffeomorphism equivalence to the static geometry of . In this course we will find a more general class of solutions including the one of , clarify issues regarding the implications of holographic stress energy tensor conservation, comment on the meaning of and the boundary cosmological constant from a bulk perspective, and classify the horizons appearing in the cosmological solution. In sec. 2 we then classify the (de)confinement properties of the Wilson loop in this background, and compare to the AdS-Schwarzschild case.
Ten-dimensional type IIB supergravity truncates to the metric , dilaton , axion and five form ,
for some constant . Under this Ansatz the metric decouples from the axion-dilaton,
The main observation of  is that the LHS of (10) is a function of only, depending on the integration function . Eq. (10) thus can be solved by introducing a (a priori time-dependent) boundary cosmological constant ,
The LHS of (10) becomes Friedmann’s equation, while the RHS can be integrated to yield as in eq. (31) of , with given by (8). Note that in this treatment the boundary cosmological constant , and hence the cosmological evolution , is essentially arbitrary: From the point of view of the bulk is, upon requiring an FRW boundary metric, a single free integration function which is not determined in the model itself. Dynamical determination of by coupling to boundary gravity or some other mechanism may be an interesting direction of future research. In the remainder of this paper we however choose to be time-independent in order to investigate the properties of the dual gauge theory in the usual cosmological backgrounds (dS/flat/AdS). It should also be noted that only for a time-independent the solution for in eq. (33) of  coincides with (9).
Holographic Stress-Energy-Tensor In the above construction, the time-dependence of was a priori unconstrained. Using the results of App. B of , the vacuum expectation value of the boundary stress-energy tensor can be found for a general , but its form is complicated and not very illuminating. For a time-independent cosmological constant it greatly simplifies to the following ideal fluid form
The form of (12) is interesting by itself: The part proportional to the dark radiation constant is actually conformal, and dilutes with , as relativistic radiation in an expanding FRW universe does. We will see below that this is not a coincidence: In the frame where (6) becomes the usual static topological black hole, is the black hole mass parameter, and contributes to (12) relativistic radiation in an expanding universe. The cosmological constant part on the other hand is not conformal, and originates from the conformal anomaly of the SYM fields on the curved FRW space-time . We note that the holographic stress energy tensor as e.g. calculated from eq. (B19) of  is covariantly conserved by construction for arbitrary time-dependent . Stress energy conservation hence does not restrict to be time-independent, as originally claimed in .
Equivalence with Static Topological Black Holes We now show that the solutions obtained above are in fact diffeomorphism equivalent to the usual static form of topological black holes in . Having explicitly constructed the function , we now consider a diffeomorphism of the form , with to be specified. Implementing such a diffeomorphism the metric (6) becomes
where . Absence of a mixed term requires that there exists a function such that
The function therefore exists iff there exists a function that satisfies the integrability condition
where . In the frame the metric is the static topological black hole
Although the existence of such a diffeomorphism has been discussed before , to our knowledge this is the first time that it is constructed explicitly. In particular, this shows that the dark radiation constant is the black hole mass parameter. This is in accordance with the form of the stress energy tensor (12), where the dark radiation contribution is conformal and dilutes in the same way as relativistic radiation ().
The Dilaton Solution Assuming spatial homogeneity the dilaton equation in the frame reads
The general static solution in these coordinates is
where and are arbitrary constants, related to the asymptotic value of the dilaton and the asymptotic D(-1) charge (which has to be positive for positive ) via
Some special time dependence can be taken into account by allowing and to be linear functions of . The general time-dependent solution of (18) can be constructed by Fourier transforming the dilaton in and solving numerically the resulting second order linear ODE.
A few comments are in order here. First, the solution obtained above in the coordinate system contains very non-trivial time dependence when transformed back into the coordinates. This
is to be contrasted with the approximate expression given in eq. (41) of , where it is assumed that
varies very slowly in time so that it can be taken to be a constant. Of course, this approximation
becomes exact in the case and . Secondly, unless , the dilaton diverges logarithmically at the static black hole horizon corresponding to the real root of , namely , which is nothing but the dilaton singularity of Liu and Tseytlin . However, as we will discuss below, the horizons in the and coordinate systems do not coincide. We will discuss the implications of the dilaton singularity after an analysis of the horizons below.
Classification of Horizons In both and coordinate systems the horizons are time dependent curves, but in fact, these curves do not coincide in the two coordinate systems. The (naive) horizons in the coordinates are defined by the condition , which yields
However, the horizons of the static black hole (17) are located at the roots of . For and there is only one real root , which in the coordinates translates to the curves
Figure 1 shows plots of these horizons for all allowed values of the constants and .
2 Wilson Loops and the Quark-Antiquark Potential
The main result of  is the calculation of the static quark-antiquark potential from a fundamental string hanging from the boundary into the bulk space-time given by (6), at any fixed time and as a function of proper distance , measured in the FRW metric (1). Following the standard procedure of calculating the embedding of a Nambu-Goto string, we found its tension
The quark-antiquark potential scales with the distance if has a finite minimum at some distance outside the horizon at a given . We now evaluate the Wilson loop tension from the general result (23) for the dilaton solution given in (19), making use of additional information on the presence of an event horizon with diverging dilaton . By plotting (23) numerically777A Mathematica notebook can be found in the supplementary material to this ArXiv submission., we find that whenever is the outermost horizon (i.e. the horizon with largest value of ) for a given “time” (all cases except and in fig. 1), diverges to due to the dilaton divergence at the horizon, and hence the Wilson loop confines.888As the tension goes . In the absence of the event horizon the Wilson loop can deconfine whenever the naive horizon at can be reached, as , being the case for in fig. 1 for a interval . If neither or horizons are present, as for e.g. , at large the Wilson loop would confine again as the tension is finite at , where space ends in these coordinates. The cases and are special: In the former case the dilaton diverges at , but the Wilson loop deconfines as this divergence is cancelled in (23) as (c.f. eq. (8)) approaches zero at the same location. The latter case, , shows a Wilson loop confinement-deconfinement transition: During the time interval ,
the Wilson loop will reach the naive horizon (i.e. the horizon given by ), , and exhibit perimeter law behaviour. For on the other hand the Wilson loop will reach the event horizon and exhibit area law behaviour, due to the dilaton singularity. We summarize the situation in table 1. We also note that the case does not allow for deconfinement transitions, despite the presence of the horizon, since cosmological evolution requires such that the Wilson loop will not reach the naive horizon. In conclusion, except for the cases and , the Wilson loop is always confined during the cosmological evolution in the model of . For the Wilson loop is always deconfined, and for we find a transition from a deconfined state at early times to a confined state at later times.
|C , D||C , D||C , D|
|D , D||C , D|
|D: , C: , D|
We revisited and extended the holographic backgrounds of  dual to SYM on an FRW type cosmology and in the presence of a gluon condensate and instanton density, following the supersymmetric Ansatz of . We found the most general solution of form (6) and introduced a (arbitrarily time-dependent) boundary cosmological constant , which is equivalent to choosing an integration function in the bulk, the scale factor . We showed that the holographic stress-energy tensor is conserved for any , clarifying conflicting claims between  and e.g. . Furthermore, we found that these backgrounds are diffeomorphic to static black holes, whose event horizons become time-dependent in the cosmological frame, corresponding to a time-dependent observer viewing the static black hole event horizon. In the Liu-Tseytlin model  the dilaton diverges at the event horizon, covering the naive horizon in most cases and in this way confining the Wilson loop. In two notable exceptions, and , the Wilson loop can reach the naive horizon and deconfine. For this gives rise to a confinement-deconfinement transition. On the other hand, in the absence of the gluon condensate, i.e. in a cosmological slicing of AdS/Schwarzschild with a constant dilaton, the Wilson loop is expected to break on the event horizon and to deconfine. Both cases are summarized in table 1.
We conclude with several possibilities for future research: First, it would be interesting to construct other models with Wilson loop (de)confinement transitions (ideally at , ), and to apply the construction used here to other cases such as . In such a model the free energy, another measure of (de)confinement, may also be calculated from holographic renormalization and linked to the Wilson loop behaviour. Finally, a further issue is the possible resolution of the dilaton singularity of .
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