Introduction to Holographic Superconductors

These lectures give an introduction to the theory of holographic superconductors. These are superconductors that have a dual gravitational description using gauge/gravity duality. After introducing a suitable gravitational theory, we discuss its properties in various regimes: the probe limit, the effects of backreaction, the zero temperature limit, and the addition of magnetic fields. Using the gauge/gravity dictionary, these properties reproduce many of the standard features of superconductors. Some familiarity with gauge/gravity duality is assumed. A list of open problems is included at the end.Comment: 34 pages, 10 figures, to appear in the proceedings of the 5th Aegean Summer School, "From Gravity to Thermal Gauge Theories: the AdS/CFT Correspondence"; v2: references adde

Stability of the non-extremal enhancon solution I: perturbation equations

We consider the stability of the two branches of non-extremal enhancon solutions. We argue that one would expect a transition between the two branches at some value of the non-extremality, which should manifest itself in some instability. We study small perturbations of these solutions, constructing a sufficiently general ansatz for linearised perturbations of the non-extremal solutions, and show that the linearised equations are consistent. We show that the simplest kind of perturbation does not lead to any instability. We reduce the problem of studying the more general spherically symmetric perturbation to solving a set of three coupled second-order differential equations.Comment: 20 pages, 1 figure, references added, typos fixed, version to appear in PR

Non Uniform Black Strings and Critical Dimensions in AdSdAdS_d

We study the equations of black strings in spacetimes of arbitrary dimensions with a negative cosmological constant and construct numerically non uniform black strings solutions. Our results suggest the existence of a localised black hole in asymptotically locally $AdS$ spacetime. We also present evidences for a dependence of the critical dimension on the horizon radius.The critical dimension represents the dimension where the order of the phase transition between uniform and non uniform black string changes. Finally, we argue that both, the regular asymptotically locally $AdS$ solution and $AdS$ black string solutions with a very small horizon radius, present a negative tension. This turns out to be an unexpected feature of the solutions.Comment: 13 pages, 5 figure

The holographic superconductors in higher-dimensional AdS soliton

We explore the behaviors of the holographic superconductors at zero temperature for a charged scalar field coupled to a Maxwell field in higher-dimensional AdS soliton spacetime via analytical way. In the probe limit, we obtain the critical chemical potentials increase linearly as a total dimension $d$ grows up. We find that the critical exponent for condensation operator is obtained as 1/2 independently of $d$, and the charge density is linearly related to the chemical potential near the critical point. Furthermore, we consider a slightly generalized setup the Einstein-Power-Maxwell field theory, and find that the critical exponent for condensation operator is given as $1/(4-2n)$ in terms of a power parameter $n$ of the Power-Maxwell field, and the charge density is proportional to the chemical potential to the power of $1/(2-n)$.Comment: LaTeX, 16 pages, 5 figures, typos corrected, one reference added, version to appear in European Physical Journal

Can we live on a D-brane? -- Effective theory on a self-gravitating D-brane --

We consider a D-brane coupled with gravity in type IIB supergravity on S^5 and derive the effective theory on the D-brane in two different ways, that is, holographic and geometrical projection methods. We find that the effective equations on the brane obtained by these methods coincide. The theory on the D-brane described by the Born-Infeld action is not like Einstein-Maxwell theory in the lower order of the gradient expansion, i.e., the Maxwell field does not appear in the theory. Thus the careful analysis and statement for cosmology on self-gravitating D-brane should be demanded in realistic models.Comment: 13 pages, accepted for publication in Physical Review

Hilbert Lattice Equations

There are five known classes of lattice equations that hold in every infinite dimensional Hilbert space underlying quantum systems: generalised orthoarguesian, Mayet's E_A, Godowski, Mayet-Godowski, and Mayet's E equations. We obtain a result which opens a possibility that the first two classes coincide. We devise new algorithms to generate Mayet-Godowski equations that allow us to prove that the fourth class properly includes the third. An open problem related to the last class is answered. Finally, we show some new results on the Godowski lattices characterising the third class of equations.Comment: 24 pages, 3 figure

On a general class of brane-world black holes

We use the general solution to the trace of the 4-dimensional Einstein equations for static, spherically symmetric configurations as a basis for finding a general class of black hole (BH) metrics, containing one arbitrary function $g_{tt} = A(r)$ which vanishes at some $r = r_h > 0$, the horizon radius. Under certain reasonable restrictions, BH metrics are found with or without matter and, depending on the boundary conditions, can be asymptotically flat or have any other prescribed large $r$ behaviour. It is shown that this procedure generically leads to families of solutions unifying non-extremal globally regular BHs with a Kerr-like global structure, extremal BHs and symmetric wormholes. Horizons in space-times with zero scalar curvature are shown to be either simple or double. The same is generically true for horizons inside a matter distribution, but in special cases there can be horizons of any order. A few simple examples are discussed. A natural application of the above results is the brane world concept, in which the trace of the 4D gravity equations is the only unambiguous equation for the 4D metric, and its solutions can be continued into the 5D bulk according to the embedding theorems.Comment: 9 pages, revtex

The effect of higher derivative correction on η/s\eta /s and conductivities in STU model

In this paper we study the ratio of shear viscosity to entropy, electrical and thermal conductivities for the R-charged black hole in STU model. We generalize previous works to the case of a black hole with three different charges. Actually we use diffusion constant to obtain ratio of shear viscosity to entropy. By applying the thermodynamical stability we recover previous results. Also we investigate the effect of higher derivative corrections.Comment: revised versio

Holographic Cosmic Quintessence on Dilatonic Brane World

Recently quintessence is proposed to explain the observation data of supernova indicating a time-varying cosmological constant and accelerating universe. Inspired by this and its mysterious origin, we look for the possibility of quintessence as the holographic dark matters dominated in the late time in the brane world scenarios. We consider both the cases of static and moving brane in a dilaton gravity background. For the static brane we use the Hamilton-Jacobi method motivated by holographic renormalization group to study the intrinsic FRW cosmology on the brane and find out the constraint on the bulk potential for the quintessence. This constraint requires a negative slowly varying bulk potential which implies an anti-de Sitter-like bulk geometry and could be possibly realized from the higher dimensional supergravities or string theory. We find the similar constraint for the moving brane cases and that the quintessence on it has the effect as a mildly time-varying Newton constant.Comment: 16pages, no figure, Latex; revised version, references added, typos corrected, abstract and comments improved; final version, will appear in PR

Conformal aspects of Palatini approach in Extended Theories of Gravity

The debate on the physical relevance of conformal transformations can be faced by taking the Palatini approach into account to gravitational theories. We show that conformal transformations are not only a mathematical tool to disentangle gravitational and matter degrees of freedom (passing from the Jordan frame to the Einstein frame) but they acquire a physical meaning considering the bi-metric structure of Palatini approach which allows to distinguish between spacetime structure and geodesic structure. Examples of higher-order and non-minimally coupled theories are worked out and relevant cosmological solutions in Einstein frame and Jordan frames are discussed showing that also the interpretation of cosmological observations can drastically change depending on the adopted frame