1. Applications of Gauge/Gravity Duality
The holographic principle relates a gravity theory to a quantum field theory (QFT) in one lesser dimension. A prototype example of this correspondence is the AdS/CFT duality, which is a conjectured equivalence between a gravity theory on Anti-de Sitter (AdS) space and a strongly coupled conformal field theory (CFT) in one lesser dimension. In some sense, the correspondence suggests that the emergent degrees of freedom of the strongly coupled field theory are described by a gravitational theory. This can be taken as a “definition” of quantum gravity given the lack of alternate definitions.
The original suggestion for the correspondence was based on studies of D-branes in flat space. In the low energy limit, one obtains an equivalence between a supersymmetric gauge theory that describes the light open string modes describing the world-volume theory on the branes and a gravity theory in AdS. The gravity theory becomes classical in the limit where the number of colors (in the gauge theory) becomes infinite. In this limit, it is possible to compute observables in the strongly coupled gauge theory by studying classical (or semi-classical) gravity. This feature is pleasing - the correspondence maps intractable computations in the field theory to simple computations in classical gravity. For example, the AdS/CFT correspondence (also known as gauge/gravity correspondence) maps the problem of computing transport coefficients to solving linear wave equations in a black hole background. It is a highly non-trivial task to obtain transport quantities of a strongly interacting many body system using conventional field theoretic techniques.
More generally, we would like to exploit the power of the holographic principle to understand features of strongly interacting systems, such as high-Tc superconductors and QCD at finite density, that are not well understood using conventional techniques. It seems important to develop new ideas to understand the concepts underlying such many body phenomena. Holography seems to provide fertile grounds for harvesting new ideas about strongly interacting theories. This motivates a study of CFTs with holographic description (holographic CFTs). String theory has a plethora of AdS vacuua and each AdS solution is dual to some CFT. This landscape of CFTs resembles the landscape of CFTs that describe the dynamics near quantum critical points arising in quantum phase transitions. The prospect of studying many body physics using holography makes the existence of the string landscape a pleasant feature.
1.1. Non-relativistic Holography
Many quantum critical points arising in condensed matter systems are described by non-relativistic theories. Scale invariance can be realized in non-relativistic theories in many ways. One freedom is the relative scale dimension of time and space, called the dynamical exponent z. Non-relativistic systems with z=2 scaling symmetry that are also Galilean invariant respect a symmetry algebra known as the Schrödinger algebra. Cold fermions at unitarity is an interesting exmaple of a system that can be described a non-relativistic CFT with z=2. One is lead to the following question: “Is it possible to find gravity duals of such non-relativistic systems that are realizable in a laboratory?” In [1] we showed that the symmetries of a Galilean invariant CFT can be realized geometrically as the isometries of a higher dimensional spacetime (now known as Schrödinger spacetime and the ). An unusual feature of this duality is that the bulk geometry has two extra dimensions than the CFT, instead of the usual one. The additional direction is a compact direction and shift symmetry along this direction corresponds to the particle number transformation. This solution can be embedded in string theory by performing a set of operations (known as the Null-Melvin twist) on AdS5 × S5 solution of type IIB supergravity [2]. This method also provides a way of finding a black hole solution which has asymptotic Schrödinger symmetries. The field theory dual of these gravity solutions happens to be a modified version of DLCQ (Discrete Light Cone Quantization) N = 4 super Yang-Mills theory. The thermodynamics of these theories is very different from that of cold atoms. This happens to be a consequence of realizing the entire Schrödinger group as isometries of the spacetime. We gave an example of a holographic realization in which the particle number symmetry is realized as a bulk gauge symmetry [3]. In this proposal, the Schrödinger algebra is realized in the bulk without the introduction of an additional compact direction. Using this proposal, we found a solution that describes a confining ground state of a non-relativistic theory at finite density. We computed the conductivity of this system and showed that it is zero for all non-zero frequencies. However, the DC conductivity of this system is infinite. It would be nice to find asymptotically AdS solutions which describe confining ground states of a relativistic theory at finite density.
In the above examples, we described gravity duals of theories that are both Galilean invariant and scale invariant. There are non-relativistic theories that are scale-invariant but not boost invariant. These theories also known as Lifshitz theories, do not have a particle number symmetry unlike the boost invariant NRCFTs. Gravity duals that realize Lifshitz scaling symmetries as isometries were found in [4]. Some difficulties in embedding these solutions in string theory were pointed out in [5]. In [6], we found explicit solutions of 10D and 11D supergravity theories with Lifshitz isometries. We showed that Lifshitz geometries with z=2 can be constructed by breaking null translation symmetry of geometries with Schrödinger symmetries. In [7], we argued that the field theory dual of a solution (of type IIB supergravity) with asymptotic Lifshitz symmetry is a non-abelian version of Lifshitz-Chern-Simon (LCS) gauge theory. In [8], we found an analytical solution for a black hole that asymptotes to vacuum Lifshitz solution. We computed the scalar response function by solving the scalar wave equation in this background. The scalar equation in this background turned out to be exactly solvable. We showed that the finite temperature correlators in the interacting theories do not exhibit ultra-local behavior which was observed in free Lifshitz theories.
1.2. Numerical Methods for Holography
Very often one is interested in understanding the behavior of a system when it is pushed out of equilibrium. It is possible to use linear response theory to analyze systems that are near-equilibrium or when the driving is small. What happens when the system is pushed far from equilibrium? When the departure from equilibrium happens at long times or length scales large compared to mean free path, hydrodynamic description becomes useful. In a hydrodynamic description, each point in space-time is treated as if it were in local equilibrium. One is interested in knowing when such a hydrodynamic description ceases to be a valid description. In many situations, hydrodynamics is not a good description. For instance, it is not clear if hydrodynamics is a valid description when you drive a turbulent state at time scales and length scales much shorter than the mean free path. In compressible flows, smooth initial conditions can evolve into shocks. In the absence of viscosity, the shock thickness is zero and it is a discontinuity in the flow. In the presence of viscous forces, this discontinuity is smeared out into a shock with thickness of the order of mean-free path. So, it is quite unlikely that first order hydrodynamics is a valid description within the shock, but it is interesting to know if hydrodynamics with higher order gradient corrections is sufficient to describe the physics within the shock. In order to understand the breakdown of hydrodynamic description it is essential to understand the non-equilibrium behavior of the underlying microscopic description. At present, it seems difficult to address such questions using traditional tools. Gauge/gravity duality seems like a promising tool for addressing questions encompassing non-equilibrium phenomena. This duality maps non-equilibrium dynamics of a quantum field theory to classical gravitational dynamics in negatively curved spacetime.
Using gauge/gravity duality, we studied the dynamics of a 2+1 dimensional conformal field theory when it is pushed away from equilibrium by a metric source [9]. Translation invariance is broken when the metric perturbation is space dependent. When the metric fluctuation is small it can be shown that the momentum of the system damps to zero at an exponential rate. In Drude's model, the electrical conductivity of a material is linearly proportional to the momentum relaxation time scale. Gravity duals of systems with continuous translation invariance at finite density exhibit infinite DC conductivity. This is a consequence of momentum relaxation time scale being infinite. In order to have a finite momentum relaxation rate, translation invariance must be broken. Translation invariance can be broken in many ways - for instance, by a scalar operator, an ionic lattice or by metric fluctuations. In [9], we assume that translation invariance is broken by the background metric. We developed a numerical scheme to solve Einstein's equations in the presence of negative cosmological constant and a spacetime dependent boundary metric. In this scheme, the spatial derivatives are discretized using pseudospectral methods. We used a combination of 3rd order Runge-Kutta method and Adams-Bashforth's method for integrating the null-evolution equations. We used some exact analytical results to test the robustness and accuracy of our numerical scheme. In the future, we plan to use our code for studying more intricate problems involving shock waves and turbulent eddies etc.
1.3. Comments
Though the holographic models are somewhat far from describing realistic condensed matter systems, it has spurred a lot of studies on gravitational instabilities. As a result of these studies a numerous instabilities of charged black holes have been identified. At this point, the implications of these instabilities for quantum gravity is not clear. The precise comparison between the experimental results and holographic theories may not be possible now, but developments in the future might yield fruitful results. As mentioned earlier, these studies help in visualizing various non-perturbative features that could arise in many body systems. The hope is that some of these features might become more transparent from experimental or theoretical studies. Before the era of holography, physicists searched for new phenomena and answers to old puzzles under the light of perturbative QFT-this was the only lamp post known till the development of the holographic principle. We have an additional lamp-post now - the holographic lamp post! We might find answers to some of the old puzzles or find new interesting phenomena by searching under this new lamp post. At the least, we can get rid of the boredom of searching under one single lamp post!
References
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D. T. Son “ Towards an AdS/cold atoms correspondence: a geometric realization of the Schrödinger symmetry ”, Phys. Rev D. 78 046003 (2008).
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C. P. Herzog, M. Rangamani, S. F. Ross “ Heating up Galilean holography ”, JHEP 11 080 (2008).
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[3] K. Balasubramanian, J. McGreevy “ The particle number in Galilean holography ”, JHEP 01 137 (2011).
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[5] W. Li, T. Nishioka, T. Takayanagi “ Some No-go Theorems for String Duals of Non-relativistic Lifshitz-like Theories ”, JHEP 10 015 (2009).
[6] K. Balasubramanian, K. Narayan “ Lifshitz spacetimes from AdS null and cosmological solutions ”, JHEP 08 014 (2010).
[7] K. Balasubramanian, J. McGreevy “ String theory duals of Lifshitz-Chern-Simons gauge theories ”, Class. Quant. Grav. 29 194007 (2012).
[8] K. Balasubramanian, J. McGreevy “ An analytic Lifshitz black hole ”, Phys. Rev. D. 80 104039 (2009).
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