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I'm afraid this question is a little too open-ended, but bear with me while I find a better formulation.

carbon allotropes (like fullerenes and graphene) are regular patterned. Conduction bands of electrons in graphene in particular, have energy-momentum relationships that make them behave as massless carriers (sticking to what is called a Dirac cone). More interestingly the propagation dynamics seem to be at first order, direction independent.

Given the above facts, is possible that the interactions of carriers on such 2D sheet is describable by the limit of some Conformal Field Theory. If that were to be the case, and I have a big closed surface of graphene material, what does the holographic principle will say about the fields in the volume bounded by the surface? What is the equivalent of the Gravitational Theory in that volume?

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You are right with your assumption - the special behaviour at the Dirac cone allows for an application of the holographic principle. But how is this possible?

It turns out that since in this region of the band structure the Fermi velocity is very large, i.e. two orders of magnitude below the speed of light, graphene behaves effectively as a relativistic fluid, which can be shown to be strongly coupled due to the large value of the fine structure constant. In this context, one speaks of a (nearly) quantum critical liquid, as can be deduced from the renormalization group flow of the coupling. This behaviour allows for a treatment of the system in terms of relativistic hydrodynamics.

This is where the holographic principle kicks in: in the context of the so-called fluid-gravity correspondence, the AdS-CFT duality allows for a description of a strongly coupled relativistic fluid in terms of a higher-dimensonal gravitational theory. In this sense, the fluid "lives" on the boundary of a space on which a theory of quantum gravity (string theory) in its weak-coupling limit can be formulated. See these lecture notes for a review of this special case of the Maldacena duality. Note that the duality is not applied to graphene directly, but to a similar framework which exhibits features of graphene. This allows for the calculation of universal quantities such as transport coefficients characterizing relativistic fluid.

Now what about the duality? What does the correspondence tell us about the "other side"?

The theory on the boundary is a conformal field theory (CFT) in $2+1$ dimensions, which represents the relativistic fluid near the quantum critical point. According to the string-theoretical realization of the holographic principle, this corresponds to a $3+1$ dimensional Anti-de Sitter (AdS) space, a geometry of negative curvature. The dynamics of the boundary system are captured by propagating gravitational waves: their analysis aids in calculating properties of the fluid, like the famous result for the lower bound of the ratio of shear viscosity and entropy density. While gravitational degrees of freedom describe dynamics, thermodynamic properties of the fluid are captured holographically by the presence of a black hole. Temperature and entropy of the fluid are mapped to the equivalent quantities of the black hole.

So what do we have now? A space of negative curvature, a thermal black hole and string theory (even if strings do not appear as degrees of freedom due to the weak-coupling limit). Does this mean when we construct a closed surface, a "graphene ball", we have to worry about negative curvature, black holes and strings?

No, it does not. The AdS/CFT correspondence is a purely mathematical tool which allows one to map a problem that is difficult or not solvable to a physical system of a higher dimension, in the hope that one can solve it there (e.g. strong coupling to weak coupling). It does not imply anything about the "physical bulk" in real world corresponding to a physical system living on a $2D$ boundary of any kind. In the case of graphene, the problem happens to be two(plus one)-dimensional, this raises the illusion that the correspondence might show us some relation to the volume encompassed by a closed surface, but it does not. The original form of the conjecture related a four-dimensional spacetime to something even higher, a five dimensional AdS space. It made its way to particle physics, nowadays one applies the concept to quantum chromodynamics in the context of the so-called AdS/QCD duality. One attempts to circumvent the breakdown of perturbation series due to strong coupling by mapping the problem to a space of higher dimension and calculating relevant quantities there. But this does not mean that there is any physical relevance to this mapping, that there are any implications for "real world physics" (like measuring the higher dimensions).

If you are interested in applications of AdS/CFT to condensed matter physics, you might consider reading this review.

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  • $\begingroup$ I have the hope that both Gauss theorem and Stokes theorem are somehow limits of complex symmetries represented by AdS/CFT. I agree that the duality is merely mathematical, but maybe there is some way to make it physical? $\endgroup$ – lurscher Mar 13 '14 at 23:19
  • $\begingroup$ For instance; what if the bulk volume inside the graphene closed surface would made of a substance with some effective field described by the limit of some gravitational theory? (I'm thinking in fluid dynamics analogs, for example). Are you saying that under no conditions would the charge carriers in the boundary couple with the dynamics in the bulk fluid such that the whole dynamics would be approximately described by AdS/CFT? $\endgroup$ – lurscher Mar 13 '14 at 23:23
  • $\begingroup$ Basically I'm interested in some analog computing mechanism to evaluate the map between the bulk states and the surface states $\endgroup$ – lurscher Mar 13 '14 at 23:24

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