The relevant description of many phenomena in heavy ion physics is governed by four-dimensional gauge theories that - as string theory teaches us - admit a dual, holographic description in terms of five-dimensional Anti de Sitter space (AdS5). The canonical theory studied in this context is SU(N), N=4 four-dimensional gauge theory (CFT4), the most popular field theory among a very large portion of theoretical physicists in 2007.
Several heavy ion physicists and their string-theoretical colleagues have realized that this simple gravitational theory in AdS5 describes many observations at RHIC and elsewhere in terms of black holes and other objects from string theory remarkably accurately and the underlying mathematics is unexpectedly simple and natural. There exist contexts where the maximally supersymmetric theory gives misleading results but the common assumption is that they can be described by a different, less symmetric background of string theory.
Many physicists - string theorists as well as heavy ion physicists - actually study some points in the "landscape" of solutions of string theory. Please, don't get irritated. When I say "landscape", it doesn't yet mean that I defend the anthropic principle or something of this kind. ;-)
Superfluids and insulators
Prof Subir Sachdev, a condensed matter physicist, had a very natural, simple, yet ingenious idea. If string theory is taking over heavy ion physics, shouldn't we expect that it will take over parts of condensed matter physics, too? As they figured out in a paper with Herzog, Kovtun, and Son, the answer is probably Yes. Subir just gave an exciting talk whose title coincided with the title of their new preprint:
One half of the audience were condensed matter physicists, the other half were string theorists.
Instead of the AdS5/CFT4 duality employed in heavy ion physics, they study a system that uses the AdS4/CFT3 duality. Just like string/M-theory contains many backgrounds whose geometry is a product of AdS5 and a compact space, it also predicts the existence of many backgrounds whose geometry is AdS4 times something.
The canonical and most symmetric example in this case is the N=8 superconformal field theory in 2+1 dimensions that is dual to the "AdS4 x S7" near-horizon geometry of M2-branes in 11-dimensional M-theory. The 2+1-dimensional theory can be constructed as the long-distance limit of the 2+1-dimensional maximally supersymmetric gauge theory describing D2-branes in type IIA string theory. The long-distance limit is equivalent to a strong coupling limit of the type IIA string theory - something that leads us to M-theory with a new, eleventh dimension and that transforms D2-branes to M2-branes. The R-symmetry is promoted from SO(7) to SO(8): the additional eighth adjoint scalar arises from the Hodge dual of the electromagnetic field. The enhancement of the R-symmetry follows from supersymmetry but it is much easier to understand it via the string/M-theoretical realization of the field theories.
This CFT with 16 supercharges and 16 additional superconformal generators plays the same important role for the stringy analysis of 2+1-dimensional systems such as those in condensed matter physics as the N=4 gauge theory in 3+1 dimensions played for situations such as heavy ion physics. It is the most symmetric, most easily calculable, and most well-established example of a theory in this universality class. Other theories or string theory backgrounds are less symmetric, less calculable, but they still share the same properties. At least qualitatively.
Time to get to real physics
So what are the physical systems from the real world that Subir et al. want to investigate? They want to look at models analogous to the boson Hubbard model in 2+1 dimensions, a canonical textbook material for condensed matter physicists. This system exhibits a very interesting phase transition between an insulator and a superfluid.
Take a thin foil of bismut whose thickness is X. Keep X fixed but send the temperature T to zero. What will happen with the conductivity? It turns out that for a vanishing temperature, the conductivity goes either to zero or to infinity. There must exist a critical thickness in between these two regimes where the conductivity remains finite. As you could expect, such phase transitions are controlled by a conformal field theory. The class of conformal field theories that you can obtain in these situations are called the Wilson-Fisher fixed points.
In the condensed matter approach, the 2+1-dimensional superconformal theory describing M2-branes is a fancy generalization of the Wilson-Fisher fixed point.
The relevant quantity that you want to measure or compute is a correlator of two conserved currents "J_mu^a(x,y,t)". Take the retarded correlator and find its Fourier transform. You will obtain a function of the frequency and the wavenumber that depends on four indices. The dependence on the Lorentz vector indices "mu,nu" splits the result into a transverse tensor and a longitudinal tensor. Schematically,
- [ J J ] = KL(omega,k) LongiTensor + KT(omega,k) TransTensor
- KL(omega,k) KT(omega,k) = N^3 / 72 pi^2
This self-duality has been viewed as a consequence of a particle-vortex duality in the past. However, Sachdev et al. derive it much more quantitatively and naturally from the AdS/CFT correspondence. The duality between the longitudinal modes and the transverse mode can be reduced to an S-duality in a limit of a dual gauge theory in the bulk.
Recall that the dual bulk description involves an SO(8) R-symmetry in the bulk arising from the isometry of the seven-sphere if we view it as a four-dimensional theory defined on AdS4. If the theory is weakly coupled, it is very close to a U(1)^{28} Abelian gauge theory - because 28 is the dimension of SO(8) - and one can apply four-dimensional S-duality on all these U(1) factors. This symmetry manifests itself as the "holographic self-duality", as they call it, or a particle-vortex duality in the three-dimensional boundary CFT description.
Although the actual two-spatial-dimensional materials they could construct are a bit different, so far there have been no really convenient and natural methods to calculate quantities in the hydrodynamical limit of these 2+1-dimensional theories. The dual gravitational description controlled by a large non-evaporating AdS4 Schwarzschild black hole seems to do the job very well: it kind of trivializes the situation.
Andy Strominger and myself were kind of surprised by Subir's statement that he could have obtained exactly Lorentz-invariant interacting theories in the infrared by adjusting one parameter only in a non-relativistic theory with particles and holes. Some of us have asked the question whether it is possible to create materials in the lab that would behave as a supersymmetric conformal theory in 2+1 dimensions. The answer was essentially No.
Nevertheless, it is becoming clear that the mathematical concepts and manipulations that were first discovered in the context of very high energy physics are recycled at many places of Nature, as Joe Polchinski puts it. It is a good opportunity for every physicist (and other scholar) to see whether
they can improve their understanding by applying a well-established method of string theory - such as the AdS/CFT correspondence - to their subject. The scientists who encounter conformal, scale-invariant theories are the first ones who should try to look what string theory can teach them.
And that's the memo.
