### The String Theory Connection of RHIC

The idea that superstring theory may help inform the physics of relativistic heavy ion collisions begins with Maldacena’s remarkable discovery in 1997 that the (*N* = 4) supersymmetric Yang-Mills theory in (3+1)-dimensional space-time and 10-dimensional superstring theory in a geometrical background composed of the product of 5-dimensional Anti-deSitter space (AdS) and a 5-dimensional sphere describe the same physics. Since the (*N* = 4) supersymmetric Yang-Mills is a conformal field theory (CFT), this isomorphism is commonly known as the *AdS/CFT duality*. What makes this duality especially useful is that the weak-coupling of the string theory, when it reduces to classical supergravity in the AdS background geometry is dual to the strong-coupling limit of the quantum field theory. Since classical supergravity is a theory one knows how to solve – it is simply a set of nonlinear partial differential equations – the duality renders the problem of strongly coupled super-Yang-Mills theory, which was formerly completely intractable, a solvable theory. The duality, which is strictly speaking still an conjecture, has been investigated and confirmed in many studies. One especially interesting realization is obtained when one adds a black hole to the AdS geometry. In this case, as first discussed by Witten, the string theory is dual to the thermal super-Yang-Mills theory with the Hawking temperature of the black hole. Remarkably, as Gubser *et al.* showed, the energy density of the thermal plasma at infinitely strong coupling is only reduced by 25% from its free value.

The reason why this somewhat esoteric development in theoretical physics became of interest to RHIC physics is that Policastro, Son and Starinets found in 2001 that the shear viscosity η, relative to the entropy density *s*, of the super-Yang Mills theory at strong coupling takes on the value η/*s* = \hbar/4π, which was later conjectured to be the lower bound for this quantity compatible with the principles of quantum mechanics and relativity. In the gravity dual the KSS bound has the simple interpretation as the absorption cross section of a graviton by the black hole, relative to the area of the horizon of the black hole. If the strongly coupled super-Yang-Mills theory is a good model for QCD near the critical temperature, then this result would provide an explanation for the small shear viscosity of the matter produced at RHIC, which is indicated by the large collective (elliptic) flow observed in the RHIC experiments. Simplified introductions into the idea underlying the AdS/CFT duality and its application to relativistic heavy ion collisions can be found in

Independent of the possible phenomenological connections to RHIC physics, the AdS/CFT duality has considerable merit by opening up a theoretical venue for rigorous studies of the predictions of a strongly coupled gauge theory, which has been widely exploited in recent years to investigate questions relevant to RHIC physics. Examples are calculations of the energy loss of a fast colored particle:

- Calculating the jet quenching parameter from AdS/CFT
- Energy loss of a heavy quark moving through
*N*= 4 supersymmetric Yang-Mills plasma

the response of the thermal medium to the passage of a fast colored particle, revealing the formation of a sonic Mach cone:

- The stress tensor of a quark moving through
*N*= 4 thermal plasma - The wake of a quark moving through a strongly-coupled
*N*= 4 supersymmetric Yang-Mills plasma - Sonic booms and diffusion wakes generated by a heavy quark in thermal AdS/CFT

screening of the color force between a heavy quark-antiquark pair:

and the propagation of a localized (“hard”) color-singlet disturbance:

A most intriguing application of the AdS/CFT duality is to the hydrodynamic expansion of the hot matter. Using a solution of AdS space with an embedded cooling black hole Janik and Peschanski constructed a solution of the 5-dimensional Einstein equations corresponding to Bjorken’s boost-invariant expansion in Minkowski space, which includes all interactions within the thermal plasma. By comparing this solution to the hydrodynamic expansion in the Bjorken scenario, it is possible to obtain exact results for the transport coefficients of the interacting matter in the strong-coupling limit, including the shear viscosity and relevant relaxation times. Whether or not these results are directly applicable to RHIC, it is noteworthy that they have helped to elucidate the correct form of relativistic viscous hydrodynamics, which was previously unknown:

The great amount of activity dedicated to study observables related to RHIC physics using the AdS/CFT duality begs the question whether the strongly coupled super-Yang-Mills theory is a good model for QCD near the critical temperature. QCD is not conformal, but has an intrinsic scale, and lattice-QCD calculations have shown that the deviations from conformality are large near the critical temperature. One unrealistic consequence of the conformal symmetry of the super-Yang-Mills theory is that the energy loss of a heavy fast colored particle grows linearly with energy and not logarithmically, as it is believed to be the case in QCD. Another qualitative difference to QCD is that the super-Yang-Mills theory does not exhibit confinement at zero temperature, even at strong coupling. This problem can, to some extent, be circumvented by modifying the geometry of the 5-dimensional AdS space:

A specific proposal of how such a scheme may be applied to describe a relativistic heavy ion collision is outlined in

However, it is important to recall that the correct gravity dual of QCD, if it exists, is not known, and thus all present models of this kind are schematic, at best.