or its relativistic analog. The viscosity term in this equation produces the damping of excitations, and it is connected with the waves falling into the black hole. Computing the viscosity in the particle theory from first principles is very difficult. However, it is very simple from the point of view of Einstein’s equations, because it is given by a purely geometric quantity: the area of the black hole horizon. This has been used to qualitatively model strongly interacting systems of quantum particles. These range from the quark-gluon fluids that are produced by heavy ion collisions (at the Relativistic Heavy Ion Collider at Brookhaven National Laboratory or at the Large Hadron Collider at Geneva) to high-temperature superconductors in condensed matter physics.
Many examples of AdS/CFT involve additional structures, such as supersymmetry. In these cases the geometry obeys special constraints, giving rise to Sasaki-Einstein spaces, which are closely related to Calabi-Yau spaces. This is merely an example of a more general trend developing connections between matrix models, algebraic curves, and supersymmetric quantum field theory.
A recent development of the past decade has been the discovery of integrability in N = 4 super-Yang-Mills. This four-dimensional quantum field theory is the most symmetric quantum field theory. The study of this highly symmetrical example is very useful since it will probably enable us to find some underlying structures common to all quantum gauge theories. Integrability implies the existence of an infinite-dimensional set of symmetries in the limit of a large number of colors. In this regime the particles of the theory, or gluons, form a sort of necklace. The symmetry acts on these states and allows us to compute their energies exactly as a function of the coupling. The deep underlying mathematical structures are only starting to be understood. Integrability in so-called (1 + 1)-dimensional systems has led to the development of quantum groups and other interesting mathematics. The way integrability appears here is somewhat different, and it is quite likely that it will lead to new mathematics. A closely related area is the computation of scattering amplitudes in this theory. A direct approach using standard methods, such as Feynman diagrams, quickly becomes very complicated. On the other hand, there are new methods showing that the actual answers are extremely simple and have a rich structure that is associated with the mathematics of Grassmanians. This has led to another fruitful collaboration.15
The connection between theoretical physics and mathematics is growing ever stronger, and it is supported by the emergence of interdisciplinary centers, such as the Simons Center for Geometry and Physics at Stony
15 See, for example, A.B. Goncharov, M. Spradlin, C. Vergu, and A. Volovich, 2010, Classical polylogarithms for amplitudes and Wilson loops. Physical Review Letters 105:151605.