N=4 supersymmetric Yang-Mills theory

Montonen-Olive duality

The first kind of S-duality relationship involved this very N=4 supersymmetric Yang-Mills theory. S-duality involves a large coupling constant and a weak coupling constant. The coupling constant determines the strength of a physical theory. For example, in string theory, the coupling constant is determined by the probability of strings splitting and reconnecting. Claus Montonen and David I. Olive, proposed that the N=4 supersymmetric Yang-Mills theory with a coupling constant g, is equivalent to the same N=4 supersymmetric Yang-Mills theory with the coupling constant 1/g. This is the oldest example of this kind of a Strong-weak duality.

AdS/CFT

In the AdS/CFT correspondence, in it's initial proposal by Juan Maldacena in 1997, there is a correspondence between a bulk theory of quantum gravity with a boundary conformal field theory. The initial proposal was that the N=4 supersymmetric Yang-Mills theory lives on the 4-dimensional boundary to type IIB string theory with 5 extended and 5 compact dimensions. This boundary theory, the N=4 supersymmetric Yang-Mills theory, has some features in common with Quantum chromodynamics, which describes how hadrons are quarks confined by gluons. For example, a baryon, is a hadron composed of 3 quarks. A proton is 2 up quarks and 1 down quark. A neutron is 2 down quarks and an 1 up quark. A meson is not a baryon, however, a hadron composed of 1 quark and 1 antiquark.