| We analyze some systematics of the coupling constant dependence of correlators in N = 4 Yang-Mills, which is the world-volume theory on D3-branes. We use the fact that the operator Ot that generates infinitesimal changes of the coupling constant in this theory sits in the same supermultiplet as the superconformal currents. We show how superconformal current Ward identities determine a class of terms in the operator product expansion of Ot with any other operator. In certain cases, this leads to constraints on the coupling dependence of correlation functions in N = 4 Yang-Mills. As an application, we demonstrate the exact non-renormalization of two and certain three-point correlation functions of BPS operators. We next approximate these integrated correlators by using a truncated OPE expansion. This leads to differential equations for the coupling dependence. When applied to a particular sixteen point correlator, the coupling dependence we find agrees with the corresponding amplitude computed via the AdS/CFT correspondence. We conjecture that this truncation becomes exact in the large N and large 't Hooft coupling limit. A geometric description of electric-magnetic duality in N = 4 Yang-Mills leads naturally to M5-branes in M theory. The self-dual string soliton in M theory given by M2-branes ending on M5-branes, gives the magnetic monopole in N = 4 Yang-Mills, given by D1-branes ending on D3-branes. We propose an equation that describes these M2-branes ending on M5-branes, and which generalizes the description of the D1--D3 system via Nahm's equation. The simplest solution to this equation constructs the transverse geometry in terms of a fuzzy three sphere. We show that the solution passes a number of consistency checks including a calculation of the energy of the system, matching to the self-dual string solution in the M5-brane world-volume, and a study of simple fluctuations about the ground state configuration. We write down certain terms in the effective action of multiple membranes, which includes a sextic scalar coupling. |
