Electric-Magnetic Duality and Supersymmetry in Four and Six Dimensions

Abstract: We study some aspects of the still very mysterious M-theory five-brane, mostly from the world-sheet field theory perspective. One aspect is the Seiberg-Witten curve in N = 2 four-dimensional super Yang-Mills theory. We also consider the six-dimensional theories which have an A-D-E classification where the A_N - 1 theories should correspond to the world-sheet theories on a stack of N parallel M five-branes. We will not have much to say about this very difficult problem. But already a single M five-brane presents some problems, and so we will also review this.