From SYM Perturbation Theory to Closed Strings in Matrix Theory

For the purpose of better understanding the AdS/CFT correspondence it is useful to have a description of the theory for all values of the 't Hooft coupling, and for all $N$. We discuss such a description in the framework of Matrix theory for SYM on D4-branes, which is given in terms of quantum mechanics on the moduli space of solutions of the Nahm equations. This description reduces to both SYM perturbation theory and to closed string perturbation theory, each in its appropriate regime of validity, suggesting a way of directly relating the variables in the two descriptions. For example, it shows explicitly how holes in the world-sheets of the 't Hooft expansion close to give closed surfaces.