Simple singularities and N = 2 supersymmetric Yang-Mills theory

We present a first step towards generalizing the work of Seiberg and Witten on N = 2 supersymmetric Yang-Mills theory to arbitrary gauge groups. Specifically, we propose a particular sequence of hyperelliptic genus n - 1 Riemann surfaces to underly the quantum moduli space of W(n) N = 2 supersymmetric gauge theory. These curves have an obvious generalization to arbitrary simply laced gauge groups, which involves the A-D-E type simple singularities. To support our proposal, we argue that the monodromy in the semiclassical regime is correctly reproduced. We also give some remarks on a possible relation to string theory.