N=2 gauge theories and degenerate fields of Toda theory

We discuss the correspondence between degenerate fields of the W_N algebra and punctures of Gaiotto’s description of the Seiberg-Witten curve of N=2 superconformal gauge theories. Namely, we find that the type of degenerate fields of the W_N algebra, with null states at level one, is classified by Young diagrams with N boxes, and that the singular behavior of the Seiberg-Witten curve near the puncture agrees with that of W_N generators. We also find how to translate mass parameters of the gauge theory to the momenta of the Toda theory.