# N=2 gauge theories and degenerate fields of Toda theory

## TL;DR

This paper explores the link between degenerate fields in W_N algebra and punctures in Seiberg-Witten curves, revealing a classification scheme and translating gauge theory parameters into Toda theory variables.

## Contribution

It establishes a classification of degenerate W_N fields by Young diagrams and connects gauge theory mass parameters to Toda theory momenta.

## Key findings

- Degenerate fields with null states at level one are classified by Young diagrams with N boxes.
- The singular behavior of Seiberg-Witten curves matches W_N generator behavior.
- Mass parameters in gauge theories correspond to Toda theory momenta.

## Abstract

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.