N=2 gauge theories and degenerate fields of Toda theory

TL;DR

This work establishes a precise link between degenerate level-1 null states of the ${\mathcal{W}}_N$ algebra and punctures in Gaiotto's SU(N) theories, showing that puncture data are classified by Young diagrams with N boxes. It demonstrates that the Seiberg-Witten differentials near a puncture reproduce the semiclassical behavior of ${\mathcal{W}}_N$ generators, and it provides a concrete mapping between gauge-theory mass parameters and Toda momenta, including a proposed real-subspace integration for physical states. The results generalize known SU(3) correspondences to general N, relate the puncture structure to null-state conditions in the Toda/CFT framework, and lay groundwork for matching Nekrasov partition functions and conformal blocks via ${\mathcal{W}}_N$ data. These findings deepen the connection between 4D N=2 gauge theories, 2D Toda/CFT, and the Hitchin/Gaiotto setup, with potential extensions to matrix-model/dynamical systems interpretations.

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.