One can associate an invariant to a large class of regular codimension two defects of the six dimensional $(0,2)$ SCFT $mathscr{X}[j]$ using the classical Springer correspondence. Such an association allows a simple description of S-duality of associ
ated Gaiotto-Witten boundary conditions in $mathcal{N}=4$ SYM for arbitrary gauge group and by extension, a determination of certain local aspects of class $mathcal{S}$ constructions. I point out that the problem of textit{classifying} the corresponding boundary conditions in $mathcal{N}=4$ SYM is intimately tied to possible symmetry breaking patterns in the bulk theory. Using the Springer correspondence and the representation theory of Weyl groups, I construct a pair of functors between the class of boundary conditions in the theory in the phase with broken gauge symmetry and those in the phase with unbroken gauge symmetry.
Codimension two defects of the $(0,2)$ six dimensional theory $mathscr{X}[mathfrak{j}]$ have played an important role in the understanding of dualities for certain $mathcal{N}=2$ SCFTs in four dimensions. These defects are typically understood by the
ir behaviour under various dimensional reduction schemes. In their various guises, the defects admit partial descriptions in terms of singularities of Hitchin systems, Nahm boundary conditions or Toda operators. Here, a uniform dictionary between these descriptions is given for a large class of such defects in $mathscr{X}[mathfrak{j}], mathfrak{j} in A,D,E$.
The role played by the Euler anomaly in the dictionary relating sphere partition functions of four dimensional theories of class $mathcal{S}$ and two dimensional nonrational CFTs is clarified. On the two dimensional side, this involves a careful trea
tment of scale factors in Liouville/Toda correlators. Using ideas from tinkertoy constructions for Gaiotto duality, a framework is proposed for evaluating these scale factors. The representation theory of Weyl groups plays a critical role in this framework.
