The Small $E_8$ Instanton and the Kraft Procesi Transition

One of the simplest $(1,0)$ supersymmetric theories in six dimensions lives on the world volume of one M5 brane at a $D$ type singularity $\mathbb{C}^2/D_k$. The low energy theory is given by an SQCD theory with $Sp(k-4)$ gauge group, a precise number of $2k$ flavors which is anomaly free, and a scale which is set by the inverse gauge coupling. The Higgs branch at finite coupling $\mathcal{H}_f$ is a closure of a nilpotent orbit of $D_{2k}$ and develops many more flat directions as the inverse gauge coupling is set to zero (violating a standard lore that wrongly claims the Higgs branch remains classical). The quaternionic dimension grows by $29$ for any $k$ and the Higgs branch stops being a closure of a nilpotent orbit for $k>4$, with an exception of $k=4$ where it becomes $\overline{{\rm min}_{E_8}}$, the closure of the minimal nilpotent orbit of $E_8$, thus having a rare phenomenon of flavor symmetry enhancement in six dimensions. Geometrically, the natural inclusion of $\mathcal{H}_f \subset \mathcal{H}_{\infty}$ fits into the Brieskorn Slodowy theory of transverse slices, and the transverse slice is computed to be $\overline{{\rm min}_{E_8}}$ for any $k>3$. This is identified with the well known small $E_8$ instanton transition where 1 tensor multiplet is traded with 29 hypermultiplets, thus giving a physical interpretation to the geometric theory. By the analogy with the classical case, we call this the Kraft Procesi transition.