The Small $E_8$ Instanton and the Kraft Procesi Transition

TL;DR

The paper analyzes 6d $\mathcal{N}=(1,0)$ theories on M5-branes at $D_k$ singularities, focusing on the Higgs branch structure at finite and infinite coupling. Using brane constructions, 3d mirror symmetry, and Brieskorn–Slodowy transverse-slice theory, it shows a 29-dimensional jump in the Higgs branch at infinite coupling, identified with the small $E_8$ instanton (Kraft–Procesi) transition, and provides a quiver-based realization of this transition. For a single M5, the infinite-coupling Higgs branch is generated by $M$ in the adjoint of $SO(4k)$ and a new spinor $S$ with $SU(2)_R$ spin $(k-2)/2$, with an explicit HWG, while for $N$ M5-branes a 3d quiver description of the infinite-coupling Higgs branch is proposed and related to compactifications to 4d and 3d. These results illuminate the interplay between nilpotent-orbit geometry, transverse slices, and quiver mutations in 6d SCFTs and point toward generalizations to other singularities and higher-rank configurations.

Abstract

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.