Branes and the Kraft-Procesi Transition

TL;DR

The paper develops a brane-based realization of Kraft–Procesi transitions between closures of nilpotent orbits, tying physical Higgsing processes to mathematical stratifications of hyperkähler singularities. It builds on Namikawa’s theorem to argue that closures of nilpotent orbits provide the simplest nontrivial Higgs and Coulomb branches for 3d $\mathcal N=4$ theories and shows how minimal singularities govern the transitions. A novel matrix formalism encodes brane data and enables efficient computation of the entire KP transition network across $\frak{sl}_N$ cases, including explicit quivers and mirrors. These results yield a concrete, algorithmic bridge between brane dynamics, hyperkähler geometry, and the Kraft–Procesi poset, with potential extensions to other Lie algebras and periodic brane configurations.

Abstract

The Coulomb and Higgs branches of certain 3d N=4 gauge theories can be understood as closures of nilpotent orbits. Recently, a new theorem by Namikawa suggests that this is the simplest possible case, thus giving this class a special role. In this note we use branes to reproduce the mathematical work by Kraft and Procesi. It studies the classification of all nilpotent orbits for classical groups and it characterizes an inclusion relation via minimal singularities. We show how these minimal singularities arise naturally in the Type IIB superstring embedding of the 3d A-type theories. The Higgs mechanism can be used to remove the minimal singularity, corresponding to a transition in the brane configuration that induces a new effective 3d theory. This reproduces the Kraft-Procesi results, endowing the family of gauge theories with a new underlying structure. We provide an efficient procedure for computing such brane transitions.

Figures (37)

• Figure 1: Hasse diagram with the partial order of all closures of nilpotent orbits of the algebra $\mathfrak{sl}_2$. The numbers $dim$ refer to the quaternionic dimension of the variety.
• Figure 2: In this picture vertical lines correspond to NS5-branes. The vertical direction corresponds to directions $\vec{m}$, spanned by the NS5-branes. The horizontal direction corresponds to $x^6$, so the different positions of $t_i$ of the two NS5-branes along this direction are evidenced in this way. The third axis, perpendicular to the paper, would correspond to directions $\vec{w}$, in this case both NS5-branes are in the picture since their $\vec{w}_i$ position coincides and a D3-brane with the same position $\vec{y}=\vec{w}_i$ can be stretched between them.
• Figure 3: In this picture the dashed vertical lines correspond to D5-branes. The vertical direction corresponds to directions $\vec{w}$, spanned by the D5-branes. The horizontal direction corresponds to $x^6$, so the different positions of $z_i$ of the two D5-branes along this direction are evidenced in this way. The third axis, perpendicular to the paper, would correspond to directions $\vec{m}$, in this case both D5-branes are in the picture since their $\vec{m}_i$ position coincides and a D3-brane with the same position $\vec{x}=\vec{m}_i$ can be stretched between them.
• Figure 4: In the phase depicted in this figure the two NS5-branes share the same position $\vec{w}$ along directions $\{x^7,x^8,x^9\}$, this makes the existence of a Couolomb branch possible. The two crosses correspond to two D5-branes stretching along the perpendicular direction to the paper, which corresponds to $\{x^7,x^8,x^9\}$. They share the same position $\vec{m}$ along directions $\{x^3,x^4,x^5\}$, which in the diagram is represented by the vertical direction, this makes the existence of a Higgs branch possible. In the special point of the moduli where the D3-brane $\vec{x}$ position coincides with the position $\vec{m}$ of the D5-branes the two hyper multiplets become massless due to fundamental strings of length zero stretching between the D5-branes and the D3-brane.
• Figure 5: The phase depicted in this picture corresponds to the Higgs branch of the $3d\ \mathcal{N}=4$ SQED theory with two flavours. The vertical dashed lines correspond to D5-branes stretching along directions $\{x^7,x^8,x^9\}$. The circled crosses correspond to NS5-branes stretching along the directions $\{x^3,x^4,x^5\}$.