Branes and the Kraft-Procesi Transition

TL;DR

This paper uses brane configurations in string theory to reproduce and extend mathematical classifications of nilpotent orbits and their transitions in 3d N=4 gauge theories, revealing new structural insights.

Contribution

It demonstrates how brane setups can reproduce Kraft-Procesi classifications and introduces an efficient method to compute brane transitions corresponding to nilpotent orbit inclusions.

Findings

Reproduces Kraft-Procesi classification using branes

Identifies minimal singularities in Type IIB brane setups

Provides a procedure to compute brane transitions

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.