Branes and the Kraft-Procesi transition: classical case

TL;DR

This paper extends the brane realization of Kraft-Procesi transitions from A-type to orthogonal and symplectic Lie algebras using Type IIB superstring theory with O3-planes, linking mathematical orbit closures to physical models.

Contribution

It generalizes previous work by incorporating orthogonal and symplectic cases into the brane framework for Kraft-Procesi transitions, introducing O3-planes and new moduli space descriptions.

Findings

Brane realizations for orthogonal and symplectic Lie algebras established.

A physical interpretation of the mathematical 'collapse' map provided.

Kraft-Procesi transitions described via orthosymplectic quiver moduli spaces.

Abstract

Moduli spaces of a large set of $3d$ $\mathcal{N}=4$ effective gauge theories are known to be closures of nilpotent orbits. This set of theories has recently acquired a special status, due to Namikawa's theorem. As a consequence of this theorem, closures of nilpotent orbits are the simplest non-trivial moduli spaces that can be found in three dimensional theories with eight supercharges. In the early 80's mathematicians Hanspeter Kraft and Claudio Procesi characterized an inclusion relation between nilpotent orbit closures of the same classical Lie algebra. We recently showed a physical realization of their work in terms of the motion of D3-branes on the Type IIB superstring embedding of the effective gauge theories. This analysis is restricted to A-type Lie algebras. The present note expands our previous discussion to the remaining classical cases: orthogonal and symplectic algebras. In order to do so we introduce O3-planes in the superstring description. We also find a brane realization for the mathematical map between two partitions of the same integer number known as "collapse". Another result is that basic Kraft-Procesi transitions turn out to be described by the moduli space of orthosymplectic quivers with varying boundary conditions.