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Branes and the Kraft-Procesi transition: classical case

Santiago Cabrera, Amihay Hanany

TL;DR

This work extends Kraft-Procesi transitions from sl(n) to the full set of classical Lie algebras by embedding Type IIB brane configurations with orientifold O3-planes, thereby constructing orthosymplectic quivers whose Higgs branches are closures of nilpotent orbits. It develops the partition-duality machinery (transpose, B/C/D-collapse, d_{LS}, d_{BV}) and translates Brieskorn-Slodowy slices into explicit brane moves, S-dualities, and quiver readouts. The authors provide a comprehensive matrix formalism that encodes brane data and systematizes KP transitions across so(2n), so(2n+1), and sp(n) sectors, including D_n, A_{2n-1}, and A_{2n-1}∪A_{2n-1} singularities and their minimal counterparts. They also illustrate the framework with detailed SO(4) and SO(5) examples, derive Hasse diagrams for the corresponding orbit closures, and discuss T_{ρ}(G) theories and mirror-symmetric aspects within orthosymplectic settings. The results unify geometric invariant theory with brane dynamics, offering a practical toolkit for identifying slices and transverse structures in moduli spaces of 3d N=4 theories and suggesting directions for refined Hilbert-series and complete mirror descriptions.

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.

Branes and the Kraft-Procesi transition: classical case

TL;DR

This work extends Kraft-Procesi transitions from sl(n) to the full set of classical Lie algebras by embedding Type IIB brane configurations with orientifold O3-planes, thereby constructing orthosymplectic quivers whose Higgs branches are closures of nilpotent orbits. It develops the partition-duality machinery (transpose, B/C/D-collapse, d_{LS}, d_{BV}) and translates Brieskorn-Slodowy slices into explicit brane moves, S-dualities, and quiver readouts. The authors provide a comprehensive matrix formalism that encodes brane data and systematizes KP transitions across so(2n), so(2n+1), and sp(n) sectors, including D_n, A_{2n-1}, and A_{2n-1}∪A_{2n-1} singularities and their minimal counterparts. They also illustrate the framework with detailed SO(4) and SO(5) examples, derive Hasse diagrams for the corresponding orbit closures, and discuss T_{ρ}(G) theories and mirror-symmetric aspects within orthosymplectic settings. The results unify geometric invariant theory with brane dynamics, offering a practical toolkit for identifying slices and transverse structures in moduli spaces of 3d N=4 theories and suggesting directions for refined Hilbert-series and complete mirror descriptions.

Abstract

Moduli spaces of a large set of 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.

Paper Structure

This paper contains 99 sections, 128 equations, 75 figures, 24 tables.

Table of Contents

  1. Introduction
  2. Summary of the brane description of Kraft-Procesi transitions for $\mathfrak{sl}$(n)
  3. Brane systems and quivers for unitary theories
  4. Brane systems and quivers for surface and minimal singularities
  5. The Kraft-Procesi transition
  6. Example
  7. General definition
  8. Orietifold planes and orthosymplectic quivers
  9. Nilpotent orbits of $\mathfrak{sp}(n)$ and $\mathfrak{so}(n)$
  10. Partitions and the dual map
  11. Transpose map
  12. X-collapse
  13. Lusztig-Spaltenstein map
  14. Barbasch-Vogan map
  15. Comment about notation
  16. ...and 84 more sections

Figures (75)

  • Figure 1: Generic brane configuration. The horizontal direction corresponds to spatial direction $x^6$. Horizontal lines are D3-branes. The vertical direction corresponds to spatial directions $\vec{x}=(x^3,x^4,x^5)$. Vertical lines are NS5-branes. The direction perpendicular to the paper corresponds to spatial directions $\vec{y}=(x^7,x^8,x^9)$. Crosses are D5-branes.
  • Figure 2: Quiver corresponding to brane configuration depicted in figure \ref{['fig:firstExample']}. The circles are gauge nodes while the squares are flavor nodes.
  • Figure 3: Model $3d$$\mathcal{N}=4$ SQED with $N$ flavors. (a) Quiver. (b) Coulomb branch brane configuration. There are $N$ D5-branes. (c) Higgs branch brane configuration. In (c) a rotation with respect to (b) has been performed: the vertical direction corresponds with spatial directions $\vec{y}$ and the direction perpendicular to the paper corresponds with spatial directions $\vec{x}$. Vertical dashed lines now represent D5-branes (there are $N$ of them) and circled crosses represent NS5-branes.
  • Figure 4: Mirror dual of $3d$$\mathcal{N}=4$ SQED with N flavors. (a) Quiver. There are $N-1$ gauge nodes, with label $n_i=1$. (b) Coulomb branch brane configuration. There is a total of $N$ NS5-branes. (c) Higgs branch brane configuration.
  • Figure 5: Self-dual quiver theory. The Higgs and the Coulomb branch of this theory are both $\bar{\mathcal{O}} _{(n)}$, i.e. the closure of the maximal nilpotent orbit of $\mathfrak{g}=\mathfrak{sl}(n)$.
  • ...and 70 more figures