Kraft Quivers & KP Transitions in 3d N=4

• Kraft quivers are 3d N=4 quiver gauge theories whose Higgs branches are closures of nilpotent orbits in classical Lie algebras, constructed via Type IIB brane setups.
• Kraft–Procesi transitions involve elementary brane moves that remove minimal singularities, leading to new quivers associated with smaller orbit closures.
• Extensions include orthosymplectic quivers for B, C, and D types and folded non-simply-laced quivers, linking brane dynamics to deep aspects of nilpotent orbit theory.

Searching arXiv for the cited work and related papers on Kraft quivers and Kraft–Procesi transitions. arXiv search: "Kraft quivers Kraft-Procesi branes nilpotent orbit closures orthosymplectic quivers" Kraft quivers are $3d\ \mathcal N=4$ quiver gauge theories whose Higgs branches are closures $\overline{\mathcal O}_\lambda$ of nilpotent orbits $\mathcal O_\lambda$ in classical Lie algebras. In the type $A_n$ setting they arise as linear unitary quivers associated with partitions of $n+1$; in the classical $B$, $C$, and $D$ settings they become orthosymplectic quivers engineered by Type IIB branes with orientifold planes. Their defining structural feature is that inclusions of orbit closures are realized by elementary brane moves, or Kraft–Procesi transitions, which remove minimal singularities and produce new quivers associated with smaller orbit closures (Cabrera et al., 2016, 1711.02378).

1. Definition and mathematical setting

For type $A_n$, one fixes $n\in\mathbb N$ and a partition $\overline{\mathcal O}_\lambda$0 of $\overline{\mathcal O}_\lambda$1,

$\overline{\mathcal O}_\lambda$2

By classical theory, $\overline{\mathcal O}_\lambda$3, and its closure $\overline{\mathcal O}_\lambda$4 is a hyperkähler cone. The associated Type IIB brane construction engineers the $\overline{\mathcal O}_\lambda$5 theory $\overline{\mathcal O}_\lambda$6, and the corresponding Kraft quiver $\overline{\mathcal O}_\lambda$7 is the quiver whose Higgs branch is $\overline{\mathcal O}_\lambda$8 (Cabrera et al., 2016).

In the broader classical case, a Kraft quiver is a $\overline{\mathcal O}_\lambda$9 orthosymplectic quiver gauge theory whose Higgs branch is the closure $\mathcal O_\lambda$0 of a nilpotent orbit $\mathcal O_\lambda$1 in a classical Lie algebra $\mathcal O_\lambda$2. Kraft–Procesi transitions are partial Higgsings that remove D3-brane subsystems engineering elementary surface or minimal singularities $\mathcal O_\lambda$3, $\mathcal O_\lambda$4, $\mathcal O_\lambda$5, $\mathcal O_\lambda$6, and $\mathcal O_\lambda$7. Each such removal produces a transverse slice $\mathcal O_\lambda$8 and a new orbit closure $\mathcal O_\lambda$9 (1711.02378).

A central motivation is supplied by Namikawa’s theorem: any $A_n$0 Higgs or Coulomb branch with only spin-1 chiral generators under $A_n$1 is exactly the closure of a nilpotent orbit. Within that class, nilpotent orbit closures are therefore identified as the simplest non-trivial moduli spaces appearing in three-dimensional theories with eight supercharges (Cabrera et al., 2016, 1711.02378).

2. Type $A_n$2 construction from branes and partitions

The type $A_n$3 construction starts with $A_n$4 NS5-branes and $A_n$5 D5-branes. All D5-branes carry identical linking number

$A_n$6

so they all sit in the same interval between NS5-branes. The NS5 linking numbers are taken to be the parts of the transpose partition $A_n$7, padded to length $A_n$8 and read in increasing order from left to right: $A_n$9 In a Coulomb-brane frame, each interval between consecutive NS5-branes contains $n+1$0 D3-branes, and these define gauge group factors $n+1$1. Because all D5-branes lie in the final interval, a single flavor node $n+1$2 attaches to the last, rightmost gauge node (Cabrera et al., 2016).

The resulting type $n+1$3 Kraft quiver is the linear quiver

$n+1$4

with the $n+1$5 determined by the NS5 linking numbers $n+1$6. Equivalently, it may be read directly from the Higgs-brane configuration after maximal splitting of D3-branes (Cabrera et al., 2016).

This construction identifies the Higgs branch of the quiver with $n+1$7. In the language of the paper, the Coulomb and Higgs branches of certain $n+1$8 gauge theories can be understood as closures of nilpotent orbits, and the type $n+1$9 brane system provides a direct realization of the Kraft–Procesi classification in physical terms (Cabrera et al., 2016).

3. Kraft–Procesi transitions and minimal singularities

To remove a minimal singularity $B$0, one performs a Higgs mechanism on the associated brane system. In the type $B$1 analysis, two families of minimal singularities appear. An $B$2 singularity, $B$3, arises when $B$4 NS5-branes coincide in one D5 interval and is generated by a single D3-brane in that interval. An $B$5 singularity arises when two single-NS5 intervals are separated by $B$6 empty intervals and is generated by $B$7 D3-branes forming a wedge (Cabrera et al., 2016).

The transition consists of three steps. First, the relevant D3-brane or D3-branes are aligned with NS5-branes and/or D5-branes and split into segments. Second, the coordinates of the newly massless vectormultiplet scalars, namely the D3 segments between NS5–NS5, are taken to infinity, thereby fully removing these D3-branes from the Higgs-brane system. Third, one reads off the new linking numbers $B$8 of the surviving NS5-branes, while $B$9 remains unchanged; this determines a new partition $C$0 and hence a new quiver $C$1 (Cabrera et al., 2016).

A local rule is provided by the Hanany–Witten effect: under a single D3 crossing an NS5, the NS5 linking number changes by $C$2, while $C$3 is untouched. In the brane-matrix description, if in interval $C$4 one has $C$5 in the first row, then an $C$6 KP transition removes one D3 by decreasing the second-row entry $C$7 by $C$8, shifts one NS5 to each adjacent interval so that $C$9 each increase by $D$0, and changes $D$1 to $D$2. A similar but slightly more involved rule holds for $D$3 singularities (Cabrera et al., 2016).

The corresponding mathematical statement is the Kraft–Procesi theorem of 1982: the partial-order, or Hasse, graph of closures $D$4 under inclusion is generated by minimal singularities of type $D$5 or $D$6. The brane realization identifies each such singularity with the moduli of a minimal D3-brane subsystem, and its removal reproduces the covering relations in the Hasse diagram (Cabrera et al., 2016).

4. Orthosymplectic Kraft quivers for $D$7, $D$8, and $D$9

The classical extension beyond type $A_n$0 uses Type IIB brane systems with half-branes and an O3-plane along $A_n$1, with all half 5-branes placed at the orientifold’s transverse origin. The linking numbers are

$A_n$2

for a half NS5 at $A_n$3, and

$A_n$4

for a half D5 at $A_n$5. Between consecutive half NS5-branes all D3-branes end on the NS5-branes; similarly, on the Higgs side, all D3-branes end on D5-branes (1711.02378).

The O3 variant determines whether a gauge node is orthogonal or symplectic and whether a flavor node is $A_n$6- or $A_n$7-type.

O3 variant	Gauge node	Flavor node
$A_n$8	$A_n$9	$n\in\mathbb N$0
$n\in\mathbb N$1	$n\in\mathbb N$2	$n\in\mathbb N$3
$n\in\mathbb N$4, $n\in\mathbb N$5	$n\in\mathbb N$6	$n\in\mathbb N$7

Nilpotent orbits are organized by constrained partitions:

• $n\in\mathbb N$8, where even parts have even multiplicity.
• $n\in\mathbb N$9, where odd parts have even multiplicity.
• $\overline{\mathcal O}_\lambda$00, with a “very-even” $\overline{\mathcal O}_\lambda$01-ambiguity (1711.02378).

Two maps play a central role. The $\overline{\mathcal O}_\lambda$02-collapse, with $\overline{\mathcal O}_\lambda$03, sends an arbitrary partition to the largest $\overline{\mathcal O}_\lambda$04-partition it dominates by successive “trim and add” steps. The Barbasch–Vogan map $\overline{\mathcal O}_\lambda$05 is an order-reversing bijection from special partitions of $\overline{\mathcal O}_\lambda$06 to those of $\overline{\mathcal O}_\lambda$07: $\overline{\mathcal O}_\lambda$08

$\overline{\mathcal O}_\lambda$09

$\overline{\mathcal O}_\lambda$10

The brane “collapse transition” is realized by pushing half-D5-branes off an $\overline{\mathcal O}_\lambda$11 without D3 creation; the interval numbers of half-NS5-branes then change exactly by $\overline{\mathcal O}_\lambda$12-collapse (1711.02378).

For the three classical families, the quivers are specified uniformly by a special partition $\overline{\mathcal O}_\lambda$13, its dual partition $\overline{\mathcal O}_\lambda$14, and the brane data:

• $\overline{\mathcal O}_\lambda$15: $\overline{\mathcal O}_\lambda$16 special, $\overline{\mathcal O}_\lambda$17, $\overline{\mathcal O}_\lambda$18, $\overline{\mathcal O}_\lambda$19, rightmost $\overline{\mathcal O}_\lambda$20. The gauge chain is

$\overline{\mathcal O}_\lambda$21

• $\overline{\mathcal O}_\lambda$22: $\overline{\mathcal O}_\lambda$23 special, $\overline{\mathcal O}_\lambda$24, $\overline{\mathcal O}_\lambda$25, $\overline{\mathcal O}_\lambda$26, rightmost $\overline{\mathcal O}_\lambda$27. The chain is $\overline{\mathcal O}_\lambda$28, with $\overline{\mathcal O}_\lambda$29-flavors on the two end $\overline{\mathcal O}_\lambda$30-nodes.
• $\overline{\mathcal O}_\lambda$31: $\overline{\mathcal O}_\lambda$32 special, $\overline{\mathcal O}_\lambda$33, $\overline{\mathcal O}_\lambda$34, $\overline{\mathcal O}_\lambda$35, rightmost $\overline{\mathcal O}_\lambda$36. The chain is $\overline{\mathcal O}_\lambda$37, with $\overline{\mathcal O}_\lambda$38-flavors at both ends (1711.02378).

5. Dimensions, explicit examples, and the transition algorithm

For type $\overline{\mathcal O}_\lambda$39, if

$\overline{\mathcal O}_\lambda$40

is the transpose partition in exponential form, then the quaternionic dimension of the Higgs branch is

$\overline{\mathcal O}_\lambda$41

Equivalently,

$\overline{\mathcal O}_\lambda$42

In the classical orthosymplectic setting, the same physical principle is stated in brane terms: $\overline{\mathcal O}_\lambda$43 equals the number of physical D3-branes in the Coulomb brane configuration, and the transverse slice has dimension equal to the number of D3-branes removed in the KP transition (Cabrera et al., 2016, 1711.02378).

Two worked $\overline{\mathcal O}_\lambda$44 examples make the construction explicit. For $\overline{\mathcal O}_\lambda$45, one has $\overline{\mathcal O}_\lambda$46 and NS5 linking numbers

$\overline{\mathcal O}_\lambda$47

read from the left. The Higgs quiver is

$\overline{\mathcal O}_\lambda$48

Its Higgs branch Hilbert series, or a direct hyperkähler quotient, gives $\overline{\mathcal O}_\lambda$49. The minimal singularity $\overline{\mathcal O}_\lambda$50, namely $\overline{\mathcal O}_\lambda$51, sits in the interval with $\overline{\mathcal O}_\lambda$52 NS5-branes; performing the $\overline{\mathcal O}_\lambda$53 KP transition removes one gauge node $\overline{\mathcal O}_\lambda$54 and merges its neighbours $\overline{\mathcal O}_\lambda$55, reproducing the quiver for $\overline{\mathcal O}_\lambda$56 (Cabrera et al., 2016).

For $\overline{\mathcal O}_\lambda$57, one has $\overline{\mathcal O}_\lambda$58 and the Higgs quiver

$\overline{\mathcal O}_\lambda$59

The minimal singularity $\overline{\mathcal O}_\lambda$60 arises from the two single-NS5 intervals at positions $\overline{\mathcal O}_\lambda$61 and $\overline{\mathcal O}_\lambda$62. Performing this $\overline{\mathcal O}_\lambda$63 transition deletes three gauge nodes in the middle and produces the trivial quiver

$\overline{\mathcal O}_\lambda$64

The Higgs branch then collapses to a point, as expected (Cabrera et al., 2016).

The orthosymplectic paper formulates a general KP-quiver algorithm. Given special partitions $\overline{\mathcal O}_\lambda$65 in the same $\overline{\mathcal O}_\lambda$66, one computes $\overline{\mathcal O}_\lambda$67 and $\overline{\mathcal O}_\lambda$68, forms $\overline{\mathcal O}_\lambda$69 according to the Lie type and rank, computes $\overline{\mathcal O}_\lambda$70 for $\overline{\mathcal O}_\lambda$71- or $\overline{\mathcal O}_\lambda$72-type or $\overline{\mathcal O}_\lambda$73 for $\overline{\mathcal O}_\lambda$74-type, constructs the Coulomb-branch brane configuration with stacked half 5-branes and fixed O3 choice, reads off the quiver from the O3 table, and then identifies and removes the D3 subsystem corresponding to the singularity. This yields new linking numbers $\overline{\mathcal O}_\lambda$75 and hence the quiver for $\overline{\mathcal O}_\lambda$76 (1711.02378).

Basic transitions are listed uniformly by Lie type. In $\overline{\mathcal O}_\lambda$77-type, $\overline{\mathcal O}_\lambda$78 removes an $\overline{\mathcal O}_\lambda$79 singularity, with

$\overline{\mathcal O}_\lambda$80

In $\overline{\mathcal O}_\lambda$81-type, $\overline{\mathcal O}_\lambda$82 removes $\overline{\mathcal O}_\lambda$83, with $\overline{\mathcal O}_\lambda$84 and $\overline{\mathcal O}_\lambda$85. In $\overline{\mathcal O}_\lambda$86-type, $\overline{\mathcal O}_\lambda$87 removes $\overline{\mathcal O}_\lambda$88, with $\overline{\mathcal O}_\lambda$89 and $\overline{\mathcal O}_\lambda$90. In $\overline{\mathcal O}_\lambda$91-type, $\overline{\mathcal O}_\lambda$92 removes $\overline{\mathcal O}_\lambda$93, with $\overline{\mathcal O}_\lambda$94 and $\overline{\mathcal O}_\lambda$95 (1711.02378).

6. Folding, non-simply-laced extensions, and Hasse diagrams

Later work extends the Kraft–Procesi framework to folded orthosymplectic quivers. In a simply-laced unitary quiver, one may fold two or more identical legs by identifying a $\overline{\mathcal O}_\lambda$96 symmetry exchanging the legs, gauging the diagonal subgroup, setting the magnetic charges of the two legs equal, and dividing by the $\overline{\mathcal O}_\lambda$97 Weyl action. The result is a quiver with a non-simply-laced edge, and physically one ungauges the diagonal $\overline{\mathcal O}_\lambda$98 on a long node so that the Coulomb branch is well-defined. For orthosymplectic quivers, the brane realization introduces an $\overline{\mathcal O}_\lambda$99 plane transverse to the D3-branes and overlapping the NS5-branes; at the intersection of $\mathcal O_\lambda$00 and $\mathcal O_\lambda$01 lies an $\mathcal O_\lambda$02 plane, and the combined projection identifies two orthosymplectic legs in the magnetic quiver (Bourget et al., 2021).

The corresponding monopole formula for non-simply-laced orthosymplectic quivers is

$\mathcal O_\lambda$03

For a non-simply-laced edge of multiplicity $\mathcal O_\lambda$04 between a $\mathcal O_\lambda$05 node with charges $\mathcal O_\lambda$06 and an $\mathcal O_\lambda$07 node with charges $\mathcal O_\lambda$08,

$\mathcal O_\lambda$09

The vector multiplet contributes $\mathcal O_\lambda$10 as usual, and for non-simply-laced orthosymplectic quivers one must include half-integers in the magnetic charges for short nodes if the non-simply-laced edge is odd (Bourget et al., 2021).

Some folded orthosymplectic quivers have Coulomb branches that are closures of minimal nilpotent orbits of exceptional algebras. The paper lists, for example, foldings that realize $\mathcal O_\lambda$11, $\mathcal O_\lambda$12, and $\mathcal O_\lambda$13. It also derives Hasse diagrams by quiver subtraction as well as by Kraft–Procesi transitions in the brane system (Bourget et al., 2021).

The quiver-subtraction rules for orthosymplectic special minimal slices are

$\mathcal O_\lambda$14

$\mathcal O_\lambda$15

$\mathcal O_\lambda$16

An explicit example is the Hasse diagram of $\mathcal O_\lambda$17: starting from the folded orthosymplectic quiver, one subtracts a minimal $\mathcal O_\lambda$18 slice by removing an $\mathcal O_\lambda$19–$\mathcal O_\lambda$20 pair, then subtracts an $\mathcal O_\lambda$21 quiver, and continues until the bottom point $\mathcal O_\lambda$22. Each subtraction matches the corresponding Kraft–Procesi transition in the D3–D5–O3 brane system, where each minimal slice is associated with moving one D3-brane across an orientifold (Bourget et al., 2021).

Taken together, these constructions place Kraft quivers at the intersection of nilpotent orbit theory, hyperkähler moduli spaces, and Type IIB brane engineering. In the type $\mathcal O_\lambda$23 case they provide a linear unitary realization of $\mathcal O_\lambda$24; in the $\mathcal O_\lambda$25, $\mathcal O_\lambda$26, and $\mathcal O_\lambda$27 cases they give a unified orthosymplectic framework based on partitions, Barbasch–Vogan duality, and orientifold data; and in folded settings they connect Kraft–Procesi transitions to non-simply-laced magnetic quivers and phase diagrams of $\mathcal O_\lambda$28 Higgs branches (Cabrera et al., 2016, 1711.02378, Bourget et al., 2021).