Nilpotent orbits and the Coulomb branch of $T^\sigma (G)$ theories: special orthogonal vs orthogonal gauge group factors

Coulomb branches of a set of $3d\ \mathcal{N}=4$ supersymmetric gauge theories are closures of nilpotent orbits of the algebra $\mathfrak{so}(n)$. From the point of view of string theory, these quantum field theories can be understood as effective gauge theories describing the low energy dynamics of a brane configuration with the presence of orientifold planes. The presence of the orientifold planes raises the question to whether the orthogonal factors of a the gauge group are indeed orthogonal $O(N)$ or special orthogonal $SO(N)$. In order to investigate this problem, we compute the Hilbert series for the Coulomb branch of $T^\sigma(SO(n)^\vee)$ theories, utilizing the monopole formula. The results for all nilpotent orbits from $\mathfrak {so} (3)$ to $\mathfrak{so}(10)$ which are special and normal are presented. A new relationship between the choice of $SO/O(N)$ factors in the gauge group and the Lusztig's Canonical Quotient of the corresponding nilpotent orbit is observed. We also provide a new way of projecting several magnetic lattices of different $SO(N)$ gauge group factors by the simultaneous action of a $\mathbb Z_2$ group.


Introduction
In [2] 3d N = 4 quiver gauge theories known as T σ ρ (G) theories [1,3] were studied for G a classical Lie group of type A, B, C and D. The Hilbert series 1 for the Coulomb branch and for the Higgs branch of these theories were computed for the case when either σ or ρ are trivial partitions.For T σ (G) on the B, C and D cases (where ρ is trivial) an interesting phenomenon occurs: there are several possible quivers for a single choice of partition σ and the Coulomb branches of these quivers differ by a projection by a discrete group, like for example Z 2 .
On the other hand, Coulomb branches of T σ (G) theories are predicted to be closures of nilpotent orbits of the algebra Lie(G ∨ ) [3,[5][6][7][8], where G ∨ is the Langlands or GNO dual group of G [9].Therefore the question arises as to which of the different quivers obtained for a single theory has the closure of the nilpotent orbit as its Coulomb branch, rather than a discrete cover of the same.
The present note constitutes a systematic approach to this problem.The Hilbert series of the Coulomb branches of all the candidate quivers are computed utilizing the monopole formula [4,10,11].Analyzing the Hilbert series of each moduli space one can discern which Coulomb branch corresponds to the closure of a nilpotent orbit.The following pages contain both the quivers that resulted to have the closure of the nilpotent orbit as their Coulomb branch and the corresponding Hilbert series.Section 2 of the paper reviews the construction of T σ (G ∨ ) orthosymplectic quivers given in [2].Section 3 overviews the monopole formula, used to compute the Hilbert series of the Coulomb branch of the quivers.Section 4 tabulates the T σ (G ∨ ) quivers with the choice of SO(N )/O(N ) gauge nodes such that their Coulomb branch is the closure of the corresponding nilpotent orbit.Results for so (3) to so (10) are presented along with their Hilbert series and plethystic logarithms.Section 5 contains an analysis of the results and provides an answer to the problem.In section 6 a new method to project several magnetic lattices by the diagonal action of a Z 2 group is introduced.This mechanism is a key part of the answer to the problem and opens the door to a new range of computations that can be performed utilizing the monopole formula.

Constructing quiver diagrams for T σ (G) theories
This section reviews the method of [2] to obtain the quiver of the 3d N = 4 superconformal field theory known as T σ (G).This quiver is determined by a partition σ and a classical Lie group G. Please see [4] for a review on 3d N = 4 quiver gauge theories and their moduli spaces.
We only consider partitions such that where d LS is the Lusztig-Spaltenstein map [12] that transposes and D-collapses the partition, see the discussion in2 [13].Partitions σ ∈ P D (2n) that satisfy equation (2.1) are called special.
The orthosymplectic quiver has a chain with a number of 2n − 1 gauge nodes that alternate between the groups SO(N j ) and U Sp(N j ), starting with U Sp(N 1 ) (we label the nodes from right to left, i.e.U Sp(N 1 ) is the rightmost node).Note that in principle O(N j ) groups can be chosen instead of SO(N j ).The parameter N j can be obtained from the partition data: where σ T = [σ T 1 , σ T 2 , . . ., σ T l ] is the transpose partition of σ and ρ = [1, 1, . . ., 1] is the trivial partition of 2n.The symbol [p] + means the smallest even integer that is larger or equal to p and [p] − is the largest even integer smaller or equal to p. Also l = 2n is the length of ρ and l is the length of σ T ."+" needs to be used when computing SO(N j ) gauge nodes whereas "−" is for U Sp(N j ) gauge nodes.
A single flavor node can be attached to each gauge node.The flavor nodes also alternate: SO(M j ) are attached to symplectic gauge nodes and U Sp(M j ) are attached to orthogonal gauge nodes.The parameters M j are also determined by the partition data: where zero parts σ T l+1 = 0, σ T l+2 = 0, etc. can be added at the end of partition σ T if necessary.
The orthosymplectic quiver has now a chain with a number of 2n gauge nodes that alternates between the groups SO(N j ) and U Sp(N j ), starting with SO(N 1 ).Note that in principle O(N j ) groups can be chosen instead of SO(N j ).The parameter N j can be obtained from the partition data with equation (2.2) (with the difference that in this case ρ = [1, 1, . . ., 1] is the trivial partition of 2n + 1).
A single flavor node can be attached to each gauge node.The flavor nodes also alternate: SO(M j ) are attached to symplectic gauge nodes and U Sp(M j ) are attached to orthogonal gauge nodes.The parameters M j are also determined by the partition data as indicated in equation (2.3).[7,8] established that that the Coulomb branch of the T σ (G) theory should be the closure of the nilpotent orbit 3Ōλ ⊂ g ∨ with g ∨ = Lie(G ∨ ) and

Coulomb branches
The map d BV is the Barbasch-Vogan map [15] defined by equation 4 (2.8) of [7], which is taken from equation ( 5) in [16].This is summarized in table 1, where σ T is the transpose partition, σ ± is defined as σ + := [σ 1 , σ 2 , . . ., σ s , 1] and σ − := [σ 1 , σ 2 , . . ., σ s − 1].σ X , with X = B, C, D, is the X-collapse of partition σ as defined in [13].P A (n) is the set of all partitions of n; P B (2n + 1) is the set of partitions of 2n + 1 where even parts have even multiplicity; P C (2n) is the set of partitions of 2n where odd parts have even multiplicity and P D (2n) is the set of partitions of 2n where odd parts have even multiplicity.This means that for G = SO(2n) the Coulomb branches of T σ (SO(2n)) for special partitions σ cover the whole set of closures of special nilpotent orbits of so(2n).For G = U Sp(2n) Coulomb branches of T σ (U Sp(2n)) for special partitions σ cover the whole set of closures of special nilpotent orbits of so(2n + 1).There are some exceptions for this statement: closures of nilpotent orbits that are non-normal have not been found as Coulomb branches of any theory5 .

2.4
Example: λ = [3, 3, 1] orbit of so (7) Let us look at an example for the closure of the nilpotent orbit Ō[3,3,1] ⊂ so (7).This means that λ = [3, 3, 1] and g ∨ = so (7).Therefore the relevant quiver is T σ (G) with: Note that λ = d BV (σ) implies σ = d BV (λ) because we are restricting σ to the set of special partitions.The quiver theory is T [2,2,2] (U Sp (6)).The necessary partition data in order to obtain the quiver is: The N j parameters for the gauge nodes are: The M j parameters for the flavor nodes are: This gives four possible quivers with different Coulomb branch C depending on the choice SO/O(N j ) for N j even6 , depicted in figure 1.The aim of this paper is to compute the Coulomb branch of the four quivers in order to determine which of them has the closure of the nilpotent orbit C = Ō [3,3,1] as its Coulomb branch, and therefore corresponds to T [2,2,2] (U Sp(6)).In this case the answer is the quiver in figure 1(a), as is shown in the third row of table 6.
3 The monopole formula

The Hilbert series
A Hilbert series H(t) can be computed in order to determine the Coulomb branch of each quiver, see [4].In this case the Hilbert series can be obtained via the monopole formula introduced in [10], which counts dressed monopole operators in the chiral ring.We shall only consider the unrefined Hilbert series H(t), given as: where G is the gauge group and the magnetic fluxes m are summed over the dominant Weyl chamber of the weight lattice of G ∨ , the GNO dual group of G [9]. m breaks G into a residual gauge group G m .In 3d N = 4, the dressing factors P G (m; t) are given by the dimensions d i of the Casimir invariants of G m : where r = rank(G).∆(m) is the conformal dimension of the bare monopole operators: where the first sum is over positive roots of the gauge group G and it is the contribution from the vector multiplets of the theory.The second sum is over weights of the representation R i under which the hypermultiplets of the theory transform (do not confuse the symbol ρ i for the weights with the previous symbol ρ for the trivial partitions).

The plethystic logarithm
Once a Hilbert series H(t) is obtained, it is useful to analyze its plethystic logarithm [19], P L(H(t)) defined as: where µ(k) is the Möbius function.The P L(H(t)) is a generating function that counts generators of the chiral ring and their relations.Since a moduli space that is the closure of a nilpotent orbit restricts the dimension of the generators of the chiral ring to ∆ = 1 (see discussion in sections 3.1 and 3.2 of [14]) the P L(H(t)) provides invaluable information for the present analysis.

Quivers for closures of nilpotent orbits
This section contains the main results of the paper.The quiver diagrams of theories T σ (G) with the right choice of SO/O(N j ) gauge nodes so their Coulomb branch C is the closure of nilpotent orbit Ōλ ⊂ so(n) are tabulated for 3 ≤ n ≤ 10.Remember that σ = d BV (λ) and vice versa.The Hilbert series and plethystic logarithms of the Coulomb branch of these quivers are also tabulated.

Tables: quivers
For the tables in this section: • All red nodes with label N denote SO(N ) groups and all blue nodes with label N denote U Sp(N ) groups.Red nodes corresponding to O(N ) groups are explicitly labeled as O(N ).
• A single quiver is provided for each closure of a special nilpotent orbit Ōλ ⊂ so(n) that is normal.
• Partitions λ corresponding to non-special orbits and non-normal orbits are indicated explicitly.No quivers are provided for these orbits as we cannot obtain quivers whose Coulomb branch is the closure of the nilpotent orbit using the prescriptions in sections 2 and 3.
• Trivial partitions λ = [1 n ] have been omitted from the analysis since the corresponding Coulomb branch is trivial.
• The very even under the adjoint action of SO(4).The corresponding Coulomb branch is isomorphic to the closure of only one of these orbits.This is also the case for the remaining very even partitions λ = [2 4 ] and λ = [4 2 ].
• The symbol ( †) denotes a quiver where the GNO magnetic lattice of two gauge groups is projected by a diagonal action of a Z 2 group.This action is described in section 6.
• The Lusztig's Canonical Quotient7 Ā(O λ ) corresponding to each orbit is shown.The choice of SO/O(N ) gauge factors in the quiver seems to be beautifully encoded in this group.This is discussed at length in section 5.
It is worthwhile to highlight that the resulting quivers with Coulomb branch C 1 being the closure of the minimal orbit of so(2n) (with partition λ = [2 2 , 1 2n−4 ]) and the resulting quivers with Coulomb branch C 2 being the closure of the next to minimal orbit of so(2n−1) (with partition λ = [3, 1 2n−4 ]) differ only by a Z 2 factor on the first node (this is the rightmost node in the tables).This is a compelling physical realization of the mathematical result by Brylinski and Kostant, that states that the two varieties are related by a Z 2 quotient [22]: Table 3: Quivers with C = Ōλ ⊂ so(4).The dual partition is σ = d BV (λ).

Tables: Hilbert series and plethystic logarithms
The Hilbert series of the Coulomb branch of the set of T σ (G ∨ ) quivers presented in the previous section are computed using the monopole formula (see section 3).
The computation of the Hilbert series H(t) is done perturbatively.H(t) is expanded in powers of t around t = 0: and the coefficients a i are computed starting with a 0 = 1.The highest order a k that can be computed is restricted by the computing power, obtaining an expression: One can then use properties of the nilpotent orbit such as its complex dimension (which in this case equals twice the rank of the gauge group of the corresponding T σ (G ∨ ) quiver).Closures of nilpotent orbits that are normal have a Hilbert series with rational form: where p is the complex dimension and P (t) is a palindromic polynomial of degree p/2 with P (1) non zero.This allows to reconstruct the rational form of H(t) after computing a finite set of coefficients a i .
For nilpotent orbits of so(n) with n < 10 the Hilbert series of closures of nilpotent orbits can be compared with the results in [23].For nilpotent orbits of so (10) we turn our attention to the plethystic logarithm defined in section 3, which provides the degrees of generators and the degrees of relations between them.This information can be compared with the constrains (such as rank or degree of nilpotency) on the nilpotent orbit when viewed as a set of matrices, i.e. linear operators acting on the fundamental representation of the Lie algebra 8 .
The following tables include the Hilbert series and the plethystic logarithms of the quivers displayed in the previous section.The plethystic logarithm is given up to an order so it at least includes the relations that show nilpotency.Partition (λ) Hilbert series Plethystic logarithm [3] ( Table 12: Hilbert series and plethystic logarithm for the Coulomb branch of quivers in table 2. Partition (λ) Hilbert series Plethystic logarithm Table 13: Hilbert series and plethystic logarithm for the Coulomb branch of quivers in table 3.

Hilbert series
Plethystic logarithm [7] (1 Non-special Table 16: Hilbert series and plethystic logarithm for the Coulomb branch of quivers in table 6.
Table 17: Hilbert series and plethystic logarithm for the Coulomb branch of quivers in table 7.

Partition (λ)
Hilbert series Plethystic logarithm Table 19: Hilbert series and plethystic logarithm for the Coulomb branch of quivers in table 10.

Partition (λ)
Hilbert series Plethystic logarithm Table 20: Hilbert series and plethystic logarithm for the Coulomb branch of quivers in table 11.

First case with Ā(O
The first nilpotent orbit with a Lusztig's Canonical Quotient of the form Ā(O λ ) = Z 2 2 is the orbit O λ ⊂ so(13) with λ = [5, 3 2 , 1 2 ].The quiver with Coulomb branch C = Ōλ is different from all the ones computed so far since it corresponds to the choice of two orthogonal gauge group factors (all previous results in section 4.1 have either zero or one orthogonal gauge group factor).It is depicted in figure 2 and it corresponds to theory T [4 2 ,2 2 ] (U Sp( 12)).
The computed Hilbert series is: Its plethystic logarithm:

Analysis of the results
Analyzing the obtained results we are able to produce a prescription that fully determines the orthosymplectic quiver corresponding to T σ (SO(n) ∨ ) theory such that its Coulomb branch is C = Ōλ ⊂ so(n) with λ = d BV (σ).
In order to present the prescription let us define a chain Ξ i of SO/O(N j ) groups as a subset of the quiver constituted by consecutive SO/O(N j ) gauge nodes (starting with SO/O(N i ), following with SO/O(N i+2 ), etc. and ending with SO/O(N i+2m−2 ), where the length or number of nodes in the chain is m) such that: • If the quiver has gauge nodes U Sp(N i−1 ) and/or U Sp(N i+2m−1 ), they have flavor nodes attached to them.
When faced with the choice SO/O(N j ): The maximal magnetic lattice is the lattice for the gauge group where all chains Ξ i are chosen to have SO(N i+2p ) as their groups for all p.Let us define this as the default lattice.This magnetic lattice is chosen when Ā(O λ ) is the trivial group.] ⊂ so (13).
When the Lusztig's Canonical Quotient is Ā(O λ ) = Z 2 the chain Ξ i with the lowest i changes.If the chain has length one, i.e. it constitutes a single node SO(N i ), this node is changed to O(N i ).From the point of view of the magnetic lattice this translates into projecting the magnetic lattice of SO(N i ) by a Z 2 action.If the chain has length m bigger than one, the magnetic lattice of the groups needs to be projected by the diagonal Z 2 acting diagonally on the individual lattices of all the groups SO(N i+2p ).This is the case of the quiver marked with the symbol ( †) in table 9. How to perform this diagonal Z 2 action is a new result that has never been discussed before as far as the authors are aware.This process is explained in detail in the next section.
When the Lusztig's Canonical Group has the form Ā(O λ ) = Z k 2 the k chains Ξ i with lowest i are changed in the same fashion9 : in each chain Ξ i all the lattices of the groups SO(N i+2p ) are projected by a diagonal Z 2 action.This is for example the case of the quiver in figure 2 2 , the first chain Ξ 1 has length one, the magnetic lattice of its gauge group SO(4) is acted by a single Z 2 , so the group changes to O(4); the second chain Ξ 3 has also length one, its gauge group SO(8) changes to O(8), i.e. its magnetic lattice is halved by a different Z 2 action.
6 Diagonal Z 2 actions on different magnetic lattices 6.1 Diagonal action on the magnetic lattices of SO(2) × SO(4) For clarity let us discuss the quiver with C = Ō[3,2 2 ,1 2 ] ⊂ so(9) depicted in table 9.There are two chains: Ξ 1 with groups SO(N 1 = 2) and SO(N 3 = 4), and Ξ 5 with groups SO(N 5 = 4) and SO(N 7 = 2).The aim is to divide the magnetic lattice of the groups in Ξ 1 by a single Z 2 action.
Let the weights and dressing factors of the magnetic lattices of gauge group factors SO(2), O(2), SO(4) and O(4), as they were discussed in [2], be described by tables 21, 22, 23 and 24.Note that for the groups O(2) and O(4) some magnetic fluxes are defined to have vanishing dressing factor, this is equivalent to removing them from the lattice.The choice to include them in the tables becomes justified in the following discussion.
A general description of the magnetic lattices and dressing factors of gauge groups SO(N ) and U Sp(2N ) can be found in appendix A. Classical Casimir contribution for classical groups of [10].For the group SO(2N ) the magnetic weights are the weights of the GNO dual group, SO(2N ).The dominant Weyl chamber of the group can be parametrized by an integer vector: For the gauge group SO(2N + 1) the magnetic weights are summed over the dominant Weyl chamber in the weight lattice of the GNO dual, U Sp(2N ).These can be parametrized by the same integer vector m in equation ( 6 [2], not only the Weyl chambers of magnetic weights of O(2N ) and SO(2N + 1) are isomorphic, but their dressing factors are also identical.
The combination of the magnetic lattices of gauge groups SO(2), O(2), SO(4) and O(4) gives four possibilities for chain Ξ 1 (the aim of this section is to present a fifth option).Let The four options are: Note that the dressing factors can be obtained with usual multiplication: Monopole fluxes m 2 P SO(4) (m 2 ; t) (0, 0) (1, ±1) (1, 0) (2, ±1) Γ 1 is the maximal lattice of the set, and the remaining ones can be constructed via a projection by the action of a finite group: Γ 1 is the maximal lattice The Z 2 × Z 2 action on Γ 1 defines three non equivalent Z 2 actions on Γ 1 .Let Z 2 2 have elements {e, a, b, ab} where e is the identity, a 2 = b 2 = e and ab = ba.Let the Z 2 generated by a be the one that gives rise to Γ 2 and the Z 2 generated by b the one that gives rise to Γ 3 .There is a lattice that has not been considered yet, whose dressing factor cannot be Figure 3: Quiver for the theory T [6,4] (U Sp (10)) with Coulomb branch C = Ō[3,2 Table 28: Hilbert series and plethystic logarithm for the Coulomb branch of quiver in Figure 3.
Second example.The Γ 5 lattice can also be used now for partition λ = [3, 2 2 , 1 4 ], the next case that requires a diagonal Z 2 action on SO(2) and SO(4).The theory is T [6,4] (SO(11) ∨ ), its quiver is given in figure 3 and its Hilbert series and plethystic logarithm are given in Table 28.The Coulomb branch of this quiver is found to be the closure of nilpotent orbit O [3,2 2 ,1 4 ] ⊂ so(11).

The general case
In general we have the diagonal Z 2 = {e, a 1 a 2 . . .a m } where Z 2 ⊂ Z m 2 and Z m 2 is generated by a k with k = 1, 2, . . ., m acting on a chain Ξ i of length m with groups SO(N i+2p ) and p = 0, 1, . . ., m − 1.The magnetic lattice of the gauge group G = × p SO(N i+2p ), projected by the Z 2 action, is defined by the dressing factor: where P O(N i+2k−2 ) (m k ; t) is extended over the weights that are in the magnetic lattice of SO(N i+2k−2 ) but not in the magnetic lattice of O(N i+2k−2 ) by defining it to be zero at those weights.

Figure 1 :
Figure1: Candidate quivers for T[2,2,2] (U Sp(6)) theory.Gauge nodes are denoted by a circle and flavor nodes are represented by a square.Red gauge nodes with label N j denote SO(N j ) groups and blue gauge nodes with label N j denote U Sp(N j ) gauge groups.The red square node with label M 2 = 3 denotes an SO(3) flavor node.When the gauge group O(N j ) is chosen instead of SO(N j ) the group is written explicitly.

Table 1 :
Barbasch-Vogan map d BV .The first column is the set of partitions P such that σ ∈ P. The last column is the set of partitions such that d BV (σ) ∈ P ∨ .

Table 14 :
Hilbert series and plethystic logarithm for the Coulomb branch of quivers in table 4.

Table 18 :
Hilbert series and plethystic logarithm for the Coulomb branch of quivers in table 8 and table 9.
[2] but this time with m 1 ≥ m 2 ≥ • • • ≥ m N .For the gauge group O(2N ) the magnetic weights and dressing factors can be found in appendix A. Monopole formula for orthogonal and special orthogonal gauge groups of[2].Starting with the dominant Weyl chamber of SO(2N ) described by (6.1) the parity symmetry Z 2 identifies (m 1 , m 2 , . . ., m N ) with (m 1 , m 2 , . . ., −m N ).This results in a Weyl chamber isomorphic to the Weyl chamber of U Sp(2N ) that is employed for the SO(2N + 1) gauge group.As explained in