Orthosymplectic Implosions

We propose quivers for Coulomb branch constructions of universal implosions for orthogonal and symplectic groups, extending the work on special unitary groups in arXiv:2004.09620. The quivers are unitary-orthosymplectic as opposed to the purely unitary quivers in the A-type case. Where possible we check our proposals using Hilbert series techniques.


Introduction
We study examples of symplectic duality, that is, duality between the Higgs and Coulomb branches of a 3-dimensional N = 4 SUSY theory. Mathematically, this is a duality between varieties with a complex-symplectic (or even hyperKähler) structure admitting a rotating circle action.
We study quivers which are candidates for the symplectic duals of universal hy-perKähler implosions for various groups. We briefly recall some relevant properties of these implosions here, referring the reader to [2][3][4] for a more detailed description.
The universal hyperKähler implosion for a complex reductive group G = K C with maximal compact subgroup K, is supposed to be a complex-symplectic variety, that is in fact hyperKähler in a suitable stratified sense. It has a complex-symplectic action of K C × T C , where T is the maximal torus of K. This action is supposed to be the complexification of a hyperKähler action of the compact group K × T . Moreover, the dimension of the implosion is equal to the dimension of K C × T C .
HyperKähler reduction of the implosion by T , or equivalently complex-symplectic reduction by the complex torus T C , will give the Kostant varieties of the Lie algebra k C , that is, the varieties we get by fixing the values of all Ad-invariant polynomials on this Lie algebra. In particular, reduction at zero gives the nilpotent cone N G . This is consistent with the dimension statements above, as each Kostant variety has dimension equal to dim K C − rank K C . The remaining K C action on the implosion then descends to the natural action on the Kostant varieties. We can thus think of the implosion as a master space which yields the Kostant varieties on reduction by the torus action at the appropriate level.
The implosion also has a description as a Geometric Invariant Theory quotient by the maximal unipotent subgroup N of G = K C . Explicitly, the implosion is the GIT quotient where n • denotes the annihilator of n = Lie N in the dual Lie algebra g * . We can view this quotient as the complex-symplectic quotient, in the GIT sense, of the cotangent bundle T * G by N . The left action of G on T * G survives, but the right action is broken to a T C action, giving the required action on the implosion. We recall that Kronheimer has shown the existence of a complete hyperKähler metric on T * G, invariant under the action of K × K.
On choosing an invariant inner product on g, we may identify n • with the Borel algebra b. Complex-symplectic reduction by T C = B/N then fixes the Cartan component of b to be zero, and replaces the N quotient by a B quotient, yielding the GIT quotient (G × n) / / B. The fact that this gives the nilpotent cone is just the statement that the Springer resolution G × B n of the nilpotent cone is an affinisation map. We can view the implosion in this way as occupying an intermediate position between the cotangent bundle T * G and the nilpotent cone-the implosion is the symplectic reduction in the GIT sense of T * G by the maximal unipotent, and further reduction by the complex torus yields the nilpotent cone.
In [3] two of the authors and Swann produced the implosion for K = SU(n) as a hyperKähler quiver variety. As a complex-symplectic variety the implosion is known to exist for general K by work of Ginzburg-Riche [5] (we note that as N is nonreductive it is a nontrivial result that the quotient exists as an algebraic variety).
Implosion spaces should admit symplectic duals -in particular as mentioned above the SU(n) implosion is a hyperKähler quotient of a linear space at level zero, so has a Sp(1) action rotating the complex structures.
In this paper we present candidates for the duals of the implosions for K = SO(2n) and SO(2n + 1), extending the discussion for SU(n) in [1]. One test that we will perform is to check that the dual spaces give the correct symmetry group of the original implosion. Another check is on the dimension, using Nakajima's observation that if the Higgs branch is the hyperKähler quotient of a linear space M by a compact group H, then the Coulomb branch should be birational to the quotient by the Weyl group of the cotangent bundle of the complexified dual maximal torus of H; hence dim R (Coulomb branch) = 4 rank H. (1.2) We also do some checks that the hyperKähler quotient of the implosion by the torus action gives the nilpotent cone, as expected.

Bouquet for Orthogonal groups
For this section, we will focus on quivers whose Coulomb branch is the nilpotent cone of a complex semisimple group G = K C . We introduce the process of explosion where a flavour node of rank n is exploded into n rank 1 gauge nodes. These unbalanced nodes generate the required Abelian symmetries in the Coulomb branch, as required for the implosion. The number of rank 1 nodes is chosen so as to preserve the balancing condition for the remaining nodes in the original quiver, so that the G symmetry in the Coulomb branch is preserved as well, so the implosion has commuting symmetries of G and its maximal torus as required.
For unitary quivers this is well understood where we start with a T [SU(n)] theory with SU(n) flavour node. We then explode the SU(n) flavour node into n U(1) gauge nodes (recall that one has to ungauge an overall U (1) in the resulting unframed quiver).

Explosion
Exploded quiver The resulting quiver has the global symmetry changed from SU(n) to SU(n) × U(1) n−1 . In [1] various computational checks for the duals of these examples were made, including calculating the Hilbert series, verifying the dimension of the global symmetry group and checking Nakajima's equality, and checking that torus reduction yielded the nilpotent cone.
We now extend this to orthosymplectic quivers following the extension in the mathematical literature in [4]. We adopt the usual convention in our diagrams that red nodes with label m denote orthogonal groups SO(m) 1 and blue nodes with label 2k denote symplectic groups USp(2k) = Sp(k). We recall that a USp(2k) gauge node is balanced if the neighboring SO(m j ) nodes satisfy: and an SO(m) node is balanced if the neighboring USp(2k j ) nodes satisfy: Note that for unframed unitary-orthosymplectic quivers, without SO(m) nodes with odd m, the gauge group G is take to be the product of the groups G i associated to the nodes i, modulo a diagonal Z 2 : This is not a choice of convention, but rather a necessity. Taking the gauge group to be i G i yields a different Coulomb branch which differs by a Z 2 quotient [7]. We start by looking at the Coulomb branch of quivers that are nilpotent cones of SO(2n) and SO(2n + 1). These are given explicitly in [8]. The unrefined Hilbert series of 1 The choice of SO or O is important here as the Coulomb branch is sensitive to discrete factors in the gauge groups. For orthosymplectic quivers that are closures of maximal nilpotent orbits of SO(2n) and SO(2n + 1), the Coulomb branch Hilbert series are computed explicitly in [6] and contains only SO and U Sp gauge groups. the nilpotent cone of G takes a simple form: where d i are the degrees of Casimir invariants of G and PE is the plethystic exponential. For quivers whose Coulomb branch is the nilpotent cone of SO(2n), the flavour node is SO(2n) which we explode into n U(1) = SO(2) gauge groups. The USp(2n − 2) node thus remains balanced, as do the nodes further down the chain. The Coulomb branch Hilbert series for the exploded quivers take a more complicated form and perturbative Hilbert series are presented in Table 1.
Let us demonstrate this for n = 3 which gives us the following explosion: Since the orthosymplectic quiver is unframed we ungauge a diagonal Z 2 , as stated in (2.4). The Coulomb branch Hilbert series can be readily computed and is found to be the same as the Coulomb branch Hilbert series of the exploded unitary quiver in (2.1) for n = 4. This is not surprising due to the isomorphism so(6) ∼ = su(4) but nevertheless validates our approach in extending the explosion procedure to orthosymplectic quivers. For general n, we thus have:

Explosion
Exploded quiver (2.7) The resulting global symmetry is the expected SO(2n) × U(1) n . The Hilbert series and plethystic logarithm for several members of this family are given in Table 1. The coefficient of t 2 is 2n 2 = n + 2n(2n − 1)/2 = rank SO(2n) + dim SO(2n) , (2.8) in agreement with the dimension of the global symmetry group. The group by which we hyperKähler quotient has rank n+2 n−1 i=1 i = n 2 , and the implosion has complex dimension 2n 2 so real dimension 4n 2 , so Nakajima's equality is satisfied.
For quivers whose Coulomb branch is the nilpotent cone of SO(2n + 1), the flavour node is an USp(2n) (the Langlands dual of SO(2n + 1)). After explosion, the flavour node explodes into n U(1) nodes. For purposes of the balancing condition at the SO(2n) node, the U(1) nodes behave the same as C 1 = USp(2) (see the discussion in [7], for example) hence the SO(2n) node remains balanced.

Explosion
Exploded quiver (2.9) The Hilbert series and plethystic logarithm for several members of this family are given in Table 2.
There might be concerns about the nature of the bouquet nodes and why they are U(1) rather than SO (2) or USp (2). For the so(2n) type quivers in (2.7), the SO(2n) flavour node explodes into n SO(2) nodes. Since SO(2) ∼ = U(1), we can use either one. If we take instead the so(2n + 1) quivers in (2.9), the flavour group is USp(2n) so in theory we can try to explode it into n USp (2) nodes. This computation is done in Table 3. Unfortunately, the global symmetry does not match the quivers we require for the explosion quiver family (since there are no U(1) n factors). However, it does give interesting results such as for n = 4, where there is a symmetry enhancement from SO(9) to F 4 .
Another argument for the explosion of the rank n flavour nodes into n U(1) gauge nodes follows from the property that taking the hyperKähler quotient of the exploded quivers by U(1) n returns the quiver for the nilpotent cone, going from right to left in (2.7) and (2.9). This is satisfied by all our exploded quivers, hence making them good candidates for unitary-orthosymplectic quiver counterparts to the well known cases with unitary quivers (2.1).
The SO(2n + 1) universal implosion should have symmetry group SO(2n + 1) × U(1) n , whose complex dimension is 2n(n + 1). We see that this appears as the t 2 coefficient in the Hilbert series for the displayed examples.
We can also check that this is consistent with Nakajima's picture. The group by which we perform the hyperKähler quotient has rank 2 n i=1 i = n(n + 1), and the implosion has real dimension 4n(n + 1), as expected.

HyperKähler quotient
Let us look at T [G], the nilpotent cone of G = SU(n), SO(2n), SO(2n + 1) and its explosion T [G]. As shown in [1], one can recover T [G] from the exploded quiver through hyperKähler quotient.
It is well known that orthosymplectic gauge groups lack the U(1) J topological symmetry that allows us to refine the Coulomb branch Hilbert series. However, we can partially refine the Coulomb branch Hilbert series by assigning fugacities z i with i = 1, . . . , n for the bouquet of n U(1) gauge nodes. The fugacities carry the U(1) J topological charge under each of the U(1) gauge nodes. On the level of the Hilbert series the hyperKähler quotient with respect to these charges takes the following form: where G = SU(n), SO(2n), SO(2n + 1). The ability to do this further justifies our choice of using a bouquet of n U(1) nodes for T [SO(2n + 1)] rather than a bouquet of n USp(2) gauge nodes. This is because USp(2) ∼ = SU(2) does not have a U(1) J topological symmetry with which we can refine our Hilbert series.
Let us demonstrate this with T [SO (5)]: An exact computation with the refined Hilbert series is time consuming so we checked and matched the results to order t 40 using perturbative computations.
Integer and half-integer magnetic lattice In [7], it has been pointed out for unframed orthosymplectic quivers made of SO(2n), USp(2k) and U(m) gauge groups, one can ungauge an overall diagonal Z 2 . The effect of ungauging this Z 2 translates to changing the magnetic lattice of the gauge groups to include half-integer magnetic charges. Therefore, we can write the Hilbert series as HS = HS Z + HS Z+ 1 2 where HS Z comes from integer magnetic charge contributions and HS Z+ 1 2 from half-integer magnetic charge contributions. One thing to notice is that all the operators from HS Z+ 1 2 carry non-trivial U(1) J charges from every U(1) node in the bouquet. Therefore, the contribution from HS Z+ 1 2 is zero after taking a hyperKähler quotient. This result is expected because taking a hyperKähler quotient over a U(1) gauge node is equivalent to turning it into a U(1) flavor node. Therefore, even if we quotient over one of the U(1) nodes in the bouquet, it changes the exploded quiver from an unframed orthosymplectic quiver to a framed orthosymplectic quiver. And for a framed orthosymplectic quiver, the magnetic lattice only contains integer magnetic charges and the Hilbert series is HS Z . 1+18t 2 +32t 3 +206t 4 +544t 5 +1993t 6 + 5344t 7 + 15531t 8 + 39040t 9 + 98672t 10 + O t 12  Table 1: Quivers whose Coulomb branches are the nilpotent cones of SO(2n) for n = 3, 4, 5, 6, 7 are fully 'exploded'. The SO(2n) flavour node turns into n U(1) gauge nodes. The Hilbert series and the plethystic logarithm are provided.  Table 2: Quivers whose Coulomb branches are the nilpotent cones of SO(2n + 1) for n = 4, 5, 6 are fully 'exploded'. The USp(2n) flavour node turns into n U(1) gauge nodes. The Hilbert series and the plethystic logarithm are provided.

Bouquet for Symplectic groups
Above, bouquets for G = SU(n), SO(2n), SO(2n + 1) are investigated. The reason we haven't investigated G = USp(2n) is because the orthosymplectic quivers are all 'bad'. In particular, some of the gauge nodes have negative balance. Therefore, the monopole formula diverges and we cannot test any of our conjectures through explicit computation. However, orthosymplectic quivers whose Coulomb branch are conjectured to be closures of G = USp(2n) nilpotent orbits had been studied through other means, in particular through brane configurations in [8][9][10]. With this, one can construct the quiver for T [USp(2n)] where the USp(2n) global symmetry can be read off following a new set of balancing conditions. Since the quivers are obtained from brane configurations which are not sensitive to the difference between SO and O gauge groups, we will stick with their algebras so in this section. For usp(2k), we need so(m j ) neighboring nodes such that 4k = j mj. For so(2m + 1), we need usp(2k j ) neighboring nodes such that 4m + 2 = j 2k j . For 2n nodes arranged linearly satisfying these balancing conditions will give a USp(2n). For T [USp(2n)], the flavor node is so(2n + 1) and the gauge node it is connected to is usp(2n). A very natural partition of the bouquet is into so(1) and n u(1)s. Here, we know that the global symmetry for U(1) with n flavors where n > 2 is U(1). Therefore, the global symmetry of the following explosion based on the balance of the gauge nodes give the expected G global = USp(2n) × U(1) n . Note that the sum of the ranks of the groups at the gauge nodes is n + n i=1 i + n−1 i=1 i = n(n + 1), as in the SO(2n + 1) case discussed in the previous section. However, the Coulomb branch for a SO(3) gauge group with n flavors has trivial global symmetry (no contribution at order t 2 in the Hilbert series) for n ≥ 3. Therefore, the global symmetry is USp(2n) × U(1) n−1 which is not the expected result for implosion. Based on the argument from global symmetry, we suggest the correct explosion is as in (3.1).
Comment on bad quivers. Since the quivers in this section are bad, and hence cannot be checked with current Hilbert series techniques, we should address what we know exactly about their moduli space. We will only consider the simplest example, the left quiver for n = 1 in (3.1). There is a choice to make, whether the leftmost gauge node is an SO (1) or O(1). The difference in moduli space is drastic. If we pick SO(1) then the theory is equivalent to USp(2) with 4 fundamental half hypermultiplets. The classical Higgs branch of this theory is the union of two cones, It is the nilpotent cone of O(4). There is a Z 2 = O(1) symmetry exchanging the two cones. The Coulomb branch of this theory is D 2 = (R 3 × S 1 )/Z 2 , where the Z 2 acts on both the R 3 and the S 1 . The resulting space has two singular points, which are both of A 1 type [11]. The O(1) symmetry exchanges the two singularities. The classical Higgs branch is distorted quantum mechanically. In the full moduli space each C 2 /Z 2 cone in the classical Higgs branch emanates from a different singularity of the Coulomb branch.
We can now gauge the O(1) symmetry, leading to the second choice in (3.1), the leftmost node now being O rather than SO. The resulting moduli space is much simpler. The two cones in the Higgs branch are identified with each other, as are the two singularities in the Coulomb branch. The Coulomb branch of this quiver is expected to be C 2 /Z 2 [8]. We can summarise this in Figure 1, depicting the moduli spaces by their Hasse diagram [12,13], using red for Coulomb branch and blue for Higgs branch directions.
Since the Coulomb branch in the latter case is a hyper-Kähler cone, its Hilbert Series is computable. However the monopole formula does not give the desired result, but rather diverges. This is because something goes wrong with the conformal dimension formula for 'bad' quivers. One could hope to adjust to this formula for certain cases, where the quiver is bad, but the Coulomb branch is a cone. We leave this for future work.

A A bouquet of USp(2) nodes
In this section, we list some interesting results for the explosion of T [SO(2n + 1)] quivers with the choice of a bouquet of USp(2) gauge nodes rather than U(1) gauge nodes that is discussed in Section 2.
For a USp(2) gauge group with n flavors, we need n ≥ 3, otherwise the gauge node has negative imbalance and the monopole formula diverges. The first such case is T [SO (7)  We see here that all the gauge nodes are balanced, yet the Coulomb branch Hilbert series diverges. This is not too surprising as a quiver can be bad even if none of the individual gauge nodes are bad. This is often observed for orthosymplectic quivers. For n ≥ 4, the Coulomb branch Hilbert series no longer diverges. Using the fact that the Coulomb branch of USp(2) gauge group with n flavors and n ≥ 3 has trivial global symmetry, we know that the global symmetry of the exploded quiver will just be SO(2n + 1). This is reported in Table 3. However, for n = 4, there is a contribution from HS Z+ 1 2 at order t 2 , hence enhancing the global symmetry. The enhancement from HS Z+ 1 2 comes in the spinor representation of SO(9) which results in F 4 of dimension 52.  Table 3: Quivers whose Coulomb branches are the nilpotent cones of SO(2n + 1) for n = 4, 5, 6 are fully 'exploded'. The USp(2n) flavour node turns into n USp(2) gauge nodes. The Hilbert series and the plethystic logarithm are provided.