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Coulomb Branches of Star-Shaped Quivers

We study the Coulomb branches of 3d N=4 `star-shaped' quiver gauge theories and their deformation quantizations, by applying algebraic techniques that have been developed in the mathematics and physics literature over the last few years. The algebraic techniques supply an abelianization map, which embeds the Coulomb-branch chiral ring into a vastly simpler abelian algebra A. Relations among chiral-ring operators, and their deformation quantization, are canonically induced from the embedding into A. In the case of star-shaped quivers -- whose Coulomb branches are related to Higgs branches of 4d N=2 theories of Class S -- this allows us to systematically verify known relations, to generalize them, and to quantize them. In the quantized setting, we find several new families of relations.


Introduction
The Coulomb branches of 3d N = 4 gauge theories have long been an object of physical and mathematical interest. Early physical studies [1,2] led to the discovery of 3d mirror symmetry [3][4][5], and related the Coulomb branch of ADE quiver gauge theories to moduli spaces of monopoles and instantons [6,7]. Unfortunately, non-perturbative corrections make the Coulomb branch difficult to analyze directly in non-abelian gauge theory. (Calculations of instanton corrections in simple non-abelian theories were carried out in e.g. [8,9], but quickly became impractical.) This difficulty was recently circumvented in a surprising confluence of physical [10][11][12][13] and mathematical [14][15][16][17][18] work, based on ideas from algebra, representation theory, and topological quantum field theory.
In this paper, we will apply some of the recent physical and mathematical techniques to study the Coulomb branch of star-shaped quiver (or simply "star quiver") gauge theories T N,k , shown in Figure 1. This gives a new, concrete perspective on generators and relations for the T N,k Coulomb branch chiral rings, supplementing known physical results and conjectures [19][20][21][22][23][24][25][26][27], as well as the recent geometric analysis in [28,29]. We also explicitly construct natural deformation quantizations of the chiral rings.
The 3d theories T N,k first came to prominence due to their relation [30] to 4d N = 2 theories of Class S [11,19]. Let T N [Σ 0,k ] denote the 4d theory of Class S defined by compactifying the 6d (2, 0) SCFT of type A N −1 on a sphere with k maximal punctures, and let T N [Σ 0,k ×S 1 ] denote its further compactification to three dimensions, a so-called "Sicilian" 3d N = 4 theory. It was argued by [30] that the star quiver theory T N,k is the 3d mirror of T N [Σ 0,k × S 1 ] (in the limit of zero S 1 radius). This implies several relations among moduli spaces: • The Coulomb branch M C of T N,k is isomorphic to the Higgs branch M 4d H of T N [Σ 0,k ] as a complex symplectic manifold. In particular, the rings of holomorphic functions on the two moduli spaces (which are particular chiral rings of local operators in the supersymmetric QFT's) agree in every complex structure (1.1) • As hyperkähler manifolds endowed with a Riemannian metric, M C and M 4d H will generally differ. In particular, M C depends on dimensionful parameters -the gauge couplings of the 3d quiver gauge theory -while M 4d H does not. However, M C should be isomorphic to M 4d H in the infrared limit where all gauge couplings are sent to infinity.
• Though not relevant for this paper, one also expects the Higgs branch of T N,k (which is easy to identify from the quiver as the hyperkähler quotient of k nilpotent cones in sl(N, C) by a diagonal SU (N ) isometry) to correspond to a particular decompactification limit of the Coulomb branch of T N [Σ 0,k × S 1 ] (which is a type-A Hitchin system on the k-punctured sphere [31]).
The 4d theories T N [Σ 0,k ] -and in particular the "trinion" theory at k = 3, which was simply called T N in [19] -are principal building blocks in the gluing construction of Class S theories. Their Higgs branches M 4d H were conjecturally used to define a "2d TQFT valued in holomorphic symplectic varieties" in [20], now fully constructed by [28] and [29]. However, despite their prominent role, it has been relatively difficult to analyze the Higgs branches of general theories T N [Σ 0,k ] directly, because for N ≥ 3 and k ≥ 3 these 4d theories are non-Lagrangian.
Some of what's known about the Higgs branches of T N [Σ 0,k ] includes their dimension dim C M 4d H = (kN + 2)(N − 1) , (1.2) and the existence of an SU (N ) k hyperkähler isometry. For low N and k, one has [19,32,33]  and for N = 2, 3 and general k ≥ 3 the Higgs branches can obtained by a gluing procedure [20,33], as hyperkähler reductions of products of (1.3). More generally, a putative set of generators and (partial) relations for the chiral rings C[M 4d H ] at k = 3 and any N ≥ 3 were uncovered in a series of papers [21][22][23][24][25][26], nicely summarized in the review [27]. These putative generators admit a natural generalization to any k ≥ 3, N ≥ 3.
One of our main motivations was to obtain new information about the structure of the Higgs branches M 4d H via a direct analysis of the corresponding Coulomb branches M C of T N,k , for general N and k. The Hilbert series of the chiral ring C[ H ] was computed in [34] with this perspective in mind. However, one can now do much better, producing actual ring elements and relations among them.
To achieve this, we will follow the "abelianization" approach of [12], which corresponds to fixed-point localization in the equivariant (co)homology of [13,15,16]. The basic idea of abelianization is to embed the Coulomb-branch chiral ring C[M C ] of a non-abelian theory into a much larger -but much simpler -abelian algebra A (1. 4) In physical terms, A is the local Coulomb-branch chiral ring near a generic point on the Coulomb branch, where the gauge group has been broken to its maximal torus. The algebra A has extremely simple generators and relations. Moreover, it has a simple Poisson structure, a simple deformation quantization, and a simple extension over twister space. Thus, embedding C[M C ] → A immediately allows one to • verify relations among elements of C[M C ] (chiral ring relations) • identify the Poisson structure on C[M C ], and its deformation quantization • extend the algebra C[M C ] over twistor space, and thereby access the hyperkähler structure on M C .
In the initial work [12], the precise image of the embedding (1.4) was only identified in a handful of examples; however, at least in principle, a complete combinatorial construction of the image has since been described by Webster [18].
In the case of T N,k theories, we will identify the putative generators of C[M C ] proposed by [21][22][23][24][25] (from a 4d Higgs-branch perspective) as elements of A. We will show how to explicitly verify and then quantize the conjectured relations among them.
An important insight in the derivation of C[M 4d H ] chiral-ring relations in [24] was that various generators could be "diagonalized," as tensors for the SU (N ) k flavor symmetry. We find that the abelian algebra A plays a surprisingly important role in this diagonalization. In particular, the eigenvalues of the generators, which are complicated algebraic functions on the actual moduli space M C ≈ M 4d H , turn out to be extremely simple monomials in the algebra A. This allows the entire diagonalization procedure to be deformation-quantized.
From the perspective of 4d Higgs branches, the fact that the chiral ring C[M 4d H ] admits a deformation quantization may not be obvious. However, this extra structure is completely natural (and physical) in 3d Coulomb branches. Indeed, in the recent mathematical/TQFT constructions of Coulomb branches [12,13,[15][16][17][18], one typically works with quantized algebras from the very beginning. In physical terms, the Poisson structure in the chiral ring of 3d N = 4 theories arises from topological descent in the Rozansky-Witten twist [35][36][37], and quantization comes from turning on an Omega background [38,39]. (See also [40,41]. An analogous quantization arising from an Omega background in four dimensions is familiar from [42][43][44][45][46]. ) We note that when k = 1 or k = 2, the expected relation between the Coulomb branch of T N,k and the Higgs branch of T N [Σ 0,k ] breaks down. Neither the 3d nor the 4d theories are CFT's in this case. Nevertheless, the Coulomb branch of T N,k is still a well-defined hyperkähler manifold, in fact a smooth manifold. We will see explicitly that the Coulombbranch chiral rings of T N,k are consistent with k = 1 : M C T * SL(N, C)// ψ N , k = 2 : M C T * SL(N, C) , (1.5) where T * SL(N, C)// ψ N is the Kostant-Whittaker symplectic reduction of the cotangent bundle. The spaces in (1.5) agree perfectly with those assigned to 1-and 2-punctured spheres by the Moore-Tachikawa TQFT [20]. 1 One of our initial goals was to prove that the finite set of generators proposed by [21][22][23][24][25][26] really do generate the entire chiral ring C[ H ]. Unfortunately, this remains an open question. It appears that identifying a finite set of generators for the Coulomb-branch chiral ring of a nonabelian 3d theory is a rather difficult problem in general. It would be useful to develop methods to address this in the future.

Other connections and future directions
Our work is related to several other ideas that would be interesting to explore. For example: 1 Taking some care with scaling limits, the spaces (1.5) can also be related to the Higgs branches of the 6d (2,0) theory compactified on one-or two-punctured spheres, even though they are not Higgs branches of 4d CFT's. 1. In upcoming work [28], Ginzburg and Kazhdan propose a geometric definition for the Higgs-branch chiral ring of 4d T N [Σ 0,k ] theories. The proposal was shown in [29] to agree with the mathematical structure of the Coulomb branch in T N,k theories. However, the proposed definition is not elementary: it involves the equivariant cohomology of a certain perverse sheaf over the affine Grassmannian for SL(N, C). It would be interesting to decipher how the relevant cohomology classes match the physically motivated generators and relations of C[M C ] discussed in [21][22][23][24][25] and in this paper.
2. There are many expected relations among geometric structures on the Higgs and Coulomb branches of 3d N = 4 theories -for example, relations among cohomology rings as in Hikita's conjecture [15,47,48], and symplectic duality of module categories associated to the Higgs and Coulomb chiral rings [49,50]. It could be interesting to investigate the structure of these relations for star quivers.
3. The methods in this paper can be extended to 3d quiver gauge theories associated to other punctured spheres in Class S: non-maximal punctures in type A, as well as various punctures in type D. The 4d N = 2 theories obtained by gluing spheres with more general punctures (and in more general types) participate in an intricate web of dualities, cf. [19,33,[51][52][53][54]; and our methods should allow a comparison of 4d Higgs branches across the dualities. 4. The deformation-quantization C ε [M C ] of the Coulomb-branch algebras of star quivers, which as explained above is natural in 3d, should define the basic building blocks for a quantized version of Moore-Tachikawa's "TQFT valued in holomorphic symplectic varieties" [20]. In particular, one should find a TQFT that assigns a quantum algebra to any punctured 2d surface, with gluing implemented by quantum symplectic reduction.

Organization
Section 2 is a brief review of known and conjectured relations in C[M C ] for the theories T N,k , as well as a summary of our main results in this paper, including an explicit presentation of a set of "diagonalized" operators in the abelianized algebra A that are expected to produce all relations in C[M C ]. We present the quantization of these operators and their relations. Section 3 reviews the general structure of Coulomb branch chiral rings in 3d N = 4 gauge theories, and the algebraic techniques used to analyze them. (In Appendix A we connect to the mathematical approach of Braverman-Finkelberg-Nakajima.) Of particular interest is the abelianized algebra A that contains the chiral ring, as in (1.4), and its quantization A ε . We also define a subalgebra W ε ⊂ A ε that, due to Webster [18], helps us characterize the image of the embedding (1.4).
We then consider some special families of T N,k theories, building our way up to the general case, working almost exclusively with quantum algebras. Section 4 analyzes "small" star quivers with N = 2 and arbitrarily many legs. For k = 3 legs we observe how the simple Coulomb branch T * C 4 ( = the familiar Higgs branch of the T 2 trinion theory) is recovered. For all k we identify the moment maps for the SU (2) k flavor symmetry and a collection of operators furnishing a k-fold fundamental representation of SU (2) k . The simultaneous diagonalization of the moment maps and the k-fold fundamental, accessible via the the quantum abelianized algebra A ε , results in a dramatic simplification of expressions.
Section 5 considers the complementary family of linear quivers with k = 1 leg but with N arbitrary. A new feature here is the appearance of antisymmetric tensors of the SU (N ) flavor symmetry. We find that the change of basis that diagonalizes the SU (N ) moment map vastly simplifies the antisymmetric tensors.
In Section 6 we then generalize to arbitrary T N,k star quivers. Many properties of their Coulomb branches may be inferred by combining the results of the previous two sections. In particular, by working with diagonalized operators, relations among moment maps and antisymmetric tensors are easy to determine from the one-legged k = 1 analysis.
We conclude with two short Sections 7, 8 that connect our general results with some important and well-studied examples. Namely, we explain how our characterization of chiral rings for k = 1 and k = 2 quivers relates to the geometric spaces (1.5) (Kostant-Whittaker reduction and the cotangent bundle of SL(N, C)); and for k = 3 we discuss the generalizations we have found of chiral-ring relations in 4d N = 2 T N trinion theories.
In the appendices, in addition to collecting various computations and interesting examples of quantum chiral-ring relations, we include a summary of the BFN construction C ε [M C ] [15,16], from the perspective of physical TQFT.

Summary of results
Before delving into the algebraic analysis of 3d N = 4 Coulomb branches, we whet the reader's appetite with some results. We review known and conjectured relations in the chiral rings of T N,k theories for k = 3. Then we summarize the general structure found in this paper for arbitrary k, including quantum generalizations of known relations, and a handful of new relations that only appear upon quantization.
The quiver gauge theory T N,3 has an SU (N ) 3 flavor symmetry acting on the Coulomb branch, which induces a holomorphic SL(N, C) 3 action on the chiral ring C[M C ]. As reviewed further below in Section 3.6, this means that there must exist a triplet of complex moment map operators in the chiral ring, µ a ∈ sl(N, C) * , a = 1, 2, 3 . (2.1) We denote the components of the moment maps as (µ a ) i j . The entire chiral ring is believed to be generated by the components of the moment maps as well as a collection of operators Q (r) ∈ ∧ r ⊗ ∧ r ⊗ ∧ r , Q (r) ∈ (∧ r ) * ⊗ (∧ r ) * ⊗ (∧ r ) * r = 1, 2, ..., N − 1 (2.2) in the r-th antisymmetric tensor representations of SL(N, C) 3 and their duals. We denote the components of Q (r) and Q (r) as Q IJK The Q (r) and Q (r) are not independent, obeying IJK . Here is the totally antisymmetric tensor of SL(N, C), normalized so that 12...N = 12...N = 1.
The chiral ring is also graded by charge under a U (1) subgroup of the SU (2) R-symmetry acting on the Coulomb-branch. For a CFT, this R-charge coincides with dimension. The Rcharges of the above generators are The most important nontrivial relations among the generators are and more generally (2.7) The first relation (2.5) says that all the Casimir operators built from the moment maps are equal. This implies that at generic points on the Coulomb branch, where the moment maps can be diagonalized, the eigenvalues of µ 1 , µ 2 , and µ 3 will all coincide. It helps to be somewhat explicit about this: at generic points on the Coulomb branch there should exist three invertible matrices (S a ) i j such that for all a = 1, 2, 3, where (m 1 , ..., m N ) is the common set of eigenvalues, satisfying i m i = 0. The second pair of relations (2.6) implies that at generic points on the Coulomb branch all the tri-fundamental and tri-antifundamental Q's can be diagonalized, by the same similarity transformation that diagonalizes the moment maps. In other words and for some "eigenvalues" q i and q i . Due to (2.7), the remaining Q (r) , Q (r) operators can be simultaneously diagonalized exactly the same way, with eigenvalues that we denote q I (r) = q [i 1 ...ir] and q (r) From the diagonalized perspective, all the information in the chiral ring has been repackaged in the eigenvalues m i , q I , q I and the three similarity transformations S a .
The remaining known chiral-ring relations may be found in [24][25][26]. They come in two basic types, contractions that relate Q IJK Q I J K to a product of moment maps; and equivalences among products of tri-fundamentals (Q (1) ) r and the higher anti-symmetric powers Q (r) . The two simplest relations are where c are coefficients of the characteristic polynomial P (x) = det(x1 − µ a ) = N =0 c x N − (due to (2.5), these are independent of the choice of a = 1, 2, or 3); and 2 (2.13) A more general version of (2.12) appears in [25,App. A]. A generalization of (2.13) was discovered 3 by [26] for N = 4, and extended to all N in [27]: and similarly for the antifundamentals. This can be written more suggestively as (2.14) In App. D we provide a list of miscellaneous relations computed for small N, k which include variants of (2.13) for different rank tensors as well as variants of (2.14) for N = 4.
We use these computations to predict relations of the very general form for any N and 1 ≤ r ≤ N . Near generic points on the Coulomb branch where diagonalization is possible, it has been conjectured [27] that all possible relations reduce to is the product of determinants of the similarity matrices.
Typically it is assumed that det(S a ) = 1, though we will find it convenient to keep the determinants generic. As mentioned in the introduction, the construction of the 3d Coulomb-branch chiral ring C[M C ] will proceed by embedding the ring into an abelianized algebra A, which is much larger but has canonical generators and relations (see Section 3.4). Somewhat miraculously, we will be able to find explicit similarity transformations S a whose entries belong to A, and eigenvalues m i , q i , q i that are simple monomials in A.

Extending to general k, and quantizing
For k = 3 we will verify that there exist operators in the Coulomb-branch chiral ring of T N,3 with all the expected properties and relations outlined above. More so, we will extend the (putative) generators and relations to general k, and deformation-quantize them. The key to extending the analysis to general k rests in implementing a uniform diagonalization procedure.
Recall that the quantized Coulomb-branch chiral ring C ε [M C ] of T N,k is an associative algebra. For general k, it has an SL(N, C) k symmetry generated by taking commutators with moment maps {µ a } k a=1 . 4 In analogy with k = 3 above, we also prove that there exist 4 In the quantum setting, saying a symmetry is generated by the moment maps means that the infinitesimal action of a Lie algebra generator T on any operator O ∈ Cε[MC ] is given by the commutator 1 ε [ T, µa , O]. As ε → 0, this reduces to a Poisson bracket. See Section 3 for more details.
operators Q I 1 ...I k (r) and Q (r) that transform in the k-fold anti-symmetric tensor representations and their duals, We strongly suspect (but do not prove) that the moment maps and the Q (r) , Q (r) are a complete set of generators. Some of the basic relations among these operators are easy to guess and to verify, even in the quantum setting. In particular, if we define ε-shifted moment map operatorsμ a = µ a − N −1 2 ε 1, the relations (2.5)-(2.7) become Thus, in a suitable quantum sense, the "eigenvalues" of the µ a are independent of the choice of leg, and we may expect to diagonalize the moment maps and all the Q (r) , Q (r) operators simultaneously.
We explicitly perform the diagonalization by constructing similarity transformations S a , one for each leg. The entries of the S a and their inverses S −1 a take values in the abelianized quantum algebra A ε , which contains operators that exist at generic points of the Coulomb branch. Applying the similarity transformations in the right order we obtain with i m i = 0, as well as The eigenvalues m i , q i and q i are again elements of the abelianized operator algebra A ε . All the higher antisymmetric tensors Q (r) and Q (r) get diagonalized in a similar way, with eigenvalues The eigenvalues m i , q I , q I generate an especially simple quantum algebra. Commutation with the moment-map eigenvalues measures charges of the q i and q i , while products of fundamental q's are related to higher tensor powers via In addition, we find ..i N | is a fully symmetric N -index tensor, and we have also introduced which in our conventions is a nontrivial operator. 5 Note that the set of relations (2.23) reduce to the classical expressions in (2.16) as ε → 0, for k = 3. Moreover, they are consistent with the R-charge assignments We also emphasize that the q's do not commute with each other for generic k (though the q i do commute amongst themselves when k = 1). Instead, their commutators are determined by (2.23a) and (2.23b). For example, In principle, relations among the actual chiral-ring operators µ a , Q (r) , Q (r) could be obtained by judiciously applying S a and S −1 a to "un-diagonalize" the simple relations (2.23) above. In practice, this is quite difficult to do -it is known to even be difficult for k = 3, in the classical ε → 0 limit. Nevertheless, we do identify a handful of nontrivial un-diagonalized relations.
For example, the quantum version of (2.12) for k = 3 is a straightforward generalization The element σ is almost central in the algebra of eigenvalues. It may also be written as σ = We prove that this quantum relation holds in Section 8. Similarly, the quantum version of the k = 3, N = 4 relation (2.14) is obtained by replacing µ 1 withμ 1 : .
We find evidence that, for general N and r, the quantized version of (2.15) is given by Some further relations among the Q's and µ's for small N and k, generalizing the products (2.13) to the quantum setting and to k = 3, are summarized in Appendix D. As a simple example, at N = 2 the relation between first and second-order antisymmetric tensors (which are just products of Levi-Civita tensors when N = 2) un-diagonalizes to , and the c 's are as in (2.26).

New quantum relations
Working in the quantized chiral ring also leads to some new identities whose classical limits vanish. The simplest of these relate commutators of k-fold fundamental tensor operators to higher tensor powers.
In Appendix C.2 we prove that when k ≤ 3 the commutators of fundamental tensors are anti-symmetric under any exchange of indices 6 and similarly for ( between tri-fundamental and tri-antisymmetric-tensor operators. We have verified (2.31) by direct computation for N ≤ 4. In the ε → 0 limit, (2.31) clearly reduces to a Poisson bracket For higher-rank tensors, the commutators must be generalized. An obvious choice would be to consider a full 3-fold antisymmetrization of r copies of Q to get rank-r tensor operators. Alternatively, based on the general features described in Appendix C.3, it is natural to consider the recursive definition σ is the r-cycle (1 2...r) and, σ n a means apply σ n to the set {i a , j a , ..., k a }. 7 We have checked directly for N ≤ 4 that both full-anti-symmetrization and (2.32) agree with the form of higher-rank tensors Q I 1 I 2 I 3 given in the main text, weighted by an appropriate number of ε's to soak up the R-charge, 3 Review: the Coulomb branch chiral ring In this section, we review the construction of the Coulomb branch chiral ring C[M C ] of a 3d N = 4 gauge theory, following recent advances in the math and physics literature. In particular, we will incorporate mathematical results of Webster's [18] into the physical understanding of the chiral ring. We keep much of the discussion general. We assume that the gauge theory is defined by a renormalizable Lagrangian, with compact gauge group G coupled to linear matter (hypermultiplets) in a quaternionic representation R. We further assume that R is a direct sum of unitary representations The only additional parameters that the theory may depend on are real gauge couplings, masses, and FI parameters.

Generalities
Recall that the Coulomb branch of a 3d N = 4 gauge theory is a component of the moduli space of vacua on which all hypermultiplet vevs vanish, and on which vectormultiplet scalars generically acquire diagonal vevs, breaking the gauge symmetry G to its maximal torus T . The Coulomb branch is a noncompact hyperkähler manifold [2,57], possibly singular, of dimension dim C M C = 2 rank(G) . In a 3d gauge theory, the Coulomb branch has an exact SU (2) C metric isometry that rotates its CP 1 of complex structures. Thus it essentially looks the same in every complex structure. The SU (2) C is part of the R-symmetry of the 3d N = 4 theory, and shows up classically as a rotation of the triplet of g-valued scalar fields in the vectormultiplet. For example, in a T N,k quiver gauge theory the gauge group is 8 The dimension of the Coulomb branch is therefore easily computed as In any fixed complex structure, the Coulomb branch is a holomorphic symplectic manifold, i.e. a Kähler manifold, possibly singular, whose smooth part is endowed with a nondegenerate holomorphic two-form Ω. For every choice of complex structure, there is a chiral ring of half-BPS local operators whose vevs are holomorphic functions on the Coulomb branch. We simply denote this ring suppressing the dependence on complex structure. The holomorphic symplectic form Ω endows the chiral ring with a Poisson bracket, thus turning C[M C ] into a Poisson algebra. Physically, the Poisson bracket of operators may be computed by topological descent [37].

Fibration: scalars and monopoles
In a fixed complex structure, the Coulomb branch moreover has the structure a complex integrable system. 9 Specifically, the Coulomb branch is a singular fibration where t C denotes the complexified Cartan subalgebra of G, W the Weyl group, and T ∨ C the complexified dual of the maximal torus. Roughly speaking, the base t C /W C rank(G) is parameterized by the 'diagonal' expectation value of a complex vectormultiplet scalar The complex scalar ϕ combines two of the three real vectormultiplet scalars, as dictated by the choice of complex structure. Classically, it is forced to take a diagonal vev due to vacuum 8 The overall U (1) quotient is standard in quivers with no "flavor nodes" or "framing"; it makes sense because none of the bifundamental hypermultiplets are charged under the diagonal U (1). 9 This integrable system is a degeneration of the Seiberg-Witten integrable system familiar from 4d N = 2 gauge theory [58][59][60].
equations [ϕ, ϕ † ] = 0. Global coordinates on the base come from G-invariant polynomials (Casimir operators) in ϕ, which are the true gauge-invariant operators in a non-abelian theory.
We call a point ϕ on the base generic if 1) it fully breaks gauge symmetry to the torus (making all W-bosons massive) and 2) gives a nonzero effective mass to every hypermultiplet. Algebraically, these criteria mean that, respectively and Mathematically, one would say that a generic point of t C /W is in the complement of all weight and root hyperplanes. The fiber of the integrable system (3.6) above any generic point of the base is a complex dual torus T ∨ C (C * ) rank(G) . It is a holomorphic Lagrangian torus with respect to the holomorphic symplectic structure. The coordinates on the fibers are vevs of chiral monopole operators. Locally, near a generic point on the base where G is broken to T , one may define half-BPS abelian monopole operators as (cf. [2,3,61]) where g is the gauge coupling, A ∈ t is a cocharacter (satisfying e 2πiA = I), σ ∈ t is the third real vectormultiplet scalar, γ ∈ t are the dual photons (with periodicity 2πg 2 ), and ( , ) is the Cartan-Killing form. The OPE of monopole operators satisfies v A v B ∼ v A+B , for any cocharacters A and B, so their vevs are just right to produce global functions on T ∨ C (C * ) rank(G) . The way that the T ∨ C fibers vary over the base of the Coulomb branch depends qualitatively on locations of the root and weight hyperplanes. Roughly speaking, • The fibers blow up (their volume diverges) above root hyperplanes, where W-bosons become massless and gauge symmetry is enhanced.
• The fibers collapse above weight hyperplanes, where hypermultiplets become massless.
The precise hyperkähler metric on the fibration acquires non-perturbative quantum corrections that are extremely difficult to compute directly.

TQFT and non-renormalization
Nevertheless, if one ignores the hyperkähler metric and focuses on M C as a complex symplectic manifold, many computations become tractable. In particular, the computation of the chiral ring C[M C ] and its Poisson structure (as well as its deformation quantization) reduces to a relatively simple algebra problem. There are two ways to think about this simplification. In [12] it was argued that the chiral ring of a 3d N = 4 gauge theory is independent of the gauge coupling, and thus cannot receive nonperturbative quantum corrections, or perturbative corrections beyond one loop.
Alternatively, one may recognize that the chiral ring C[M C ] belongs to a topological subsector of the 3d gauge theory. Specifically, the chiral-ring operators are in the cohomology of a topological supercharge Q, which was discussed long ago in [35], and may equivalently be characterized as (cf. [13,15,37]) -the 3d reduction of the 4d N = 2 Donaldson supercharge -one of the scalars under a diagonal subgroup of SU (2) Lorentz × SU (2) H (where SU (2) H is the R-symmetry that rotates hypermultiplet scalars) -the "twisted Rozansky-Witten" supercharge, as it plays the same role on the Coulomb branch that the Rozansky-Witten twist plays for the Higgs branch .
Then the product of chiral-ring operators is topologically protected, and may be computed using standard TQFT methods. Perhaps surprisingly, the Poisson bracket and deformation quantization (via Omega background) of chiral-ring operators are also topological in nature [37]. The TQFT perspective motivated the initial mathematical work [15,16] on Coulomb branches. In Appendix A, we explain how the mathematical characterization of Coulombbranch operators relates to the physics of 3d N = 4 theories. The TQFT perspective has some important computational consequences, which we draw on in what follows.

The abelianized algebra A
The TQFT derivation of the ring C[M C ] (in Appendix A) proceeds via reduction to 1d quantum mechanics, where C[M C ] is identified as the equivariant cohomology (or more technically, Borel-Moore homology) of a certain moduli space. Fixed-point localization embeds the chiral ring into a much simpler "abelianized" algebra A, (3.10) Physically speaking, one may think of A as a local algebra of operators near generic points on the Coulomb branch, where the gauge theory is effectively abelian; this is how the abelian algebra A arose in [12]. 10 Similarly, in an Omega-background both C[M C ] and A are deformationquantized, and one finds an embedding of associative algebras All the computations in this paper will take place in A or A ε . We review their structure here. Since A can be recovered from A ε by sending ε → 0, it would be sufficient to describe A ε . However, some relations are simpler and more intuitive for A, so we shall start with the commutative case.
The algebra A can be defined as the local chiral ring, in the neighborhood of a generic point ϕ on the base of the Coulomb branch, in the sense of (3.8). To make this precise, we denote the loci on the base of the Coulomb branch were W-bosons and hypermultiplets become massless as Then we define M abel as the open subset of the Coulomb branch sitting above the complement of ∆ and ∆ R in the fibration (3.6); and define M abel C to be the trivial W -cover of M abel C (undoing the quotient by the Weyl group on the base). Then (3.14) This definition of A makes it obvious that there is an embedding (3.10), since any global function on M C defines a W -invariant local function on M abel C . The algebra A has two types of generators: 1. Rational functions in the components of the abelian complex scalar ϕ ∈ t C , whose denominators vanish only on ∆ and ∆ R .
2. Abelian monopole operators v A as in (3.9), for every cocharacter These operators satisfy relations that are essentially the expected product relations v A v B ∼ v A+B among monopole operators, with one-loop corrections from hypermultiplets and Wbosons.
To write down the relations, we first recall that there is a natural integer-valued product λ, A ∈ Z (3. 16) between weights λ and cocharacters. Then the classical relation v A v −A = 1 among abelian monopole operators is corrected by hypermultiplets and W-bosons to and more generally The abelianized algebra A simply contains polynomials in ϕ, 1/M α , and v A , modulo these relations: 11

Quantization
The quantized algebra A ε is similar. It is generated by 1. The components of ϕ, and ε. The parameter ε is central; and the components of ϕ (and the (M α,λ + nε) −1 ) all commute with each other. Otherwise, the generators satisfy two basic sets of relations: First, note that the components of ϕ can all be picked out by contraction with weights, e.g. λ, ϕ . All linear functions in ϕ arise this way. The commutator of any such linear function and a monopole operator is It follows from (3.20) that the inverted masses also satisfy (e.g.) v A 1 Mα+nε = 1 Mα+(n− α,A )ε v A . Second, the product of two abelianized monopole operators is given by an appropriately ordered and shifted version of (3.18) : is a quantum-corrected power. These relations were derived using abelian mirror symmetry in [12], but also follow from an equivariant cohomology (TQFT) computation [13,15]. Altogether, the quantized algebra is  Nevertheless, the precise image of C ε [M C ] in A ε can be tricky to identify. A few structural properties of the embedding map were discussed in [12]. For example: • The image of C ε [M C ] must sit in the Weyl-invariant subalgebra A W ε ⊂ A ε , since local operators in the full non-abelian gauge theory are gauge invariant.
• In A ε one finds arbitrarily large negative powers of the masses M α,λ + nε. In the case of W-boson masses, this is unavoidable, due to denominators in the products v A v −A .
In contrast, the image of C ε [M C ] in A ε cannot contain any of the elements 1 M α,λ +nε themselves, since operators in C ε [M C ] must define (as ε → 0) global functions on the Coulomb branch that extend smoothly across the discriminant locus.
It is also known how a basis for C[M C ] as an infinite-dimensional vector space should be indexed [10]. Physically, one expects that the elements of C[M C ] are monopole operators V A,p(ϕ) labelled by dominant cocharacters A (equivalently, by Weyl orbits in the cocharacter lattice) and dressed by polynomials p(ϕ) of ϕ ∈ t C that are invariant under the stabilizer W A of A in the Weyl group. For example, if A = 0, the "dressing factors" are just standard Weyl-invariant polynomials C[ϕ] W . Formally, we might write It is unclear whether these structural properties alone can determine how elements of C[M C ] (or C ε [M C ]) embed in A (or A ε ). However, much stronger constraints on the embedding come from the mathematical/TQFT perspective. In fact, the TQFT construction of the chiral ring gives -in principle -a complete answer to the embedding problem. Namely, elements of C ε [M C ] are identified with equivariant cohomology classes on a certain moduli space; and the embedding C ε [M C ] → A ε just expresses these classes in terms of equivariant fixed points.
It can still be very difficult to explicitly analyze equivariant cohomology classes in practice. Fortunately, Webster [18] recently outlined a combinatorial calculus that accomplishes this task for Coulomb branches. We will discuss the physical meaning of Webster's calculus in [65]. In the current paper, we take a pragmatic approach and use one simple consequence of Webster's combinatorics: the image of C ε [M C ] in A ε must always contain a particular subalgebra W ε (defined momentarily), In the case of star quivers T N,k , we will identify all expected generators of C ε [M C ] as elements of W ε . We in fact suspect that though this is not guaranteed. 12 The algebra W ε is defined as follows. One begins with a subalgebra of A ε generated by polynomials in ϕ and by rescaled monopole operators (3.28) These u A monopole operators, carrying additional factors associated to the W-boson masses, have the nice property that their products never generate denominators: we simply have with one-loop corrections from the hypermultiplets alone. Otherwise, the usual relations continue to hold for any weight λ and cocharacter A. In addition, for each root α, let s α ∈ W denote the corresponding simple reflection. Recall that the Weyl group is generated by the s α 's. We may adjoin the s α to the algebra of 12 Unfortunately, some rather complicated combinatorics are required to make up the difference between Wε . In any pure gauge theory, this equality follows from results of [66,67]. In linear quiver gauge theories, all the generators and relations of Cε[MC ] are known explicitly [12,17,68], and equality Wε = Cε[MC ] is also easy to establish. We thus have some hope that equality may hold for star quivers as well.
ϕ's and u A 's, in such a way that the s α 's satisfy the standard Weyl-group relations among themselves, and natural commutation relations where A α is the reflected cocharacter, and ϕ α is the reflected element of t C . Finally, for each α, introduce the BGG-Demazure operator 13 The algebra W ε is defined as the Weyl-invariant part of an algebra generated by 1) polynomials in ϕ; 2) the u A monopole operators; and 3) the BGG-Demazure operators: The relations, which we leave implicit, are of the form (3.29), (3.30), (3.31). Notice that once Weyl-invariance is imposed, all the s α 's are all projected out, so W ε does become an actual subalgebra of A ε . Practically speaking, the role of the Demazure operators θ α is to introduce a few denominators 1 Mα , in a controlled way, so that the structural properties of the Coulomb branch discussed above are actually satisfied. In Appendix B we will work through how (3.33) reproduces the chiral ring W ε = C ε [M C ] in the elementary example of pure G = P SU (2) gauge theory.

Flavor symmetry and R-symmetry
We finally comment briefly on symmetries of 3d N = 4 theories, in particular those applicable to star quivers.
Flavor symmetries act either on the Higgs branch or on the Coulomb branch, as tri-Hamiltonian isometries. The symmetry group F acting on the Higgs branch is easy to identify in a gauge theory, as the normalizer of G in U Sp(R) i.e. the group that acts on hypermultiplets independently of G. For star quivers, there is no Higgs flavor symmetry at all, F = 1. In general, complex mass parameters associated to the Higgs flavor symmetry (scalars in the F vector multiplet) can deform the Coulomb-branch chiral ring; but for star quivers such deformations are absent, and Coulomb branches are rigid.
In contrast, star quivers have a rich Coulomb-branch flavor symmetry. In the UV, the Coulomb-branch flavor group K is the Pontryagin dual of π 1 (G) which is an abelian group with the same rank as the center of G. In the case of star quivers, we easily find In the IR the group K may undergo a nonabelian enhancement, controlled by the "balanced" nodes in a given quiver [61,69,70], i.e. nodes N c that are coupled to exactly N f = 2N c hypermultiplets. For star quivers, the nodes on the legs are always balanced, so there is an In addition, in two special cases the central node is balanced as well, leading to Since the chiral ring C[M C ] is insensitive to RG flow, the fully enhanced IR symmetry group K will act on it. More so, since C[M C ] is a holomorphic object, the complexification K C will actually act. This action is generated by complex moment map operators µ ∈ C[M C ] ⊗ Lie(K) * , which are related to the K currents by supersymmetry. In particular, for star quivers, (3.37) implies that one is guaranteed to find k separate sl(N, C) * -valued moment maps µ = (µ 1 , µ 2 , ..., µ k ) in the chiral ring. They generate the action via Poisson brackets.
The K C action extends to the quantized chiral ring C ε [M C ], where it is generated by taking commutators (rather than Poisson brackets) with moment maps. Explicitly, if T ∈ sl(N, C) k is a generator of the Lie algebra, and we denote by µ T = T, µ ∈ C ε [M C ] the contraction of T and µ, there must be commutation relations 39) and the infinitesimal T action on any other operator O is In addition to flavor symmetries, 3d N = 4 gauge theories with linear matter also have an SU (2) C × SU (2) H R-symmetry. The two factors act on the Coulomb and Higgs branches, respectively, but in a way that rotates the hyperkähler CP 1 's of complex structures rather than as tri-holomorphic isometries. The SU (2) C acting on the Coulomb branch is important to us. Any fixed complex structure on the Coulomb branch is preserved by a U (1) R subgroup of SU (2) C , which induces into a complexified C * action on the chiral ring C[M C ]. The C * action extends to the quantized C ε [M C ], in such a way that the quantization parameter ε and all moment maps canonically have charge 14 The R-charges of some other expected operators were summarized in (2.24).
In the abelianized chiral ring A ε , the complex ϕ scalars also necessarily have [ϕ] = 1. It then follows from monopole products (3.22) (or in fact the simpler commutative (3.17)) that This is consistent with physical expectations for monopole charges [61,69,70]. If a 3d N = 4 gauge theory flows to a CFT, the C * charges of chiral-ring operators coincide with their conformal dimensions, and must therefore be strictly positive. Star quivers flow to CFT's when N ≥ 2 and k ≥ 3; in this case the positivity of R-charges is manifest in (2.24).

Short quivers (N = 2)
We now begin chiral-ring computations in earnest. Many features of general T N,k star quiver theories already appear in "short" quivers that have N = 2, i.e. an arbitrary number k of legs of length one surrounding a central node. (See Figure 2 for k = 4.) These theories are especially computationally friendly, and we work through them in detail as a warm-up for later material.

k = 3: M C C 8
We begin with the three-legged quiver T 2,3 . In this case the dual 4d theory of Class S is the basic A 1 trinion theory, i.e. a theory of free half-hypermultiplets in the tri-fundamental representation of the SU (2) 3 flavor symmetry [19]. Correspondingly, we expect to find a simple 3d Coulomb branch The way this arises from a 3d perspective turns out to be rather nontrivial. Naively, the gauge group of T 2,3 is U (2) × U (1) 3 . The hypermultiplets sit in three fundamental representations of U (2), each charged under a separate U (1). As discussed in Section 3.1, the a diagonal U (1) diag subgroup of U (2) × U (1) 3 acts trivially on the hypermultiplets, so the true gauge group is actually a quotient Correspondingly, the cocharacter lattice that will label monopole charges is which we may understand as 5-tuples of integers modulo the 1-dimensional sublattice generated by A diag = (1, 1; 1, 1, 1). In other words, two cocharacters A, A ∈ Z 5 are equivalent if they differ by an integer multiple of A diag . Dually, the weight lattice of G may be identified with 5-tuples of integers that sum to zero Note that there is a well-defined product , : weights(G) × cochar(G) → Z. In particular, λ, A diag = 0 for any weight λ. The matter representation may now be written as R = R⊕R * , with weights of R chosen to be

The A ε and W ε algebras
Our first step in constructing the Coulomb branch is to identify the abelianized algebra A ε from Section 3.4, which contains all putative Coulomb-branch operators. We work from the outset with its quantized version. As described in Section 3.4, there are three types of generators: 1. Polynomials in Omega-background parameter ε, the complex scalars ϕ a 1 , a = 1, 2, 3 corresponding to the U (1) factors in G, and the diagonal components (ϕ 21 , ϕ 22 ) of the complex scalar corresponding to the U (2) factor.
Due to the U (1) diag quotient, we should restrict to polynomials that are invariant under a simultaneous translation of all the ϕ's. It is natural to think of such polynomials as generated by weights of G, i.e. by the linear functions The constraint λ 1 + λ 2 + λ 1 + λ 2 + λ 3 = 0 guarantees that λ, ϕ is invariant under translations.
3. The abelian monopole operators v A labelled by cocharacters A ∈ cochar(G) as above.
All monopole operators with diagonal cocharacter v nA diag = v (n,n;n,n,n) are central in the algebra, and we impose the relations v (n,n;n,n,n) = 1 ∀ n . (4.9) The next intermediary step is to construct the subalgebra W ε ⊂ A ε from Section 3.5. It will help us decide which elements of A ε are actual chiral-ring operators.
To this end, we introduce the rescaled monopole operators u A as in (3.28), whose products contain no denominators. For example, we have We also introduce the single Weyl reflection s that generates the Weyl group Z 2 . It satisfies s 2 = 1 and acts on monopoles by reflecting their cocharacters: Similarly, sϕ a 1 = ϕ a 1 s and sϕ 21 = ϕ 22 s. The corresponding BGG-Demazure operator is Some important elements of W ε , which are assured to belong to the the full chiral ring (4.12) These are the undressed nonabelian monopole operators labelled by a fundamental cocharacter on the central U (2) node. The dressed nonabelian monopoles are simply In addition, W ε contains monopoles charged only under the legs (which are trivially Weyl- and all Weyl-invariant polynomials in the ϕ's. These are all the operators we will need to generate C ε [M C ]!

Moment maps
The theory T 2,3 has an SU (2) 3 flavor symmetry acting on its Coulomb branch (described in Section 3.6), and a corresponding SL(2, C) 3 symmetry in the chiral ring. This symmetry should be generated by three sl(2, C) * -valued complex moment maps µ a , a = 1, 2, 3. Each of these moment maps is associated to a leg of the quiver. Each leg looks like a copy of T [SU (2)] theory, and effectively treats the central node as a flavor symmetry. We may therefore import well known results from the chiral ring of T [SU (2)] (studied e.g. in [12,70]) to identify the moment maps. The raising and lowering operators in the moment maps turn out to be instances of (4.14) We may check expected sl(2, C) commutation relations. A quick application of (3.22) yields as well as can be identified as three mutually commuting sl(2, C) triples. These operators fit into moment maps as The shift by ε 2 is included for later convenience. It does not affect the action generated by the moment map; in particular, letting H = 1 0 0 −1 ∈ sl(2, C) be the Cartan element we find H, µ a = H,μ a = Tr(Hµ T a ) = Tr(Hμ T a ) = 2ϕ a 1 − ϕ 21 − ϕ 22 . Note that, using the general R-charge formula (3.42), we have as required for moment maps.

Tri-fundamentals
Having identified the moment maps, we may organize the chiral ring into SL(2, C) 3 representations. It is easy to check using (3.22) that the operator which is of type (4.12), is a "tri"-lowest-weight vector. Namely, By acting with raising operators on Q 222 we then produce an entire eight-dimensional trifundamental representation. For example, etc. The complete list of operators in this representation is summarized in Table 1.
Operator Table 1. Expressions for the eight operators furnishing a tri-fundamental representation of the SL(2, C) 3 action on the chiral ring of the T 2,3 star quiver. These eight operators generate the entire chiral ring. The first expression of the operator is in terms of the Weyl symmetrized image of a rescaled monopole operator under the BGG-Demazure operator θ. The second two expressions are related to one another by adding a diagonal cocharacter A diag .
Alternatively, we could have observed that ) is a trihighest-weight vector, which generates an eight-dimensional tri-antifundamental representation. However, it is equivalent to the tri-fundamental above. In particular, since cocharacters (4.4) that differ by a multiple of A diag are equivalent, we actually have Q 222 = −Q 111 , and more generally We also note that the R-charge formula (3.42) quickly implies that (4.26)

Relations
Using the above expressions for the tri-fundamental operators, it is a straightforward application of (3.20) and (3.22) to find additional relations satisfied by the Q's and µ's. For example, there are commutation relations (4.28) and more generally Sending ε → 0, these recover the Poisson brackets We may also consider contractions between moment maps and Q's. Schematically writing we use (3.22) to find (4.31) More generally, for all n ≥ 0, we may contract with powers of the moment maps to get (4.32) Note that the RHS of (4.31) contains an alternative dressed version of the fundamental nonabelian monopole operators.
Finally, we can recover the moment maps themselves as contractions of Q's, and similarly for ( It is straightforward but tedious to show that the Q i 1 i 2 i 3 operators generate all of W ε . In this case we know from duality with free half-hypermultiplets in 4d that these tri-fundamental operators really generate the entire chiral ring as desired in (3.27).

Diagonalization for k = 3
The T 2,3 theory provides a first, nontrivial example of the diagonalization procedure that was previewed in Section 2.2.

Moment maps
Let's first consider the moment maps associated to each leg of the quiver. In the "classical" limit ε → 0, we expect the Casimir operators built out of any of the moment maps to depend only on the ϕ 2i scalars on the central node. Indeed, viewing each leg as a copy of T [SU (2)] theory as in (4.15), we know from [12,70] that the Casimirs depend only on the mass parameters associated to the 2 flavor node, and these masses become the ϕ 2i scalars upon gauging the U (2) symmetry to tie the quiver together. In particular, in the ε → 0 limit we expect This relation is also well known from the dual perspective of 4d N = 2 trinion theories. It implies in particular that at generic points on the Coulomb branch all the moment maps should have coincident eigenvalues.
In the current quantum setting, we can similarly compute traces of powers of the quantum moment-map operators (4.20). Using the ε-shifted moment mapsμ a , we find (4.36) This similarly suggests that we may "generically" be able to diagonalize the quantum moment maps, and find coincident eigenvalues. In the quantum setting, "generically" will mean working in the abelianized algebra A ε . In order to diagonalizeμ a , it is helpful to find its eigenvectors. Using (3.22) we find that the vectors 16 which are related by the Z 2 Weyl symmetry, satisfy (with no summation over a) We thus arrange them into a matrix suggestively denoted S −1 a = (η a 1 η a 2 ). With a little bit of work, it is possible to find a (two-sided) inverse of this matrix: which, not too surprisingly, takes the form of two row vectors that are related by action of the Weyl group. With these matrices, it follows that It is worth noting that by choosing an ordering of the eigenvectors η a 1 , η a 2 the Weyl symmetry is explicitly broken and so any objects obtained by acting with S a or S −1 a should not be expected to be Weyl invariant. It is also worth noting that the components of S a and S −1 An immediate application of this diagonalization is to obtain the polynomial invariants of the moment map Tr[(μ a ) n ] for all n ≥ 0. Using the S matrices we now have Unfortunately, when ε = 0, the trace is no longer cyclic. Nonetheless, it is straightforward to compute the diagonal components: quickly leading to These explicitly only dependent on ε and the scalars on the central node for all n and a, so

Tri-fundamentals
Having diagonalized the moment maps, we may hope to diagonalize the tri-(anti)fundamentals Q i 1 i 2 i 3 and Q i 1 i 2 i 3 with the same S a , S −1 a matrices. Indeed, the quantum relation (4.31), which generalizes the well-known 4d trinion relations (2.6), strongly suggests that this is possible.
Let us therefore define Somewhat amazingly, a straightforward computation reveals where are simple abelian monopole operators in A ε , charged only under the central node! This parallels the fact that the moment-map eigenvalues (4.40) were scalars on the central node. Similarly, where Notice that the magnetic charge of q i , q i under the central U (2) node matches the charges of the Q i 1 i 2 i 3 , Q i 1 i 2 i 3 operators as representations of the SL(2, C) 3 flavor symmetry. In particular, the q i 's have magnetic charges corresponding to weights of the fundamental representation of U (2), and the q i 's have magnetic charges corresponding to the antifundamental representation. 17 The lack of perfect symmetry between q i and q i (in particular, the more complicated prefactors for q i ) is due to our particular choice of similarity matrices S a , S −1 a . In our conventions, the determinants are nontrivial, 18 49) and similarly for the other S a and S −1 a . We could have chosen the determinants to be 1, but only at the expense of introducing roots in the matrix entries. We strongly prefer instead to use matrix entries that are manifestly elements of the algebra A ε .
Despite the determinants being nontrivial, they do satisfy Moreover, the products of determinants are relatively simple expressions, only involving monopoles charged under the central node.

Algebra of eigenvalues
Let us end the discussion of T 2,3 by explicitly describing some relations in the algebra of eigenvalues, giving a simple example of the structure in Section 2.2.
We emphasize that (4.56) is a diagonalized form of the chiral-ring relation (4.25). In the diagonalized basis, the product of determinants σ is required to pass between fundamental and antifundamental operators.
The structures we found above at k = 3 generalize in a surprisingly straightforward way to general k. At k = 3, the algebra C ε [M C ] was freely generated by the tri-fundamental operators Q i 1 i 2 i 3 (in particular, moment maps could be recovered as products of Q's), in correspondence with the fact that the Coulomb branch itself is a flat space M C = C 8 . For k < 3 there are additional relations among the Q's; while for k > 3 we will need both Q's and moment maps as independent generators. However, the embedding of C ε [M C ] into the abelian algebra A ε and the process of diagonalization look almost identical.
We note that, in contrast, the Higgs branches of the corresponding 4d Class S theories T 2 [Σ 0,k ] are not usually described in a manner that is uniform for all k. By decomposing the k-punctured sphere into k − 2 pairs of pants, one finds that the 4d Higgs branch is a holomorphic symplectic quotient However, there are many different ways to choose a decomposition (4.57), and so many ways to express the same quotient (4.58); none is canonical, and none makes the generators and relations of the chiral ring manifest. (The combinatorics of the spaces (4.58) were studied carefully by [71].) The current 3d-Coulomb-branch perspective thus has some marked advantages.
The gauge group of the T 2,k theory is a diagonal quotient (U (2) × U (1) k )/U (1) diag . Correspondingly, the cocharacter and weight lattices are As before, we denote by (ϕ 21  The abelian algebra A ε is then generated by The subalgebra W ε that is contained in the chiral ring (and which we conjecture is equal to the chiral ring) is generated just as in Section 4.1.1 by the rescaled monopoles u A , the Weyl reflection s (s 2 = 1) and the BGG-Demazure for any B = (B 1 , ..., B k ).

Moment maps and k-fold fundamentals
There is now an SL(2, C) k action on the Coulomb-branch chiral ring, generated by moment maps associated to each "T [SU (2)]" leg (4.15). Since the moment maps on separate legs are decoupled, they take exactly the same form (4.20) as for k = 3 (but now with a = 1, ..., k).
In particular, the sl(2, C) triplets are  Table 1. Explicitly, and similarly for Q i 1 i 2 ...i k . The fundamental and antifundamental operators are manifestly related by (4.67)

Diagonalization
Since the moment maps for general k look identical to those at k = 3, we easily generalize  More explicitly, for each leg a = 1, ..., k we may introduce pairs of matrices satisfying S a S −1 a = S −1 a S a = 1 and the components of S ± a commute with the components of S ± b ,μ b andμ diag for a = b. These matrices diagonalize the moment maps tõ Now consider applying the operator S 1 ⊗ ... ⊗ S k to Q i 1 ...i k (resp. S −1 1 ⊗ ... ⊗ S −1 k to Q i 1 ...i k ). In complete analogy with the k = 3 result, we find with eigenvalues Again, rather beautifully, the eigenvalues involve abelian monopole operators charged only under the central node.
Let us analyze the subalgebra of A ε that is generated by the eigenvalues, as we did for T 2,3 above. With moment-map eigenvalues m 1 = −m 2 = (ϕ 21 − ϕ 22 )/2, we find that and similarly For odd k, we may write (4.75) as but for even k we require a slight modification As before, the product of determinants is very close to central. We may write σ = (ϕ 22 − ϕ 21 ) k w + , where w + = v (1,1; 0) is actually central in the algebra of eigenvalues.

Un-diagonalization
The chiral ring relation i 1 j 1 ... i k j k Q j 1 ...j k = Q i 1 ...i k and the diagonalized relations (4.77) are intertwined by acting with the similarity transformations S 1 ⊗ ... ⊗ S k . This relationship is neatly summarized in the commutative diagram of Figure 3.
We may also consider applyingμ diag to δ i 1 i ...δ i k i q i and then un-diagonalizing. By choosing to contact S −1 1 withμ diag , we find However, since contracting S −1 1 withμ diag was an arbitrary choice (i.e. we could have contracted any of the S −1 a ), it follows that A similar argument can also be used in applyingμ a to Q i 1 ...i k to obtain When analyzing 4d Higgs branches of trinion theories, equations (4.80) are usually a starting point, used to deduce the fact that the Q tensors can be diagonalized. Here, in the 3d Coulomb-branch analysis, it is actually easier to describe the explicit diagonalization first, and then deduce (4.80).
With the T 2,k quivers well understood, we now consider the other extreme: the T N,1 star quivers that have just a single leg, of arbitrary length: This is a linear quiver, with gauge group and flavor symmetry SU (N ). The Coulomb branch of such quivers was studied in [7], where it was related to a moduli space of P SU (N ) monopoles. The quantized Coulomb-branch chiral ring of linear quivers was analyzed in detail in [12]. Our main (novel) goal here is to describe the diagonalization of the moment map associated to the flavor symmetry of this quiver, we will then upgrade our analysis to general T N,k theories in Section 6. We note that the T N,1 star quiver is a small modification of the T [SU (N )] quiver [70]: In Section 7.1, we will use our understanding of the algebra C ε [M C ] to explain how (5.4) comes about.

Conventions
Abstractly, the cocharacter lattice of G is Z 1 2 N (N +1) /Z diag . We will write cocharacters as A = ( a N ; a N −1 , . . . , a 1 ) , a n ∈ Z n , where each a n = (a n1 , ..., a nn ) corresponds to the U (n) cocharacter on the n-th node; and we identify any two cocharacters whose difference is a multiple of A diag = ( 1; 1, ..., 1).
It is also useful to introduce basis elements for the cocharacter lattice (prior to the quotient by A diag ). For 1 ≤ n ≤ N and 1 ≤ α ≤ n, let constrained so that the total sum of entries in the λ vector is zero. We can use differences of the e nα 's from (5.6) (now re-interpreted as weights) as a basis for the weight lattice. The hypermultiplets sit in bifundamental representations associated to the edges of the T N,1 quiver. We choose to split the representation as R = R ⊕ R * , such that the weights of R are weights(R) = e nα − e n−1β , (5.8) with 2 ≤ n ≤ N , 1 ≤ α ≤ n, and 1 ≤ β ≤ n − 1.
Generalizing the notation of Section 4.1, we also introduce complex scalars ϕ n = (ϕ n1 , ..., ϕ nn ) for each U (n) node. They may be assembled in a vector The contractions of ϕ with weights (or roots) are then given by and are naturally invariant under simultaneous shifts of each component of ϕ.

Moment map and diagonalization
The Coulomb-branch flavor symmetry of T N,1 is SU (N ). Correspondingly, the chiral ring must contain a complex moment map µ ∈ sl(N, C) * generating a complexified SL(N, C) action. Borrowing results of [12] for the T [SU (N )] quiver (5.3), we find that the Chevalley-Serre generators of sl(N, C) are are Weyl-invariant sums of the scalars on the n-th node. It is clear that Φ n belong to W ε ; and we can see that V ± n also belong to W ε by writing them as where [...] W as usual denotes symmetrization over the Weyl group. The Chevalley-Serre generators can be used to construct the complex moment map operator, which altogether takes the form It turns out that, in the abelianized algebra A ε , the moment map can be explicitly diagonalized. Recall that working in A ε is the quantum equivalent of working at "generic" points in the Coulomb branch. With a bit of work, we find right eigenvectors for i = 1, ..., n, which generalize the N = 2 formulas (4.37). Note that the eigenvectors and their eigenvalues are permuted by the S N Weyl symmetry associated to the terminal N node of the quiver. We assemble these eigenvectors into a similarity transformation Its two-sided inverse is given by with rows The diagonal form of the moment map thus becomes It is worth noting that the determinants of the similarity transformations are given by There seems to be no way to restore symmetry between det S and det S −1 (or make them unimodular) while keeping S and S −1 valued in A ε -restoring symmetry would introduce roots.

Tensor operators
As mentioned at the beginning of this section, the chiral ring of T 1,N contains additional operators in arbitrary antisymmetric tensor powers of the fundamental and antifundamental representations of SL(N, C). We would like to identify them as elements of the abelianized algebra A ε (more precisely, as element of W ε ), and then investigate how the similarity matrices act on them.
In building the moment map, we used monopole operators charged under all nodes of the quiver except the first U (N ) node. So, let us now consider with fundamental magnetic charge on the U (N ) node. Comparing with the N = 2 analysis from Section 4, it is natural to guess that the operator Q N generates a fundamental representation. 19 We prove in Appendix C.1 that it is indeed a lowest-weight vector, in the sense that [F n , V + N ] = 0 ∀ n (5.29) 19 Without knowledge of the N = 2 results, another easy way to obtain the fundamental and antifundamental operators is by decomposing the adjoint representation of SU (N + 1) into SU (N ) irreducibles. By inspection of (5.18), it follows that the fundamental (resp. antifundamental) representation is generated by the operator The remaining operators in the fundamental representation may be expressed as Weylsymmetric sums In other words, the Q i are nonabelian monopole operators with fundamental magnetic charge on the U (N ) node through the U (i) node. The antifundamental representation is similar. It is generated by the highest-weight vector Q N = −V − N , and more generally contains Weyl-symmetric sums In order to construct higher antisymmetric tensor representations ∧ r , we consider monopole operators whose magnetic charges on the various nodes are those of antisymmetric tensors. For example, a collection of operators Q [i 1 i 2 ] (1 ≤ i 1 < i 2 ≤ N ) furnishing the representation ∧ 2 may be constructed as Weyl-symmetric sums . (5.37) With the overall signs chosen above, this translates to a general operator relation ...θ n(n−k) θ n(n−k−1) )(θ n(n−1) ...θ n(n−k+1) θ n(n−k) ) (5.39) and n = 1 Θ from which it follows that Q I and Q I belong to W ε . Alternatively, this follows from the facts that 1) Q N = V + N and Q N = −V − N are elements of W ε due to (5.17); 2) the remaining Q i , Q i in the (anti)fundamental representations are elements of W ε because they are obtained from Q N , Q N by taking commutators with moment maps; and 3) the higher tensors Q [i 1 ...ir] , Q [i 1 ...ir] are in W ε because they can be written as products of the Q i , Q i -see (5.51) below.

New basis for the tensors
For k = 1, the Q operators are already "diagonalized." Nevertheless, it is natural to ask how they are represented in the basis of eigenvectors of the moment map. The results will help us later in Section 6 to diagonalize the Q's for general T N,k quivers.
Let us define and more generally In this basis, the magnetic charge has been fully shifted to the U (N ) node! The (anti)fundamental q i (q i ) are magnetically charged in a (anti)fundamental representation of U (N ). Similarly, we find for higher tensors that and, with the same general form but a more complicated prefactor, (5.45b) Once again, all magnetic charge has shifted to the U (N ) node, and has been abelianized. The base-changed ∧ r operators now have magnetic charges in the ∧ r representations of the central U (N ). 21 Just as in the N = 2 theories of Section 4,, the asymmetry in prefactors of (5.45) is a result of asymmetry in our choice of similarity transformations. In the new basis, the relation between fundamental and antifundamental powers is expressed as 46) 20 We have checked (5.43) and instances of (5.45) explicitly for N ≥ 5, though do not yet have a general proof for all N . 21 We note again that when referring to magnetic charge (i.e. a cocharacter) as a representation, we mean a representation of the Langlands-dual group. Here U (N ) is its own Langlands-dual, so there is no confusion.
where I is a multi-index of length r and I is of length N − r. Thus, the "undiagonalized" relation (5.38) is corrected by determinants.
In the new basis it is fairly easy to relate the fundamental q i , q i to higher tensor powers. Letting I = {i 1 , ..., i r }, we find by direct computation that and where I c = {1, ..., N }\I is the complementary subset, and e I ∈ Z N is a vector with 1 entries for positions in I and zeroes elsewhere. 22 Combining these formulas with (5.45), we arrive at These expressions will be used momentarily to determine how to express Q I and Q I in terms of products of Q i and Q i .

Relations
As mentioned at the beginning of this section, we expect the chiral ring C ε [M C ] of the onelegged quiver T N,1 to be generated by the moment map and the fundamental Q i 's alone. In principle, the relations between fundamental Q i 's and higher tensors can be obtained by carefully applying quantum similarity transformations to the simple relations (5.49) above. As a preliminary step, it is helpful to move the denominators on the right side of (say) (5.49a) to the left side. By multiplying both sides on the left and commuting the factors ϕ N j − ϕ N i through some of the q i 's, we can bring the relation to the form This is suggestive of an un-diagonalized relation where (μ n Q) i := (μ n ) i j Q j . At least classically, (5.51) diagonalized precisely to (5.50). By explicit computation for N ≤ 4 and r ≤ N , we find that (5.51) holds even at the quantum level and we conjecture that this relation holds for all N and r. (The explicit computations appear in Appendix D.) By working out the explicit commutation relations between the components of S, S −1 , q i , q i , one could imagine deriving (5.51) from (5.50) in full generality. We do not do this here. The analogous formula for antifundamentals is given by where (Qμ n ) i := Q j (μ n ) j i . We also expect additional relations obtained from (5.51) and (5.52) via (signed) permutations of the (μ n Q) i and (Qμ n ) i factors. 23

The general case
The structure we found for "short" k-legged quivers in Section 4 and "long" single-legged quivers in Section 5 generalizes in a straightforward way to general T N,k star quivers. Indeed, analyzing general T N,k quivers is largely a matter of bookkeeping.
We shall describe the general results here, starting with the identification of moment maps and fundamental/antifundamental tensors (and higher tensors) as elements of the abelianized algebra W ε ⊂ A ε ; then diagonalizing the tensors; and finally using diagonalized relations to derive (or motivate) a general collection of relations in C ε [M C ].

Conventions
The gauge group of T N,k is Correspondingly, the cocharacter lattice is (Z N + k 2 N (N −1) )/Z diag . When needed, we write explicit cocharacters as vectors 2) 23 To better understand these formulae, consider the contraction ofμ with Q i . Taking the form ofμ given above and diligently using (3.22), one finds that this contraction is a telescoping sum, and A nearly identical formula appears for the antifundamental operator (μQ)j :=μ j j Q j ; one simply tacks on an appropriate factor of ϕNα N,j − 1 N ΦN − (N − 1)ε to each abelianized monopole found in Qj. The main feature of note is that the coefficients in each of these expressions only depend on the scalars on the central node. When moving onto multi-legged quivers this will greatly simplify computations. In particular, the operation of contracting a moment map (with its respective tensor index) will have the same result regardless of which moment map is used. subject to an identification A ∼ A if A − A is a multiple of A diag = ( 1; ....; 1) (i.e. the vector with every entry equal to 1). Here A N = (A N 1 , ..., A N N ) is the cocharacter associated to the central U (N ) node of the quiver, while A a n = (A a n1 , ..., A a nn ) is the cocharacter associated to the U (n) node on the a-th leg, with 1 ≤ n ≤ N − 1 and 1 ≤ a ≤ k. In a similar way, we denote the diagonal gauge scalars as where ϕ N = (ϕ N α ) N α=1 and ϕ a n = (ϕ a nα ) n α=1 for 1 ≤ n ≤ N − 1 . As before, it is helpful to introduce basis vectors for the unquotiented cocharacter lattice. Let e N α be the cocharacter with A N α = 1 and all remaining entries (A N β and all A a n ) set to zero. Similarly, let e a nα be the cocharacter with A a nα = 1 and all remaining entries set to zero. The weight lattice of G is Z N + k 2 N (N −1)−1 , consisting of vectors whose entries sum to zero. We can interpret differences of the e N α , e a nα as elements of the weight lattice. The weights of the hypermultiplet representation R = R ⊕ R * are taken to be The abelianized algebra A ε is generated in the usual way by the entries of ϕ, monopole operators v A , inverted roots and inverted weights The subalgebra W ε , in which we will find all the components of our moment maps and k-fold tensors, is generated by ϕ, rescaled monopoles u A , simple Weyl reflections s N α , s a nα (on the central U (N ) node and the U (n) a nodes on the a-th leg, respectively), and corresponding BGG-Demazure operators θ N α , θ a nα .

Moment maps and tensor operators
The chiral ring now has a k a=1 SL(N, C) a action, generated by moment maps µ a , a = 1, ..., k. These moment maps are associated to each leg of the star quiver, and are decoupled from each other. Indeed, each µ a may be identified as a copy of (5.18) from Section 5.2. Explicitly, for each n = 1, ..., N − 1 and a = 1, ..., k we may define nonabelian monopole operators with (anti)fundamental magnetic charge for U (n) a . Since V a± n ∝ θ a nn−1 ...θ a n2 θ a n1 u ±e a n1 W , it is an element of W ε . Similarly, we introduce Weyl-symmetric scalars Φ a n = n α=1 ϕ a nα (6.9) on each node. Then the Chevalley-Serre generators of SL(N, C) a are E a n = V a+ n , F a n = −V a− n , H a n = 2Φ a n − Φ a n−1 − Φ a n+1 (n = 1, ..., N − 1) , (6.10) with the convention that Φ a N ≡ Φ N and Φ 0 = 0. They fit into the ε-shifted moment maps µ a just as in (5.18). Since eachμ a only contains monopoles (and scalars) charged under nodes on the a-th leg, the components of the moment maps commute with each other, . Generalizing Sections 4 and 5, the operators in k-fold fundamental and antifundamental representations now arise from adding magnetic charge on the central node. Let be the nonabelian monopoles of (anti)fundamental magnetic charge on the central node. Then we show in Appendix C.1 that are simultaneous lowest-weight and highest-weight vectors (respectively) for every copy of SL(N, C) a . By taking repeated commutators with the moment-map components E a n (resp. F a n ), they generate k-fold fundamental and (resp. antifundamental) representations of SL(N, C) k . We denote the remaining operators in these representations as Following the pattern of Section 5.3, we find that Q i 1 ...i k is a nonabelian monopole operator with fundamental magnetic charge for the central U (N ) node as well as the U (n − 1) a , ..., U (i a ) a nodes of each a-th leg. The operators Q i 1 ...i k are similar, involving antifundamental magnetic charges instead. To find the higher k-fold antisymmetric tensor representations, we repeat the same process, starting with nonabelian monopole operators that have the corresponding k-fold antisymmetric magnetic charge on the central node. Namely, for any 1 ≤ r ≤ N , we may consider the the Weyl-averaged operator This is again a simultaneous lowest-weight vector for every SL(N, C) a action, and generates an entire k-fold r-th antisymmetric power representation ∧ r 1 ⊗ · · · ⊗ ∧ r k . The remaining operators in the representation may be denoted where which is a natural genealization of (5.38). By a generalization of (5.41), it follows that Q I 1 ...I k and Q I 1 ...I k are all elements of W ε . We will assume that the chiral ring C ε [M C ] is entirely generated by the components of the moment mapsμ a , and the operators in the various tensor representations Q I 1 ...I k and Q I 1 ...I k . Since these operators are all in W ε they must (by the discussion in Section 3.5) belong to C ε [M C ]. We do not have a general proof that they generate.
Let us also take a moment to summarize R-charges of various operators. Using the general formula (3.42), we find that where in the final row we have the ∧ r 1 ⊗ · · · ⊗ ∧ r k operators and their duals.

Diagonalization and diagonalized relations
Since the moment mapsμ a on different legs decouple from each other, the quantum similarity transformations S a , S −1 a that diagonalizeμ a are simply obtained by specializing the formulas (5.22), (5.23) to the a-th leg. (In other words, starting from (5.22), (5.23) we simply replace ϕ nα → ϕ a nα and e nα → e a nα to get S a and S −1 a .) Then for every a, Note that the eigenvalues m i := ϕ N i − 1 N Φ N are independent of the choice of leg, as they only contain scalars on the central node.
Just as in the T 2,k theories of Section 4, the matrix elements of the S a and S −1 a commute with the components ofμ b , S b and S −1 b for b = a as well as with the components ofμ diag . We can now use the similarity transformations to diagonalize the k-fold tensor operators. By generalizing explicit computations for low values of k and N , we find that This is a rather amazing simplification. One could have guessed that diagonalization should be possible from the chiral-ring relations (6.32) and (6.33) below. A nice surprise, however, is that the eigenvalues q i , q i are extremely simple elements of the abelianized algebra A ε , containing abelian monopoles charged only under the central node. Similarly, we can use S a and S −1 a to diagonalize the operators in the ∧ r 1 ⊗· · ·⊗∧ r k and ∧ r 1 ⊗ · · · ⊗ ∧ r k . Letting I = [i 1 ...i r ] be an antisymmetric multi-index and I c = {1....N }\I its complement (as a set), we find eigenvalues (6.24) Again, these are proportional to abelian monopole operators on the central node alone, with magnetic charges corresponding to a weight space in a single copy of the ∧ r or ∧ r representation. The eigenvalues m i , q I , q I satisfy the relations that were anticipated back in Section 2.2. It follows easily from the part of A ε associated to the central node that Moreover, introducing the product of row-determinants (noting that σ is not quite central among the eigenvalues, though w + := v ( 1; 0;...; 0) is), we obtain the simple relations These are the diagonalized versions of (6.18). The passage between fundamental and antifundamental representations, before and after diagonalization, is summarized in the commutative diagram of Figure 4. Straightforward algebra in A ε also gives us (6.28) and relates products of (anti)fundamental q i , q i to higher tensor powers as (6.29) We also point out that the R-charges of the eigenvalues m i , q I , q I are the same as the Rcharges of the original operators µ, Q I 1 ...I k , Q I 1 ...I k . This follows from the general computation of R-charges (3.42) in A ε , but is not entirely obvious from the formulas above. In particular,

Un-diagonalized relations
Many of the relations in the quantum chiral ring C ε [M C ] of T N,k follow immediately from the diagonalization above, and from discussions in the previous sections.  for all n ≥ 0. 24 Another set of relations that follows immediately from the earlier comments, in particular the derivation of (4.80), is (6.32) as well as These relations naturally extend to any rank and one finds that  where I a = {i a,1 , ..., i a,r }. Finally, in analogy with (5.51) at k = 1, we expect to be able to relate products of fundamental Q's to higher tensor powers. Heuristically, this should come from un-diagonalizing (6.29). We can write (say) the first equation in (6.29) more suggestively as Just as in the k = 1 case, the LHS classically arises (up to a numerical factor) from diagonalizing 25 ( while the RHS classically arises (up to a numerical factor) from diagonalizing together these imply that, at least classically, there should be a relation of the form ..I k (6.39a) and similarly for the antifundamentals (6.39b) where the constants of proportionality are purely numerical.
The quantum corrections to these relations are highly nontrivial. In the quantum case, we expect a general form and for some tensors T 1 , T 1 constructed out ofμ 1 , with R-charge 1 2 (k−1)r(r−1). The tensors T 1 , T 1 can be quite complicated and we have not yet found a general formula for them. Appendix D contains examples of relations (6.40) for several small values of N , k and r. See also (2.29).
We do expect several additional relations among the moment maps and tensor operators. They are increasingly difficult to guess; though once a putative relation is written down, it is straightforward to verify using the abelianized algebra A ε . (In principle, all relations in C ε [M C ] are induced from the embedding of generators into A ε .) One family of additional relations should correspond to un-diagonalizing (6.28). They should relate partial contractions of the multi-index operators Q I 1 ...I k Q I 1 ...I k to moment maps. Specific examples of such relations at k = 2 and k = 3 are discussed in Sections 7 and 8. For k = 3 we will also propose a special set of relations between products of fundamentals and higher antisymmetric tensor operators, anticipated in (2.28).

Geometry of one-and two-legged quivers
As mentioned in the introduction, the Coulomb branches of one-and two-legged star quivers coincide with well known geometric spaces. Namely, the Coulomb branch of T N,2 is the cotangent bundle of the complex group SL(N, C), while the Coulomb branch of T N,1 is the so-called Kostant-Whittaker reduction of T * SL(N, C), These are precisely the spaces assigned to one-and two-punctured spheres by the Moore-Tachikawa TQFT [20]. (Taking some care with scaling limits, these spaces can also be related to Higgs branches of the 6d (2,0) theory compactified on one-or two-punctured spheres.) In this section, we use the generators and relations of chiral rings from Sections 5-6 to explain how (7.1) and (7.2) come about.

Kostant-Whittaker reduction
To discuss the geometry of one-legged quivers, we begin by recalling what the Kostant-Whittaker reduction T * SL(N, C)// ψ N means. In physics, this space has played a major in the analysis of "punctures" in 4d N = 2 theories of class S, and in the Nahm pole boundary condition of 4d N = 4 theories.
First, T * SL(N, C) is the cotangent bundle of the complex group SL(N, C). It is canonically a complex symplectic space. The cotangent bundle is trivial, so we have T * SL(N, C) sl(N, C) * × SL(N, C). However, it is naturally trivial in not one but two different ways, which correspond to identifying the fiber at the identity 1 ∈ SL(N, C) with either leftinvariant or right-invariant one-forms on the group. A convenient way to parameterize points of T * SL(N, C) is in terms of triples Here µ L and µ R are the complex moment maps for the action of left and right multiplication of SL(N, C) on itself, extended to T * SL(N, C) as complex Hamiltonian actions. Either moment map can be used to parameterize the fiber at 1; µ L , µ R correspond precisely to the left-invariant and right-invariant trivializations of the cotangent bundle. Under the action of left and right multiplication, the triplet (µ L , g, µ R ) transforms as (7.4) The group N ⊂ SL(N, C) is a subgroup of unipotent matrices, i.e. the exponential of a nilpotent subalgebra of sl(N, C). We take it to contain upper-triangular matrices of the form We assume that N acts on SL(N, C) on the right. Then the moment map for the induced Hamiltonian N action on T * SL(N, C) may be identified as the strictly-lower-triangular part of µ R ∈ sl(N, C) * . The holomorphic symplectic quotient T * SL(N, C)// ψ N is now constructed in two steps. First, one fixes the complex moment map for the N action to a generic character ψ. Explicitly, this means fixing the form of µ R to be µ R =         * * * · · · * ψ 1 * * · · · * 0 ψ 2 * · · · * . . . . . . . . .
with fixed nonzero complex numbers ψ 1 , ..., ψ N −1 below the diagonal. Note that the form (7.6) is invariant under the right N action, which takes µ R → n −1 µ R n. Second, one quotients by the N action, obtaining the space With the restriction (7.6), the N action is free, so M ψ is smooth.
Note that Moreover, M ψ is complex symplectic, with holomorphic symplectic structure induced from T * SL(N, C). And there is still a Hamiltonian SL(N, C) action on M ψ , induced from the left action on T * SL(N, C); its moment map is µ L . The complex symplectic geometry of M ψ is independent of the precise value of the ψ i 's as long as they are nonzero, so it is convenient to take ψ = (ψ 1 , ..., ψ N −1 ) = (1, ..., 1). We want to check that M ψ=1 is equivalent to the Coulomb branch of T N,1 .
There are several ways to describe the ring of functions on M ψ . Since C[M ψ ] the ring of functions on the quotient (7.7), it is equivalent to N-invariant functions on the restriction The N-invariants are generated by • the entries of the moment map µ L , and • the entries in the first column g (1) of the SL(N, C) matrix g (since g → gn preserves g (1) ).
Now compare this to the quantized chiral ring C ε [M C ] of the T N,1 quiver from Section 5. The chiral ring is generated by a moment mapμ = µ − N −1 2 ε1 for the SL(N, C) flavor symmetry and a set of operators {Q i } N i=1 in the fundamental representation. (All antifundamental Q i and higher tensor powers Q I , Q I can be constructed from µ and Q i by using (5.38) and (5.51).) We identify Dimension-counting shows that there should be a relation among the entries of µ and g (1) . We find it as follows. Note that the first column of every matrix ((µ L ) k g) (1) = (g(µ R ) k ) (1) is N-invariant. Moreover, due to the form (7.6) of µ R , the first column of g(µ R ) k is a linear combination of the k-th column of g and some of the previous columns This implies that if we construct an N × N matrix X with columns This is the relation. Specializing to ψ = 1, we simply have det X = det g (1) µ L g (1) ... (µ L ) N −1 g (1) = 1 . (7.14) Rather beautifully, upon identifying µ L = µ and g (1) = Q • , this coincides with the ε → 0 limit of the quantum relation (5.51) we found in Section 5.5. Namely, (5.51) with r = N implies An alternative way to describe the ring of functions on M ψ involves taking a slice through the orbits of the (free) N action on T * SL(N, C)| µ R =(7.6) , i.e. gauge fixing. This description often appears in the literature, particularly in [20]. We can completely gauge-fix the N action by forcing the upper-diagonal part of µ R in (7.6) to take the form where most entries above the diagonal, except the first N − 1 entries of the last column, are set to zero. This is the so-called Kostant slice (or maximal Slodowy slice) in sl(N, C) * . The entries x 1 , ..., x N −1 are independent functions on M ψ . Using this parameterization of µ R and moreover specializing ψ = 1 we find that the matrix X from (7.12) simply becomes X = g . Unfortunately, this description makes the holomorphic symplectic structure of M ψ rather obscure.
Comparing to the Coulomb branch of T N,1 from Section 5, some quick computations indicate that the chiral-ring operators given there are already adapted to a gauge-fixed version of M ψ . In particular, defining the matrix (which satisfies det X = 1, and which we expect to equal g after gauge-fixing), we find that in the ε → 0 limit where c i are the Casimir operators ofμ, cf. Appendix D.1. The expression (7.20) is not the Kostant slice (7.16) on the nose, but is related to it by a simple transformation, namely transposing and conjugating with the longest element of the Weyl group. The same transformation appears in the analysis of two-legged quivers below, cf. (7.30), and accounts for a difference in conventions in defining the right SL(N, C) action on T * SL (N, C). The computation of X −1μ X can be carried out even at ε = 0, but the result does not follow an easily recognizable pattern. For small values of N we find:

The cotangent bundle
For two-legged (k = 2) quivers, we expect the Coulomb branch to be even simpler: classically, M C T * SL(N, C). We gave a description of this cotangent bundle in (7.3), in terms of an SL(N, C) element g and moment maps µ L , µ R for the left and right actions of SL(N, C) on itself, extended to T * SL(N, C) as Hamiltonian actions. Here we'll explain how holomorphic functions on T * SL(N, C) match the structure of the Coulomb-branch chiral ring. In the quantum chiral ring C ε [M C ] of a two-legged quiver, we already have two SL(N, C) moment mapsμ 1 ,μ 2 . It is natural to identify these with µ L , µ R -modulo a slight reparameterization that we'll explain below. In addition, the chiral ring contains bi-fundamental tensors Q ij and bi-antifundamental tensors Q ij for left and right SL(N, C) actions. We expect to identify one or the other of these with the group element g.
The tensor operators in k = 2 theories turn out to satisfy some special relations that are the key to making this identification. First we note that the general R-charge formula (6.19) implies that the bi-(anti)fundamentals and all other higher antisymmetric tensors have R-charge exactly zero This is consistent with the natural C * scaling action on T * SL(N, C), which acts on g with weight zero and the cotangent fibers µ L (or µ R ) with weight 1, also scaling the holomorphic symplectic form with weight 1. The R-charges (7.22) allow for simple contraction relations on a single index that un-diagonalize (6.28) with no need for a moment map on the RHS. In addition, all the higher tensor operators can be built in a simple way from the bi-(anti)fundamentals, again without involving moment maps. Namely, if we let I = [i 1 ...i r ] and J = [j 1 ...j r ] denote fully antisymmetric multi-indices, then 24) where the LHS has a full antisymmetrization on each set of indices. We have checked these relations explicitly in the quantum algebra for N ≤ 4 (see Appendix D.2.3), and infer the general form shown here.
Thanks to (7.24) and the universal relations (6.18), we find that all antifundamentals Q ij and all higher tensors Q IJ , Q IJ can be written in terms of the basic bifundamental operators Q ij . Under the general assumption (made throughout this paper) that the chiral ring C ε [M C ] is generated by moment mapsμ a and the entire collection of tensor operators, this observation implies that for k = 2 quivers the generatorsμ 1 ,μ 2 and Q ij are actually sufficient. Now, if we set r = N in (7.24), we get the specialization In the classical ε → 0 limit, this ensures that det Q ij = (−1) . At ε = 0 we must be more careful about defining the determinant. It is convenient to introduce anti-diagonal (This is the longest element of the SL(N, C) Weyl group.) Then, defining we find that g naturally has a unit row-determinant, even at ε = 0 We identify this g with (a quantization of) the SL(N, C) matrix in the expected Coulomb branch T * SL(N, C). We also note that, due to (7.22), g has a two-sided inverse given by It remains to carefully identify the left and right moment maps µ L , µ R for T * SL(N, C) (in the presentation give by (7.3)) in terms ofμ 1 andμ 2 . We propose to set, both for ε = 0 and ε = 0, In other words, (µ R ) i j =μ N +1−j 2 N +1−i . The universal relations (6.32) involving moment-map contractions with Q's then imply which conveniently translates (using (7.27), (7.30)) to and in the ε → 0 limit reproduces the relation µ L g = gµ R in (7.3). For ε = 0, we have explicitly recovered a quantization of T * SL(N, C). We remark that the matrix K here, together with transposition, plays the role of intertwining the SL(N, C) k action on C ε [M C ] that is most natural for general T N,k quivers (namely, with every SL(N, C) acting as a left multiplication) with the SL(N, C) L ×SL(N, C) R action by left and right multiplication that is more natural when k = 2.

Relations for trinion theories
In Section 6, we identified moment maps and antisymmetric tensor operators as elements of the abelianized algebra A ε , for general T N,k star quivers. We then derived and/or verified several universal families of relations among the moment maps and tensor operators, which hold for all N and k. In this section, we specialize to k = 3, i.e. the star quivers related to trinion theories T N of Class S. We quickly compare the universal relations to know T N relations in the literature, and then discuss how to quantize and generalize several additional relations from the literature that are special for k = 3.
We mainly follow [24] and [26] as references for known T N relations, at ε = 0.

Universal relations
Relations ( as well as 2) and more generally, for higher-rank tensors with identical formulas for contractions from the right. As ε → 0, the shifted moment maps µ a become ordinary moment maps µ a , and these relations reduce to the well known Eqs.
(2.4)-(2.6) of [24]. As argued in Section 6 via diagonalization, relations (8.2)-(8.3) still hold if the moment maps are replaced by their n-th powers. We also always have (6.18) relating fundamental and antifundamental tensors, namely where I, J, K are multi-indices of length r and I , J , K are multi-indices of length N − r.

Contractions and Casimirs
One of the special relations we find for k = 3 is a quantized version of (2.7) in [24]. It takes the form where c ,a are coefficients the characteristic equation ofμ a P a (t) = det(t +μ a + ( This Capelli-like determinant is used to ensure that the c ,a are central elements; they can be expressed in terms of the Tr[(μ a ) n ] just as with ordinary determinants. Several of the c are listed in Appendix D.1, as explicit elements of the abelianized algebra A ε . As noted in [24], and so P 1 (t) = P 2 (t) = P 3 (t) . (8.8) The verification of (8.5) will closely mirror [24]. Consider the operator R ij i j = Q ijk Q i j k . This operator clearly transforms as a bi-adjoint (with possible traces) of SL(N, C) 2 and trivially under the third SL(N, C). From (6.32) and (6.33) we have and similarly forμ 2 ,μ 3 . At the classical level this would imply that R commutes withμ 1 andμ 2 as matrices but that is not necessarily the case for ε = 0. In order to show that this indeed is the case, note that since Q ijk is a fundamental and Q ijk is an antifundamental it follows that Putting these together we find that as desired. An identical computation shows that R commutes withμ 2 . From the fact that R commutes with bothμ 1 andμ 2 and its tensor structure, it must be that R is a polynomial iñ µ 1 ,μ 2 with some prefactors that commute with everything. Furthermore, the R-charge of Q implies that this can be a polynomial of degree at most N − 1. Equation (6.32) can be written as (μ 1 ⊗ 1 − 1 ⊗μ 2 )R = 0 and so, since P (μ 1 ) = P (μ 2 ), one finds that such a polynomial is given by We claim the constant of proportionality is (−1) N , which has been verified explicitly for low values of N , resulting in the proposed formula (8.5). In the classical limit, the modified Capelli determinant becomes the usual determinant and so the coefficients c take their classical values, in agreement with (2.7) of [24].

Fundamentals and higher tensors
At k = 3 there are also a beautiful set of relations between products of tri-fundamental operators and higher-rank tensors, which un-diagonalize (6.29).
For the N = 4 trinion theory, a generalization of (8.13) appears in Table 3 of [26] (discovered by the chiral-algebra analysis of [55,56].) With our normalization conventions, the relation takes the form Notice that if we were to read this as an N=3 relation, it would reduce to (8.13) after contractions with Levi-Civita tensors and an application of (8.4). We propose a uniform generalization of (8.13) and (8.14). Namely, from explicit computations at low values of N , we find that for appropriate tensors T (N,r) built out of the moment mapμ 1 . We do not yet have a general formula for T (N,r) , for all N and r. Some additional quantum relations among the Q's with vanishing classical limits are summarized in Section 2.3, and detailed in Appendix C.

A C ε [M C ] and A ε in the BFN construction
The abelianized algebra A ε is related to fixed-point localization in the mathematical definition of the Coulomb branch proposed by Braverman, Finkelberg, and Nakajima [15,16]. To provide some context, we review some aspects of the physical interpretation of the BFN construction and the appearance of A ε therein. For further discussion see [13] (where an analogous construction of C[M C ] via equivariant cohomology appears) and the upcoming [65].
The analysis here is directly analogous to physical constructions of the category of 't Hooft (or Wilson-'t Hooft) operators in topologically twisted 4d N = 4 (or N = 2) super-Yang-Mills theory. Line operators in 4d are the dimensional lifts of local operators in 3d. The categories were studied in [72] and [73] (generalized in [42]), and identified as categories of sheaves on particular versions of the space M [C∪C] that appears below.
The BFN construction arises physically by considering a 3d N = 4 gauge theory on spacetime C × R, with a particular half-BPS boundary condition B near spatial infinity on C We recall that the full hypermultiplets are in a quaternionic representation (3.1) that is split (not necessarily uniquely) into two unitary representations R ⊕ R * ; this boundary condition thus sets half of the hypermultiplets to zero. This setup may roughly be imagined as a 3d theory on a solid cylinder ( Figure 5). We may also think of this 3d theory as a 1d N = 4 quantum mechanics whose fields are various sections of bundles on C, obeying the boundary condition. If we further work in the cohomology of the twisted Rozansky-Witten supercharge Q RW (whose local operators contain elements of C[M C ]), we may restrict (localize) the fields in the quantum mechanics to sections of bundles that solve the Q RW BPS equations. After a bit of work, and a translation to algebraic language, this yields gauged N = 4 QM on and X is a holomorphic section of an associated R-bundle with a gauge group 27 The Hilbert space of the quantum mechanics, in the Q RW twist, should be the G-equivariant de Rham cohomology of M [C] [74]. 28 Turning on the Omega background further corresponds to working equivariantly with respect to the U (1) ε spatial rotation group of C, which is an ordinary symmetry of the moduli space M [C] . We find a Hilbert space While this may look foreboding, a bit of thought shows that M [C] is contractible to a point where E is the trivial bundle and X is the zero-section. Similarly, G is contractible to G, so just consists of Weyl-invariant polynomials in the equivariant weights ϕ ∈ t C and ε. Now consider local operators of the 3d theory, inserted at a point (0, 0) ∈ C × R. A stateoperator correspondence in TQFT would identify the vector space of local operators with the Hilbert space on a sphere S 2 surrounding the point (0, 0) ∈ C × R. From the perspective of N = 4 QM, it is convenient to deform S 2 to the boundary of a cylindrical slab, as on the right side of Figure 5. We then obtain a description of the Hilbert space as the cohomology of a moduli space of solutions to the Q RW -BPS equations on this deformed S 2 , i.e.  27 In the mathematics literature, C is usually replaced by a formal disc (so that holomorphic sections look like Taylor series rather than polynomials), and the gauge group is usually denoted ] is the ring of formal Taylor series. We will not be careful about such distinctions here, as we are merely trying to give an overview. 28 Since M [C] is infinite-dimensional, physics would dictate that L 2 de Rham cohomology be used. In the mathematics literature, Borel-Moore homology, i.e. "homology with closed support," is employed instead, as it is better behaved on spaces such as M [C∪C] /G encountered below, which are generalizations of affine Grassmannians. It is fairly clear by now that mathematical computations in Borel-Moore homology, such as (A.7), agree with physical expectations about the structure of local operators. Nevertheless, a fundamental understanding of why Borel-Moore homology is natural in QFT still seems to be missing. Figure 5. Left: 3d N = 4 theory on C × R with a boundary condition B at infinity, reinterpreted as 1d N = 4 quantum mechanics. Right: surrounding a local operator with a topological S 2 that has been deformed to two closely separated copies of C (the boundary of a cylindrical slab).
Here g is a holomorphic gauge transformation on C * , which is allowed to be singular at the origin, where a putative local operator is inserted. We then obtain the vector space of local operators in Q RW -cohomology as where G, G are the groups of regular (holomorphic) gauge transformations on the top and bottom copies of C, and U (1) ε as usual is the spatial rotation group. In a nutshell, (A.7) is the BFN construction. There are several highly nontrivial aspects of the formal definition (A.7). In contrast to (A.5), the space M [C∪C] /G is not contractible; it has highly nontrivial topology, as it must in order for (A.7) to contain monopole operators. Moreover, the OPE of local operators in the algebra C ε [M C ] does not correspond to a cup product in cohomology. Rather, it naturally corresponds to an operation known as the convolution product, which results from colliding and merging two of the "spheres" surrounding local operators in Figure 5. Both the cohomology classes in (A.7) and their convolution product can be difficult to describe explicitly.
A useful tool in equivariant cohomology is fixed-point localization. Letting T ⊂ G denote the maximal torus as usual, one finds that the T ×U (1) ε fixed points of M [C∪C] /G are isolated and actually quite easy to describe: they are points (E, X), (E , X ) where X = X = 0 are zero-sections, E is trivial, and E is obtained from E by a gauge transformation A ∈ cochar(G) .
. Its "localized" version inverts all weights λ, ϕ + nε (for any λ in the weight lattice of G), from which we see that our abelianized algebra A ε from (3.24) sits inside The only difference between A ε and H * T ×U (1)ε (F) loc (as vector spaces) is that in A ε we only inverted roots M α + nε and weights M λ + nε where λ ∈ weights(R); whereas the localized cohomology indiscriminately inverts all weights. The localization theorem provides the map When this is carefully interpreted using Borel-Moore homology (see Footnote 28), one finds that the image actually lies inside A ε , In other words, only the roots M α + nε need to be inverted. It is hardly obvious mathematically that the maps (A.11), (A.12) are embeddings of algebras (under the convolution product) rather than just vector spaces. The compatibility of fixed-point localization with the convolution product was proved by [16].

B P SU (2) Coulomb branch via Demazure operators
Here we wish to give the simplest possible example of how the algebra W ε from Section 3.5 fully reproduces the quantized Coulomb branch chiral ring. We consider 3d N = 4 pure gauge theory with G = P SU (2) (and R = ). The mathematics of this example (in fact of any pure gauge theory) first appeared in [66], and were connected with the physics of 3d N = 4 theories in [14].
The Coulomb branch of P SU (2) gauge theory is the centered moduli space of two SU (2) monopoles [2], known as the Atiyah-Hitchin manifold [75]. As a holomorphic symplectic manifold, it is cut out of C 3 by the equation and has a holomorphic symplectic form induced from the holomorphic 3-form on C 3 , In terms of the 3d N = 4 theory, Φ = 1 2 Tr(φ 2 ) is the generator of gauge-invariant polynomials in the complex scalar φ ∈ g C ; V is a nonabelian monopole operator of fundamental charge (labelled by the fundamental cocharacter '1'), and U is a dressed monopole operator.
The quantized ring of functions C ε [M C ] was described in [12]. It is generated by operators U, V, Φ with commutation relations and a central constraint Let's see how these arise from abelianization. The abelianized complex scalar φ is ϕ 0 0 −ϕ ∈ t C , and the full abelianized algebra can be generated from ϕ, ϕ −1 , and the abelian monopole operators v + = v 1 and v − = v −1 of unit cocharacters, following from (3.20), (3.22). Note that here the W-boson masses are M α = ±ϕ, and it is sufficient to invert ϕ because all other denominators 1 ϕ+nε can be obtained by commutation with v ± : The subalgebra W ε is obtained by considering polynomials in ϕ and the rescaled monopole operators (3.28) u which obey the extremely simple algebra We then throw in the Demazure operator where s is the simple reflection that generates the Weyl group, and acts as su ± = u ∓ s, sf (ϕ) = f (−ϕ)s. The definition of the W ε algebra (3.33) says to take the Weyl-invariant part of the polynomials in ϕ, u ± , and θ: With the Demazure operator, we may also construct an operator The Weyl-invariant part of the Demazure operator itself is zero; and we cannot apply the Demazure operator multiple times because it is nilpotent θ 2 = 0. It turns out that the entire algebra W ε is generated by the three Weyl-invariant operators above, with relations among them induced from the relations in A ε . The reader may check that the these relations perfectly match (B.3) and (B.4) above, so that W ε C ε [M C ] as claimed.

C Some basic quantum identities
In this appendix, we collect some basic results on commutation relations between moment maps and the tensor operators Q (r) , Q (r) , as well as among the tensor operators themselves. We work exclusively in the quantized chiral rings.

C.1 Highest and lowest weight vectors
We first show, by means of (3.20) and (3.22), that Q N N...N (resp. Q N N...N ) is a lowest (resp. highest) weight vector of a fundamental (resp. antifundamental) representation for each SU (N ) action, and then generalize to higher antisymmetric powers. Due to the permutation symmetry between the different legs, it suffices to consider the case of k = 1. As described in Section 6, Q N = V + N , i.e.
which precisely matches the weight of the lowest weight vector of a fundamental representation.
Now consider the lowering operators V − n , which can be expressed as where v ± nα := v ±enα is the abelian monopole with cocharacter ±e nα , i.e. a unit of fundamental magnetic charge (in the α-th weight space) for the U (n) node. To see that Q N commutes with these operators, it suffices to notice that for all n, β, α. For n < N this follows from (3.20) by noting that the set of weights that pair nontrivially with both cocharacter is completely empty. For n = N , it is suffices to note that the only weight that pairs nontrivially with both cocharacters yield +1 for both cocharacters. Thus, Q N is a lowest weight vector of SU (N ). A completely analogous proof shows that Q N is a highest weight vector of a antifundamental representation. Now consider the operator Q [N N −1...N −r+1] described in the main body of the text, we show that this operator is indeed a lowest weight vector of a rank r antisymmetric tensor for each SU (N ) factor. We will express cocharacters of the gauge group as A = ( a N ; a N −1 , ..., a 1 ) where a n ∈ Z n . Choose a subset I r n ⊆ {1, 2, ..., n} of size r ≤ n and set e I r n ∈ Z n to be the vector with 1 for each i ∈ I r n and 0 otherwise. With this notation, we can write It is worth noting that for r = 1 this reduces to Q N and for r = N the only cocharacter is A diag . From this presentation it should be clear that so this operator certainly has the proper weight to be a lowest weight vector of a rank r antisymmetric tensor representation of SU (N ). To see that it is a lowest weight vector, we appeal to (5.51); in particular we write ) To see that this extends to the remaining n we compute: (C.10) where the last line follows because n−1 appears twice in the anti-symmetrization. In principle it should be possible to show that Q [N N −1...N −r+1] is a lowest weight vector directly from applying (3.22) but this would be a rather nontrivial process. It is worth noting that a completely analogous proof shows that Q [N N −1...N −r+1] is a highest weight vector of a rank r antifundamental tensor representation, as desired.

C.2 Index swaps
We next investigate a series of identities of the form capturing the antisymmetry of the commutator of two fundamental tensors under a swap of any two indices. (Here it is shown for i 1 ↔ j 1 , but by symmetry of the legs of the star quiver, any swap i n ↔ j n behaves the same way.) For k = 3 this will motivate a conjectured quantum relation between Q (1) and the antisymmetric tensor Q (2) in Appendix C.3. Whenever an identity (C.11) holds, it should also hold upon replacing Q i 1 i 2 ...i k with Q i 1 i 2 ...i k ; but we shall focus on the relations with Q i 1 i 2 ...i k for simplicity. A quick computation shows (C.11) relation cannot hold for all k. For example in the (N, k) = (2, 4) theory we have Nonetheless, we will prove that (C.11) does hold for k ≤ 3. We go about this in several steps. First we show that antisymmetry (C.11) holds if for all i 1 , i 2 , ...., i k , j 2 , ..., j k . Then we show that (C.13) holds for all i 1 , i 2 , ...., i k , j 2 , ..., j k if for all i 2 , ...., i k , j 2 , ..., j k . We finish the proof by showing that, indeed, (C.14) holds for k ≤ 3. It is worth noting that (C.11), (C.13), and (C.14) hold trivially for k = 1. Furthermore, for k = 2 (C.11) implies that all of those commutators must vanish.
Step 1: We show (C.13) implies (C.11) via induction on |i 1 − j 1 |. Without loss of generality assume i 1 < j 1 , the base case is then j 1 − i 1 = 0 which follows from the assumption that [Q i 1 i 2 ...i k , Q i 1 j 2 ...j k ] = 0. Now, for j 1 − i 1 = n we compute: 15) where the second line follows form the Jacobi identity and the third follows from the inductive hypothesis and the action of V 1+ i 1 +n . Applying the Jacobi identity a second time yields (C.16) where the last line follows form the action of V 1+ i 1 +n .
Step 2: To show that (C.14) implies (C.13), we use induction on N − i 1 , the base case i 1 = N holds assuming (C.14) is true. We compute: (C.17) where only the Jacobi identity (lines 1 → 2, 3 → 4, 5 → 6), the SU (N ) action (lines 1, 2 → 3, 3, 4 → 5, 6 → 7) and the inductive hypothesis (line 4 → 5) were used. Therefore Step 3: We now move on to showing that (C.14) holds for k = 2, 3, starting with k = 2. For k = 2 we can assume j 2 < N and since Q N N is only charged under the central node it suffices to check j 2 = N − 1. We are then interested in the commutator [Q N N , Q N N −1 ]. This commutator takes the following (schematic) form: where B a β is the analog of the A α but for the N − 1 node on the a-th leg of the quiver. When α 1 = α 2 this commutator must vanish, so we are interested in (C. 19) for α 1 = α 2 . A quick application of (3.22) yields and therefore for α 1 = α 2 and therefore [Q N N , Q N j 2 ] = 0 for all j 2 . To see this extends to [Q N j 1 , Q N j 2 ], we induct on N − j 1 . We can further assume j 2 ≤ j 1 , this yields where the second line follows from the Jacobi identiy and the third by the SU (N ) action and the inductive hypothesis. Now consider k = 3, we hope to show that via induction in a similar fashion as the k = 2 computation, it suffices to check the base cases as the induction will follow identically to the above. We can assume that j 1 ≤ j 2 giving us two cases to check, depending on which of k 1 , k 2 is larger. When k 1 ≤ k 2 , the base case of induction translates to [Q N N N , Q N j 2 k 2 ] = 0 , (C. 24) which follows from the case where j 2 = k 2 = N − 1. We compute: ] . (C.25) Another quick application of (3.22) yields as desired.
When k 2 ≤ k 1 , the base case of induction similarly translates to [Q N N k 1 , Q N j 2 N ] = 0 (C. 28) which follows from k 1 = j 2 = 0. We compute: ]. (C. 29) and by applying (3.22) we find that v Aα 1 +B 3 which implies that as desired.

The k ≥ 4 cases
The above analysis shows why antisymmetry (C.11) cannot hold as written for k ≥ 4. Consider the commutator [Q N N...N , Q N N −1...N −1 ], this has the schematic form ]. (C.32) (C.33) The numerator in the above expression vanishes when ϕ N α 1 − ϕ N α 2 = ε and so the vanishing of [Q N N...N , Q N N −1...N −1 ] hinges on the fact that is invariant under α 1 ↔ α 2 . Unfortunately, for k ≥ 4 this need not be the case.
Triviality of k = 1 What happens for k = 1? In this case, just as above, induction reduces the problem to computing one commutator: [Q N , Q N −1 ]. We compute: where σ is the r-cycle (12...r) and σ n a means apply σ n to the set {i a , j a , ..., k a }. Clearly, the resulting operator is totally antisymmetric in {i a , j a , ..., k a } and if we require this operation to be linear we come to the definition Although the above operators furnish the appropriate representation to agree with the 3-fold antisymmetric tensor operators found in the primary text, it is not clear whether these operators are identical, up a simple prefactor, or whether there is a nontrivial relationship between the two (e.g. some contraction of moment maps with the indices of . What can be said is that which follows the antisymmetry of the commutators found above. The general form of this relationship could provide a systematic approach to constructing the higher tensor operators from the fundamentals. It would also be worthwhile to understand how this procedure compares to completely antisymmetrizing a product of fundamentals. This agreement is obvious for r = 2, since Z 2 S 2 and so these processes are identical, but for higher r the relationship is much less clear.

D Characteristic polynomials and higher tensors
In this appendix we recall the definition of the Capelli determinant that is used to define characteristic polynomials and Casimir operators in a non-commutative algebra. Then we collect some computations of relations among (quantum) k-fold fundamental operators Q (r) and higher tensors that should correspond to un-diagonalizing relations of the form as in (2.23a), (2.23b).

D.1 Quantum characteristic polynomials
Here we review the Capelli determinant that is used to compute the characteristic polynomial of quantum moment maps. At ε = 0, the moment-map operators µ a for the SL(N, C) k action on the Coulomb branch chiral ring have characteristic polynomials det(t + µ 1 ) = det(t + µ 2 ) = . . . = det(t + µ k ) = In the quantum case, at ε = 0, we would similarly like to identify a characteristic polynomial whose coefficients are SL(N, C) invariants, in that they commute with all individual components of the µ a . The naive determinant det(t + µ a ) does not have this property. (It is also not well defined.) The solution, however, is well known: we must use a Capelli shifted determinant instead. We recall how this works.
Given a matrix E = (E ij ) whose elements are noncommutative operators that satisfy the gl(N, C) commutation relations,  Table 2. Coefficients of (Capelli corrected) characteristic equation for the SL(N, C) moment maps, for N = 2, 3, 4, in terms the scalars ϕ N α on the central node. Keeping with the notation of the main text, Φ N is the sum of the ϕ N α .
A particularly natural way to parameterize the coefficients c of the quantum characteristic polynomial is as where, just as in the main body of the text, Notice that c 0 = 1 and c N = detE. We use (D.6) to define the c 's for the quantum moment map operatorsμ a , namely . Just as in the classical (ε = 0) case, the coefficients c do not depend on which leg a we choose. This is consistent with the fact that the eigenvalues of the quantum moment maps, described in the main text, are independent of a.
We write down some explicit expressions for the c 's in T N,k theories with N ≤ 4 in Table 2. Note that they depend only on the scalars ϕ N α associated to the central node of the star quiver.
for any number z and (D. 25) For k = 4 the form of the tensors becomes (T 1 (2,4,2)

D.2.3 N = 4
There are three ranks of tensor relations of interest for N = 4 theories, r = 2, 3, 4. We only have complete data for k = 1. In this case, at r = 2 we find The r = 3 relations are given by Finally, the r = 4 relations are given by For k = 2 we have found that as well as (D.57) and (4!) 2 Q [i 1 [i 2 Q j 1 j 2 Q k 1 k 2 Q l 1 ]l 2 ] = 24 i 1 j 1 k 1 l 1 i 2 j 2 k 2 l 2 (4!) 2 Q [i 1 [i 2 Q j 1 j 2 Q k 1 k 2 Q l 1 ]l 2 ] = 24 i 1 j 1 k 1 l 1 i 2 j 2 k 2 l 2 .
(D.58) Additionally, we find the relations corresponding to the tensors: (D.59) Finally, for k = 3 we find relations with (D.60)