Nonabelian Mirror Symmetry Beyond the Chiral Ring

Mirror symmetry is a type of infrared duality in 3D quantum field theory that relates the low-energy dynamics of two distinct ultraviolet descriptions. Though first discovered in the supersymmetric context, it has far-reaching implications for understanding nonperturbative physics in general 3D quantum field theories. We study mirror symmetry in 3D N = 4 supersymmetric field theories whose Higgs or Coulomb branches realize Dand E-type Kleinian singularities in the ADE classification, generalizing previous work on the A-type case. Such theories include the SU(2) gauge theory coupled to fundamental matter in the D-type case and non-Lagrangian generalizations thereof in the E-type case. In these cases, the mirror description is given by a quiver gauge theory of affine Dor E-type. We investigate the mirror map at the level of the recently identified 1D protected subsector described by topological quantum mechanics, which implements a deformation quantization of the corresponding ADE singularity. We give an explicit dictionary between the monopole operators and their dual mesonic operators in the D-type case. Along the way, we extract various operator product expansion (OPE) coefficients for the quantized Higgs and Coulomb branches. We conclude by offering some perspectives on how the topological subsectors of the E-type quivers might shed light on their non-Lagrangian duals. ar X iv :1 91 2. 05 10 8v 2 [ he pth ] 5 J an 2 02 0


Introduction
Three-dimensional gauge theories are strongly coupled at low energies due to the positive mass dimension of the Yang-Mills coupling. Consequently, they exhibit a wide range of interesting nonperturbative phenomena, including monopole operators and confinement/deconfinement transitions. A powerful tool for elucidating complicated gauge dynamics is duality, which states that two distinct ultraviolet (UV) field theory descriptions give rise to the same theory in the deep infrared (IR) [1][2][3][4][5][6][7]. A key merit of duality is that there often exists a manifestly weakly coupled dual description. Thus duality provides an efficient language for tackling the problem of strong coupling.
While duality, by definition, requires a map between all observables of the dual quantum field theories, most known dualities are motivated by matching quantities that are insensitive to dynamical details of the theories, such as 't Hooft anomalies. 1 On one hand, this is precisely what makes duality an efficient and elegant way to extract physical information. On the other hand, this procedure can be misleading in cases where it fails to pinpoint the fate of the renormalization group (RG) flow (see [8][9][10] for examples in 4D). Fortunately, for a subclass of dualities known as mirror symmetry of supersymmetric gauge theories [11][12][13][14], we have increased analytic control over the dynamics thanks to supersymmetry even while many features of generic three-dimensional gauge dynamics remain.
Supersymmetry, especially the localization method, allows us to extract nontrivial dynamical data from quantum field theories, such as their protected operator spectrum, lowenergy effective action, and supersymmetric partition functions, which all play important roles in testing and refining the duality maps. Once a supersymmetric dual pair passes such tests, one can consider supersymmetry-breaking deformations to generate a larger class of dualities. Indeed, many recently formulated 3D dualities are motivated by mirror symmetry and supported by SUSY-breaking deformations thereof, including the important bosonization dualities of [15,16].
Much recent progress has been made in better understanding 3D gauge theories with N = 4 supersymmetry, which is the original context in which mirror symmetry was discovered [11]. These theories are characterized by an SU (2) H × SU (2) C R-symmetry, and they have a vacuum moduli space consisting of (singular) hyperkähler manifolds that can be labeled as Coulomb branch M C , Higgs branch M H , or mixed branches, depending on which combination of R-symmetries is broken. The Coulomb and Higgs branches appear very different at first sight: the Coulomb and mixed branches embody the complicated dynamics of 3D gauge theories, whereas the Higgs branch is protected by supersymmetric non-renormalization theorems and thus has a rigid structure [11]. The nontriviality of mirror symmetry amounts to the statement that there exist mirror-dual pairs of 3D N = 4 theories where the roles of the Coulomb and Higgs branches, as well as classical and quantum effects, are interchanged. A particularly simple example is that of 3D N = 4 U (1) super-QED with one charged hypermultiplet, which is dual to a free hypermultiplet. For suitable matter content, a gauge theory can flow to a superconformal fixed point in the IR whose operator spectrum naturally has a description in terms of the elementary degrees of freedom in the UV gauge theory. In this case, mirror symmetry amounts to interchanging the descriptions of the CFT operators. Namely, under mirror symmetry, an order-type mesonic operator written in terms of the fundamental fields in one UV description is mapped to a disorder-type operator such as a monopole in the dual description. A particular class of operators in the SCFT is that of chiral ring operators, which are half-BPS and whose vacuum expectation values give rise to the Coulomb branch, Higgs branch, and mixed branches. The matching of the chiral rings (equivalently, the moduli spaces) [17][18][19][20][21][22] provides a first check for mirror duality proposals beyond anomaly matching.
A more refined protected subsector in 3D N = 4 SCFTs was discovered in [23,24]. It is described by a one-dimensional topological quantum mechanics (TQM) associated to either the Higgs or Coulomb branch. 2 The relevant operators are twisted translations of the Higgs (resp. Coulomb) branch chiral primaries by SU (2) H (resp. SU (2) C ) R-symmetry rotations along a line in R 3 . Their correlation functions depend only on the ordering of the insertions. The TQM contains nontrivial information about the operator product expansion (OPE) data of the full SCFT, which can be computed systematically from supersymmetric localization after mapping the TQM to a great S 1 on S 3 [31][32][33], and plays an important role in determining the full OPE data of the SCFT using the conformal bootstrap technique [23,[34][35][36][37]. In recent work [38], these TQMs are formalized as noncommutative associative algebras equipped with an even and positive short star product -equivalently, a (twisted) trace or bilinear form. The latter is essential for mapping the TQM data to CFT correlators. The action of mirror symmetry in the TQM sectors has been studied to a limited extent in [31][32][33], focusing mainly on the case where the corresponding SCFTs arise from abelian gauge theories such as SQED and abelian quivers. 3 The bootstrap analysis for the particular case of SQED 2 , or equivalently the T [SU (2)] theory, was carried out in [37], where nontrivial evidence for the (self-)mirror symmetry beyond the TQM sector was found.
In this paper, we initiate the systematic study of nonabelian mirror symmetry in the TQM sectors of 3D N = 4 SCFTs. Beautifully, the most well-studied examples of mirror symmetry fall into an ADE classification [11], with those of A-and D-type admitting higherrank generalizations [12,13]. We focus on the simple class of theories T g that have rank-one Coulomb branches given by the ADE singularities where g labels an ADE Lie algebra and Γ g is the corresponding discrete subgroup of SU (2) under the McKay correspondence, which can equivalently be represented as a hypersurface singularity in C 3 (see Table 1). The latter description makes explicit the Coulomb branch chiral ring of the theory T g , which is nothing but the coordinate ring of the hypersurface singularity.
From the quotient structure of the Coulomb branch, it is obvious that the free 3D N = 4 theory with a single twisted hyper in which the discrete symmetry Γ g ⊂ SU (2) F is gauged realizes this Coulomb branch as its vacuum moduli space (similarly for its mirror in terms of a free hypermultiplet). A more interesting theory with Coulomb branch (1.1) in the A n−1 case is super-QED with n hypermultiplets of unit charge, which we denote by SQED n . Similarly, for the D n case, an interacting theory with Coulomb branch (1.1) is SU (2) SQCD with n fundamental hypermultiplets. The exceptional cases of (1.1) do not appear to have gauge theory realizations: however, there are mirror dual descriptions which instead realize (1.1) as their Higgs branch M H (T mirror In general, they are given by 3D N = 4 quiver gauge theories of affine ADE type. The associative algebras associated to the ADE singularities are in general given by the spherical symplectic reflection algebras of complex dimension two [38]. This ADE series of theories T g (resp. T mirror g ) has, in addition, a Higgs branch (resp. Coulomb branch) given by where O min (g) denotes the minimal nilpotent orbit of g. This Higgs branch has a rigid structure in the TQM sector thanks to the g flavor symmetry. The g-equivariant deformation quantization of these hyperkähler cones was solved in [37], where unique star products were obtained except in the case of A 1 = sl 2 [24]. Returning to the Coulomb branches in (1.1), the deformation quantization of the A n−1 case was studied in [24], where the extra U (1) flavor symmetry played an important role in simplifying the analysis. The mirror symmetry between SQED and the cyclic abelian quiver was then spelled out in [32].
The primary goal of this paper is to carry out the analysis of mirror symmetry at the level of the TQM for the D-and E-type cases, where the relevant (mirror) gauge theories are nonabelian. In these cases, there are no continuous global symmetries at our disposal, although there do exist discrete Z 2 symmetries for D n (enhanced to S 3 for D 4 ) and E 6 , which still place some constraints on the TQM. We start by solving the algebraic problem of deformation quantization for D n singularities. We then compute the correlators in the TQM from both the SQCD description and the affine D-type quiver description. By studying the explicit form of the matrix models and insertions that are obtained from supersymmetric localization, we establish the precise mirror map for the operators in the TQM that preserves the short star product. For the E-type cases, we solve the deformation quantization of the E 6,7,8 singularities and present some preliminary observations from the affine quiver side, leaving a complete analysis to future work. We also include results for free theories that realize (1.3) via discrete gauging.
Here is an outline of the rest of the paper. We start by providing some relevant background on TQM sectors in 3D N = 4 theories in Section 2. We then give a brief review of mirror symmetry for the abelian A-type theories in Section 3. We move on to deformation quantizations of D-type singularities in Section 4 and explain how they are realized in 3D N = 4 gauge theories. In Section 5, by explicitly computing TQM correlators, we infer the mirror map for TQM operators that quantize the D n singularity: our results are summarized in (5.30) for n > 4 and (5.40) for n = 4. We carry out a similar deformation quantization of E-type singularities in Section 6 and, in Section 7, present some motivating remarks toward understanding the non-Lagrangian theories whose Coulomb branches realize (1.1) with g = E n through the lens of the TQM. Some details of our Higgs branch TQM computations, which tend to be more convoluted than their Coulomb branch counterparts, are gathered in Appendix A. In Appendix B, we consider additional quantizations of the A-and D-type singularities via field theory (outside the context of mirror symmetry), generalizing some examples from [32,33]. In Appendix C, we give a self-contained exposition of the Higgs branch chiral rings of the affine D-and E-type quivers, filling some gaps in the literature.
Notation. Throughout this paper, we adhere to the following notational conventions: • Straight O denotes an abstract chiral ring generator.
• HattedÔ denotes an abstract quantum algebra generator.
• Curly O denotes the realization ofÔ as an SCFT operator (with suitable mixing).
We also introduce the shorthand

Topological Quantum Mechanics
In this section, we briefly review the prescriptions of [31][32][33] for computing observables within certain protected operator algebras of 3D N = 4 theories on the sphere. 4 Combining these formalisms gives a way to derive precise maps between half-BPS operators across nonabelian 3D mirror symmetry, and to compute previously unknown quantizations of Higgs and Coulomb branch chiral rings.
We consider 3D N = 4 gauge theories of cotangent type, namely with gauge group G and matter representation R ⊕ R. We denote by g the Lie algebra of G, t a fixed Cartan subalgebra of g, W the Weyl group, ∆ the set of roots, Λ W the weight lattice, and Λ ∨ W the coweight lattice.
3D N = 4 SCFTs have two one-dimensional protected subsectors that each take the form of a topological quantum mechanics (TQM) [23,24]. The associative operator algebra of the TQM is a deformation quantization of either the Higgs or Coulomb branch chiral ring, and as such, it encodes detailed information about the geometry of the vacuum manifold. When the SCFT arises from an RG flow with a Lagrangian description in the UV, the Higgs branch sector is directly accessible by supersymmetric localization [31], but the Coulomb branch sector includes monopole operators, which are disorder operators that cannot be represented in terms of the Lagrangian fields [32,33]. For this reason, the known methods for computing OPE data within these two sectors look qualitatively different.
Each 1D sector can be described as the equivariant cohomology of an appropriate supercharge. The corresponding cohomology classes are called twisted Higgs or Coulomb branch operators (HBOs or CBOs). 5 They are realized as Higgs or Coulomb branch chiral ring operators which, when translated along a chosen line in R 3 or a chosen great circle S 1 ϕ on S 3 , are simultaneously twisted by SU (2) H or SU (2) C rotations. The OPE within each sector takes the form of a noncommutative star product where, for theories placed on S 3 , the quantization parameter ζ is the inverse radius of the sphere: ζ = 1/r. In addition to associativity, the star product inherits several conditions from the physical SCFT, namely [24]: truncation or shortness (the sum in (2.1) terminates after the term of order ζ 2 min(∆ i ,∆ j ) ) due to the SU (2) H or SU (2) C selection rule, evenness (swapping O i and O j in (2.1) takes ζ → −ζ) inherited from the symmetry properties of the 3D OPE, and positivity from unitarity (reflection positivity) of the 3D SCFT.

Higgs Branch Formalism
Assuming a UV Lagrangian, the operators that comprise the Higgs branch topological sector are gauge-invariant polynomials in antiperiodic scalars Q(ϕ),Q(ϕ) on S 1 ϕ , which are twisted versions of the hypermultiplet scalars q a ,q a transforming in the fundamental of su(2) H and in R, R of G. The correlation functions of these twisted HBOs O i (ϕ) can be computed within a 1D Gaussian theory [31] with path integral in terms of which the S 3 partition function is Namely, an n-point correlation function O 1 (ϕ 1 ) . . . O n (ϕ n ) on S 3 can be written as in terms of an auxiliary correlator O 1 (ϕ 1 ) . . . O n (ϕ n ) σ at fixed σ. The latter is computed via Wick contractions with the 1D propagator

Coulomb Branch Formalism
The operators in the Coulomb branch topological sector, in terms of a UV gauge theory Lagrangian, consist of a scalar Φ(ϕ) (a twisted combination of the vector multiplet scalars 6 Wick contractions between elementary operators at coincident points are performed using to resolve normal-ordering ambiguities. Φ˙a˙b transforming in the adjoint of su(2) C and of G), bare monopoles M b (ϕ), and dressed monopoles P (Φ)M b (ϕ). The coweight b breaks the gauge group at the insertion point to G b , the centralizer of b, and the corresponding monopole may be dressed by a G b -invariant polynomial P (Φ) in Φ(ϕ) [29].
In [32,33], a method for computing all observables within the Coulomb branch TQM was obtained for 3D N = 4 gauge theories of cotangent type by constructing a set of "shift operators," acting on functions of σ ∈ t and B ∈ Λ ∨ W , whose algebra is a representation of the 1D OPE. 7 We find that Φ(ϕ) is represented by a simple multiplication operator The shift operator describing a dressed monopole has a more intricate definition: it is constructed as where W b is the stabilizer of b in W, with the Weyl sum reflecting the fact that a physical magnetic charge is labeled by the Weyl orbit of a coweight b. For a given coweight b, we define the abelianized (non-Weyl-averaged) monopole shift operator where the sum is taken over all coweights shorter than b and the rational functions Z ab b→v (Φ), dubbed abelianized bubbling coefficients in [33], account for nonperturbative effects in nonabelian gauge theories in which the GNO charge of a singular monopole is screened away from the insertion point by smooth monopoles of vanishing size [40]. 8 Finally, M b is an abelianized monopole shift operator that represents a bare monopole singularity in the absence of monopole bubbling: where (x) + ≡ max(x, 0), (x) n ≡ Γ(x + n)/Γ(x), and powers of r encode scaling dimensions.
With the above shift operators in hand, the S 3 correlator of twisted CBOs O i (ϕ i ), inserted at points ϕ i along S 1 ϕ with 0 < ϕ 1 < · · · < ϕ n < π, can be computed as where the operators on the right are understood to be the shift operators corresponding to O i and 1 S 3 = 1. Above, we have introduced the empty hemisphere wavefunction as well as the gluing measure (2.13) While the matrix model (2.11) converges only for theories with a sufficiently large matter representation (i.e., "good" and "ugly" theories [41]), the shift operators can always be used to compute star products in the Coulomb branch TQM. Finally, in the commutative limit r → ∞, the algebra of shift operators reduces to the Coulomb branch chiral ring and we recover the abelianization description of the Coulomb branch proposed in [29]. In this limit, the operators e −b·( i 2 ∂σ+∂ B ) turn into generators e[b] of the group ring C[Λ ∨ W ], which act trivially on functions of Φ but satisfy the relations (2.14) We find that M b itself has a well-defined r → ∞ limit, 9 as do the abelianized bubbling coefficients Z ab b→v (Φ).

Review: A-Series
We begin by reviewing the deformation quantization of A n−1 singularities, independently of quantum field theory realizations. See [24] for discussions of A 1,2,3 and [32] for discussions of A n .
For general A n−1 singularities defined as the coordinate ring together with the holomorphic symplectic two-form gives rise to a Z ≥0 -graded Poisson algebra where the generators have degrees and the Poisson bracket (equivalently, ω) has degree The (filtered) deformation quantization of this graded Poisson algebra is easy to work out (see [42]). The quantum algebra is given by the central quotient where the noncommutative algebra C[X,Ŷ ,Ẑ] is defined by the commutators the deformation parameter ζ has degree deg(ζ) = 2, (3.7) and the center is generated by Physically, Ω A n−1 (X,Ŷ ,Ẑ) = 0 is the quantum chiral ring relation in the TQM.
Here, P (t) and Q(t) are polynomials of degree n − 1 and n with leading terms nt n−1 and t n , respectively. They satisfy Thus Q(t) is fixed by P (t) except for the constant term. Expanding P (t) as and denoting the constant term of Q(t) by α 0 , we see that the space of quantizations of the A n−1 singularity is n-dimensional and parametrized by {α 0 , α 1 , . . . , α n−1 }.
Imposing the evenness condition amounts to picking out terms in P (t) that have even degree in ζ. Thus we end up with a space of even quantizations A A n−1 of dimension n 2 .

A-Type Mirror Symmetry
A detailed TQM analysis of the abelian mirror duality between the affine A N −1 quiver gauge theory and SQED N was given in [32]. Here, we summarize the results for the "rank-one" side of this duality, namely that between the Higgs branch of the former theory and the Coulomb branch of the latter.
We denote by SU (2) R the relevant SU (2) R-symmetry (either for the Higgs or Coulomb branch) of the TQM sector. The degree of an element O in the quantum algebra is related to the R-symmetry spin (taking values in half-integers) by since the holomorphic symplectic form ω must transform as an SU (2) R triplet (corresponding to the three independent complex structures of the hyperkähler cone). Moreover, superconformal representation theory requires that the scaling dimensions of the corresponding operators satisfy ∆(O) = R(O). (3.12) From this, we conclude thatX,Ŷ ,Ẑ must be associated to chiral ring operators of dimension n 2 , n 2 , 1, respectively, in the physical theory. Below, we give their explicit realizations in terms of mesonic and monopole operators in 3D N = 4 theories.
The necklace quiver gauge theory has gauge group U (1) N /U (1), bifundamental hypermultiplets (Q I ,Q I ) for I = 1, . . . , N , and Higgs branch C 2 /Z N . The Higgs branch chiral ring generators are On the other hand, the Coulomb branch TQM operators of SQED N are products of the twisted vector multiplet scalar Φ and monopole operators of charge b ∈ Z. The corresponding shift operators act on functions of σ ∈ R and B ∈ Z. 10 The Coulomb branch of this theory is also isomorphic to C 2 /Z N , and its chiral ring is generated by (3.14) On either side of the duality, the above operators obey X Y = Z N + O(1/r), have identical correlation functions, and generate all other gauge-invariant operators in the corresponding TQM. Correlation functions of composite operators can also be matched using the OPE.
In this example, the U (1) top symmetry prohibits operator mixing and thus we have unambiguous identifications X =X, Y =Ŷ , Z =Ẑ (3.15) in both mirror-dual descriptions. This simplifying feature will no longer be present in the D case. The deformation quantization is again easy to work out (see [43]). The quantum algebra is given by the central quotient where the algebra C[X,Ŷ ,Ẑ] is defined by the commutators and the center is generated by Here, P (t) and Q(t) are polynomials of degree n − 1 and n with leading terms nt n−1 and t n , respectively. They satisfy Q(−t(t/ζ 2 − 1)) − Q(−t(t/ζ 2 + 1)) = (t − ζ 2 )P (−t(t/ζ 2 − 1)) + (t + ζ 2 )P (−t(t/ζ 2 + 1)). (4.6) Thus Q(t) is fixed by P (t) except for the constant term. Expanding P (t) as and denoting the constant term of Q(t) by α 0 , we see that the space of quantizations of the D n+1 singularity is (n + 1)-dimensional and parametrized by {α 0 , α 1 , . . . , α n−1 , γ}. Imposing the evenness condition, we see that γ = 0 for n even and is unconstrained for n odd. Thus we conclude that the space of even quantizations for D n+1 singularities is n-dimensional for n even and (n + 1)-dimensional for n odd.
To pin down the TQM, one needs to further specify the short product structure (which is equivalent to specifying a trace) of the associative algebra A D n+1 . We will analyze how combining discrete symmetry and physical input from 3D N = 4 SCFTs allows us to determine the short product and to provide the deformed mirror map for dual observables.

Periods and Associativity
We would like to quantize the D 4 singularity which merits special attention due to its extra symmetry. We start by writing down the most general deformed commutators compatible with the Jacobi identity (4.4): as well as the central element Notice that in (4.9), the leading-order terms in ζ are simply the Poisson bracket associated with the singularity, coming from the symplectic two-form ω. The deformation quantization of the D 4 singularity falls into isomorphism classes [44] (see also [38]) that are parametrized by so-called "periods" taking values in , (4.11) which is simply the root lattice of D 4 modulo the Weyl group. Here, the periods that label the quantizations are {A, B, C, γ}.
The D 4 singularity has an S 3 symmetry (preserving the holomorphic symplectic two-form ω) that becomes manifest in the coordinates 12) in terms of which the singularity becomes This is invariant under generating an S 3 symmetry that preserves the hyperkähler structure.
If we insist on having an S 3 symmetry upon deformation quantization, 11 then the periods are constrained. Up to a redefinition, the most general S 3 -preserving (deformed) commutators are (4.15) From here, we can work out the most general even short star product structureŝ where all of the operators U α V β W δ are normal-ordered products that are assumed (with suitable shifts) to have vanishing one-point functions, and nonvanishing two-point functions only with their conjugates (conjugation being defined by the Z 2 generator in (4.14)).
The parameters that appear above in the star product are further constrained by associativity as follows: Consequently, the only free parameters are α 1 , α 2 , α 4 , and A.
We can determine these parameters for specific deformation quantizations. One nontrivial example comes from the Coulomb branch of N = 4 SU (2) SQCD with four fundamental hypermultiplets, or by mirror symmetry, the Higgs branch of the affine D 4 quiver theory. In either case, by explicit computation, we find as well as , is the natural deformation parameter that arises in the derivation of the 1D TQM from the 3D N = 4 SCFT on S 3 (r is the sphere radius). 12 Combined with (4.16), these data determine a large class of correlators in the 1D TQM for either the SQCD or the affine D 4 quiver theory.

Realizations in Lagrangian 3D SCFTs
Let us define R to be the ring of holomorphic functions on the D 4 singularity, We now discuss 3D N = 4 SCFTs that realize R as their Higgs or Coulomb branch chiral ring.
In the next section, we will extract the 1D TQM that gives the deformation quantization of R by the corresponding SCFT.
We denote by SU (2) R the relevant SU (2) R-symmetry under which R is charged. We can fix the SU (2) R representations of R using the fact that the holomorphic symplectic form ω must transform as an SU (2) R triplet: Consequently, Moreover, superconformal representation theory fixes the scaling dimensions to be ∆ = R.
Γ D 4 gauged free hyper. In this case, the generators X , Y, Z can be written in terms of the complex scalars Q,Q in a single hypermultiplet 13 which is acted upon by the gauge symmetry Q 8 = Γ D 4 ⊂ SU (2) as Hence a basis of gauge-invariant operators is given by which satisfy the constraint 13 Alternatively, we denote the scalars in a hyper by Q aA where a is the SU (2) H index and i is the SU (2) flavor index. They obey the reality condition where Q = q 1 2 andQ = q 1 1 . (2) is O 48 , the binary octahedral group of order 48. Thus the global symmetry group of the discretely gauged hypermultiplet is 14

Now the normalizer of Q 8 in SU
the permutation group of order 6.
More explicitly, O 48 is defined by generators and relations The Z 2 and Z 3 generators of S 3 are identified with C and B in the quotient, respectively, which act as Affine D 4 quiver. In this case, the gauge theory is described as a 3D N = 4 quiver with SU (2) × U (1) 4 gauge group and four bifundamental hypermultiplets (Q A ,Q A ) where A = 1, 2, 3, 4. The quiver Lagrangian has an obvious S 4 global symmetry that acts naturally on the A index, but its action is not faithful on the Higgs branch chiral ring after we take into account the D-term relations. In fact, the Higgs branch chiral ring R is organized into faithful representations of S 3 . Without loss of generality, we choose an S 3 subgroup of S 4 to be the one permuting A = 1, 2, 3. Then by identifying the generators we find that (where the contraction of SU (2) gauge indices is pairwise from the left) transform under S 3 in the expected manner and satisfy Alternatively, we have See Appendix C.1 for a derivation.
The mirror dual of the affine D 4 quiver theory is known to be SU (2) SQCD with four fundamental hypermultiplets. The Coulomb branch chiral ring of the latter theory now realizes R. More explicitly, the relevant Coulomb branch operators consist of monopole operators M ±2 and the Cartan scalar Φ, which are subject to the quantum ring relation (one-loop effect) and transform under the Z 2 Weyl group as R is generated by the gauge-invariant combinations (We discuss the algebra of these operators in more detail in the next section, using a slightly different normalization.)

n > 4
For n > 4, we focus on two specific Lagrangian realizations of the quantized D n singularity, namely those that participate in 3D mirror symmetry.

Higgs Branch of Affine D n Quiver
This quiver takes the shape of an affine D n Dynkin diagram. We label the four U (1) × U (2) bifundamental hypermultiplets by Q A ,Q A , as before. But now we have, in addition, n − 4 U (2)×U (2) bifundamental hypers K I ,K I connecting the n−3 U (2) gauge nodes. Nontrivial gauge-invariant elements of the Higgs branch chiral ring correspond to closed paths ending on one of the univalent U (1) nodes.
For this theory, we use the conventions where the overall quotient by U (1) is implemented in the matrix model by making the first U (2) node SU (2). We label the hypermultiplets by (so that the first index of K I and the last index ofK I are associated with node I), where i, j = 1, 2. With contractions implicit, the Higgs branch chiral ring generators are for n ∈ 2Z (as in [45,46]) and for n ∈ 2Z + 1, which satisfy the ring relation The Z 2 symmetry is induced by the 1 ↔ 3 flip of the quiver, or (Q 1 ,Q 1 ) ↔ (Q 3 ,Q 3 ) at the Lagrangian level. See Appendix C.2 for derivations.
So far, our discussion has been at the level of the ("classical") chiral ring. In the next section, we will see through TQM computations how these operators become "quantized." Consider SU (2) SQCD with N f ≥ 0 fundamental flavors. Using the slightly more compact notation from Section 2.2, the Coulomb branch chiral ring is generated by the Weyl-invariant operators M 2 , ΦM 2 , Φ 2 with where M 2 is the monopole of minimal charge and ΦM 2 is a dressed monopole. (Again, we discuss the algebra of these operators in more detail in the next section.)

D-Type Mirror Symmetry
Let us now see how the kinematical considerations of the previous section translate into dynamical information about quantum field theories. Specifically, by computing TQM correlators in the theories whose Higgs and Coulomb branches are exchanged by D-type mirror symmetry, we derive the mirror map at the level of quantized Higgs and Coulomb branch chiral rings. As before, we examine the cases n = 4 and n > 4 separately due to the extra symmetry of the former case.

Higgs Branch of Affine
We summarize the relevant Higgs branch TQM OPE data for the gauge-invariant operators The one-point functions are The diagonal two-point functions are where the c subscript denotes a connected correlator. The mixed two-point function is We then set as required by the S 3 global symmetry of the theory, all of which can be checked explicitly using the Higgs branch TQM path integral (A.16).

n > 4
To illustrate how Higgs branch computations work for arbitrary n, we start with the S 3 partition function of the affine D n quiver theory: (note the 1/2 n−4 prefactor rather than 1/2 n−3 ), 15 which reduces to the expected for n = 4.
Consider insertions of the operator Z = −Q 1 Q 3Q3 Q 1 . In Appendix A.4, we prove that these can be rewritten as insertions of a function of the Coulomb branch scalar VEV s into the rank-one SU (2) SQCD Coulomb branch matrix model. The SU (2) matrix model takes the form (5.13) 15 As explained in Appendix A.4, accounting for this factor of two is crucial for mirror symmetry to work at the level of S 3 partition functions, namely for matching the partition function to that of SU (2) SQCD. This corrects a number of errors in [47].
Roughly speaking, the reason is that the gauge group of the quiver theory is [U (2) n−3 × U (1) 4 ]/U (1), and one of the SU (2) gauge nodes is really SO(3) due to the identification. Since the volume of SO(3) is halved relative to that of SU (2), we have a Weyl factor of 1/2 only for n − 4 of the n − 3 nonabelian gauge nodes.
where "SU (2) + n" is shorthand for "SU (2) with n flavors" (see (5.39)). By manipulations on the Higgs branch side of the D n theory, we find that which provides strong evidence for the mirror map for all n > 4.
To go further, we now set up the computation of more general Higgs branch correlation functions for D n . We can write and u I = diag(u 1 I , u 2 I ). Hence we have As an example, consider for ϕ 1 > · · · > ϕ p . The exponential factors cancel in all full contractions. There are clearly (p!) 2 different contractions (p! for each of A = 1 and A = 3): This reduces to previous results in Appendix A.4 for p = 1, 2. On the other hand, the following involves only one contraction: Similarly, To simplify these insertions in the matrix model ( From the symmetries of the resulting integral, one can see many of the expected equivalences between chiral ring representatives at the level of u .
For illustration, consider n = 4. By the Z 2 symmetry, we expect To demonstrate the latter (which is not obvious in the matrix model), we use that . (5.27) into (5.27) are the same, and insertions of into (5.27) are the same. So we have the unnormalized correlator where we have written the integrand in such a way that the insertion contains only −u 1 1 in the arguments. This is useful because one can then write and use standard Fourier transform identities, including (A.7), to reduce this expression to a single-variable integral (along the lines of Appendix A.4) that vanishes because the integrand is odd.
The lesson that we draw from the above discussion is that Higgs branch computations for general n are hard. For example, while the vanishing of X and Y follows simply from the Z 2 symmetry for all D n , this fact seems to be highly non-obvious in the Higgs branch matrix model: according to (4.40)-(4.45), X and Y are given by inserting some linear combination of (5.22) and (5.20) or (5.21) and (5.20) into (5.10), respectively, depending on whether n is even or odd. Thus the Z 2 symmetry of the affine quiver Lagrangian is no longer manifest when we insert X and Y into the matrix model. Nonetheless, the non-symmetric part of the integrand of the matrix model with X or Y insertions should be a total derivative. In other words, the F-term relations that imply that X and Y are Z 2 -odd (see Appendix C.2) become integration-by-parts identities in the matrix model.
In the end, we would like to derive the mirror map for all of the chiral ring generators, including quantum corrections (which we have, so far, for the generator of smallest dimension in (5.15)). While such a task seems daunting from the point of view of Higgs branch correlators, we can evade most difficulties by passing to the Coulomb branch of the mirror theory, which we do next. The Coulomb branch analysis is significantly simpler because the gluing formula (2.11)-(2.13) contains a delta function that forces vanishing of magnetic flux, from which one sees that correlators of an odd number of monopoles vanish without even doing any integration.
In fact, symmetries already take us a long way toward rounding out the mirror map. Using our knowledge of how Z maps to the Coulomb branch of the mirror dual, including normalization and quantum corrections, we can deduce the normalization of the mirror map for X and Y by demanding that the chiral ring relation be satisfied (this obviates the need to compute, e.g., X X on the Higgs branch side). Furthermore, we know that the quantum correction to Y vanishes by the Z 2 symmetry, while X can only mix with Y. This completely fixes the quantum mirror map for Y. Finally, we use the Coulomb branch results of the next subsection regarding the orthogonality of bare and dressed monopoles (particularly (5.56)) to write down the remaining entries in the quantized D-type mirror symmetry dictionary. Combined with (5.15), we arrive at the complete map where the Higgs branch operators X , Y, Z are given in (4.40)-(4.45) and we have accounted for operator mixing by setting which satisfies X Y = 0. 16 The two-point functions in (5.31) can in principle be computed explicitly from the matrix model for the affine quiver. 17 16 Strictly speaking, there is a sign ambiguity in (one of) the first two entries of (5.30). This is because knowing that Z ↔ C Z C in the chiral ring (i.e., ignoring 1/r corrections), where C = 1/(8π) 2 ∈ R >0 and C subscripts denote Coulomb branch operators, implies only that at the level of the chiral ring, where ± and ± are distinct signs. This follows from the Higgs and Coulomb branch chiral ring relations (4.46) and (5.67). While the overall sign is inherently ambiguous due to the Z 2 global symmetry (X , Y) → (−X , −Y), the relative sign can be fixed by computing suitable nonvanishing mixed correlators involving X and Y (e.g., numerically). 17 In fact, Y Y can also be read off from the SQCD Coulomb branch matrix model via (5.84).

Coulomb Branch of SU (2) SQCD with N f Flavors
On the SU (2) SQCD side, the basic non-Weyl-invariant shift operators are where M ±2 are related by the Z 2 Weyl group. These act on functions f (σ, B) where σ ∈ R and B ∈ 2Z. For N f ≥ 1, the monopole bubbling terms are necessarily polynomials and can therefore be removed by operator mixing. In other words, there exists an operator basis in which the bubbling terms for M 2 and ΦM 2 are zero; all other bases are related to this one by operator mixing. This means that we can write, without loss of generality, These shift operators already allow us to compute the star product in the Coulomb branch TQM. For N f ≥ 3, we can further compute correlators of twisted CBOs as follows. Define the vacuum wavefunction and the gluing measure Since |W| = 2, the partition function is (the 1/2 in the measure dσ accounts for the half-integer normalization of the weights). Then where the LHS of (5.38) means O 1 (ϕ 1 ) · · · O n (ϕ n ) with the ϕ i in descending order. 18 It is also convenient to define Computing correlators of Coulomb branch chiral primary operators (CPOs) from monopole shift operators is particularly straightforward when N f = 4, as we now show. This allows for a precise match to the results of Section 5.1.1.
SU (2) SQCD with N f = 4 is mirror to the affine D 4 quiver theory; the Higgs branch of the latter has a global S 3 symmetry, which we reproduce. The mirror map is as follows: where the Higgs branch operators U, V, W are given by (5.6)-(5.8) with one-point functions subtracted. We present the derivation below, using C subscripts to distinguish Coulomb branch operators from Higgs branch operators.
The Coulomb branch chiral ring generators (in our normalization, following from (5.33)) are These operators have dimensions ∆(X C ) = 3 and ∆(Y C ) = ∆(Z C ) = 2. At the level of the chiral ring, we have X 2 Using the Coulomb branch formalism, 19 we compute the one-point functions This is a special case of (5.73) (note that the RHS is real because it vanishes unless n is even). Some other useful integrals are These formulas can be used, for example, to give alternative derivations of (A.34).
(the monopole one-point functions vanish automatically in the absence of bubbling). We also have the two-point functions To define the operators U C , V C , W C including 1/r corrections, note that by dimensional analysis, U C and V C can only mix with the identity, while W C can mix with U C , V C , and the identity. In addition, we would like all correlation functions of U C , V C , W C , and composites thereof to respect the S 3 symmetry. In particular, we must have Using (5.46), the requirement that U C = V C = 0 fixes (note that conjugation flips the sign of the monopole). Next, requiring that W C = 0 shows that W C = 4ΦM 2 + O(1/r) cannot mix with the identity, so for some dimensionless constants u, v. To respect the S 3 symmetry, we must also impose that U C W C = V C W C = 0. More simply, we have the following ansatz and requirements for W C : Using 20 respectively, which implies that The commutativity M 2 ΦM 2 = ΦM 2 M 2 is required for consistency of the deformation quantization.
which holds for all N f , fixes a = −4i and b = 0: Having fixed the exact definitions of U C , V C , W C in (5.50) and (5.57), we check that 58) so these correlators respect the full S 3 symmetry. 21 Specifically, the Higgs branch computation gives (for which we reproduce on the Coulomb branch side by identifying thus substantiating the mirror map (5.40). Note that such a rescaling by powers of c ∈ R >0 preserves the chiral ring relation.
As a further check, (5.40) implies that (5.62) Using (5.61) and (5.62), we reproduce all of the Higgs branch correlators of theQ i Q 3Q3 Q i (i = 1, 2) using the Coulomb branch formalism. These identifications make sense because the Z 2 switches 1 ↔ 2 on the Higgs branch side.
The integral manipulations that led to (5.14) are equally valid when n = 4. So how do we reconcile the conclusion (5.15) with the known mirror map (5.61) in this case? It turns out that there is no contradiction. By writing sh(s) 2 ch(s/2) 2n = 1 ch(s/2) 2n−4 − 4 ch(s/2) 2n−2 (5.63) 21 One can go on to define additional composite operators. For example, U 2 C (defined as a shift of U C U C ) can mix with V C and V 2 C can mix with U C , which is consistent with the S 3 symmetry. and using the trick of differentiating ds e 2πist ch(s) # , 22 we derive below that Φ 2 = 2 π 2 r 2 ψ (1) (n − 2) + 2 n − 2 (5.64) in the SU (2) + n theory. For each n ≥ 3, there exists a q n ∈ Q such that In fact, we have q 3 = 0 and q 4 = −1, whereas q n≥5 ∈ Q \ Z. Hence n = 4 is special in that Φ 2 = 1/3r 2 is simply a rational number with no factors of π 2 . In particular, we see that so an insertion of 3s 2 is equivalent to a trivial insertion! Hence the results (5.14) cannot be used directly to read off the mirror map when n = 4: they are ambiguous. Specifically, (5.66) implies that the one-point function of the operator (5.61) in SU (2) + 4 is equivalent to an insertion of −(s 2 − 1)/(8πr) 2 , despite appearances. Somewhat miraculously, a similar statement holds for all p-point functions despite the mixing with the monopole. Namely, the insertion corresponding to p copies of (5.61) can always be written as a polynomial of degree p in s 2 plus a multiple of (4 + s 2 ) −1 , and one can check numerically for any given p that this gives the same result as an insertion of [−(s 2 − 1)/(8πr) 2 ] p . It would be interesting to construct a proof of this fact. Finally, for n ≥ 5, there is no mixing with the monopole and we can read off the mirror map directly from (5.14).
We finish with some conceptual comments. At fixed ϕ, the twisted CBOs in (5.41) represent nontrivial elements of the chiral ring. However, only after operator mixing is properly accounted for do they correspond to twisted translations of scalar conformal primaries in the CFT (hence CPOs). Namely, we must choose a basis in which their one-point functions vanish and they are orthogonal to all lower-dimension operators (this is the "CFT gauge" of [24]). In this basis, the monopoles correspond to the primary operators constructed in [48]. Usually, such a basis is obtained by diagonalizing the matrix of two-point functions. However, to respect the S 3 symmetry, that is not what we do here: rather, we impose that composite operators have vanishing one-point functions and nonvanishing two-point functions only with their conjugates, where conjugation is defined by the Z 2 subgroup of S 3 .

N f > 4
For N f > 4, we do not expect any mixing between the scalar and the monopole(s), by dimension-counting. Correspondingly, we lose the S 3 symmetry and are left with only the Z 2 of charge conjugation.
In our normalization, the Coulomb branch chiral ring generators are where the above equalities hold at the level of the classical chiral ring. The dimensions are as in (4.48). The Z 2 symmetry takes ( . We wish to determine the "quantum corrections" to X C , Y C , Z C . First note that by the duplication formula Γ(z)Γ(z + 1 2 ) = 2 1−2z √ πΓ(2z), so the partition function (5.37) is .

(5.70)
More generally, it is convenient to write (5.39) as By differentiating (5.68) with respect to τ , we get which vanishes for odd p and can be written in terms of polygamma functions for even p (it seems challenging to obtain a closed-form expression for this integral, but it can be evaluated for fixed p and arbitrary N ). 23 In particular, we have .

(5.74)
We can then evaluate 23 Likewise, one derives the simple formula where we have used the recurrence relation for simplification (the other one-point functions are trivial: M 2 = ΦM 2 = 0). Now note that the Z 2 symmetry requires that X C = Y C = 0, but does not restrict Z C . Therefore, including 1/r corrections, the most general mixing pattern is To constrain these coefficients, we consider two-point functions. The Z 2 symmetry requires that and does not restrict We clearly have (by flux conservation) that the two-point functions of M 2 or ΦM 2 with Φ 2 vanish. Using we compute that On general grounds, we have the relation (5.56) where We also compute that The monopole two-point functions are difficult to evaluate analytically, unless one fixes N f . Requiring But now any correlator containing odd numbers of X C , Y C obviously vanishes by flux conservation, so higher-point functions automatically respect the Z 2 symmetry and do not fix x, z. That is, the Z 2 symmetry does not completely determine X C , Z C at the quantum level (we have more freedom than when N f = 4). However, we can still map M 2 , ΦM 2 , Φ 2 individually to the Higgs branch side by matching correlators. 24 6 Deformation Quantization of C 2 /Γ E 6,7,8

Periods and Associativity
We now move on to the E-type singularities which are hyperkähler quotients of the type C 2 /Γ E 6,7,8 .
In 3D N = 4 SCFTs that realize these on the Higgs or Coulomb branch, X, Y, Z are half-BPS chiral primaries of scaling dimension (= SU (2) R spin) The equations in (6.1) then correspond to the chiral ring relations.
As usual, the dynamics of the SCFT gives rise to a deformation quantization of the Higgs or Coulomb branch. In particular, the chiral ring relations (6.1) are deformed. The truncation property of the Higgs or Coulomb branch algebra (TQM) implies that the deformations are all relevant, the sense of which should be obvious below.
where the β i are dimensionless parameters. 25 The most general even deformations of the commutators that satisfy the Jacobi identities are given by [Ŷ ,Ẑ] = 2ζX.
Here, we have used the freedom in operator redefinitions to put the last two commutators above in simpler forms. Furthermore, consistency requires Ω to be in the center of the algebra This puts constraints on the coefficients β i . Indeed, all of the β i except for one are uniquely determined by α 1,2,3 in (6.4) as follows: The four parameters {α 1 , . . . , α 3 , γ} label the even quantizations of the E 6 singularity.
The undeformed E 6 singularity has a nontrivial Z 2 symmetry that acts as X → −X, Z → −Z. Note that while the general Coulomb branch algebra presented in (6.4) and (6.6) is conveniently invariant under this Z 2 , the constraints of Z 2 will show up in specifying the short products on this algebra.
For the E 7 case, by solving the Jacobi identities and the center condition (6.5), we find a seven-parameter family of even quantizations labeled by {α 1 , . . . α 6 , γ}, where the commutation relations are [Ŷ ,Ẑ] = 2ζX and the center element is Ω E 7 =X 2 +Ŷ 3 +ŶẐ 3 25 Note that we have partially fixed the gauge redundancy in defining the operators to put the deformed chiral ring relation in this form.

Discretely Gauged Free Hyper
In the following free theories, the Higgs branch chiral ring generators are easily deduced from the known polynomial invariants of the binary tetrahedral, octahedral, and icosahedral groups (see, e.g., [49]). Γ E 6 gauged free hyper. In this case, which satisfy X 2 + Y 3 + Z 4 = 0. Γ E 7 gauged free hyper. In this case, which satisfy X 2 + Y 3 + YZ 3 = 0.

Star-Shaped Quivers
The only known realizations of E 6,7,8 singularities by interacting SCFTs are through the Higgs branches of affine E 6,7,8 quiver theories.
Affine E 6 quiver. This theory looks as follows: The quiver has an obvious S 3 symmetry acting on the Higgs branch, but at the operator level, only a Z 2 subgroup acts faithfully. The latter corresponds to the nontrivial Z 2 acting as (X , Y, Z) → (−X , Y, −Z) on the Higgs branch CPOs.
The affine E 6 quiver is obtained by gauging the diagonal SU

Recall the T [SU (3)] quiver
and denote the two bifundamental hypers by (q i ,q i ) and (Q A i ,Q i A ), with i = 1, 2 and A = 1, 2, 3 being the fundamental indices for U (2) and SU (3), respectively. The Higgs branch chiral ring is generated by the meson (moment map operator) ).
The precise expressions are given in Appendix C.3.1. In (6.17), we give a particular way to represent the CPOs X , Y, Z in terms of the hypermultiplet scalars. All other representatives differ by terms involving the D-term relations.
Affine E 7 quiver. Our conventions are where subscripts label mesons for each leg. One can use the same reasoning as for E 6 to find the invariants at various ∆; the result, as summarized in [50] and [46], is that the basic invariants are .

E-Type Mirror Symmetry
The main appeal of the Higgs branch topological sectors in the affine E-type quivers is that they might shed light on the non-Lagrangian Coulomb branch algebras (not associated to a nonabelian gauge theory with matter) to which they are mirror dual. One hope is that applying suitable manipulations and Fourier transform identities to the Higgs branch matrix models for the E-series partition functions might give hints about the mirror duals.
Since the E-type (and D-type) theories can be built from T [SU (N )] theories (which are realized on S-duality domain walls of 4D N = 4 SU (N ) SYM [41]) by diagonal gauging, it is natural to use the massive TQM of the constituent T [SU (N )] theories to determine the operator algebras of the full quiver theories. 26

E n Matrix Models
While we leave an in-depth examination of the Higgs branch matrix models of the affine E n quiver theories to future work, we briefly make some comments on the most tractable case, E 6 . The partition function of the affine E 6 quiver is given by (3)] partition function. 27 The partition function of a single leg can be evaluated explicitly [47]:

is the Higgs branch mass-deformed T [SU
.

(7.4)
Thus by integrating over the T [SU (3)] variables u 1,2 and then taking Fourier transforms, one can rewrite (7.1) in a form reminiscent of a rank-one matrix model. Namely, using dy e 2πixy y n sh(y) = dy e 2πixy 4 − ch(2y) ch(y) 4 (7.5) and a cyclic convolution identity from [32] gives 28 To mimic the one-loop determinants in a rank-one Lagrangian theory, one might wish to write the integrand in the form sh··· sh ch··· ch , but it remains to be seen whether this rewriting has any physical significance.
Note that the E 7 theory contains two copies of T [SU (4)] and one copy of T [2,2] [SU (4)], whereas the E 8 theory contains one copy each of T [SU (6)], T [3,3] [SU (6)], and T [2,2,2] [SU (6)]. 29 Hence one can use same strategy of combining the convolution identity (7.7) with the results of [47,51] for T [SU (N )] and the results of [52] for the partition functions of the other legs 27 The prefactor of 1/2! rather than 1/3! is due to our convention of defining the affine E-type quivers by making the central node P SU as opposed to SU ; see Footnote 15.
F j (τ ). (7.7) 29 Here, we use the notation of [52] where T [SU (N )] ≡ T (in the limit of vanishing FI parameters) to rewrite the E 7,8 partition functions as onedimensional integrals.

Returning to T [SU (3)], we have
where 3 a=1 u a 3 = 0 and = Dq Dq DQ DQ e 4πr dϕ L (7.10) We have yet to find a way to write these insertions in an enlightening way.
Finally, the Z 2 symmetry of the E 6 theory may help identify which chiral ring generators map to monopoles and which to scalars, assuming that this Z 2 is realized as charge conjugation in the mirror theory. 30 By this logic, X and Z in the E 6 theory (which flip sign under charge conjugation, and whose one-point functions must vanish) should map to monopoles in the non-Lagrangian dual. This is contrary to the D-case, where Z maps to a scalar. In the E 7 and E 8 cases, we no longer have a Z 2 symmetry, so the circumstantial vanishing of one-point functions can no longer be used as evidence of mapping to monopoles (for instance, one cannot rule out mixing with the Cartan scalar, after subtracting one-point functions).

E n Monopoles
Putting aside the structure of the (known) matrix models, it is interesting to ask whether the structure of the would-be shift operators themselves reveals any information about the monopole spectrum of the non-Lagrangian duals to the E n quiver theories. Some hints that we can use to answer this question are Lagrangian intuition, the commutative limit, and scaling dimensions (for constraining bubbling coefficients).
Let us make a few preliminary comments that can hopefully be clarified in future work. We make the following assumptions: • The fact that the mirror dual theories have rank one means that their monopole charges belong to a one-dimensional vector space.
• The hypothetical dual gauge group is "semisimple," meaning that the dimensions of (dressed) monopoles are fully accounted for by powers of the vector multiplet scalar in the commutative limit (see Footnote 9).
• One of the Coulomb branch chiral ring generators is constructed from the vector multiplet scalar and therefore takes the form Φ d , where d is a positive integer determined by the hypothetical Weyl group.
The second assumption is motivated by the fact that the dimensions (6.2) of the E n chiral ring generators are known to be integers, just as the dimensions of monopoles in a Lagrangian theory with semisimple gauge group are integers (otherwise, they could be half-integers, or conceivably even other fractions in a non-Lagrangian theory). The third assumption is perhaps most plausible in the case of E 6 , which has a Z 2 symmetry.
We now work out the consequences of these assumptions. In the commutative limit, a primitive monopole [33] of dimension ∆ and charge q can only bubble to the identity: )e[w(q)] + b(w(q))), (7.15) where b, c are complex numbers. By the rank-one assumption, a given Weyl group element w can only act via multiplication by a constant c w , so If ∆ ≥ 0, then the bubbling term is a "Weyl-invariant" polynomial (monomial in the commutative limit) and can be removed by a change of basis: Note that e[c w q] = e[q] cw . Now recall the relevant singularities (below, we omit the subscript C for "Coulomb"): We wish to determine the Coulomb branch operators in rank-one theories that satisfy these relations. For A N and D N , we think of X and Y as (dressed) monopoles and Z as the vector multiplet scalar. Since overall factors of Φ must cancel for dimensional reasons, to solve the above relations, we may set Z = 1 and replace X and Y by Laurent polynomials P and Q in a single variable x ∼ e[q] (by the rank-one assumption): This equation is easily solved by For the E-series, there are more possibilities to consider for which operators are scalars and which are monopoles, but let us restrict our attention to the possibility that Y is the scalar (which is most plausible in the case of E 6 ). Then we obtain the equation where m, n are positive integers and P, Q are single-variable Laurent polynomials with coefficients in C. The cases of E 6,7,8 correspond to (m, n) = (2, 4), (2, 3), (2,5), respectively, while the A-and D-series correspond to (m, n) = (2, 2). We wish to find nontrivial solutions to the above polynomial Diophantine equations, i.e., solutions with neither P nor Q constant (if no such solutions exist, then the assumptions should be relaxed). 31 In general, one can ask for which m, n there exist nontrivial solutions to (7.25) (WLOG, we may restrict our attention to 2 ≤ m ≤ n, where we impose the first inequality because 31 Recall that the A-series has two independent monopoles and trivial Weyl group, while the D-series has one independent monopole and nontrivial Weyl group. For the E-series, we assume that two of the generators are monopoles, but assuming only one independent monopole (so that the other is simply a dressed version of it) would imply that P and Q have the same powers of x and differ only in their coefficients; then P and Q (and the corresponding monopole operator) would need infinitely many terms, since the degrees could not match otherwise. So we are led to postulate two independent monopoles for the E-series.
solutions are trivial to obtain if either m, n is 1). We have not been able to find or rule out nontrivial solutions beyond (m, n) = (2, 2). 32 One possibility is that we should abandon semisimplicity, so that the monopoles have dimensions not accounted for by Φ.

Summary and Future Directions
This paper presents the results of precision studies of nonabelian ADE mirror symmetry beyond the chiral ring, using the recently developed TQM techniques in [31][32][33]. As a byproduct, we extend the construction of deformation quantizations of [24] to the D-and Eseries. We focus on D-type quivers, in particular synthesizing OPE data (structure constants) for the chiral ring generators of the D-series, but we also comment on possible implications for the monopole spectrum of the non-Lagrangian theories whose Coulomb branches are We find the precise map between quantized Higgs branch chiral ring generators in D-type quivers and quantized Coulomb branch chiral ring generators in SU (2) SQCD. Our results provide additional entries in the mirror symmetry dictionary for nonabelian 3D N = 4 gauge theories beyond, e.g., the matching of supersymmetric partition functions 33 [55][56][57][58] and chiral rings [17].
It is safe to say that the range of applications of the Higgs and Coulomb branch TQM has yet to be fully explored. For one thing, it would be interesting to incorporate the additional constraints of N = 6 or N = 8 SUSY [59] into the TQM analysis. For another, the OPE data that we have computed can be fed into the bootstrap machine to study the full CFT spectrum and (self-)mirror symmetry beyond the TQM sector,à la [37]. Finally, the connection between these techniques and protected operator algebras in one dimension higher [60] (several aspects of which have recently been derived from localization [61,62]) via dimensional reduction [63][64][65] leads us to wonder whether the TQM contains tractable lessons about line operators in 4D gauge theories.
A technical detail that we have glossed over is the following. To define the star-shaped quivers of interest, we start with all nodes unitary (U ) and quotient by the diagonal U (1), as suggested by their brane constructions. (See [66] for Coulomb branch computations in these theories.) As a computational matter, it is convenient to implement the quotient simply by making one of the nodes SU . More precisely, we should make one of the nodes P SU . 32 However, the existence of nontrivial solutions for small (m, n) is not immediately ruled out by the abc inequality for polynomials (Mason-Stothers theorem). We thank J. Silverman for this comment. 33 Matrix models for sphere partition functions of affine A-type quiver theories of arbitrary rank have been studied in [55], leading to a derivation of the mirror map between mass and FI parameters. The corresponding analysis for D-type quivers was performed in [56,57], and in this case, a free-fermion representation for the partition function (with vanishing mass and FI parameters) was derived in [58].
The distinction between SU and P SU is irrelevant to normalized correlation functions of local operators (in particular, TQM observables). However, the precise normalization of the partition function depends on which U node we make P SU : a P SU (N ) node introduces a factor of N in the partition function relative to an SU (N ) node because the volumes of these groups differ, and the inverse volume enters into the gauge-fixed path integral. We found that to match the partition function of the affine D-type quiver to that of SU (2) SQCD, it suffices to make one of the U (2) nodes SO(3). The situation is less clear for the affine E-type quivers since their mirrors are non-Lagrangian, but one can in principle match partition functions (including discrete factors) by reducing the 4D index of the E n Minahan-Nemeschansky theories [67,68]. It would be interesting to clarify the general procedure for decoupling the overall U (1) and to understand better the global structure of the gauge group in the affine quiver when comparing to the mirror theory.

A.1 Fourier Transform Identities
The basic Fourier transform identities that we will need are By further differentiating (A.3), we obtain analogous formulas for y n th(y)/ ch(y), e.g., dy e 2πixy y th(y) ch(y) We have in addition that dy e 2πixy th(y) 2 ch(y) and by differentiating (A.5), we obtain analogous formulas for y n th(y) 2 / ch(y), e.g., dy e 2πixy y th(y) 2 ch(y) One can go on to derive similar identities. Finally, we note that . (A.7)

A.2 Γ D 4 Gauged Free Hyper
Recall that in this theory, where we use the 0 subscript to denote a "bare" Higgs branch CPO. Canonically normalized CPOs without 0 subscripts have vanishing one-point functions and diagonal two-point functions (in a real basis).
We would like to compute correlation functions of the CPOs. To proceed, we need the two-point function of Q,Q, which is (recall that = −4πr from [31]). The correlator at coincident points is 0. In particular, and consequently the normalized CPOs are There is no further "gauge ambiguity" in this case.
Doing simple Wick contractions, we obtain Thus, comparing to (4.16), we have . (A.14) In particular, the S 3 symmetry is obvious in the TQM.

A.3 Affine D 4 Quiver
Recall that the Higgs branch chiral ring generators are given by in terms of gauge-invariant combinations of the hypermultiplets.
We adopt the normalization (5.12) for the S 3 partition function of the affine D 4 quiver, which we can write as and t = σ 3 . Thus the propagators are where ϕ 12 ≡ ϕ 1 − ϕ 2 and the ± sign is + when i = j = 1 and − when i = j = 2. We emphasize that the 1D TQM path integral has an explicit S 4 symmetry permuting the A indices (as explained before, only an S 3 subgroup acts faithfully on CPOs). At coincident points, we use the symmetrized expression and in computing correlation functions, we always assume the ϕ i are ordered as Note that (incomplete) self-contractions of a composite operator can also contribute to connected correlators. The correlators that we compute below are normalized by (A.16).
Computation of TQM correlators. To compute the (normalized) two-point function U V , we need to compute U 0 V 0 as well as the one-point functions U 0 and V 0 .
We start by recording the Wick contractions in the 1D TQM on the Higgs branch: for ϕ 1 > ϕ 2 , where cross-contractions give and self-contractions give as well as Doing the matrix integral, we get Similarly, has only self-contractions. We get We also have the one-point functions Thus the connected two-point functions ( Putting everything together, we have Similarly, one can check explicitly that At the level of the TQM, this is a simple consequence of the exact S 3 symmetry of (A.16).
We summarize the results of similar computations for various correlators below: These correspond to (4.18) with ζ = 1. Note that we have performed Gram-Schmidt diagonalization to define the composite operators.

A.4 Affine D n Quiver
Partition function. We start by simplifying the Higgs branch matrix model of the affine D n quiver, while also reviewing the mirror equivalence to SQCD at the level of S 3 partition functions. The Cauchy determinant formula where M ij = 1/ ch(x i − y j ) and i, j = 1, . . . , N , proves useful for removing "sh" factors from the integrand.
We first check that accounting for the volume factor of two (for P SU versus SU gauge group) is necessary to match the partition function to that of SU (2) SQCD. 34 Relabeling σ 1,3 = u 1,2 0 and σ 2,4 = u 1,2 n−2 , we write . Now note that the integrand is even under swapping the integration variables u 1 I and u 2 I for I = 0, . . . , n − 2; by swapping these variables in turn, this becomes simply 34 Alternatively, the affine D n quiver can be realized by gluing a non-affine D 3 quiver to a D n−3 quiver by gauging the SU (2) flavor node(s). Each D k quiver takes the form . (A.36) Using (A.1) and simplifying, we get which coincides with the partition function of SU (2) with n flavors up to our conventional factor of 1/r 2 (see (5.37)). 35 Computation of TQM correlators. We now compute the one-and two-point functions Z and Z Z . Recall that It is helpful to note that integration by parts can be used to simplify Z · · · Z HB : in the matrix model (5.10) for Z Dn , an insertion of the form (where σ is σ 1 or σ 3 and u ≡ u 1 1 = −u 2 1 ) is a total derivative. In particular, taking (p, q) to be (0, 0) or (1,0) shows that the expressions are total derivatives. This observation is useful because before simplification, Z · · · Z HB is a polynomial in th(σ ± u), while dropping total derivatives allows it to be written as a polynomial in th(σ − u); this facilitates manipulation of the resulting integrals because it allows for shifts of σ by u, thus decoupling the u integral. 35 Another typo in [47] can be found in their formula (3.3). The correct version is the (−1) N f +1 on the RHS having been overlooked. The LHS is the partition function of SU (2) SQCD with N f flavors and mass parameters, which reduces to our expression (5.37) (up to the 1/r 2 ) when the m i = 0. However, the above equality holds only when the mass parameters are distinct.
To set the stage, we determine the expression for the partition function after integrating over all Higgs branch variables except for u i 0 (i.e., σ 1,3 ) and u i 1 . This yields a simplified Higgs branch matrix model with all scalar VEVs integrated out, apart from those relevant to an insertion of Z · · · Z HB . Starting from the first line of (A.37), we derive that: .
We now compute by taking Wick contractions that Setting u ≡ u 1 1 = −u 2 1 and integrating by parts allows us to write We also have Z Z HB = 1 8πr 4 (I c + I s + I ss ) (A. 46) where, as in the D 4 case, Setting u ≡ u 1 1 = −u 2 1 and integrating by parts gives (A.50) To evaluate the multidimensional integrals for Z and Z Z , our main tool for simplification is to take a Fourier transform whenever an argument of "sh," "ch," or "th" involves a combination of two variables: this allows us to decouple the single-variable integrals. For instance, we have where we have shifted u i 0 → u i 0 + u and integrated over u using (A.1). Taking the Fourier transform of the 1/ sh(u 1 0 − u 2 0 ), and iterating this process as necessary (possibly with the help of various identities from Appendix A.1), leaves us with nested single-variable integrals that can be evaluated sequentially to yield a single integral: For Z Z , we must evaluate three additional integrals (call them I 1 , I 2 , I 12 ). First, we have sh(s) 2 ch(s/2) 2n (s 2 + 1). (A.52) Second, we have (A.53) Third, we have ds sh(s) 2 ch(s/2) 2n (s 4 + 10s 2 + 9) − (which we deduce from our result for Z ) gives (A.56) Combining the above gives (5.14).

B Quantized Coulomb Branches for AD
Here, we consider some realizations of deformation quantizations of the C 2 /Γ A N and C 2 /Γ D N singularities by Lagrangian quantum field theories, namely the Coulomb branches of 3D N = 4 U (1) and SU (2) gauge theories with arbitrary matter representations. In these cases, the choice of basis is strongly constrained by U (1) and Z 2 flavor symmetries, respectively.
We expect both the Coulomb branch chiral ring (the "classical" Coulomb branch) and its Poisson structure to depend only on N , because N determines the holomorphic symplectic form. 36 We also expect the number of distinct quantizations realized by these theories to be related to partitions of N . An interesting question that one might ask, which we do not attempt to answer here, is: do there exist examples of different Lagrangian theories with the same "quantum" Coulomb branch, to higher orders in ∼ 1/r beyond O( 1 )?
Nondegenerate short star products for quotient singularities, including Kleinian singularities, have been classified in [38]. For example, even nondegenerate short star products for A n singularities depend on n e + n s parameters where the first n e = (n + 1)/2 parameters determine the corresponding quantum algebra A up to isomorphism (i.e., the period of the quantization) and the remaining n s = n e + ((−1) n − 1)/2 parameters determine maps from the associated graded algebra gr(A) (the "commutative limit" of the associative algebra A) into A, corresponding to physical gauge fixings. This agrees with the counting of free parameters in [24] for the examples of A n≤4 before imposing unitarity (i.e., positivity), which is a stronger condition than nondegeneracy. In the examples below, fixing a Lagrangian SCFT should be understood as fixing a particular value of the period for the quantization.

B.1 U (1)
Consider U (1) for some set of charges {q} with multiplicities {N q }, where q ∈ Z \ {0} and N q ∈ Z ≥0 (uncharged matter does not contribute, but we may consider the pure case). The shift operators for the Coulomb branch chiral ring generators are: Setting N = q |q|N q and (this normalization being natural from the point of view of correlation functions), we find that X Y = Z N in the chiral ring. Accounting for sign, we obtain distinct quantizations from these theories for fixed N . At finite r and to subleading order in 1/r, we compute that Equivalently, Hence the Poisson structure, like the chiral ring, depends only on N (as expected). These quantizations are distinguished by the coefficients of the subleading terms in the polynomial P (Z) (computing the commutator is simpler than directly computing three-point functions because various gauge-fixing ambiguities cancel in the former).
The structure constants for the deformation quantizations corresponding to the Higgs branch of the affine A 2,3 quivers were originally bootstrapped in [24] and later derived from localization in [31]. By using the above techniques for the Coulomb branch of the mirror dual, we obtain these results and more with very little effort.

B.2 SU (2)
Consider SU (2) SQCD with matter specified by some set of spins {j} with multiplicities {N j }, where j ∈ 1 2 Z >0 and N j ∈ Z ≥0 (uncharged matter does not contribute, but we may consider the pure case). In conventions where the weights of SU (2) are half-integers and monopole charges are even integers (b ∈ 2Z), we have where m j ∈ {−j, −j + 1, . . . , j}. Indicating the commutative limit with an ∞ subscript (for r → ∞), we have (B.11) Set N = j S j N j . Then in particular, we see that On dimensional grounds, the bubbling coefficient for M 2 ∞ is a monomial in Φ for N ≥ 2, which we can eliminate by a change of basis. So for N ≥ 2: (B.14) Using e [2]e[−2] = 1 gives Then setting yields the equation of a D N singularity: For N = 1 (i.e., N 1/2 = 1), one can show that the relevant bubbling coefficient vanishes by a polynomiality computation at finite r [33], but let us not assume this. We have Equivalently, Unless c = 0, this is a nonsingular deformation of a D 1 singularity (as can be seen from the nonvanishing of the partial derivatives at (0, 0, 0)). For N = 0 (the pure case), we have and therefore Equivalently, The degree of the relation is reduced when c = ±1 (the sign ambiguity is present even when using polynomiality [33]): where we have slightly redefined the variables. This gives an alternative way to fix c 2 . Note that the theory is good for N ≥ 3, we expect the D N equation to hold for N ≥ 1, and we expect it to be modified as above

C Higgs Branch Chiral Rings for DE
In this appendix, we derive the Higgs branch chiral rings of the D-and E-type quivers considered in the main text. We discuss the D n chiral ring in some detail, since a comprehensive derivation seems to be missing from the literature (see [50], Section 5 of [45], and Appendix A.1 of [46] for earlier discussions). In the cases of E 6,7,8 , we also fill in some details regarding existing derivations (useful references include [50] and Appendix A.2 of [46]).
Note that for 3D N = 4 theories containing only vector multiplets and hypermultiplets, there exists no distinction between the D-term and F-term relations in 3D N = 2 language because the auxiliary fields combine into an SU (2) R triplet (equivalently, the Kähler potential fixes the superpotential). Hence we may equivalently write the D-term relations in the TQM, which take the form (QR(T )Q)(ϕ) = 0 (C.1) for all T ∈ g [31], or derive the F-term relations from the superpotential, as we do below. 37 37 While (C.1) holds at the level of the chiral ring, it may be modified by contact terms at the level of correlation functions. Additionally, the RHS of (C.1) receives contributions from FI parameters, which we have set to zero.

C.1 Affine D 4 Quiver
The affine D 4 quiver contains hypermultiplets (Q A ) i , (Q A ) i with A = 1, . . . , 4 and i = 1, 2. The superpotential is where Φ and φ A are adjoint chirals for the SU (2) and U (1) gauge nodes, respectively. We introduce the notation AB ≡Q A Q B ; then the F-term relations give

C.2 Affine D n>4 Quiver
For the affine D n quiver, we have adjoint chirals φ A and Φ I (I = 1, . . . , n − 3) for the U (1) and U (2) nodes, respectively. The superpotential is 14) The signs keep track of orientation in the N = 2 sense (the legs are unoriented in the N = 4 sense). The F-term relations are It should be kept in mind that the U (2) indices are associated with different nodes. Below, gauge indices are appropriately contracted between pairs of hypers when suppressed.
To conclude, we remark that the basis (4.40)-(4.45) (which we refer to as the "alternate basis") differs from the earlier one that we used when n = 4, namely (4.33), and that the Z 2 acts differently in the two cases. When n = 4, we have in the alternate basis that where we have rescaled the generators so that they satisfy X 2 + ZY 2 + Z 3 = 0 and so that X is the same as in (4.33). At the level of the chiral ring, this alternate basis maps to on the Coulomb branch of SU (2) SQCD with N f = 4, where we have set C ≡ 3 1/4 (4π) −1 .
A short calculation with the corresponding commutative shift operators shows that these operators likewise satisfy X 2 C + Z C Y 2 C + Z 3 C = 0. By the same reasoning as in Section 5.2.1, the enhanced S 3 symmetry requires that 40 at the quantum level, where we have defined Φ 2 ≡ Φ 2 − 1/3r 2 (which satisfies Φ 2 = 0). In the alternate basis, the Z 2 symmetry therefore acts as: This should be contrasted with the Z 2 symmetry acting on (4.33), which is more natural from the Coulomb branch point of view in that it simply flips the signs of monopoles. The Z 2 is only ambiguous when n = 4 because it can be conjugated by elements of S 3 : otherwise, it is unique.

C.3 Affine E n Quivers
We now turn to the E-type quiver theories. In all cases, the fundamental "meson" operators satisfy M I (I) = 0 where I is the length of leg I. To derive the chiral ring relation for E 6 , we need only the U (1) Schouten identity: following [46], the trick is to write the generators containing squares of mesons as U (1)×U (1) bifundamentals. For E 7,8 , we instead employ the U (2) Schouten identity: following [50], we define auxiliary operators with only U (2) indices uncontracted. We present the derivations for E 6,7,8 in decreasing amounts of detail.
Deriving a Schouten identity for tensors of given rank involves antisymmetrizing over an appropriate number of indices and then contracting a subset of these indices. For instance, the Schouten identity for two-component vectors follows from contracting any two indices in x [i y j z k] = 0. A Schouten identity for matrices [50] following from M k 1 [i 1 N k 2 i 2 K k 3 where the indices range over {1, 2}. 40 At the level of the quantized chiral ring, we know that (C. 43) as well as (5.61)  C D · · · = · · · (Q (I) ) A i (Q (I) ) i B (Q (I) ) B j (Q (I) ) j C (Q (I) ) C k (Q (I) ) k D · · · = · · · (Q (I) ) A i (q (I) ) i (q (I) ) j (q (I) ) j (q (I) ) k (Q (I) ) k D · · · = 0, (C.52) since (q (I) ) j (q (I) ) j = 0.