PHYSICAL REVIEW D 101, 085008 (2020)

Non-Abelian mirror symmetry beyond the chiral ring

† Yale Fan 1,* and Yifan Wang 1,2,3, 1Department of Physics, Princeton University, Princeton, New Jersey 08544, USA 2Center of Mathematical Sciences and Applications, Harvard University, Cambridge, Massachusetts 02138, USA 3Department of Physics, Harvard University, Cambridge, Massachusetts 02138, USA

(Received 1 March 2020; accepted 29 March 2020; published 20 April 2020)

Mirror symmetry is a type of infrared duality in three-dimensional (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 ¼ 4 supersymmetric field theories whose Higgs or Coulomb branches realize D- and E-type Kleinian singularities in the ADE classification, generalizing previous work on the A-type case. Such theories include the SUð2Þ 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 D-orE-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 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.

DOI: 10.1103/PhysRevD.101.085008

I. INTRODUCTION ’t Hooft anomalies.1 On the one hand, this is precisely what makes duality an efficient and elegant way to extract physical Three-dimensional gauge theories are strongly coupled information. On the other hand, this procedure can be at low energies due to the positive mass dimension of the misleading in cases where it fails to pinpoint the fate of Yang-Mills coupling. Consequently, they exhibit a wide the renormalization group (RG) flow [see [8–10] for exam- range of interesting nonperturbative phenomena, including ples in four dimensions (4D)]. Fortunately, for a subclass of monopole operators and confinement/de-confinement tran- dualities known as mirror symmetry of supersymmetric sitions. A powerful tool for elucidating complicated gauge gauge theories [11–14], we have increased analytic control dynamics is duality, which states that two distinct ultra- over the dynamics thanks to supersymmetry even while violet (UV) field theory descriptions give rise to the same many features of generic three-dimensional (3D) gauge theory in the deep infrared (IR) [1–7]. A key merit dynamics remain. of duality is that there often exists a manifestly weakly Supersymmetry (SUSY), especially the localization coupled dual description. Thus duality provides an efficient method, allows us to extract nontrivial dynamical data from language for tackling the problem of strong coupling. quantum field theories, such as their protected operator While duality, by definition, requires a map between all spectrum, low-energy effective action, and supersymmetric observables of the dual quantum field theories, most known partition functions, which all play important roles in testing dualities are motivated by matching quantities that are and refining the duality maps. Once a supersymmetric dual insensitive to dynamical details of the theories, such as 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 *[email protected] † [email protected] by mirror symmetry and supported by SUSY-breaking

Published by the American Physical Society under the terms of the Creative Commons Attribution 4.0 International license. Further distribution of this work must maintain attribution to 1For a class of large-N Chern-Simons-matter theories, the the author(s) and the published article’s title, journal citation, relevant dualities have been checked at the level of local and DOI. Funded by SCOAP3. correlation functions [1–4].

2470-0010=2020=101(8)=085008(41) 085008-1 Published by the American Physical Society YALE FAN and YIFAN WANG PHYS. REV. D 101, 085008 (2020) deformations thereof, including the important bosonization computed systematically from supersymmetric localization dualities of [15,16]. after mapping the TQM to a great S1 on S3 [31–33], and Much recent progress has been made in better under- plays an important role in determining the full OPE data standing 3D gauge theories with N ¼ 4 supersymmetry, of the SCFT using the conformal bootstrap technique which is the original context in which mirror symmetry was [23,34–37]. In recent work [38], these TQMs are formalized discovered [11]. These theories are characterized by an as noncommutative associative algebras equipped with an 2 2 SUð ÞH × SUð ÞC R-symmetry, and they have a vacuum even and positive short star product—equivalently, a moduli space consisting of (singular) hyperkähler mani- (twisted) trace or bilinear form. The latter is essential for folds that can be labeled as Coulomb branch MC, Higgs mapping the TQM data to CFT correlators. The action of branch MH, or mixed branches, depending on which mirror symmetry in the TQM sectors has been studied to a combination of R-symmetries is broken. The Coulomb limited extent in [31–33], focusing mainly on the case where and Higgs branches appear very different at first sight: the the corresponding SCFTs arise from Abelian gauge theories Coulomb and mixed branches embody the complicated such as super-quantum electrodynamics (SQED) and dynamics of 3D gauge theories, whereas the Higgs branch Abelian quivers.3 The bootstrap analysis for the particular is protected by supersymmetric nonrenormalization theo- case of SQED2, or equivalently the T½SUð2Þ theory, was rems and thus has a rigid structure [11]. The nontriviality of carried out in [37], where nontrivial evidence for the (self-) mirror symmetry amounts to the statement that there exist mirror symmetry beyond the TQM sector was found. mirror-dual pairs of 3D N ¼ 4 theories where the roles of In this paper, we initiate the systematic study of non- the Coulomb and Higgs branches, as well as classical and Abelian mirror symmetry in the TQM sectors of 3D N ¼ 4 quantum effects, are interchanged. A particularly simple SCFTs. Beautifully, the most well-studied examples of example is that of 3D N ¼ 4 Uð1Þ super-QED with one mirror symmetry fall into an ADE classification [11], with charged hypermultiplet, which is dual to a free hyper- those of A- and D-type admitting higher-rank generaliza- multiplet. For suitable matter content, a gauge theory can tions [12,13]. We focus on the simple class of theories T g flow to a superconformal fixed point in the IR whose that have rank-one Coulomb branches given by the ADE operator spectrum naturally has a description in terms of the singularities elementary degrees of freedom in the UV gauge theory. In M T C2 Γ this case, mirror symmetry amounts to interchanging the Cð gÞ¼ = g; ð1:1Þ descriptions of the conformal field theory (CFT) operators.

Namely, under mirror symmetry, an order-type mesonic where g labels an ADE Lie algebra and Γg is the operator written in terms of the fundamental fields in one corresponding discrete subgroup of SUð2Þ under the UV description is mapped to a disorder-type operator such as McKay correspondence, which can equivalently be repre- a monopole in the dual description. A particular class of sented as a hypersurface singularity in C3 (see Table I). operators in the superconformal field theory (SCFT) is that of The latter description makes explicit the Coulomb branch chiral ring operators, which are half-BPS (Bogomol'nyi- chiral ring of the theory T g, which is nothing but the Prasad-Sommerfield) and whose vacuum expectation values coordinate ring of the hypersurface singularity. give rise to the Coulomb branch, Higgs branch, and mixed From the quotient structure of the Coulomb branch, it is branches. The matching of the chiral rings (equivalently, the obvious that the free 3D N 4 theory with a single twisted – ¼ moduli spaces) [17 22] provides a first check for mirror hyper in which the discrete symmetry Γ ⊂ SUð2Þ is duality proposals beyond anomaly matching. g F gauged realizes this Coulomb branch as its vacuum moduli A more refined protected subsector in 3D N 4 ¼ space (similarly for its mirror in terms of a free hyper- SCFTs was discovered in [23,24]. It is described by a multiplet). A more interesting theory with Coulomb branch one-dimensional (1D) topological quantum mechanics (1.1) in the A −1 case is super-QED with n hypermultiplets (TQM) associated with either the Higgs or Coulomb n of unit charge, which we denote by SQED . Similarly, for branch.2 The relevant operators are twisted translations n the D case, an interacting theory with Coulomb branch of the Higgs (respectively, Coulomb) branch chiral pri- n (1.1) is SUð2Þ super-quantum chromodynamics (SQCD) maries by SU 2 [respectively, SU 2 ] R-symmetry ð ÞH ð ÞC n R3 with fundamental hypermultiplets. The exceptional cases rotations along a line in . Their correlation functions of (1.1) do not appear to have gauge theory realizations: depend only on the ordering of the insertions. The TQM however, there are mirror dual descriptions which instead contains nontrivial information about the operator product realize (1.1) as their Higgs branch expansion (OPE) data of the full SCFT, which can be

3Specifically, the Abelian A-type mirror symmetries were 2A variation of the Ω background [25–27] leads to related analyzed in [32], and a simple N ¼ 8 non-Abelian A-type mirror deformation quantizations of the Higgs and Coulomb branches symmetry was analyzed in [33]. The D3 case was also discussed [28–30]. It would be interesting to spell out the explicit relation to in Appendix F. 2 of [33], but this belongs to the A-series the TQM sector. (D3 ≅ A3).

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TABLE I. ADE Lie algebras along with their corresponding results for free theories that realize (1.3) via discrete SUð2Þ discrete subgroups and hypersurface singularities. Here, gauging. 2 2 2 2 Q4n and T; O; I are SUð Þ lifts of the familiar dihedral (D2n), Here is an outline of the rest of the paper. We start by tetrahedral (A4), octahedral (S4), and icosahedral (A5) subgroups providing some relevant background on TQM sectors in 3D of SOð3Þ. N ¼ 4 theories in Sec. II. We then give a brief review of A g Γg fgðX;Y; ZÞ mirror symmetry for the Abelian -type theories in Sec. III. 2 2 n We move on to deformation quantizations of D-type An−1 Zn X þ Y þ Z 2 2 n singularities in Sec. IV and explain how they are realized Dnþ1 Q4ðnþ1Þ X þ ZY þ Z 2 3 4 in 3D N ¼ 4 gauge theories. In Sec. V, by explicitly E6 2T X þ Y þ Z 2 3 3 computing TQM correlators, we infer the mirror map E7 2O X þ Y þ YZ 2 3 5 for TQM operators that quantize the Dn singularity: our E8 2I X þ Y þ Z results are summarized in (5.30) for n>4 and (5.40) for n ¼ 4. We carry out a similar deformation quantization of E-type singularities in Sec. VI and, in Sec. VII, present M T mirror C2=Γ : : Hð g Þ¼ g ð1 2Þ some motivating remarks toward understanding the non- Lagrangian theories whose Coulomb branches realize (1.1) In general, they are given by 3D N ¼ 4 quiver gauge with g ¼ En through the lens of the TQM. Some details of theories of affine ADE-type. The associative algebras our Higgs branch TQM computations, which tend to be associated with the ADE singularities are in general more convoluted than their Coulomb branch counterparts, given by the spherical symplectic reflection algebras of are gathered in Appendix A. In Appendix B, we consider complex dimension two [38]. This ADE-series of theories additional quantizations of the A- and D-type singularities T T mirror g (respectively, g ) has, in addition, a Higgs branch via field theory (outside the context of mirror symmetry), (respectively, Coulomb branch) given by generalizing some examples from [32,33]. In Appendix C, we give a self-contained exposition of the Higgs branch M T M T mirror O¯ Hð gÞ¼ Cð g Þ¼ minðgÞð1:3Þ chiral rings of the affine D- and E-type quivers, filling some gaps in the literature. O where minðgÞ denotes the minimal nilpotent orbit of g. This Higgs branch has a rigid structure in the TQM sector A. Notation thanks to the g flavor symmetry. The g-equivariant defor- Throughout this paper, we adhere to the following mation quantization of these hyperkähler cones was solved notational conventions: in [37], where unique star products were obtained except in (i) Straight O denotes an abstract chiral ring generator. ˆ the case of A1 ¼ sl2 [24]. Returning to the Coulomb (ii) Hatted O denotes an abstract quantum algebra branches in (1.1), the deformation quantization of the generator. ˆ An−1 case was studied in [24], where the extra Uð1Þ flavor (iii) Curly O denotes the realization of O as an SCFT symmetry played an important role in simplifying the operator (with suitable mixing). analysis. The mirror symmetry between SQED and the We also introduce the shorthand cyclic Abelian quiver was then spelled out in [32]. The primary goal of this paper is to carry out the analysis shðxÞ ≡ 2 sinhðπxÞ; chðxÞ ≡ 2 coshðπxÞ; of mirror symmetry at the level of the TQM for the D- and shðxÞ E-type cases, where the relevant (mirror) gauge theories are thðxÞ ≡ : ð1:4Þ ch x non-Abelian. In these cases, there are no continuous global ð Þ symmetries at our disposal, although there do exist discrete II. TOPOLOGICAL QUANTUM MECHANICS Z2 symmetries for Dn (enhanced to S3 for D4) and E6, which still place some constraints on the TQM. We start by In this section, we briefly review the prescriptions of solving the algebraic problem of deformation quantization [31–33] for computing observables within certain protected N 4 4 for Dn singularities. We then compute the correlators in the operator algebras of 3D ¼ theories on the sphere. TQM from both the SQCD description and the affine D- Combining these formalisms gives a way to derive precise type quiver description. By studying the explicit form of the maps between half-BPS operators across non-Abelian 3D matrix models and insertions that are obtained from super- mirror symmetry, and to compute previously unknown symmetric localization, we establish the precise mirror map quantizations of Higgs and Coulomb branch chiral rings. for the operators in the TQM that preserves the short star We consider 3D N ¼ 4 gauge theories of the cotangent- product. For the E-type cases, we solve the deformation type, namely with gauge group G and matter representation quantization of the E6;7;8 singularities and present some preliminary observations from the affine quiver side, 4See [39] for a complementary perspective on these protected leaving a complete analysis to future work. We also include correlation functions.

085008-3 YALE FAN and YIFAN WANG PHYS. REV. D 101, 085008 (2020) Z Z ¯ R ⊕ R. We denote by g the Lie algebra of G, t a fixed ˜ ˜ Zσ ≡ DQDQ exp 4πr dφQð∂φ þ σÞQ ; ð2:2Þ Cartan subalgebra of g, W the Weyl group, Δ the set Λ Λ∨ of roots, W the weight lattice, and W the coweight lattice. in terms of which the S3 partition function is The 3D N ¼ 4 SCFTs have two one-dimensional Z protected subsectors that each take the form of a TQM 1 3 μ σ ZS ¼ d ð Þ; [23,24]. The associative operator algebra of the TQM is a jWj t deformation quantization of either the Higgs or Coulomb det0 ðshðσÞÞ 0 adj branch chiral ring, and as such, it encodes detailed dμ σ ≡ dσ σ Zσ dσ : : ð Þ detadjðshð ÞÞ ¼ σ ð2 3Þ information about the geometry of the vacuum manifold. detRðchð ÞÞ When the SCFT arises from an RG flow with a Lagrangian O φ description in the UV, the Higgs branch sector is directly Namely, an n-point correlation function h 1ð 1Þ O φ 3 accessible by supersymmetric localization [31], but the nð nÞi on S can be written as Coulomb branch sector includes monopole operators, hO1ðφ1ÞO ðφ Þi which are disorder operators that cannot be represented Zn n in terms of the Lagrangian fields [32,33]. For this reason, 1 ¼ dμðσÞhO1ðφ1ÞOnðφnÞiσ ð2:4Þ the known methods for computing OPE data within these W 3 j jZS t two sectors look qualitatively different.

Each 1D sector can be described as the equivariant in terms of an auxiliary correlator hO1ðφ1ÞOnðφnÞiσ at cohomology of an appropriate supercharge. The corre- fixed σ. The latter is computed via Wick contractions with sponding cohomology classes are called twisted Higgs the 1D propagator or Coulomb branch operators (HBOs or CBOs).5 They are realized as Higgs or Coulomb branch chiral ring operators sgnφ12 þ thðσÞ 3 φ ˜ φ ≡ φ ≡ − −σφ12 which, when translated along a chosen line in R or a hQð 1ÞQð 2Þiσ Gσð 12Þ 8π e ; 1 3 r chosen great circle Sφ on S , are simultaneously twisted by φ12 ≡ φ1 − φ2; ð2:5Þ 2 2 SUð ÞH or SUð ÞC rotations. The OPE within each sector takes the form of a noncommutative star product derived from (2.2).6 X ΔiþΔj−Δk k Oi ⋆ Oj ¼ ζ cij Ok; ð2:1Þ k B. Coulomb branch formalism

3 The operators in the Coulomb branch topological sector, where, for theories placed on S , the quantization parameter in terms of a UV gauge theory Lagrangian, consist of a ζ is the inverse radius of the sphere: ζ ¼ 1=r. In addition to scalar ΦðφÞ [a twisted combination of the vector multiplet associativity, the star product inherits several conditions Φ 2 scalars a_ b_ transforming in the adjoint of suð ÞC and of from the physical SCFT, namely [24]: truncation or short- G], bare monopoles MbðφÞ, and dressed monopoles ness [the sum in (2.1) terminates after the term of order Φ Mb φ 2 Δ Δ Pð Þ ð Þ. The coweight b breaks the gauge group at ζ minð i; jÞ] due to the SUð2Þ or SUð2Þ selection rule, H C the insertion point to Gb, the centralizer of b, and the evenness [swapping O and O in (2.1) takes ζ → −ζ] i j corresponding monopole may be dressed by a Gb-invariant inherited from the symmetry properties of the 3D OPE, and polynomial PðΦÞ in ΦðφÞ [29]. positivity from unitarity (reflection positivity) of the In [32,33], a method for computing all observables 3D SCFT. within the Coulomb branch TQM was obtained for 3D N ¼ 4 gauge theories of cotangent-type by constructing a A. Higgs branch formalism set of “shift operators,” acting on functions of σ ∈ t and B ∈ Λ∨ , whose algebra is a representation of the Assuming a UV Lagrangian, the operators that comprise W 1D OPE.7 We find that ΦðφÞ is represented by a simple the Higgs branch topological sector are gauge-invariant ˜ 1 multiplication operator polynomials in antiperiodic scalars QðφÞ; QðφÞ on Sφ, which are twisted versions of the hypermultiplet scalars ˜ 2 6Wick contractions between elementary operators at coinci- qa; qa transforming in the fundamental of suð ÞH and in R, R¯ of G. The correlation functions of these twisted HBOs dent points are performed using O φ ið Þ can be computed within a 1D Gaussian theory [31] ˜ thðσÞ hQðφÞQðφÞiσ ≡ Gσð0Þ¼− ð2:6Þ with path integral 8πr to resolve normal-ordering ambiguities. 5Mixed-branch operators are not in the cohomology of either 7All expressions are given in the “North” picture. See [32,33] supercharge. for details.

085008-4 NON-ABELIAN MIRROR SYMMETRY BEYOND THE CHIRAL … PHYS. REV. D 101, 085008 (2020) 1 i as well as the gluing measure Φ ¼ σ þ B ∈ tC ¼ t ⊗ C: ð2:7Þ r 2 Y α · σ 2 α · B 2 μ σ;B −1 α·B The shift operator describing a dressed monopole has a ð Þ¼ ð Þ þ 2 α∈Δþ r r more intricate definition: it is constructed as Y 1 jρ·Bj jρ·Bj−ρ·B Γð2 þ iρ · σ þ 2 Þ 1 X × ð−1Þ 2 : ð2:13Þ −1 ˜ 1 jρ·Bj Φ Mb Φ w·b ρ∈ Γð − iρ · σ þ Þ Pð Þ ¼ W Pðw · ÞM ; ð2:8Þ R 2 2 j bj w∈W While the matrix model (2.11) converges only for theories W W where b is the stabilizer of b in , with the Weyl sum with a sufficiently large matter representation (i.e., “good” reflecting the fact that a physical magnetic charge is labeled and “ugly” theories [41]), the shift operators can always by the Weyl orbit of a coweight b. For a given coweight b, be used to compute star products in the Coulomb we define the Abelianized (non-Weyl-averaged) monopole branch TQM. shift operator Finally, in the commutative limit r → ∞, the algebra X of shift operators reduces to the Coulomb branch chiral ˜ b b ab Φ v M ¼ M þ Zb→vð ÞM ; ð2:9Þ ring, and we recover the Abelianization description of jvj

8It was proposed in [33] that the abelianized bubbling 9An important caveat is that the expression (2.15) holds coefficients are fixed by algebraic consistency of the OPE within for semisimple G. Otherwise, it would have some residual the Coulomb branch topological sector. r-dependence.

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degðX; Y; ZÞ¼ðn; n; 2Þð3:3Þ Higgs branch of the former theory and the Coulomb branch of the latter. 2 2 and the Poisson bracket (equivalently, ω) has degree We denote by SUð ÞR the relevant SUð Þ R-symmetry (for either the Higgs or the Coulomb branch) of the degðωÞ¼2: ð3:4Þ TQM sector. The degree of an element O in the quantum algebra is related to the R-symmetry spin (taking values in The (filtered) deformation quantization of this graded half-integers) by Poisson algebra is easy to work out (see [42]). The quantum 1 algebra is given by the central quotient O O Rð Þ¼2 degð Þ; ð3:11Þ C½X;ˆ Y;ˆ Zˆ A ¼ ; ð3:5Þ since the holomorphic symplectic form ω must transform as An−1 Ω h An−1 i 2 an SUð ÞR triplet (corresponding to the three independent complex structures of the hyperkähler cone). Moreover, C ˆ ˆ ˆ where the noncommutative algebra ½X; Y;Z is defined by superconformal representation theory requires that the the commutators scaling dimensions of the corresponding operators satisfy ˆ ˆ ζ ˆ ½X; Y¼i PðZÞ; ΔðOÞ¼RðOÞ: ð3:12Þ ½X;ˆ Zˆ ¼2iζY;ˆ ˆ ˆ ˆ ˆ ˆ ˆ From this, we conclude that X;Y;Z must be associated with ½Y;Z¼−2iζX; ð3:6Þ n n chiral ring operators of dimension 2 ; 2 ; 1, respectively, in the physical theory. Below, we give their explicit realiza- the deformation parameter ζ has degree tions in terms of mesonic and monopole operators in 3D N ¼ 4 theories. degðζÞ¼2; ð3:7Þ The necklace quiver gauge theory has gauge group N ˜ Uð1Þ =Uð1Þ, bifundamental hypermultiplets ðQI; QIÞ for and the center is generated by 2 I ¼ 1; …;N, and Higgs branch C =ZN. The Higgs branch Ω ˆ 2ζ ˆ − 2 ˆ 2 ˆ 2 chiral ring generators are An−1 ¼ QðZ þ ÞþQðZÞ ðX þ Y Þ: ð3:8Þ X Y ˜ ˜ ˜ Ω ˆ ˆ ˆ 0 ¼ Q1Q2 QN; ¼ Q1Q2 QN; Physically, A −1 ðX; Y;ZÞ¼ is the quantum chiral ring n ˜ ˜ relation in the TQM. Z ¼ Q1Q1 ¼¼QNQN: ð3:13Þ Here, PðtÞ and QðtÞ are polynomials of degree n − 1 and n with leading terms ntn−1 and tn, respectively. They satisfy On the other hand, the Coulomb branch TQM operators of SQEDN are products of the twisted vector multiplet scalar QðZˆ þ 2ζÞ − QðZˆ Þ¼2ζPðZˆ Þ: ð3:9Þ Φ and monopole operators of charge b ∈ Z. The corre- sponding shift operators act on functions of σ ∈ R 10 Thus QðtÞ is fixed by PðtÞ except for the constant term. and B ∈ Z. The Coulomb branch of this theory is also 2 Expanding PðtÞ as isomorphic to C =ZN, and its chiral ring is generated by

Xn−1 1 1 X M−1 Y M1 Z − i Φ n−1 α ζn−i i−1 ¼ 2 ; ¼ 2 ; ¼ : PðtÞ¼nt þ i t ð3:10Þ ð4πÞN= ð4πÞN= 4π i¼1 ð3:14Þ and denoting the constant term of QðtÞ by α0, we see that the space of quantizations of the An−1 singularity is On either side of the duality, the above operators obey X ⋆ Y ZN 1 n-dimensional and parametrized by fα0; α1; …; αn−1g. ¼ þOð =rÞ, have identical correlation functions, Imposing the evenness condition amounts to picking out and generate all other gauge-invariant operators in the terms in PðtÞ that have an even degree in ζ. Thus we end up corresponding TQM. Correlation functions of composite A n with a space of even quantizations A −1 of dimension b2c. operators can also be matched using the OPE. n 1 In this example, the Uð Þtop symmetry prohibits operator B. A-type mirror symmetry mixing, and thus we have unambiguous identifications A detailed TQM analysis of the Abelian mirror duality X ¼ X;ˆ Y ¼ Y;ˆ Z ¼ Zˆ ð3:15Þ between the affine AN−1 quiver gauge theory and SQEDN was given in [32]. Here, we summarize the results for the “rank-one” side of this duality, namely that between the 10See Appendix B1 for details and generalizations.

085008-6 NON-ABELIAN MIRROR SYMMETRY BEYOND THE CHIRAL … PHYS. REV. D 101, 085008 (2020) in both mirror-dual descriptions. This simplifying feature input from 3D N ¼ 4 SCFTs allows us to determine the will no longer be present in the D case. short product and to provide the deformed mirror map for dual observables. C2 Γ IV. DEFORMATION QUANTIZATION OF = Dn A. n =4 For general Dnþ1 singularities defined as 1. Periods and associativity M ∶ f X; Y; Z X2 ZY2 Zn 0; : Dnþ1 ð Þ¼ þ þ ¼ ð4 1Þ We would like to quantize the D4 singularity the degrees of the generators are given by M ∶ 2 2 3 0 D4 fðX; Y; ZÞ¼X þ ZY þ Z ¼ ; ð4:8Þ degðX; Y; ZÞ¼ð2n; 2n − 2; 4Þ: ð4:2Þ which merits special attention due to its extra symmetry. The deformation quantization is again easy to work out (see We start by writing down the most general deformed [43]). The quantum algebra is given by the central quotient commutators compatible with the Jacobi identity (4.4):

ˆ ˆ ˆ 2 ˆ 2 2 ˆ 4 C½X;ˆ Y;ˆ Zˆ ½X; Y¼ζðY þ 3Z þ ζ ð2A þ 8ÞZ þ ζ ð2A þ B þ 8ÞÞ; A ¼ ; ð4:3Þ Dnþ1 Ω ˆ ˆ ˆ ˆ 2 ˆ 5 h Dnþ1 i ½X; Z¼−2ζZ Y −2ζ X þ γζ ; ˆ ˆ 2ζ ˆ where the algebra C½X;ˆ Y;ˆ Zˆ is defined by the commutators ½Y;Z¼ X; ð4:9Þ

½X;ˆ Yˆ ¼ζYˆ 2 þ ζPðZˆ Þ; as well as the central element ½X;ˆ Zˆ ¼−2ζZˆ Yˆ −2ζ2Xˆ þð−1Þnþ1γζnþ2; Ω ˆ 3 ζ2 ˆ 2 ζ4 ˆ ζ6 ˆ 2 D4 ¼ Z þ A Z þ B Z þ C þ X ˆ ˆ ˆ ½Y;Z¼2ζX; ð4:4Þ þ Zˆ Yˆ 2 þ 2ζXˆ Yˆ −γζ4Y:ˆ ð4:10Þ and the center is generated by Notice that in (4.9), the leading-order terms in ζ are simply the Poisson bracket associated with the singularity, coming Ω Q Zˆ Xˆ 2 Zˆ Yˆ 2 2ζXˆ Yˆ −1 nγζnþ1Y:ˆ : Dnþ1 ¼ ð Þþ þ þ þð Þ ð4 5Þ from the symplectic two-form ω. The deformation quan- tization of the D4 singularity falls into isomorphism classes Here, PðtÞ and QðtÞ are polynomials of degree n − 1 and n [44] (see also [38]) that are parametrized by so-called −1 with leading terms ntn and tn, respectively. They satisfy “periods” taking values in

Qð−tðt=ζ2 − 1ÞÞ − Qð−tðt=ζ2 þ 1ÞÞ 2 Mreg C H ð D4 ; Þ 2 2 2 2 ; ð4:11Þ ¼ðt − ζ ÞPð−tðt=ζ − 1ÞÞþðt þ ζ ÞPð−tðt=ζ þ 1ÞÞ: WðD4Þ ð4:6Þ which is simply the root lattice of D4 modulo the Weyl group. Here, the periods that label the quantizations Thus QðtÞ is fixed by PðtÞ except for the constant term. are fA; B; C; γg. Expanding PðtÞ as The D4 singularity has an S3 symmetry (preserving the ω Xn−1 holomorphic symplectic two-form ) that becomes mani- n−1 2ðn−iÞ i−1 fest in the coordinates PðtÞ¼nt þ αiζ t ð4:7Þ i¼1 1 Y 1 Y 1 U ¼ Z þpffiffiffi ;V¼ Z −pffiffiffi ;W¼ X; ð4:12Þ and denoting the constant term of QðtÞ by α0, we see that the 2 3 2 3 2 1 space of quantizations of the Dnþ1 singularity is (n þ )- dimensional and parametrized by fα0; α1; …; αn−1; γg. in terms of which the singularity becomes Imposing the evenness condition, we see that γ ¼ 0 for n even and is unconstrained for n odd. Thus we conclude that M ∶ 3 3 2 0 D4 fðU; V; WÞ¼U þ V þ W ¼ : ð4:13Þ the space of even quantizations for Dnþ1 singularities is n-dimensional for n even and (n þ 1)-dimensional for n odd. This is invariant under To pin down the TQM, one needs to further specify the short product structure (which is equivalent to speci- Z2∶ ðU; V; WÞ ↦ ðV;U;−WÞ; A fying a trace) of the associative algebra D 1 . We will 4πi 2πi nþ Z ∶ ↦ 3 3 analyze how combining discrete symmetry and physical 3 ðU; V; WÞ ðUe ;Ve ;WÞ; ð4:14Þ

085008-7 YALE FAN and YIFAN WANG PHYS. REV. D 101, 085008 (2020) generating an S3 symmetry that preserves the hyperkähler 1 Að2α1 − 1Þþ2ðα1 þ 6α2 − 2Þα4 A1 ¼ − pffiffiffi ;A2 ¼ ; structure. 3 2α1ð4 þ A þ 6α1Þ If we insist on having an S3 symmetry upon deformation Að1 − 2α1Þ − 2ðα1 þ 6α2 − 2Þ quantization,11 then the periods are constrained. Up to a A3 ¼ 12 ; redefinition, the most general S3-preserving (deformed) 2 1 commutators are B1 ¼ − pffiffiffi ;B2 ¼ − − A1 þ A2; 3 2 2 ffiffiffi 4 1 ˆ ˆ ˆ ˆ ˆ p ˆ 2 þ A 3 ˆ ½U; V¼pffiffiffi ζW; ½U; W¼− 3ζV − pffiffiffi ζ U; B3 ¼ − α1ð4 þ A þ 6α1Þ: ð4:17Þ 3 2 3 6 pffiffiffi 4 þ A ½V;ˆ Wˆ ¼ 3ζUˆ 2 þ pffiffiffi ζ3V:ˆ ð4:15Þ Consequently, the only free parameters are α1, α2, α4, 2 3 and A. We can determine these parameters for specific defor- From here, we can work out the most general even short mation quantizations. One nontrivial example comes from star product structures the Coulomb branch of N ¼ 4 SUð2Þ SQCD with four fundamental hypermultiplets, or by mirror symmetry, the ˆ ˆ c2 2 ˆ Higgs branch of the affine D4 quiver theory. In either case, U ⋆ U U α1ζ V; ¼ þ by explicit computation, we find ˆ ˆ c2 2 ˆ V ⋆ V ¼ V þ α1ζ U; 32π4 − 2835 2π4 − 135 ζ α α ˆ ˆ d ˆ 4 1 ¼ 4 ; 2 ¼ 4 ; U ⋆ V ¼ UV þ pffiffiffi W þ α2ζ ; 42π − 2835 180π 3 1 −1 pffiffiffi A ¼ −6; l ¼ 34ζ ; ð4:18Þ 3ζ c ð6α1 þ A þ 4Þ Uˆ ⋆ Wˆ ¼ UWd − V2 − pffiffiffi Uˆ ζ3; 2 4 3 as well as pffiffiffi 3ζ c ð6α1 þ A þ 4Þ 4 8 4 Vˆ ⋆ Wˆ ¼ VWd þ U2 þ pffiffiffi Vˆ ζ3; 20π ð1376π −178185π þ4365900Þ 2 4 3 α4 ¼ ; ð4:19Þ 231ð2π4 −135Þð128π8 −12180π4 −14175Þ c c A þ 4ðα1 þ α4Þ Wˆ ⋆ Wˆ ¼ −U3 − V3 − ζ2UVd 2 where 6α 4 α ð 1 þ A þ Þ 2 ζ6 þ 4 ; l ¼ −4πr ð4:20Þ α ˆ c2 c3 2d 1 3 ˆ U ⋆ U ¼ U þ α4ζ UV − pffiffiffi ζ W; is the natural deformation parameter that arises in the 3 3 α derivation of the 1D TQM from the 3D N ¼ 4 SCFT on S ˆ c2 c3 2d 1 3 ˆ 12 V ⋆ V ¼ V þ α4ζ UV þ pffiffiffi ζ W; (r is the sphere radius). Combined with (4.16), these data 3 determine a large class of correlators in the 1D TQM for ˆ c2 d2 d 2 c2 4 ˆ U ⋆ V ¼ UV − B1VWζ þ ζ B2U þ B3ζ V; either the SQCD or the affine D4 quiver theory. ˆ c2 d2 d 2 c2 4 ˆ V ⋆ U ¼ VU þ B1UWζ þ ζ B2V þ B3ζ U; 2. Realizations in Lagrangian 3D SCFTs ˆ d d2 d 2 ˆ2 4 ˆ R U ⋆ UV ¼ VU − A1UWζ þ ζ A2V þ A3ζ U; Let us define to be the ring of holomorphic functions on the D4 singularity, ˆ d d2 d 2 c2 4 ˆ V ⋆ UV ¼ UV þ A1VWζ þ ζ A2U þ A3ζ V; ð4:16Þ M ∶ 2 2 3 0 3 3 2 0 D4 X þ ZY þ Z ¼ or U þ V þ W ¼ : d where all of the operators UαVβWδ are normal-ordered ð4:21Þ products that are assumed (with suitable shifts) to have vanishing one-point functions, and nonvanishing two-point We now discuss 3D N ¼ 4 SCFTs that realize R as their functions only with their conjugates [conjugation being Higgs or Coulomb branch chiral ring. In the next section, defined by the Z2 generator in (4.14)]. we will extract the 1D TQM that gives the deformation The parameters that appear above in the star product are quantization of R by the corresponding SCFT. 2 2 further constrained by associativity as follows: We denote by SUð ÞR the relevant SUð Þ R-symmetry R 2 under which is charged. We can fix the SUð ÞR 11From the perspective of 3D N ¼ 4 SCFT, this is equivalent 12 to insisting that S3 be a global symmetry of the Higgs or Coulomb In deriving these results, we use the results of Appendix A3, branch operator algebra. particularly (A34).

085008-8 NON-ABELIAN MIRROR SYMMETRY BEYOND THE CHIRAL … PHYS. REV. D 101, 085008 (2020) representations of R using the fact that the holomorphic 01 1 1 þ i −1 þ i ω 2 A ¼ ;B¼ ; symplectic form must transform as an SUð ÞR triplet: −10 2 1 þ i 1 − i ω 1 1 1 þ i 0 R½ ¼ : ð4:22Þ C ¼ pffiffiffi : ð4:30Þ 2 01− i Consequently, The Z2 and Z3 generators of S3 are identified with C and B R½U¼R½V¼R½X¼R½Y¼2;R½W¼R½Z¼3: in the quotient, respectively, which act as

ð4:23Þ C∶ ðX; Y; ZÞ ↦ ð−X; −Y; ZÞ; Y þ 3iZ Z þ iY Moreover, superconformal representation theory fixes the B∶ ðX; Y; ZÞ ↦ X; − ; − : ð4:31Þ scaling dimensions to be Δ ¼ R. 2 2

Γ — X Y D4 gauged free hyper. In this case, the generators , , Z ˜ Affine D4 quiver.—In this case, the gauge theory is can be written in terms of the complex scalars Q; Q in a N 4 2 1 4 single hypermultiplet13 which is acted upon by the gauge described as a 3D ¼ quiver with SUð Þ × Uð Þ Γ ⊂ 2 gauge group and four bifundamental hypermultiplets symmetry Q8 ¼ D4 SUð Þ as ðQ ; Q˜ Þ where A ¼ 1, 2, 3, 4. The quiver Lagrangian A A i 0 01 has an obvious S4 global symmetry that acts naturally on rZ ¼ ;sZ ¼ : ð4:25Þ the A index, but its action is not faithful on the Higgs 4 0 − 2 −10 i branch chiral ring after we take into account the D-term relations. In fact, the Higgs branch chiral ring R is Hence a basis of gauge-invariant operators is given by organized into faithful representations of S3. Without Z −2 2 ˜ 2 Y 4 ˜ 4 loss of generality, we choose an S3 subgroup of S4 to be ¼ Q Q ; ¼ iðQ þ Q Þ; 1 pffiffiffi the one permuting A ¼ , 2, 3. Then by identifying the X ¼ 2iQQ˜ ðQ4 − Q˜ 4Þ; ð4:26Þ generators 123 12 which satisfy the constraint sZ3 ¼ð Þ;rZ2 ¼ð Þ; ð4:32Þ

X 2 þ ZY2 þ Z3 ¼ 0: ð4:27Þ we find that ffiffiffi p ˜ ˜ ˜ ˜ Now the normalizer of Q8 in SUð2Þ is O48, the binary Z ¼ 3ðQ1Q3Q3Q1 þ Q2Q3Q3Q2Þ; ffiffiffi octahedral group of order 48. Thus the global symmetry p ˜ ˜ ˜ ˜ Y 3i Q1Q3Q3Q1 − Q2Q3Q3Q2 ; group of the discretely gauged hypermultiplet is14 ¼ ð Þ 3=4 ˜ ˜ ˜ X ¼ 2 · 3 iQ1Q2Q2Q3Q3Q1 ð4:33Þ O Γ S3 ¼ 48= D4 ; ð4:28Þ [where the contraction of SUð2Þ gauge indices is pairwise the permutation group of order 6. from the left] transform under S3 in the expected manner More explicitly, O48 is defined by generators and and satisfy relations X 2 þ ZY2 þ Z3 ¼ 0: ð4:34Þ O Γ 2 3 4 2 2 − 48 ¼ E7 ¼hA; B; CjA ¼ B ¼ C ¼ðC AÞ ¼ I2; 2 3 6 Alternatively, we have ðC BÞ ¼ðABÞ ¼ I2i; ð4:29Þ iπ=6 ˜ ˜ −iπ=6 ˜ ˜ U ¼ e Q1Q3Q3Q1 þ e Q2Q3Q3Q2; where −iπ=6 ˜ ˜ iπ=6 ˜ ˜ V ¼ e Q1Q3Q3Q1 þ e Q2Q3Q3Q2; 13 3=4 ˜ ˜ ˜ Alternatively, we denote the scalars in a hyper by QaA where W ¼ 3 iQ1Q2Q2Q3Q3Q1: ð4:35Þ 2 2 a is the SUð ÞH index and i is the SUð Þ flavor index. They obey the reality condition See Appendix C1 for a derivation. a i ϵ ϵij b ϵ12 ϵ 1 ðqi Þ ¼ qa ¼ ab qj ; ¼ 12 ¼ ; ð4:24Þ SUð2Þ SQCD with Nf ¼ 4.—The mirror dual of the affine 1 ˜ 1 2 where Q ¼ q2 and Q ¼ q1. D4 quiver theory is known to be SUð Þ SQCD with four 14In general, the global symmetry of G gauged hypers is given fundamental hypermultiplets. The Coulomb branch chiral R by the quotient group NUSpð2nH ÞðGÞ=G. ring of the latter theory now realizes . More explicitly, the

085008-9 YALE FAN and YIFAN WANG PHYS. REV. D 101, 085008 (2020)

˜ ˜ relevant Coulomb branch operators consist of monopole Z ¼ −Q1Q3Q3Q1; ð4:40Þ operators M2 and the Cartan scalar Φ, which are subject to ˜ ˜ ˜ ˜ n=2−1 the quantum ring relation (one-loop effect) Y ¼ 2Q3K1 Kn−4Q2Q2Kn−4 K1Q3 þð−ZÞ ; ð4:41Þ 4M2M−2 ¼ Φ4 ð4:36Þ ˜ ˜ ˜ ˜ ˜ X ¼ 2Q1K1 Kn−4Q2Q2Kn−4 K1Q3Q3Q1 ð4:42Þ and transform under the Z2 Weyl group as for n ∈ 2Z (as in [45,46]) and Φ → −Φ 2 → ∓2 ;M M : ð4:37Þ ˜ ˜ Z ¼ −Q1Q3Q3Q1; ð4:43Þ R is generated by the gauge-invariant combinations ˜ ˜ ˜ ˜ Y ¼ 2Q3K1 Kn−4Q2Q2Kn−4 K1Q3; ð4:44Þ 2 2 −2 2 −2 Z ¼ Φ ; Y ¼ iðM þ M Þ; X ¼ ΦðM − M Þ: ˜ ˜ ˜ ˜ ˜ ðn−1Þ=2 X ¼ 2Q1K1 Kn−4Q2Q2Kn−4 K1Q3Q3Q1 −ð−ZÞ ð4:38Þ ð4:45Þ ∈ 2Z 1 (We discuss the algebra of these operators in more detail in for n þ , which satisfy the ring relation the next section, using a slightly different normalization.) X 2 þ ZY2 − Zn−1 ¼ 0: ð4:46Þ

B. n > 4 The Z2 symmetry For n>4, we focus on two specific Lagrangian Z ∶ X Y Z ↦ −X −Y Z realizations of the quantized Dn singularity, namely those 2 ð ; ; Þ ð ; ; Þð4:47Þ that participate in 3D mirror symmetry. ˜ is induced by the 1 ↔ 3 flip of the quiver, or ðQ1; Q1Þ ↔ ˜ Q3; Q3 1. Higgs branch of affine D quiver ð Þ at the Lagrangian level. See Appendix C2 for n derivations. This quiver takes the shape of an affine Dn Dynkin So far, our discussion has been at the level of the diagram. We label the four Uð1Þ × Uð2Þ bifundamental (“classical”) chiral ring. In the next section, we will see ˜ hypermultiplets by QA; QA, as before. But now we have, in through TQM computations how these operators become ˜ addition, n − 4 Uð2Þ × Uð2Þ bifundamental hypers KI; KI “quantized.” connecting the n − 3 Uð2Þ gauge nodes. Nontrivial gauge- invariant elements of the Higgs branch chiral ring corre- 2. Coulomb branch of SUð2Þ SQCD with Nf =n spond to closed paths ending on one of the univalent 2 ≥ 0 1 Consider SUð Þ SQCD with Nf fundamental fla- Uð Þ nodes. vors. Using the slightly more compact notation from For this theory, we use the conventions Sec. II B, the Coulomb branch chiral ring is generated by the Weyl-invariant operators M2, ΦM2, Φ2 with

2 2 2 ΔðM Þ¼Nf − 2; ΔðΦM Þ¼Nf − 1; ΔðΦ Þ¼2; ð4:39Þ ð4:48Þ

where M2 is the monopole of minimal charge and ΦM2 is where the overall quotient by Uð1Þ is implemented in the a dressed monopole. (Again, we discuss the algebra of matrix model by making the first Uð2Þ node SUð2Þ.We these operators in more detail in the next section.) label the hypermultiplets by V. D-TYPE MIRROR SYMMETRY ˜ i 1 … 4 ðQAÞi; ðQAÞ for A ¼ ; ; ; Let us now see how the kinematical considerations of the j ˜ j 1 … − 4 ðKIÞi ; ðKIÞi for I ¼ ; ;n previous section translate into dynamical information about quantum field theories. Specifically, by computing TQM (so that the first index of KI and the last index of correlators in the theories whose Higgs and Coulomb ˜ KI are associated with node I), where i, j ¼ 1, 2. With branches are exchanged by D-type mirror symmetry, we contractions implicit, the Higgs branch chiral ring gener- derive the mirror map at the level of quantized Higgs and ators are Coulomb branch chiral rings. As before, we examine the

085008-10 NON-ABELIAN MIRROR SYMMETRY BEYOND THE CHIRAL … PHYS. REV. D 101, 085008 (2020) cases n ¼ 4 and n>4 separately due to the extra symmetry We then set of the former case. −iπ=6 ˜ ˜ iπ=6 ˜ ˜ U0 ¼ e Q1Q3Q3Q1 þ e Q2Q3Q3Q2; ð5:6Þ A. Higgs branch of affine Dn quiver

1. n = 4 iπ=6 ˜ ˜ −iπ=6 ˜ ˜ V0 ¼ e Q1Q3Q3Q1 þ e Q2Q3Q3Q2; ð5:7Þ We summarize the relevant Higgs branch TQM OPE ˜ ˜ data for the gauge-invariant operators Q1Q3Q3Q1 and ˜ ˜ W 33=4 ˜ ˜ ˜ Q2Q3Q3Q2, extracted from localization computations 0 ¼ iQ1Q2Q2Q3Q3Q1; ð5:8Þ (see Appendix A3). The one-point functions are pffiffi 3 where hU0i¼hV0i¼ 2 2. (The connected n-point func- ˜ ˜ ˜ ˜ 1 96π r hQ1Q3Q3Q1i¼hQ2Q3Q3Q2i¼ : ð5:1Þ tion of operators O0 is the n-point function of the 96π2r2 normalized operators O ≡ O0 − hO0i with hOi¼0.) We The diagonal two-point functions are find that

˜ ˜ ⋆ ˜ ˜ ˜ ˜ ⋆ ˜ ˜ hQ1Q3Q3Q1 Q1Q3Q3Q1i¼hQ2Q3Q3Q2 Q2Q3Q3Q2i hUi¼hVi¼hWi¼hU ⋆ Ui¼hV ⋆ Vi π4 − 30 U ⋆ W V ⋆ W 0 ¼ ð5:2Þ ¼h i¼h i¼ ; 5120π8r4 pffiffiffi 2π4 − 135 3 3ðπ4 − 105Þ U ⋆ V ; W ⋆ W ; ⇒ ˜ ˜ ⋆ ˜ ˜ h i¼ 60π4l4 h i¼ 4 6 hQ1Q3Q3Q1 Q1Q3Q3Q1ic 140π l 4 5:9 ˜ ˜ ˜ ˜ 2π − 135 ð Þ ¼hQ2Q3Q3Q2 ⋆ Q2Q3Q3Q2i ¼ ; ð5:3Þ c 23040π8r4 as required by the S3 global symmetry of the theory, all of where the c subscript denotes a connected correlator. The which can be checked explicitly using the Higgs branch mixed two-point function is TQM path integral (A16). 4 ˜ ˜ ˜ ˜ π þ 45 hQ1Q3Q3Q1 ⋆ Q2Q3Q3Q2i¼ ð5:4Þ 15360π8r4 2. n > 4

4 To illustrate how Higgs branch computations work for ˜ ˜ ˜ ˜ 2π − 135 arbitrary n, we start with the S3 partition function of the ⇒ Q1Q3Q3Q1 ⋆ Q2Q3Q3Q2 − : : h ic ¼ 46080π8 4 ð5 5Þ r affine Dn quiver theory,

Z 1 Y4 Yn−3 Y2 Yn−3 i 1 2 1 2 2 Z ¼ dσ du δðu1 þ u1Þ shðu − u Þ Zσ ; ð5:10Þ Dn 2n−4 A I I I ;u A¼1 I¼1 i¼1 I¼1

1 Zσ ≡ Q Q Q Q Q ð5:11Þ ;u 2 σ − i σ − i n−4 2 i − j i¼1½ A¼1;3chð A u1Þ A¼2;4chð A un−3Þ I¼1 i;j¼1 chðuI uIþ1Þ

Z 1 2n−4 1 2n−3 15 Y4 2 (note the = prefactor rather than = ), which shð2uÞ Z ¼ du dσ Q ð5:12Þ reduces to the expected D4 A 4 σ A¼1 A¼1 chð A uÞ

15As explained in Appendix A4, accounting for this factor of 2 is crucial for mirror symmetry to work at the level of S3 partition for n ¼ 4. functions, namely for matching the partition function to that of ˜ ˜ SUð2Þ SQCD. This corrects a number of errors in [47]. Consider insertions of the operator Z ¼ −Q1Q3Q3Q1. Roughly speaking, the reason is that the gauge group of the In Appendix A4, we prove that these can be rewritten as 2 n−3 1 4 1 2 quiver theory is ½Uð Þ × Uð Þ =Uð Þ, and one of the SUð Þ insertions of a function of the Coulomb branch scalar gauge nodes is really SOð3Þ due to the identification. Since the 2 volume of SOð3Þ is halved relative to that of SUð2Þ, we have a vacuum expectation value (VEV) s into the rank-one SUð Þ Weyl factor of 1=2 only for n − 4 of the n − 3 non-Abelian gauge SQCD Coulomb branch matrix model. The SUð2Þ matrix nodes. model takes the form

085008-11 YALE FAN and YIFAN WANG PHYS. REV. D 101, 085008 (2020) Z 1 2 1 2 1 shðsÞ Z ↔ Φ2 − Z 2 ½fðsÞ ¼ ds fðsÞ; ð5:13Þ 2 ð5:15Þ SUð Þþn 4r2 chðs=2Þ2n 8π r where “SUð2Þþn” is shorthand for “SUð2Þ with n for all n>4. flavors” [see (5.39)]. By manipulations on the Higgs branch To go further, we now set up the computation of more side of the Dn theory, we find that general Higgs branch correlation functions for Dn. We can write 2 − 1 8π 2 ZSUð2Þþn½ðs Þ=ð rÞ Z hZi¼ ; Y4 Yn−4 R ZSUð2Þþn ˜ ˜ 4πr dφL Zσ;u ¼ DQADQA DKIDKI e ; 2 2 4 ZSU 2 n½ðs − 1Þ =ð8πrÞ A¼1 I¼1 hZ ⋆ Zi¼ ð Þþ ; ð5:14Þ ZSUð2Þþn ð5:16Þ which provides strong evidence for the mirror map where

X X ˜ i δj ∂ σ − j ˜ i δj ∂ σ − j L ¼ ðQAÞ ð i ð φ þ AÞ ðu1Þi ÞðQAÞj þ ðQAÞ ð i ð φ þ AÞ ðun−3Þi ÞðQAÞj A¼1;3 A¼2;4 Xn−4 ˜ j δi0 δi ∂ i0 δi − δi0 i j0 þ ðKIÞi ð j j0 φ þðuIÞj j0 j ðuIþ1Þj0 ÞðKIÞi0 ð5:17Þ I¼1

1 2 and uI ¼ diagðuI ;uI Þ. Hence we have

i sgn φ12 th σ − u i ˜ j j ð Þþ ð A 1Þ −ðσ −u Þφ12 hðQ Þ ðφ1ÞðQ Þ ðφ2Þiσ ¼ −δ δ e A 1 ðA; B ¼ 1; 3Þ; A i B ;u AB i 8πr i sgn φ12 th σ − u i ˜ j j ð Þþ ð A n−3Þ −ðσ −u Þφ12 hðQ Þ ðφ1ÞðQ Þ ðφ2Þiσ ¼ −δ δ e A n−3 ðA; B ¼ 2; 4Þ; A i A ;u AB i 8πr φ i − i0 0 j0 j sgnð 12ÞþthðuI uI 1Þ − i − i0 φ j ˜ j þ ðuI uI 1Þ 12 hðK Þ ðφ1ÞðK Þ 0 ðφ2Þiσ ¼ −δ δ 0 e þ : ð5:18Þ I i I i ;u i i 8πr As an example, consider

˜ i1 ˜ j1 ˜ i ˜ j Q1 Q3 Q3 Q1 φ1 Q1 p Q3 Q3 p Q1 φ 5:19 h½ð Þ ð Þi1 ð Þ ð Þj1 ð Þ½ð Þ ð Þip ð Þ ð Þjp ð pÞiσ;u ð Þ

2 (by convention, we take φ1 < < φp). The exponential factors cancel in all full contractions. There are clearly ðp!Þ different contractions (p! for each of A ¼ 1 and A ¼ 3): 1 2p X2 X2 X2 X2 X X ˜ ˜ p hðQ1Q3Q3Q1Þ iσ;u ¼ 8π r 1 1 1 1 ρ∈ 0 i1¼ ip¼ j1¼ jp¼ Sp ρ ∈Sp Yp Yp ik ik jρðlÞ il × δ ð−sgnðφ ρ Þþthðσ1 − u1 ÞÞ δ ðsgnðφlρ l Þþthðσ3 − u1 ÞÞ: ð5:20Þ jρðkÞ k ðkÞ il ð Þ k¼1 l¼1

This reduces to previous results in Appendix A4 for p ¼ 1, 2. On the other hand, the following involves only one contraction: 1 n−2 X Yn−4 ˜ ˜ ˜ ˜ i1 in−3 iI iIþ1 hQ3K1 K −4Q2Q2K −4 K1Q3iσ ¼ − thðσ3 − u Þthðσ2 − u Þ thðu − u Þ: ð5:21Þ n n ;u 8πr 1 n−3 I Iþ1 i1;…;in−3 I¼1

Similarly,

085008-12 NON-ABELIAN MIRROR SYMMETRY BEYOND THE CHIRAL … PHYS. REV. D 101, 085008 (2020)

˜ ˜ ˜ ˜ ˜ hQ1K1 Kn−4Q2Q2Kn−4 K1Q3Q3Q1iσ;u 1 n−1 X Yn−4 i1 i1 i −3 i iIþ1 ¼ − thðσ1 − u Þthðσ3 − u Þthðσ2 − u n Þ thðu I − u Þ: ð5:22Þ 8πr 1 1 n−3 I Iþ1 i1;…;in−3 I¼1

σ 1;2 σ 1;2 To simplify these insertions in the matrix model (5.10), let us relabel 1;3 ¼ u0 and 2;4 ¼ un−2. As explained in 1 2 1 … − 2 Appendix A4, using the Cauchy determinant formula and swapping uI and uI for I ¼ ; ;n results in a simplified matrix model (A36) with no hyperbolic functions in the numerator. Note that (5.21) and (5.22) are symmetric under 1 2 1 … − 3 0 … − 3 swapping uI and uI for I ¼ ; ;n and I ¼ ; ;n , respectively. So to evaluate these insertions in the original 1 2 matrix model (5.10), it suffices to insert them into (A36) after symmetrizing over un−2 and un−2: 1 1 n−2 X Yn−4 ˜ ˜ ˜ ˜ 2 i1 in−2 in−3 iI iIþ1 hQ3K1 K −4Q2Q2K −4 K1Q3i ∼ − thðu − u Þthðu − u Þ thðu − u Þ; ð5:23Þ n n u 2 8πr 0 1 n−2 n−3 I Iþ1 i1;…;in−2 I¼1 ˜ ˜ ˜ ˜ ˜ hQ1K1 Kn−4Q2Q2Kn−4 K1Q3Q3Q1iu 1 1 n−1 X Yn−4 ∼ − thðu1 − ui1 Þthðu2 − ui1 Þthðuin−2 − uin−3 Þ thðuiI − uiIþ1 Þ: ð5:24Þ 2 8πr 0 1 0 1 n−2 n−3 I Iþ1 i1;…;in−2 I¼1

From the symmetries of the resulting integral, one can see many of the expected equivalences between chiral ring representatives at the level of hiu. For illustration, consider n ¼ 4. By the Z2 symmetry, we expect ˜ ˜ ˜ ˜ ˜ hQ3Q2Q2Q3i ≠ 0; hQ1Q2Q2Q3Q3Q1i¼0: ð5:25Þ

To demonstrate the latter (which is not obvious in the matrix model), we use that 1 1 3X ˜ ˜ ˜ ∼ − 1 − i 2 − i j − i hQ1Q2Q2Q3Q3Q1iu 2 8π thðu0 u1Þthðu0 u1Þthðu2 u1Þð5:26Þ r i;j when inserted into Z Y2 Y2 1 2 δðu1 þ u1Þ Z ¼ 4 dui Q : ð5:27Þ D4 I 1 − 2 1 − 2 2 i − i i − i I¼0 i¼1 shðu0 u0Þshðu2 u2Þ i¼1½chðu0 u1Þchðu1 u2Þ

1 2 1 2 1 2 By simultaneously swapping u0 ↔ u0, u1 ↔ u1, u2 ↔ u2, we see that insertions of

1 1 2 1 1 1 1 2 2 2 2 2 thðu0 − u1Þthðu0 − u1Þthðu2 − u1Þ; thðu0 − u1Þthðu0 − u1Þthðu2 − u1Þ into (5.27) are the same, and insertions of

1 2 2 2 1 2 1 1 2 1 2 1 thðu0 − u1Þthðu0 − u1Þthðu2 − u1Þ; thðu0 − u1Þthðu0 − u1Þthðu2 − u1Þ into (5.27) are the same. So we have the unnormalized correlator

Z 2 2 1 3 Y Y δ u1 u2 ˜ ˜ ˜ 4 − i ðQ1 þ 1Þ ZD4 ½hQ1Q2Q2Q3Q3Q1iu¼ duI 1 2 1 2 2 i i i i 8πr shðu0 − u0Þshðu2 − u2Þ 1½chðu0 − u1Þchðu1 − u2Þ X I¼0 i¼1 i¼ 1 1 2 1 i 1 × thðu0 − u1Þthðu0 − u1Þthðu2 − u1Þ; ð5:28Þ i

1 where we have written the integrand in such a way that the insertion contains only −u1 in the arguments. This is useful because one can then write

085008-13 YALE FAN and YIFAN WANG PHYS. REV. D 101, 085008 (2020) Z 1 3 1 2 1 2 1 2 1 2 ˜ ˜ ˜ 4 − dudu0du0du2du2thðu0Þthðu0Þðthðu2Þþthðu2ÞÞ ZD4 ½hQ1Q2Q2Q3Q3Q1iu¼ 1 2 1 2 1 2 1 2 ð5:29Þ 8πr shðu0 − u0Þshðu2 − u2Þchðu0Þchðu0 þ 2uÞchðu2Þchðu2 þ 2uÞ

and use standard Fourier transform identities, including from the point of view of Higgs branch correlators, we can (A7), to reduce this expression to a single-variable integral evade most difficulties by passing to the Coulomb branch (along the lines of Appendix A4) that vanishes because the of the mirror theory, which we do next. The Coulomb integrand is odd. branch analysis is significantly simpler because the gluing The lesson that we draw from the above discussion is that formula (2.11)–(2.13) contains a delta function that forces Higgs branch computations for general n are hard. For vanishing of magnetic flux, from which one sees that example, while the vanishing of hXi and hYi follows correlators of an odd number of monopoles vanish without simply from the Z2 symmetry for all Dn, this fact seems to even doing any integration. be highly nonobvious in the Higgs branch matrix model: In fact, symmetries already take us a long way toward according to (4.40)–(4.45), hXi and hYi are given by rounding out the mirror map. Using our knowledge of how inserting some linear combination of (5.22) and (5.20) or Z maps to the Coulomb branch of the mirror dual, (5.21) and (5.20) into (5.10), respectively, depending on including normalization and quantum corrections, we whether n is even or odd. Thus the Z2 symmetry of the can deduce the normalization of the mirror map for X affine quiver Lagrangian is no longer manifest when we and Y by demanding that the chiral ring relation be satisfied insert X and Y into the matrix model. Nonetheless, the (this obviates the need to compute, e.g., hX ⋆ Xi on the nonsymmetric part of the integrand of the matrix model Higgs branch side). Furthermore, we know that the quan- with X or Y insertions should be a total derivative. In other tum correction to Y vanishes by the Z2 symmetry, while X words, the F-term relations that imply that X and Y are Z2 can only mix with Y. This completely fixes the quantum odd (see Appendix C2) become integration-by-parts iden- mirror map for Y. Finally, we use the Coulomb branch tities in the matrix model. results of the next subsection regarding the orthogonality of In the end, we would like to derive the mirror map for all bare and dressed monopoles [particularly (5.56)] to write of the chiral ring generators, including quantum corrections down the remaining entries in the quantized D-type mirror [which we have, so far, for the generator of smallest symmetry dictionary. Combined with (5.15), we arrive at dimension in (5.15)]. While such a task seems daunting the complete map

i i 2M2 1 2 1 Xˆ ↔ ΦM2 − M2 ; Y ↔ ; Z ↔ Φ2 − ; ð5:30Þ ð4πÞn−1 r ð4πÞn−2 8π r2 where the Higgs branch operators X, Y, Z are given in (4.40)–(4.45) and we have accounted for operator mixing by setting

hX ⋆ Yi Xˆ ≡ X − Y; ð5:31Þ hY ⋆ Yi which satisfies hXˆ ⋆ Yi¼0.16 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 ↔ CZC in the 2 chiral ring (i.e., ignoring 1=r corrections), where C ¼ 1=ð8πÞ ∈ R>0 and C subscripts denote Coulomb branch operators, implies only that

ðn−1Þ=2 0 n=2−1 X ↔ iC X C; Y ↔ iC YC ð5:32Þ at the level of the chiral ring, where and 0 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 Z2 global symmetry ðX; YÞ → ð−X; −YÞ, the relative sign can be fixed by computing suitable nonvanishing mixed correlators involving X and Y (e.g., numerically). 17In fact, hY ⋆ Yi can also be read off from the SQCD Coulomb branch matrix model via (5.84).

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B. Coulomb branch of SUð2Þ SQCD with Nf flavors Since jWj¼2, the partition function is On the SUð2Þ SQCD side, the basic non-Weyl-invariant Z 1 σ shift operators are d μ σ 0 Ψ σ 0 2 Z ¼ 2 2 ð ; Þ 0ð ; Þ −1 Z 1 Nf 1 1 Φ Nf 2 2 ð ir Þ ∓2 i∂ ∂ 1 σ − ð2 σ þ BÞ shð Þ M ¼ −2 e ; σ Nf ¼ 2 d 2 ð5:37Þ 2 r ðirΦÞ Nf 4r chðσ=2Þ 1 i Φ ¼ σ þ B ; ð5:33Þ r 2 (the 1=2 in the measure dσ accounts for the half-integer normalization of the weights). Then 2 where M are related by the Z2 Weyl group. These act on Z functions fðσ;BÞ where σ ∈ R and B ∈ 2Z.ForN ≥ 1, 1 dσ f hO1 ⋆ ⋆ O i¼ μðσ; 0ÞΨ0ðσ; 0Þ the monopole bubbling terms are necessarily polynomials n 2Z 2 and can therefore be removed by operator mixing. In other × ½ðO1 O Ψ0Þðσ; 0Þ; ð5:38Þ words, there exists an operator basis in which the bubbling n M2 ΦM2 terms for and are zero; all other bases are related where the left-hand side (LHS) of (5.38) means to this one by operator mixing. This means that we can hO1ðφ1ÞOnðφnÞi with the φi in ascending order. It is write, without loss of generality, also convenient to define Z M2 ¼ M2 þ M−2; ΦM2 ¼ ΦðM2 − M−2Þ: ð5:34Þ 1 σ 2 σ ≡ σ σ shð Þ 1 Z½fð Þ 2 d fð Þ 2 ;Z¼ Z½ ; ð5:39Þ 4r chðσ=2Þ Nf These shift operators already allow us to compute the star ≥ 3 2 n −1 2 2 n product in the Coulomb branch TQM. For Nf , we can so that, for example, hðΦ Þ i¼Z Z½ðσ =r Þ where further compute correlators of twisted CBOs as follows. ðΦ2Þn ¼ Φ2 ⋆ ⋆ Φ2 is understood. Define the vacuum wave function

1 1− σ 1 σ Γ i Γ þi Nf 1. Nf =4 Ψ σ δ ½2π ð 2 Þ ð 2 Þ 0ð ;BÞ¼ B;0 1 Computing correlators of Coulomb branch chiral pri- 2π Γð1 − iσÞΓð1 þ iσÞ mary operators (CPOs) from monopole shift operators is σ shð Þ particularly straightforward when N ¼ 4,aswenow ¼ δB;0 ð5:35Þ f σchðσ=2ÞNf show. This allows for a precise match to the results of Sec. VA1. and the gluing measure SUð2Þ SQCD with Nf ¼ 4 is mirror to the affine D4 2 quiver theory; the Higgs branch of the latter has a global S3 1 2 2 B μ σ −1 NfB= σ symmetry, which we reproduce. The mirror map is as ð ;BÞ¼ 2 ð Þ þ : ð5:36Þ r 4 follows:

pffiffiffi 3 1 i 33=4 i U; V ↔ Φ2 − ∓ M2; W ↔ ΦM2 − M2 ; ð5:40Þ 128π2 3r2 16π2 128π3 r

where the Higgs branch operators U, V, W are given These operators have dimensions ΔðX CÞ¼3 and – by (5.6) (5.8) with one-point functions subtracted. ΔðYCÞ¼ΔðZCÞ¼2. At the level of the chiral ring, X 2 Z Y2 Z3 0 ⇔ U3 V3 W2 0 We present the derivation below, using C subscripts we have C þ C C þ C ¼ C þ C þ C ¼ to distinguish Coulomb branch operators from Higgs where branch operators. The Coulomb branch chiral ring generators [in our normalization, following from (5.33)] are 1 Y 1 Y 1 U ¼ Z þpCffiffiffi ; V ¼ Z −pCffiffiffi ; W ¼ X : C 2 C 3 C 2 C 3 C 2 C 2 2 2 X C ¼ 8ΦM ; YC ¼ −8iM ; ZC ¼ Φ : ð5:41Þ ð5:42Þ

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Using the Coulomb branch formalism,18 we compute the (note that conjugation flips the sign of the monopole). one-point functions Next, requiring that hWCi¼0 shows that WC ¼ 4ΦM2 þ Oð1=rÞ cannot mix with the identity, so 1 hM2i¼hΦM2i¼0; hΦ2i¼ ð5:46Þ 3r2 uU þ vV W ¼ 4ΦM2 þ C C ð5:51Þ C r (the monopole one-point functions vanish automatically in the absence of bubbling). We also have the two-point for some dimensionless constants u, v. To respect the functions S3 symmetry, we must also impose that hUC ⋆ WCi¼ hVC ⋆ WCi¼0. More simply, we have the following 2 2 2 2 2 2 2 2 W hΦ ⋆M i¼hM ⋆Φ i¼hΦ ⋆ΦM i¼hΦM ⋆Φ i¼0; ansatz and requirements for C: 5:47 ð Þ a b 1 W ¼ 4ΦM2 þ M2 þ Φ2 − ; C r r 3r2 7π4 −360 2π4 −135   2 2 2 2 1 Φ ⋆Φ ; M ⋆M : : 2 2 h i¼ 15π4 4 h i¼ 120π4 4 ð5 48Þ hM ⋆ W i¼ Φ − ⋆ W ¼ 0: ð5:52Þ r r C 3r2 C To define the operators U , V , W including 1=r C C C Using19 corrections, note that by dimensional analysis, UC and VC can only mix with the identity, while WC can mix with U V 2 2 2 2 i 2 2 C, C, and the identity. In addition, we would like all hM ⋆ ΦM i¼hΦM ⋆ M i¼ hM ⋆ M i; ð5:56Þ r correlation functions of UC, VC, WC, and composites thereof to respect the S3 symmetry. In particular, we must which holds for all N , fixes a ¼ −4i and b ¼ 0: have f U V W 0 2 i 2 h Ci¼h Ci¼h Ci¼ : ð5:49Þ W ¼ 4 ΦM − M : ð5:57Þ C r Using (5.46), the requirement that hU i¼hV i¼0 fixes C C Having fixed the exact definitions of U , V , W in (5.50) C C C 1 1 4i and (5.57), we check that U ¼ Φ2 − − pffiffiffi M2; C 2 3r2 3 hU ⋆ W i¼hV ⋆ W i¼hU ⋆ U i¼hV ⋆ V i¼0; 1 1 4 C C C C C C C C V Φ2 − iffiffiffi M2 C ¼ 2 þ p ð5:50Þ hU ⋆ V i ≠ 0; hW ⋆ W i ≠ 0; ð5:58Þ 2 3r 3 C C C C

19 18 This equation can be derived as follows. Let IM2M−2 and Specializing to Nf ¼ 4, we have IM−2M2 denote the insertions in the Coulomb branch matrix Z 2 −2 −2 2 1 2 πσ 2 1 model corresponding to M M and M M . Then the correla- σ σ σ tanh ð = Þ M2 ⋆ M2 ΦM2 ⋆ M2 M2 ⋆ ΦM2 Z½fð Þ ¼ 2 d fð Þ 4 ;Z¼ 2 : ð5:43Þ tors h i, h i, h i correspond to 64r cosh ðπσ=2Þ 120πr the insertions A useful formula is σ I 2 −2 þ I −2 2 ; ðI 2 −2 − I −2 2 Þ; M M M M r M M M M 1 dn τ 1 − τ4 σn ð Þ σ þ 2i σ − 2i Z½ ¼ lim −2 2 − 2 −2 n 2 τ→0 τn πτ IM M IM M ; ð5:53Þ 120ðπiÞ r d sinhð Þ r r 1 7π4 − 360 ⇒ Z½σ2¼ ;Z½σ4¼ ; …: ð5:44Þ respectively, which implies that 360πr2 1800π5r2 2i This is a special case of (5.73) (note that the right-hand side hM2 ⋆ ΦM2iþhΦM2 ⋆ M2i¼ hM2 ⋆ M2ið5:54Þ (RHS) is real because it vanishes unless n is even). Some other r useful integrals are     2 2 i 2 2 i 2 2 ⇔ M ⋆ ΦM − M þ ΦM − M ⋆ M ¼ 0: 1 120 − π4 1 2920 − 27π4 r r Z ¼ ;Z ¼ : ð5:45Þ 4 þ σ2 120π5r2 16 þ σ2 5832π5r2 ð5:55Þ

These formulas can be used, for example, to give alternative The commutativity hM2 ⋆ ΦM2i¼hΦM2 ⋆ M2i is required derivations of (A34). for consistency of the deformation quantization.

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20 2 so these correlators respect the full S3 symmetry. π ψ ð1Þ − 2 Specifically, the Higgs branch computation gives (for ðn Þ¼qn þ 6 : ð5:65Þ U ¼ U0 − hU0i, V ¼ V0 − hV0i, etc.) In fact, we have q3 ¼0 and q4 ¼ −1, whereas qn≥5 ∈ QnZ. 4 4 Φ2 1 3 2 4 2π −135 Hence n¼ is special in that h i¼ = r is simply a hU ⋆ Vi¼α2ζ ¼ ; 2 15360π8r4 rational number with no factors of π . In particular, 6α 4 α 33=2 π4 − 105 we see that W ⋆ W ð 1 þ A þ Þ 2 ζ6 ð Þ h i¼ ¼ 10 6 ; ð5:59Þ 4 573440π r 3 2 1 ZSUð2Þþ4½s ¼ZSUð2Þþ4½ ; ð5:66Þ which we reproduce on the Coulomb branch side by 3 2 identifying so an insertion of s is equivalent to a trivial insertion. Hence the results (5.14) cannot be used directly to read off the mirror pffiffiffi 4 3 map when n ¼ : they are ambiguous. Specifically, (5.66) U V W ↔ U V 3=2W ≡ implies that the one-point function of the operator (5.61) in ð ; ; Þ ðc C;c C;c CÞ;c64π2 ; ð5:60Þ SUð2Þþ4 is equivalent to an insertion of −ðs2 − 1Þ=ð8πrÞ2, despite appearances. Somewhat miraculously, a similar thus substantiating the mirror map (5.40). Note that such a statement holds for all p-point functions despite the mixing rescaling by powers of c ∈ R 0 preserves the chiral ring > with the monopole. Namely, the insertion corresponding to p relation. copies of (5.61) can always be written as a polynomial of As a further check, Eq. (5.40) implies that 2 4 2 −1 degree p in s plus a multiple of ð þ s Þ ,andonecan 1 1 1 check numerically for any given p that this gives the same ˜ ˜ 2 2 Q1Q3Q3Q1 ↔ Φ M ; : 2 2 p 128π2 þ 2 þ 16π2 ð5 61Þ result as an insertion of ½−ðs − 1Þ=ð8πrÞ . It would be r ≥ 5 interesting to construct a proof of this fact. Finally, for n , 1 1 1 ˜ ˜ 2 2 there is no mixing with the monopole and we can read off the Q2Q3Q3Q2 ↔ Φ þ − M : ð5:62Þ 128π2 r2 16π2 mirror map directly from (5.14). We finish with some conceptual comments. At fixed φ, Using (5.61) and (5.62), we reproduce all of the Higgs the twisted CBOs in (5.41) represent nontrivial elements of ˜ ˜ branch correlators of the QiQ3Q3Qi (i ¼ 1, 2) using the the chiral ring. However, only after operator mixing is Coulomb branch formalism. These identifications make properly accounted for do they correspond to twisted sense because the Z2 switches 1 ↔ 2 on the Higgs translations of scalar conformal primaries in the CFT branch side. (hence CPOs). Namely, we must choose a basis in which The integral manipulations that led to (5.14) are equally their one-point functions vanish and they are orthogonal to valid when n ¼ 4. So how do we reconcile the conclusion all lower-dimension operators (this is the “CFT gauge” (5.15) with the known mirror map (5.61) in this case? of [24]). In this basis, the monopoles correspond to the It turns out that there is no contradiction. By writing primary operators constructed in [48]. Usually, such a basis is obtained by diagonalizing the matrix of two-point 2 shðsÞ 1 4 functions. However, to respect the S3 symmetry, that is ¼ − ð5:63Þ chðs=2Þ2n chðs=2Þ2n−4 chðs=2Þ2n−2 not what we do here: rather, we impose that composite R operators have vanishing one-point functions and non- e2πist 21 vanishing two-point functions only with their conjugates, ds # and using the trick of differentiating chðsÞ , we derive where conjugation is defined by the Z2 subgroup of S3. below that 2 2 2. Nf > 4 Φ2 ψ ð1Þ − 2 h i¼ 2 2 ðn Þþ ð5:64Þ 4 π r n − 2 For Nf > , we do not expect any mixing between the scalar and the monopole(s), by dimension counting. in the SUð2Þþn theory. For each n ≥ 3, there exists a Correspondingly, we lose the S3 symmetry and are left with Z qn ∈ Q such that only the 2 of charge conjugation. In our normalization, the Coulomb branch chiral ring 20 generators are One can go on to define additional composite operators. For c2 U U ⋆ U V Nf−1 2 Nf−1 2 2 example, C (defined as a shift of C C) can mix with C and ðX C; YC; ZCÞ¼ð2 ΦM ; −i2 M ; Φ Þ Vc2 U C can mix with C, which is consistent with the S3 symmetry. 2 2 N −1 21 ⇒ X Z Y Z f 0 In other words, Z½σn can be evaluated analytically by C þ C C þ C ¼ ; ð5:67Þ writing the SUð2Þ SQCD partition function as a sum of SQED partition functions and differentiating with respect to the Fayet- where the above equalities hold at the level of the classical Iliopoulos (FI) parameter. chiral ring. The dimensions are as in (4.48). The Z2

085008-17 YALE FAN and YIFAN WANG PHYS. REV. D 101, 085008 (2020) symmetry takes ðX C; YC; ZCÞ ↦ ð−X C; −YC; ZCÞ.We We can then evaluate wish to determine the “quantum corrections” to X C, YC, 2 ZC. First note that 2 2 2 hΦ i¼ ðz2 −2 ½σ − 4z2 −1 ½σ Þ r4Z ðNf Þ ðNf Þ Z 2πiτσ Γ N − τ Γ N τ 2 2 e ð2 i Þ ð2 þ i Þ ψ ð1Þ − 2 dσ ¼ ¼ π2 2 ðNf Þþ − 2 ; ð5:75Þ chðσÞN 2πΓðNÞ r Nf Z dσ ΓðNÞ ⇒ ¼ pffiffiffi 2 ð5:68Þ where we have used the recurrence relation σ N N Nþ1 chð Þ 2 πΓð 2 Þ −1 mm! ffiffiffi ψ ðmÞ 1 ψ ðmÞ ð Þ 1 1−2zp ðz þ Þ¼ ðzÞþ mþ1 ð5:76Þ by the duplication formula ΓðzÞΓðz þ 2Þ¼2 πΓð2zÞ, z so the partition function (5.37) is for simplification (the other one-point functions are trivial: Z Z 2 2 1 σ 2 σ hM i¼hΦM i¼0). Now note that the Z2 symmetry d − d Z ¼ 2 2 −2 2 2 −1 ð5:69Þ requires that hX Ci¼hYCi¼0, but does not restrict hZCi. 2r σ ðNf Þ r σ ðNf Þ chð Þ chð Þ Therefore, including 1=r corrections, the most general mixing pattern is

1 ΓðNf − 2Þ ffiffiffi : 5:70 0 ¼ 2 2ðN −1Þp 1 ð Þ −1 2 x 2 x 2 2 r 2 f πΓ − X 2Nf ΦM M Φ − Φ ðNf 2Þ C ¼ þ þ −3 ð h iÞ; ð5:77Þ r rNf

More generally, it is convenient to write (5.39) as −1 2 y 2 2 Y − 2Nf M Φ − Φ C ¼ i þ −4 ð h iÞ; ð5:78Þ rNf 1 1 Z½fðσÞ ¼ z2 −2 ½fð2σÞ − 2z2 −1 ½fð2σÞ ; z r2 2 ðNf Þ ðNf Þ Z ¼ Φ2 þ ð5:79Þ Z C r2 fðσÞ z ½fðσÞ ≡ dσ : ð5:71Þ N chðσÞN for x; x0;y;z∈ C. To constrain these coefficients, we consider two-point functions. The Z2 symmetry requires By differentiating (5.68) with respect to τ, we get that

X ⋆ Z Y ⋆ Z 0 Z Γ N− τ Γ N τ h C Ci¼h C Ci¼ ð5:80Þ ð2πiσÞp dp eln ð 2 i Þþln ð 2þi Þ dσ ; 5:72 σ N ¼ τp 2πΓ ð Þ chð Þ d ðNÞ τ¼0 and does not restrict hX C ⋆ X Ci; hYC ⋆ YCi; hZC ⋆ ZCi; hX C ⋆ YCi. We clearly have (by flux conservation) that 2 2 2 which vanishes for odd p and can be written in terms the two-point functions of M or ΦM with Φ vanish. of polygamma functions for even p (it seems challenging to Using obtain a closed-form expression for this integral, but it Z 1 2 3 can be evaluated for fixed p and arbitrary N).22 In σ4 ΓðNÞð6ψ ð ÞðNÞ þ ψ ð ÞðNÞÞ dσ ¼ 2 2 2 ; ð5:81Þ σ N Nþ3 9=2 Nþ1 particular, we have chð Þ 2 π Γð 2 Þ Z σ2 ΓðNÞψ ð1ÞðNÞ we compute that dσ ¼ 2 2 : ð5:74Þ σ N Nþ1 5=2 Nþ1 chð Þ 2 π Γð 2 Þ 2 2 8 4 4 hΦ ⋆ Φ i¼ ðz2 −2 ½σ − 4z2 −1 ½σ Þ ð5:82Þ r6Z ðNf Þ ðNf Þ 22 Likewise, one derives the simple formula 2 n ψ ð3Þ − 2 1 d ¼ 4 4 ðNf Þ Z½σn¼ lim π r ðπiÞnr2 τ→0 dτn 4 − 2 τ2 1 Γ − 2 − τ Γ − 2 τ ð1Þ ð1Þ ðNf ð þ ÞÞ ðNf i Þ ðNf þ i Þ þ 6ψ ðNf −2Þ ψ ðNf − 2Þþ : × ; Nf − 2 2πΓð2ðNf − 1ÞÞ ð5:73Þ ð5:83Þ which can be evaluated on a case-by-case basis. On general grounds, we have the relation (5.56) where

085008-18 NON-ABELIAN MIRROR SYMMETRY BEYOND THE CHIRAL … PHYS. REV. D 101, 085008 (2020) 2 −1 2 −1 1 σ 2 σ − ðNf Þ σ − 2 σ ðNf Þ M2 ⋆ M2 −1 ð þ iÞð iÞ þð iÞð þ iÞ h i¼Z Z 2 2 −2 2 : ð5:84Þ 2 Nf r ðNf Þ σðσ þ 4Þ

We also compute that E6∶ Δ ¼ð6; 4; 3Þ;

E7∶ Δ ¼ð9; 6; 4Þ; hΦM2 ⋆ ΦM2i 2 −1 2 −1 E8∶ Δ ¼ð15; 10; 6Þ: ð6:2Þ σ − ðNf Þ σ ðNf Þ −1 − ð iÞ þð þ iÞ ¼ Z Z 2 2 −1 : ð5:85Þ 2 Nf ðNf Þ r Equations (6.1) then correspond to the chiral ring relations. As usual, the dynamics of the SCFT gives rise to a The monopole two-point functions are difficult to evaluate X ⋆ Z deformation quantization of the Higgs or Coulomb branch. analytically, unless one fixes Nf. Requiring h C Ci¼ In particular, the chiral ring relations (6.1) are deformed. Y ⋆ Z 0 h C Ci¼ gives The truncation property of the Higgs or Coulomb branch algebra (TQM) implies that the deformations are all 0 Φ2 ⋆ Φ2 Φ2 ⋆ Φ2 0 ⇔ 0 0 x h ic ¼ yh ic ¼ x ¼ y ¼ ; ð5:86Þ relevant, the sense of which should be obvious below. For the E6 singularity, we start by writing down the most Φ2 ⋆ Φ2 Φ2 ⋆ Φ2 − Φ2 2 ≠ 0 where h ic ¼h i h i ,sowehave Ω ˆ ˆ ˆ 0 general deformed chiral ring relation as E6 ðX; Y;ZÞ¼ determined that with

−1 2 x 2 −1 2 X 2Nf ΦM M Y − 2Nf M ˆ 2 ˆ 3 ˆ 4 2 ˆ ˆ 2 3 ˆ ˆ 4 ˆ 2 C ¼ þ ; C ¼ i ; Ω X Y Z β1ζ YZ β2ζ Y Z β3ζ Z r E6 ¼ þ þ þ þ þ 2 z β ζ6 ˆ β ζ8 ˆ β ζ12 Z ¼ Φ þ : ð5:87Þ þ 4 Y þ 5 Z þ 6 ; ð6:3Þ C r2 where the β are dimensionless parameters.24 The most But now any correlator containing odd numbers of X , Y i C C general even deformations of the commutators that satisfy obviously vanishes by flux conservation, so higher-point the Jacobi identities are given by functions automatically respect the Z2 symmetry and do Z not fix x, z. That is, the 2 symmetry does not completely ˆ ˆ ˆ 3 3 ˆ ˆ ˆ ˆ 6 ˆ ½X; Y¼4ζZ þ α1ζ ðY Z þZ YÞþα2ζ Z; determine X C, ZC at the quantum level (we have more 4 ˆ ˆ ˆ 2 3 ˆ 2 9 freedom than when Nf ¼ ). However, we can still map ½X; Z¼−3ζY þ α1ζ Z þ α3ζ ; M2; ΦM2; Φ2 individually to the Higgs branch side by ½Y;ˆ Zˆ ¼2ζX:ˆ ð6:4Þ matching correlators.23 Here, we have used the freedom in operator redefinitions to C2 Γ VI. DEFORMATION QUANTIZATION OF = E6;7;8 put the last two commutators above in simpler forms. Ω A. Periods and associativity Furthermore, consistency requires to be in the center of the algebra C½X;ˆ Y;ˆ Zˆ with commutators (6.4), so that We now move on to the E-type singularities ˆ Ω ˆ Ω ˆ Ω 0 M ∶ 2 3 4 0 ½X; ¼½Y; ¼½Z; ¼ : ð6:5Þ E6 fðX; Y; ZÞ¼X þ Y þ Z ¼ ; M ∶ f X; Y; Z X2 Y3 YZ3 0; E7 ð Þ¼ þ þ ¼ This puts constraints on the coefficients βi. Indeed, all of 2 3 5 M ∶ 0 the βi except for one are uniquely determined by α1;2;3 in E8 fðX; Y; ZÞ¼X þ Y þ Z ¼ ; ð6:1Þ (6.4) as follows: which are hyperkähler quotients of the type C2=Γ . E6;7;8 Ω ˆ 2 ˆ 3 ˆ 4 12 − α ζ2 ˆ ˆ 2 − 2ζ3 ˆ ˆ In 3D N ¼ 4 SCFTs that realize these on the Higgs or E6 ¼ X þ Y þ Z þð 1Þð YZ X ZÞ 24α α Coulomb branch, X, Y, Z are half-BPS chiral primaries of 4 ˆ 2 1 þ 2 6 ˆ 2 2 þ 4ð6 − α1Þζ Y þ ζ Z scaling dimension [¼ SUð ÞR spin] 2 8 ˆ 12 − ðα2 þ α3Þζ Y þ γζ : ð6:6Þ 23The matching of Z across mirror symmetry in (5.15) determines z in (5.87). A scheme for fixing x is given in 24 (5.30). Namely, imposing that X C and YC be orthogonal (a Note that we have partially fixed the gauge redundancy in natural choice of basis, in lieu of additional symmetry) gives defining the operators to put the deformed chiral ring relation in −1 x ¼ −i2Nf ,by(5.56). this form.

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The four parameters fα1; …; α3; γg label the even quanti- even quantizations labeled by fα1; …; α6; γg, where the zations of the E6 singularity. commutation relations are The undeformed E6 singularity has a nontrivial Z2 ˆ ˆ ˆ ˆ 2 2 ˆ ˆ 3 ˆ 3 ˆ 2 symmetry that acts as X → −X, Z → −Z. Note that while ½X;Y¼3ζYZ − 6ζ X Z þζ ðα4Z − 6Y Þ 5 ˆ ˆ 6 ˆ 7 ˆ 2 11 ˆ 15 the general Coulomb branch algebra presented in (6.4) and − 2ζ α6Y Z þ2ζ α6X þ ζ Z α3 þ ζ α2Z þ ζ α1; (6.6) is conveniently invariant under this Z2, the constraints ˆ ˆ −ζ 3 ˆ 2 ˆ 3 α ζ5 ˆ 2 α ζ13 of Z2 will show up in specifying the short products on this ½X;Z¼ ð Y þ Z Þþ 6 Z þ 5 ; algebra. ½Y;ˆ Zˆ ¼2ζX;ˆ ð6:7Þ For the E7 case, by solving the Jacobi identities and the center condition (6.5), we find a seven-parameter family of and the center element is

Ω ¼ Xˆ 2 þ Yˆ 3 þ Yˆ Zˆ 3 E7 α − 3ζ ˆ ˆ 2 ζ2 −12 ˆ 2 ˆ 4 ˆ 4 18ζ3 ˆ ˆ − 36 3α α ζ4 ˆ ˆ 2 − 2ζ ˆ ˆ XZ þ Y Z þ 4 Z þ X Y ð þ 4 þ 6Þ ðYZ X ZÞ 1 ζ6 6 36 3α 2α ˆ 2 α − 27α ˆ 3 − 2ζ8 α − 12α ˆ ˆ −ζ ˆ þ 3 ð ð þ 4 þ 6ÞY þð 3 4ÞZ Þ ð 3 6ÞðY Z XÞ 1 ζ10 α − 6 4α α α ˆ 2 − ζ12 α α ˆ ζ14 α − 12α ˆ γζ18 þ 2 ð 2 ð 3 þ 4 6ÞÞZ ð 2 þ 5ÞY þ ð 1 2ÞZ þ : ð6:8Þ

Similarly, for the E8 singularity, we find an eight-parameter family of even quantizations labeled by fα1; α2; …; α7; γg with commutators

ˆ ˆ ˆ 4 3 ˆ ˆ 2 ˆ ˆ 2 ˆ 2 7 ˆ 3 9 ˆ ˆ ˆ 13 ˆ 2 19 ˆ 25 ½X; Y¼5ζZ þ 3α7ζ ð−YZ þ 2ζX Z þ2ζ Y Þþζ α4Z − 2ζ α6ðY Z −ζXÞþζ α3Z þ ζ α2Z þ ζ α1; ˆ ˆ ˆ 2 3 ˆ 3 9 ˆ 2 21 ½X; Z¼−3ζY þ ζ Z α7 þ ζ Z α6 þ ζ α5; ½Y;ˆ Zˆ ¼2ζXˆ ð6:9Þ and center element

960 þ α4 Ω ¼ Xˆ 2 þ Yˆ 3 þ Zˆ 5 − 20Yˆ Zˆ 3ζ2 þ 60Xˆ Zˆ 2ζ3 þ 120Yˆ 2Zˆ ζ4 − 120Xˆ Yˆ ζ5 þ ζ6Zˆ 4 E8 4 α − ζ8 3α α ˆ ˆ 2 − 2ζ ˆ ˆ 2ζ10 3α 2α ˆ 2 ζ12 3 48α − 56α ˆ 3 ð 4 þ 6ÞðYZ X ZÞþ ð 4 þ 6ÞY þ 3 þ 4 6 Z α − 2ζ14 α 60α ˆ ˆ −ζ ˆ ζ18 2 48α − 3α α ˆ 2 ð 3 þ 6ÞðY Z XÞþ 2 þ 3 4 6 Z 20 ˆ 24 ˆ 30 − ζ tðα2 þ α5ÞY þ ζ ðα1 þ 48ðα2 − α5ÞÞZ þ γζ : ð6:10Þ

Z Γ — Note that for E7;8, there is no hyperkähler 2 isometry: thus E6 gauged free hyper. In this case, the operators X;ˆ Y;ˆ Zˆ are all self-conjugate. ffiffiffi 3p ˜ 4 ˜ 4 B. Realizations in Lagrangian 3D SCFTs Z ¼ 34 2QQðQ − Q Þ; Y − 8 ˜ 8 14 4 ˜ 4 1. Discretely gauged free hyper ¼ ðQ þ Q þ Q Q Þ; 12 ˜ 12 4 ˜ 4 4 ˜ 4 In the following free theories, the Higgs branch chiral X ¼ Q þ Q − 33Q Q ðQ þ Q Þ; ð6:11Þ ring generators are easily deduced from the known poly- nomial invariants of the binary tetrahedral, octahedral, and icosahedral groups (see, e.g., [49]). which satisfy X 2 þ Y3 þ Z4 ¼ 0.

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Γ — 2 3 E7 gauged free hyper. In this case, whose quarks are Uð Þ × SUð Þ bifundamentals. It has Δ 1 −2 −1 8 8 4 4 dimension ¼ and transforms in the adjoint representa- Z −2 93 3 ˜ 14 ˜ ¼ ðQ þ Q þ Q Q Þ; tion of SUð3Þ. 2 2 ˜ 2 8 ˜ 8 4 ˜ 4 Y ¼ −3 × 23Q Q ðQ þ Q − 2Q Q Þ; Let us denote the generators of the Higgs branch algebra of the three copies of T½SUð3Þ by M A . By contracting X ˜ 16 − ˜ 16 − 34 5 ˜ 5 8 − ˜ 8 ðaÞ B ¼ iðQQðQ Q Þ Q Q ðQ Q ÞÞ; ð6:12Þ the SUð3Þ indices, we can construct the Higgs branch which satisfy X 2 þ Y3 þ YZ3 ¼ 0. algebra of the E6 theory. Recall that the dimensions of the CPOs are ΔðXÞ¼6, ΔðYÞ¼4, and ΔðZÞ¼3. Thus Γ — E8 gauged free hyper. In this case, X ∝ 2 2 2 2 trðMð1ÞMð2ÞMð3ÞÞþZ ; Z ¼QQ˜ ðQ10 −Q˜ 10 þ11Q5Q˜ 5Þ; Y ∝ trðM2 M2 Þ; 1 ð1Þ ð2Þ 20 15 ˜ 5 10 ˜ 10 5 ˜ 15 ˜ 20 Y ¼ ðQ −228Q Q þ494Q Q þ228Q Q þQ Þ; Z ∝ 2 12 trðMð1ÞMð2ÞÞ: ð6:17Þ i X ¼ pffiffiffiðQ30 þ522Q25Q˜ 5 −10005Q20Q˜ 10 24 3 The precise expressions are given in Appendix C. 3. a.In X Y −10005 10 ˜ 20 −522 5 ˜ 25 ˜ 30 (6.17), we give a particular way to represent the CPOs , , Q Q Q Q þQ Þ; ð6:13Þ Z in terms of the hypermultiplet scalars. All other repre- which satisfy X 2 þ Y3 þ Z5 ¼ 0. sentatives differ by terms involving the D-term relations.

2. Star-shaped quivers Affine E7 quiver.—Our conventions are The only known realizations of E6;7;8 singularities by interacting SCFTs are through the Higgs branches of affine E6;7;8 quiver theories.

Affine E6 quiver.—This theory looks as follows: ð6:18Þ

where subscripts label mesons for each leg. One can use the same reasoning as for E6 to find the invariants at various Δ; ð6:14Þ the result, as summarized in [46,50], is that the basic invariants are

Z 3 ¼ trðMð1ÞMð3ÞÞ; Y − 3 3 The quiver has an obvious S3 symmetry acting on the Higgs ¼ trðMð1ÞMð2ÞÞ; Z branch, but at the operator level, only a 2 subgroup acts X 2 3 3 ¼ trðM 1 M 2 M 1 Mð3ÞÞ: ð6:19Þ faithfully. The latter corresponds to the nontrivial Z2 acting ð Þ ð Þ ð Þ as ðX; Y; ZÞ → ð−X; Y; −ZÞ on the Higgs branch CPOs. See Appendix C. 3 b for details. The affine E6 quiver is obtained by gauging the diagonal SUð3Þ Higgs branch flavor symmetry of three T½SUð3Þ linear quiver theories. Hence the Higgs branch CPOs of the Affine E8 quiver.—Our conventions are E6 theory are conveniently described as SUð3Þ-invariant combinations of those of T½SUð3Þ. Recall the T½SUð3Þ quiver

ð6:15Þ ð6:20Þ

The basic invariants (again, see [46]) are ˜ i and denote the two bifundamental hypers by ðqi; q Þ and A ˜ i 1 1 5 ðQi ; QAÞ, with i ¼ , 2 and A ¼ , 2, 3 being the Z ¼ trðM 1 Mð2ÞÞ; fundamental indices for Uð2Þ and SUð3Þ, respectively. ð Þ Y 5 2 2 The Higgs branch chiral ring is generated by the meson ¼ trðMð1ÞMð2ÞMð1ÞMð2ÞÞ; (moment map operator) X 5 2 2 3 2 ¼ trðM 1 M 2 Mð1ÞM 2 M 1 M 2 Þ: ð6:21Þ 1 ð Þ ð Þ ð Þ ð Þ ð Þ MA ≡ QAQ˜ i − QCQ˜ i δA; ð6:16Þ B i B 3 i C B See Appendix C. 3. c for details.

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VII. E-TYPE MIRROR SYMMETRY Thus by integrating over the T½SUð3Þ variables u1;2 and then taking Fourier transforms, one can rewrite (7.1) in a The main appeal of the Higgs branch topological sectors in the affine E-type quivers is that they might shed light on form reminiscent of a rank-one matrix model. Namely, the non-Lagrangian Coulomb branch algebras (not asso- using ciated with a non-Abelian gauge theory with matter) to Z yn i ∂nthðxÞ which they are mirror dual. One hope is that applying dy e2πixy ¼ x suitable manipulations and Fourier transform identities to shðyÞ 2 ð2πiÞn Z the Higgs branch matrix models for the E-series partition x3 4 − chð2yÞ functions might give hints about the mirror duals. ⇒ ¼ dy e2πixy ð7:5Þ sh x y 4 Since the E-type (and D-type) theories can be built from ð Þ chð Þ T SU N theories [which are realized on S-duality domain ½ ð Þ 27 walls of 4D N ¼ 4 SUðNÞ SYM [41] ] by diagonal and a cyclic convolution identity from [32] gives gauging, it is natural to use the massive TQM of the Z Y3 Y 3 constituent T½SUðNÞ theories to determine the operator 1 ðua − ubÞ 25 Z ¼ duaδðu1 þ u2 þ u3Þ 3 3 algebras of the full quiver theories. E6 16 3 3 3 3 a − b 1 shðu3 u3Þ Z a¼ a

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1 E quiver theories. Some hints that we can use to answer Z ðuaÞ¼Q Q Q n u 3 2 − i 2 3 i − a this question are Lagrangian intuition, the commutative i¼1 chðu1 u2Þ i¼1 a¼1 chðu2 u3Þ Z R limit, and scaling dimensions (for constraining bubbling 4π φ ¼ DqDqDQD˜ Qe˜ r d L ð7:10Þ coefficients). Let us make a few preliminary comments that can with hopefully be clarified in future work. We make the following assumptions: i j j (i) The fact that the mirror dual theories have rank one L ¼ q˜ ðδ ð∂φ þ u1Þ − ðu2Þ Þq i i j means that their monopole charges belong to a one- ˜ i δjδA∂ jδA − δj A B þ QAð i B φ þðu2Þi B i ðu3ÞB ÞQj ð7:11Þ dimensional vector space. (ii) The hypothetical dual gauge group is “semisimple,” 1 2 1 2 3 and u2 ¼ diagðu2;u2Þ, u3 ¼ diagðu3;u3;u3Þ. Hence we meaning that the dimensions of (dressed) monopoles have are fully accounted for by powers of the vector multiplet scalar in the commutative limit (see φ − i j j sgnð 12Þþthðu1 u2Þ Footnote 9). hq ðφ1Þq˜ ðφ2Þi ¼ −δ i u i 8πr (iii) One of the Coulomb branch chiral ring generators is − − i φ constructed from the vector multiplet scalar and × e ðu1 u2Þ 12 ; ð7:12Þ therefore takes the form Φd, where d is a positive φ i − A integer determined by the hypothetical Weyl group. A ˜ j j A sgnð 12Þþthðu2 u3 Þ hQ ðφ1ÞQ ðφ2Þi ¼ −δ δ The second assumption is motivated by the fact that the i B u i B 8πr dimensions (6.2) of the En chiral ring generators are known − i − A φ × e ðu2 u3 Þ 12 : ð7:13Þ to be integers, just as the dimensions of monopoles in a Lagrangian theory with semisimple gauge group are These two-point functions allow us to compute the OPE integers (otherwise, they could be half-integers, or con- within the TQM and hence the quantization of the E6 chiral ceivably even other fractions in a non-Lagrangian theory). ring relation, along the lines of Sec. 6 of [32]. Recalling that The third assumption is perhaps most plausible in the case A A ˜ i Z ðMðIÞÞ B ¼ðQðIÞÞi ðQðIÞÞB, we can also consider insertions of E6, which has a 2 symmetry. a of operators built from these mesons into ZT½SUð3Þðu3Þ We now work out the consequences of these assump- written in the simplified form (7.3). We have yet to find a tions. In the commutative limit, a primitive monopole [33] way to write these insertions in an enlightening way. of dimension Δ and charge q can only bubble to the Finally, the Z2 symmetry of the E6 theory may help identity: identify which chiral ring generators map to monopoles and q Δ gq Δ which to scalars, assuming that this Z2 is realized as charge M∞ ¼ Φ cðqÞe½q ⇒ M∞ ¼ Φ ðcðqÞe½qþbðqÞÞ ð7:14Þ 29 X Z conjugation in the mirror theory. By this logic, and X δ q −1 Δ δ in the E6 theory (which flip sign under charge conjugation, ⇒ Φ M∞ ¼ w ðΦÞ þ ðcðwðqÞÞe½wðqÞ þ bðwðqÞÞÞ; and whose one-point functions must vanish) should map to w∈W monopoles in the non-Lagrangian dual. This is contrary to ð7:15Þ the D-case, where Z maps to a scalar. In the E7 and E8 cases, we no longer have a Z2 symmetry, so the circum- where b, c are complex numbers. By the rank-one stantial vanishing of one-point functions can no longer be assumption, a given Weyl group element w can only act used as evidence of mapping to monopoles (for instance, via multiplication by a constant c ,so one cannot rule out mixing with the Cartan scalar, after w subtracting one-point functions). X Δþδ δ q Φ Φ M∞ ¼ ðcðcwqÞe½cwqþbðcwqÞÞ: ð7:16Þ w∈W cw B. En monopoles Putting aside the structure of the (known) matrix models, If Δ ≥ 0, then the bubbling term is a “Weyl-invariant” it is interesting to ask whether the structure of the would-be polynomial (monomial in the commutative limit) and can shift operators themselves reveals any information about be removed by a change of basis: the monopole spectrum of the non-Lagrangian duals to the X Δþδ δ q Φ 29 Φ M∞ ¼ cðcwqÞe½cwq: ð7:17Þ The Coulomb branches in the A, D, and E6 cases all have a w∈W cw Z2 symmetry (S3 in the case of D4) that commutes with the hyperkähler structure, whereas the E7 and E8 cases do not have cw any symmetries. For A and D, it is natural to identify the Z2 with Note that e½cwq¼e½q . Now recall the relevant singu- charge conjugation, which acts on monopoles. larities (below, we omit the subscript C for “Coulomb”):

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2 2 Nþ1 AN∶ X þ Y þ Z ¼ 0; ð7:18Þ the first inequality because solutions are trivial to obtain if either m, n is 1). We have not been able to find or rule out 2 2 N−1 nontrivial solutions beyond m; n 2; 2 .31 One pos- DN∶ X þ ZY þ Z ¼ 0; ð7:19Þ ð Þ¼ð Þ sibility is that we should abandon semisimplicity, so that 2 3 4 the monopoles have dimensions not accounted for by Φ. E6∶ X þ Y þ Z ¼ 0; ð7:20Þ

2 3 3 E7∶ X þ Y þ YZ ¼ 0; ð7:21Þ VIII. SUMMARY AND FUTURE DIRECTION This paper presents the results of precision studies of 2 3 5 E8∶ X þ Y þ Z ¼ 0: ð7:22Þ non-Abelian ADE mirror symmetry beyond the chiral ring, using the recently developed TQM techniques in [31–33]. We wish to determine the Coulomb branch operators in As a by-product, we extend the construction of deformation rank-one theories that satisfy these relations. For AN and quantizations of [24] to the D- and E-series. We focus on DN, we think of X and Y as (dressed) monopoles and Z as D-type quivers, in particular synthesizing OPE data (struc- the vector multiplet scalar. Since overall factors of Φ must ture constants) for the chiral ring generators of the D-series, cancel for dimensional reasons, to solve the above rela- but we also comment on possible implications for the tions, we may set Z ¼ 1 and replace X and Y by Laurent monopole spectrum of the non-Lagrangian theories whose polynomials P and Q in a single variable x ∼ e½q (by the C2 Γ Coulomb branches are = E6;7;8 . We find the precise map rank-one assumption): between quantized Higgs branch chiral ring generators in D-type quivers and quantized Coulomb branch chiral ring ∶ −1 2 −1 2 1 0 AN and DN Pðx;x Þ þ Qðx;x Þ þ ¼ : ð7:23Þ generators in SUð2Þ SQCD. Our results provide additional entries in the mirror symmetry dictionary for non-Abelian This equation is easily solved by 3D N ¼ 4 gauge theories beyond, e.g., the matching of supersymmetric partition functions32 [55–58] and chiral x − x−1 iðx þ x−1Þ Pðx;x−1Þ¼ ;Qðx;x−1Þ¼ : ð7:24Þ rings [17]. 2 2 It is safe to say that the range of applications of the Higgs and Coulomb branch TQM has yet to be fully explored. For For the E-series, there are more possibilities to consider for one thing, it would be interesting to incorporate the which operators are scalars and which are monopoles, but additional constraints of N ¼ 6 or N ¼ 8 SUSY [59] into Y let us restrict our attention to the possibility that is the the TQM analysis. For another, the OPE data that we have scalar (which is most plausible in the case of E6). Then we computed can be fed into the bootstrap machine to study obtain the equation the full CFT spectrum and (self-)mirror symmetry beyond the TQM sector, ala` [37]. Finally, the connection between −1 m −1 n 1 0 Pðx; x Þ þ Qðx; x Þ þ ¼ ; ð7:25Þ these techniques and protected operator algebras in one dimension higher [60] (several aspects of which have where m, n are positive integers and P, Q are single- recently been derived from localization [61,62]) via dimen- C variable Laurent polynomials with coefficients in . The sional reduction [63–65] leads us to wonder whether the 2 4 2 3 2 5 cases of E6;7;8 correspond to ðm; nÞ¼ð ; Þ; ð ; Þ; ð ; Þ, TQM contains tractable lessons about line operators in 4D respectively, while the A- and D-series correspond to gauge theories. ðm; nÞ¼ð2; 2Þ. We wish to find nontrivial solutions to A technical detail that we have glossed over is the the above polynomial Diophantine equations, i.e., solutions following. To define the star-shaped quivers of interest, we with neither P nor Q constant (if no such solutions exist, start with all nodes unitary (U) and quotient by the diagonal 30 then the assumptions should be relaxed). Uð1Þ, as suggested by their brane constructions. (See [66] In general, one can ask for which m, n there exist for Coulomb branch computations in these theories.) As a nontrivial solutions to (7.25) (without loss of generality, we computational matter, it is convenient to implement the may restrict our attention to 2 ≤ m ≤ n, where we impose 31However, the existence of nontrivial solutions for small 30Recall that the A-series has two independent monopoles and ðm; nÞ is not immediately ruled out by the abc inequality for trivial Weyl group, while the D-series has one independent polynomials (Mason-Stothers theorem). We thank J. Silverman monopole and nontrivial Weyl group. For the E-series, we for this comment. assume that two of the generators are monopoles, but assuming 32Matrix models for sphere partition functions of affine A-type only one independent monopole (so that the other is simply a quiver theories of arbitrary rank have been studied in [55], dressed version of it) would imply that P and Q have the same leading to a derivation of the mirror map between mass and FI powers of x and differ only in their coefficients; then P and Q parameters. The corresponding analysis for D-type quivers was (and the corresponding monopole operator) would need infinitely performed in [56,57], and in this case, a free-fermion represen- many terms, since the degrees could not match otherwise. So we tation for the partition function (with vanishing mass and FI are led to postulate two independent monopoles for the E-series. parameters) was derived in [58].

085008-24 NON-ABELIAN MIRROR SYMMETRY BEYOND THE CHIRAL … PHYS. REV. D 101, 085008 (2020) Z quotient simply by making one of the nodes SU. More y thðyÞ 1 − πx thðxÞ dy e2πixy ¼ : ðA4Þ precisely, we should make one of the nodes PSU. The chðyÞ π chðxÞ distinction between SU and PSU is irrelevant to normal- ized correlation functions of local operators (in particular, We have in addition that TQM observables). However, the precise normalization of Z the partition function depends on which U node we make thðyÞ2 1 − 4x2 PSU:aPSUðNÞ node introduces a factor of N in the dy e2πixy ¼ ; ðA5Þ ch y 2ch x partition function relative to an SUðNÞ node because the ð Þ ð Þ volumes of these groups differ, and the inverse volume enters into the gauge-fixed path integral. We found that to and by differentiating (A5), we obtain analogous formulas n 2 match the partition function of the affine D-type quiver to for y thðyÞ =chðyÞ, e.g., that of SUð2Þ SQCD, it suffices to make one of the Uð2Þ Z 2 8 π 1 − 4 2 nodes SOð3Þ. The situation is less clear for the affine 2π y thðyÞ ið x þ ð x ÞthðxÞÞ dy e ixy ¼ : ðA6Þ E-type quivers since their mirrors are non-Lagrangian, but chðyÞ 4π chðxÞ one can in principle match partition functions (including discrete factors) by reducing the 4D index of the En One can go on to derive similar identities. Finally, we Minahan-Nemeschansky theories [67,68]. It would be note that interesting to clarify the general procedure for decoupling Z the overall Uð1Þ and to understand better the global dσ u1 − u2 structure of the gauge group in the affine quiver when ¼ : ðA7Þ chðσ − u1Þchðσ − u2Þ shðu1 − u2Þ comparing to the mirror theory.

ACKNOWLEDGMENTS Γ 2. D4 Gauged free hyper We thank M. Dedushenko for useful discussions about Recall that in this theory, short star products and S. Pufu for interesting comments that inspired Appendix B. The work of Y. F. was supported 2 ˜ 2 4 ˜ 4 Z0 ¼ −2Q Q ; Y0 ¼ iðQ þ Q Þ; in part by the NSF GRFP under Grant No. DGE-1656466 ffiffiffi p ˜ 4 ˜ 4 and by the Graduate School at Princeton University through X 0 ¼ 2iQQðQ − Q Þ: ðA8Þ a Centennial Fellowship. The work of Y. W. was supported in part by the U.S. NSF under Grant No. PHY-1620059 and Thus by the Simons Foundation Grant No. 488653. 2 ˜ 2 i 4 ˜ 4 U0; V0 ¼ −Q Q pffiffiffi ðQ þ Q Þ; 2 3 APPENDIX A: DETAILS OF TQM i ˜ 4 ˜ 4 COMPUTATIONS W0 ¼ pffiffiffi QQðQ − Q Þ; ðA9Þ 2 1. Fourier transform identities The basic Fourier transform identities that we will where we use the 0 subscript to denote a “bare” Higgs need are branch CPO. Canonically normalized CPOs without 0 Z Z 2π subscripts have vanishing one-point functions and diagonal 1 e ixy 1 i −2πixy two-point functions (in a real basis). ¼ dy ; ¼ 2 dye thðyÞ: ðA1Þ chðxÞ chðyÞ shðxÞ We would like to compute correlation functions of the CPOs. To proceed, we need the two-point function of Q; Q˜ , Other useful identities include which is Z Z e2πixy x e2πixy 1 4x2 þ φ φ dy 2 ¼ ; dy 3 ¼ : ðA2Þ ˜ sgnð 12Þ sgnð 12Þ chðyÞ shðxÞ chðyÞ 8chðxÞ hQðφ1ÞQðφ2Þi ¼ − ¼ ðA10Þ 8πr 2l By differentiating (A1), we obtain (recall that l ¼ −4πr from [31]). The correlator at coinci- Z thðyÞ 2ix dent points is 0. In particular, dy e2πixy ¼ : ðA3Þ chðyÞ chðxÞ hU0i¼hV0i¼hW0i¼0; ðA11Þ By further differentiating (A3), we obtain analogous for- mulas for ynthðyÞ=chðyÞ, e.g., and consequently the normalized CPOs are

085008-25 YALE FAN and YIFAN WANG PHYS. REV. D 101, 085008 (2020)

U U V V W W φ ˜ j φ ¼ 0; ¼ 0; ¼ 0: ðA12Þ hQiAð 1ÞQBð 2Þiσ;u

j sgnðφ12ÞþthðσA uÞ − σ φ “ ” ¼ −δ δ e ð AuÞ 12 ; ðA18Þ There is no further gauge ambiguity in this case. AB i 8πr Doing simple Wick contractions, we obtain where φ12 ≡ φ1 − φ2 and the sign is þ when i ¼ j ¼ 1 1 15 − 2 U ⋆ V W ⋆ W − and when i ¼ j ¼ . We emphasize that the 1D TQM h i¼ 4 ; h i¼ 6 ; 2l 8l path integral has an explicit S4 symmetry permuting the A hU ⋆ Wi¼hV ⋆ Wi¼0; indices (as explained before, only an S3 subgroup acts pffiffiffi faithfully on CPOs). At coincident points, we use the 3 5 6 U ⋆ U ⋆ U V ⋆ V ⋆ V ; U ⋆ V ⋆ W − ; symmetrized expression h i¼h i¼l6 h i¼ 2l7 15 122 4!2 8! 2 21 σ u 2 2 × þ × φ ˜ j φ −δ δj thð A Þ hU ⋆ V i¼ ; hUV⋆ UVi¼ ¼ : hQiAð ÞQBð Þiσ;u ¼ AB i ; ðA19Þ 2l8 122ð2lÞ8 4l8 8πr ðA13Þ and in computing correlation functions, we always assume 33 the φi are ordered as Thus, comparing to (4.16), we have φ φ φ pffiffiffi 1 < 2 < 3 < : ðA20Þ 4 2 179 ζ − ¼ l ;A¼ 32 ; Note that (incomplete) self-contractions of a composite 3 1 5 operator can also contribute to connected correlators. The α α α 1 ¼ 16 ; 2 ¼ 2048 ; 4 ¼ 8 : ðA14Þ correlators that we compute below are normalized by (A16).

In particular, the S3 symmetry is obvious in the TQM. a. Computation of TQM correlators

3. Affine D4 quiver To compute the (normalized) two-point function U ⋆ V U ⋆ V Recall that the Higgs branch chiral ring generators are h i, we need to compute h 0 0i as well as the U V given by one-point functions h 0i and h 0i. We start by recording the Wick contractions in the 1D iπ=6 ˜ ˜ −iπ=6 ˜ ˜ TQM on the Higgs branch: U0 ¼ e Q1Q3Q3Q1 þ e Q2Q3Q3Q2; −iπ=6 ˜ ˜ iπ=6 ˜ ˜ ˜ ˜ ˜ ˜ V0 ¼ e Q1Q3Q3Q1 þ e Q2Q3Q3Q2; hðQ1Q3Q3Q1Þðφ1ÞðQ1Q3Q3Q1Þðφ2Þiσ 3=4 ˜ ˜ ˜ W0 ¼ 3 iQ1Q2Q2Q3Q3Q1 ðA15Þ 1 ¼ ðI þ I þ I ÞðA21Þ ð2lÞ4 c s ss in terms of gauge-invariant combinations of the hypermultiplets. for φ1 < φ2, where cross-contractions give We adopt the normalization (5.12) for the S3 partition function of the affine D4 quiver, which we can write as Ic ¼ðð1 þ thðσ1 − uÞÞð1 − thðσ3 − uÞÞ Z Y þðu ↔ −uÞÞ × ðσ1 ↔ σ3Þ 2 1 Z ¼ du dσ shð2uÞ Zσ ¼ ; ðA16Þ 1 1 2 D4 A ;u 120π A ¼ 16 þ chðσ1 − uÞchðσ3 − uÞ chðσ1 þ uÞchðσ3 þ uÞ where ðA22Þ Z Y and self-contractions give i ˜ Zσ;u ¼ DQADQiA A σ − 2 σ − 2 − 1 ↔ − Z X Is ¼ðthð 1 uÞ ðthð 3 uÞ Þþðu uÞÞ 4π φ ˜ ∂ δi σ δi i j × exp r d QiAð φ j þ A j þ ut jÞQA þðσ1 ↔ σ3ÞðA23Þ A ðA17Þ 33Thus our conventions are that operator insertions in the expression hO1 ⋆ ⋆ Oni are understood to be in ascending and t ¼ σ3. Thus the propagators are order; compare to (2.11) and (5.38).

085008-26 NON-ABELIAN MIRROR SYMMETRY BEYOND THE CHIRAL … PHYS. REV. D 101, 085008 (2020) as well as At the level of the TQM, this is a simple consequence of the exact S3 symmetry of (A16). 2 Iss ¼ðthðσ1 − uÞthðσ3 − uÞþðu ↔ −uÞÞ : ðA24Þ We summarize the results of similar computations for various correlators below: Doing the matrix integral, we get 2π4 − 135 π4 − 30 hU ⋆ Vi¼ ; ˜ ˜ φ ˜ ˜ φ 60π4l4 hðQ1Q3Q3Q1Þð 1ÞðQ1Q3Q3Q1Þð 2Þi ¼ 4 4 : ðA25Þ 20π l 32π4 − 2835 U ⋆ U ⋆ U V ⋆ V ⋆ V ffiffiffi h i¼h i¼ p 4 6 ; Similarly, 420 3π l pffiffiffi 3 3ðπ4 − 105Þ ˜ ˜ φ ˜ ˜ φ W ⋆ W ; hðQ1Q3Q3Q1Þð 1ÞðQ2Q3Q3Q2Þð 2Þiσ h i¼ 140π4l6 2 pffiffiffi ∝ ðthðσ1 − uÞthðσ2 − uÞðthðσ3 − uÞ − 1Þþðu ↔ −uÞÞ 3 4 3 π4 − 105 U ⋆ V ⋆ W ð Þ h i¼ 4 7 ; þðthðσ1 − uÞthðσ3 − uÞþðu ↔ −uÞÞ 140π l π4 − 105 32π4 − 2835 σ − σ − ↔ − 2 2 ð Þð Þ × ðthð 2 uÞthð 3 uÞþðu uÞÞ ðA26Þ hU ⋆ V i¼− ; 490π4ð2π4 − 135Þl8 has only self-contractions. We get 14175 12180π4 − 128π8 UV ⋆ UV − þ h i¼ 8 8 : ðA34Þ 4 2800π l ˜ ˜ ˜ ˜ 45þ π hðQ1Q3Q3Q1Þðφ1ÞðQ2Q3Q3Q2Þðφ2Þi ¼ : ðA27Þ 60π4l4 These correspond to (4.18) with ζ ¼ 1. Note that we have performed Gram-Schmidt diagonalization to define the We also have the one-point functions composite operators. ˜ ˜ ˜ ˜ 1 hQ1Q3Q3Q1i¼hQ2Q3Q3Q2i¼ 2 : ðA28Þ 6l 4. Affine Dn quiver O O a. Partition function Thus the connected two-point functions (h 1 2ic ¼ hO1O2i − hO1ihO2i) are We start by simplifying the Higgs branch matrix model of the affine Dn quiver, while also reviewing the mirror ˜ ˜ φ ˜ ˜ φ equivalence to SQCD at the level of S3 partition functions. hðQ1Q3Q3Q1Þð 1ÞðQ1Q3Q3Q1Þð 2Þic ˜ ˜ ˜ ˜ The Cauchy determinant formula ¼hðQ2Q3Q3Q2Þðφ1ÞðQ2Q3Q3Q2Þðφ2Þi c Q ˜ ˜ ˜ ˜ − − X ρ ¼ −2hðQ1Q3Q3Q1Þðφ1ÞðQ2Q3Q3Q2Þðφ2Þi i

Similarly, one can check explicitly that

U ⋆ W V ⋆ W 0 h i¼h i¼ : ðA33Þ with k − 2 Uð2Þ gauge nodes.

085008-27 YALE FAN and YIFAN WANG PHYS. REV. D 101, 085008 (2020) Z Q 1 Yn−2 Y2 n−3 1 − 2 2 i 1 2 I¼1 shðuI uI Þ Z ¼ du δðu1 þ u1Þ Q Q Dn 2n−4 I n−3 2 i − j I¼0 i¼1 I¼0 i;j¼1 chðuI uIþ1Þ Z Q Yn−2 Y2 1 2 n−3 1 2 1 2 1 δðu1 þ u1Þ 0 shðu − u Þshðu 1 − u 1Þ ¼ dui I¼Q QI I Iþ Iþ 2n−4 I sh u1 − u2 sh u1 − u2 n−3 2 i − j I¼0 i¼1 ð 0 0Þ ð n−2 n−2Þ I¼0 i;j¼1 chðuI uIþ1Þ and then use (A35) in the form Q i j i j X ρ shðu − u Þshðu 1 − u 1Þ ð−1Þ i

Using (A1) and simplifying, we get Z Yn−2 Y2 Yn−3 Y2 1 2 Yn−3 Y2 2πisi ðui −ui Þ δðu1 þ u1Þ e I I Iþ1 Z ¼ 4 dui dsi Dn I I 1 − 2 1 − 2 si I¼0 i¼1 I¼0 i¼1 shðu0 u0Þshðun−2 un−2Þ I¼0 i¼1 chð IÞ ¼ Z 1 1 Z 2 1 ds0ds1 1 shðsÞ ¼ thðs1Þthðs1Þδðs1 − s1Þ¼ ds ; ðA37Þ 2 1 2 1 2ðn−3Þ 0 1 0 1 4 2n chðs0Þ chðs1Þ chðs=2Þ which coincides with the partition function of SUð2Þ with n flavors up to our conventional factor of 1=r2 [see (5.37)].35

b. Computation of TQM correlators We now compute the one- and two-point functions hZi and hZ ⋆ Zi. Recall that

Z ≡ − ˜ ˜ − ˜ i ˜ j Q1Q3Q3Q1 ¼ ðQ1Þ ðQ3ÞiðQ3Þ ðQ1Þj: ðA39Þ Z Z It is helpful to note that integration by parts can be used to simplify h iHB: in the matrix model (5.10) for ZDn,an insertion of the form

35Another typo in [47] can be found in their formula (3.3). The correct version is Z 2 XNf 1 shð2xÞ 1 mishð2miÞ −1 Nf þ Q dx QN ¼ð Þ 2 2 ; ðA38Þ 2 f − ≠ ðshðm Þ − shðm Þ Þ i¼1 chðx miÞchðx þ miÞ i¼1 j i i j

Nf þ1 the ð−1Þ on the RHS having been overlooked. The LHS is the partition function of SUð2Þ SQCD with Nf flavors and mass 2 parameters, which reduces to our expression (5.37) (up to the 1=r ) when the mi ¼ 0. However, the above equality holds only when the mass parameters are distinct.

085008-28 NON-ABELIAN MIRROR SYMMETRY BEYOND THE CHIRAL … PHYS. REV. D 101, 085008 (2020) p q 2 thðσ − uÞ thðσ þ uÞ 1 1 1 chðσ − uÞchðσ þ uÞ∂σ ðA40Þ hZi ¼ − ðthðσ1 þ u1Þthðσ3 þ u1Þ chðσ − uÞchðσ þ uÞ HB 8πr 2 2 1 2 þ thðσ1 þ u1Þthðσ3 þ u1ÞÞ: ðA44Þ (where σ is σ1 or σ3 and u ≡ u1 ¼ −u1) is a total derivative. In particular, taking ðp; qÞ to be (0, 0) or (1, 0) shows that Setting u ≡ u1 −u2 and integrating by parts allows us to the expressions 1 ¼ 1 write σ − σ thð uÞþthð þ uÞ; 1 2 1–2 σ − 2 − σ − σ hZi ∼−2 thðσ1 − uÞthðσ3 − uÞ: ðA45Þ thð uÞ thð uÞthð þ uÞðA41Þ HB 8πr are total derivatives. This observation is useful because We also have before simplification, hZ Zi is a polynomial in HB thðσ uÞ, while dropping total derivatives allows it to 1 4 be written as a polynomial in thðσ − uÞ; this facilitates hZ ⋆ Zi ¼ ðI þ I þ I Þ; ðA46Þ HB 8πr c s ss manipulation of the resulting integrals because it allows for shifts of σ by u, thus decoupling the u integral. where, as in the D4 case, To set the stage, we determine the expression for the partition function after integrating over all Higgs branch 1 1 i i Ic ¼ðð1 þ thðσ1 þ u1ÞÞð1 − thðσ3 þ u1ÞÞ variables except for u0 (i.e., σ1;3) and u1. This yields a 1 2 simplified Higgs branch matrix model with all scalar VEVs þðu1 ↔ u1ÞÞ × ðσ1 ↔ σ3Þ; ðA47Þ integrated out, apart from those relevant to an insertion of hZ Zi . Starting from the first line of (A37),we 1 2 1 2 1 2 HB Is ¼ðthðσ1 þ u1Þ ðthðσ3 þ u1Þ − 1Þþðu1 ↔ u1ÞÞ derive that þðσ1 ↔ σ3Þ; ðA48Þ Z 1 2 2 2πisðu1−u1Þ Y ds e thðsÞ i i Z ¼ −2i du0du1 1 1 1 2 2 Dn 2n−6 Iss ¼ðthðσ1 þ u1Þthðσ3 þ u1Þþðu1 ↔ u1ÞÞ : ðA49Þ chðsÞ i¼1 δ 1 2 1 2 ðu1 þ u1Þ Setting u ≡ u1 ¼ −u1 and integrating by parts gives × 1 − 2 1 − 1 2 − 2 shðu0 u0Þchðu0 u1Þchðu0 u1Þ Z Y X 4π Y2 1 ds e isuthðsÞ I þ I þ I ∼4 3 − 2 thðσ − uÞ − 16 ¼ −2i dui c s ss A σ − 2 2n−6 0 A¼1;3 A¼1;3 chð A uÞ chðsÞ i 1 ¼ Y 1 du 96 : A50 × : ðA42Þ þ σ − 2 ð Þ 1 2 1 2 1 3 chð A uÞ shðu0 − u0Þchðu0 − uÞchðu0 þ uÞ A¼ ;

Let us write To evaluate the multidimensional integrals for hZi and hZ ⋆ Zi, our main tool for simplification is to take a 1 Fourier transform whenever an argument of “sh,”“ch,” or h i ¼ Z ½h i : ðA43Þ Dn HB “th” involves a combination of two variables: this allows us ZDn to decouple the single-variable integrals. For instance, We now compute by taking Wick contractions that we have

Z 4π Y2 2i ds e isuthðsÞ du hZi¼− dui hZi Z ch s 2n−6 0 sh u1 − u2 ch u1 − u ch u2 u HB Dn ð Þ i¼1 ð 0 0Þ ð 0 Þ ð 0 þ Þ Z 2 4πisu Y2 1 2 4i 1 ds e thðsÞ thðu0 − uÞthðu0 − uÞ ¼ dui du Z 8πr ch s 2n−6 0 sh u1 − u2 ch u1 − u ch u2 u Dn ð Þ i¼1 ð 0 0Þ ð 0 Þ ð 0 þ Þ Z 2 Y2 1 2 −2π 2 2i 1 thðsÞ thðu Þthðu Þe isu0 ¼ ds dui 0 0 ; Z 8πr s 2n−5 0 sh u1 − u2 ch u1 Dn chð Þ i¼1 ð 0 0Þ ð 0Þ

085008-29 YALE FAN and YIFAN WANG PHYS. REV. D 101, 085008 (2020) Z i i 2 2 where we have shifted u0 → u0 þ u and integrated 1 1 shðsÞ hZi¼ ds ðs2 − 1Þ: ðA51Þ over u using (A1). Taking the Fourier transform of the 4Z 8πr ch s=2 2n 1 2 Dn ð Þ 1=shðu0 − u0Þ, and iterating this process as necessary (possibly with the help of various identities from Appen- dix A1), leaves us with nested single-variable integrals that For hZ ⋆ Zi, we must evaluate three additional integrals can be evaluated sequentially to yield a single integral: (call them I1, I2, I 12). First, we have

Z 4πisu Y2 1 ds e thðsÞ i du I 1 ≡ du 2n−6 0 1 − 2 1 − 2 1 − 2 chðsÞ i¼1 shðu0 u0Þchðu0 uÞchðu0 þ uÞ chðu0 uÞ Z Y2 −2π 2 1 thðsÞ e isu0 ¼ ds dui 2 2n−5 0 1 − 2 1 3 chðsÞ 1 shðu0 u0Þchðu0Þ Z i¼ i shðsÞ2 ¼ ds ðs2 þ 1Þ: ðA52Þ 64 chðs=2Þ2n

Second, we have Z 4πisu Y2 1 ds e thðsÞ i du I 2 ≡ du 2n−6 0 1 − 2 1 − 2 2 − 2 chðsÞ i¼1 shðu0 u0Þchðu0 uÞchðu0 þ uÞ chðu0 uÞ Z Y2 −2π 2 1 thðsÞ e isu0 ¼ ds dui 2 2n−5 0 1 − 2 1 2 2 chðsÞ 1 shðu0 u0Þchðu0Þchðu0Þ Z i¼ Z i shðsÞ2 i s shðsÞ ¼ ds ðs2 þ 1Þ − ds : ðA53Þ 64 chðs=2Þ2n 16π chðs=2Þ2n−2

Third, we have Z 4πisu Y2 1 ds e thðsÞ i du I 12 ≡ du Q 2n−6 0 1 − 2 1 − 2 2 i − 2 chðsÞ i¼1 shðu0 u0Þchðu0 uÞchðu0 þ uÞ i¼1 chðu0 uÞ Z Y2 −2π 2 1 thðsÞ e isu0 ¼ ds dui 2 2n−5 0 1 − 2 1 3 2 2 chðsÞ 1 shðu0 u0Þchðu0Þ chðu0Þ Z i¼ Z i shðsÞ2 i s shðsÞ ¼ ds ðs4 þ 10s2 þ 9Þ − ds : ðA54Þ 3072 chðs=2Þ2n 96π chðs=2Þ2n−2

Combining the results (A52), (A53), (A54), and Z Z 4π Y2 2 dse isuthðsÞ du i shðsÞ dui ½thðu1 −uÞthðu2 −uÞ ¼ − ds ðs2 − 1ÞðA55Þ 2n−6 0 1 − 2 1 − 2 0 0 16 2 2n chðsÞ i¼1 shðu0 u0Þchðu0 uÞchðu0 þ uÞ chðs= Þ

(which we deduce from our result for hZi)gives Z 4π Y2 2i ds e isuthðsÞ du hZ ⋆ Zi¼− dui hZ ⋆ Zi 2n−6 0 1 − 2 1 − 2 HB ZD chðsÞ 1 shðu0 u0Þchðu0 uÞchðu0 þ uÞ n Z i¼ 1 1 4 shðsÞ2 ¼ ds ðs2 − 1Þ2: ðA56Þ 4 8π 2 2n ZDn r chðs= Þ

Combining the above gives (5.14).

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Y APPENDIX B: QUANTIZED COULOMB ∓1 1 1 M ⋆ M − Φ jqjNq BRANCHES FOR AD ¼ ð iq Þ þ O : ðB2Þ q r Here, we consider some realizations of deformation P C2 Γ C2 Γ Setting N q N and quantizations of the = AN and = DN singularities by ¼ q j j q Lagrangian quantum field theories, namely the Coulomb branches of 3D N ¼ 4 Uð1Þ and SUð2Þ gauge theories 1 1 X ¼ M−1; Y ¼ M1; with arbitrary matter representations. In these cases, the ð4πÞN=2C1=2 ð4πÞN=2C1=2 1 Z choice of basis is strongly constrained by Uð Þ and 2 i Y Z − Φ ≡ jqjNq flavor symmetries, respectively. ¼ 4π ;C q ðB3Þ We expect both the Coulomb branch chiral ring (the q “classical” Coulomb branch) and its Poisson structure to depend only on N, because N determines the holomorphic (this normalization being natural from the point of view of symplectic form.36 We also expect the number of distinct correlation functions), we find that XY ¼ ZN in the chiral quantizations realized by these theories to be related to ring. Accounting for sign, we obtain partitions of N. An interesting question that one might ask, X which we do not attempt to answer here, is, do there exist 2#partsðPÞ >pðNÞðB4Þ examples of different Lagrangian theories with the same P∈fpartitions of Ng “quantum” Coulomb branch, to higher orders in ℏ ∼ 1=r beyond Oðℏ1Þ? distinct quantizations from these theories for fixed N.At Nondegenerate short star products for quotient singu- finite r and to subleading order in 1=r, we compute that larities, including Kleinian singularities, have been classi- Y N fied in [38]. For example, even nondegenerate short star ∓1 1 1 jqjq − 1 q M ⋆ M ¼ −iqrΦ þ products for A singularities depend on n n parameters jqj 2 n e þ s q r jqj where the first ne ¼bðn þ 1Þ=2c parameters determine the corresponding quantum algebra A up to isomorphism (i.e., ðB5Þ the period of the quantization) and the remaining n ¼ s Y n þðð−1Þn − 1Þ=2 parameters determine maps from the e ¼ ð−iqΦÞjqjNq associated graded algebra gr A (the “commutative limit” ð Þ q of the associative algebra A) into A, corresponding to X physical gauge fixings. This agrees with the counting of 1 i jqj − × þ Φ Nq 2 þ q sgnðqÞ free parameters in [24] for the examples of A ≤4 before r n q imposing unitarity (i.e., positivity), which is a stronger 1 condition than nondegeneracy. In the examples below, þ O 2 ; ðB6Þ fixing a Lagrangian SCFT should be understood as fixing r a particular value of the period for the quantization. so that X Y 1. Uð1Þ −1 1 i M M − Φ jqjNq 1 ½ ; ⋆ ¼ Φ jqjNq ð iq Þ Consider Uð Þ for some set of charges fqg with r q q multiplicities N , where q ∈ Z 0 and N ∈ Z≥0 f qg nf g q 1 (uncharged matter does not contribute, but we may con- þ O 2 : ðB7Þ sider the pure case). The shift operators for the Coulomb r branch chiral ring generators are Equivalently, Y 2 −1 ðjqjqÞ= 1− Nq 1 ð Þ qB ∓ i∂ ∂ M ¼ þiqσ e ð2 σ þ BÞ; 1 N 1 jqj=2 2 X Y Z ZN−1 r ðjqjqÞ=2 ½ ; ⋆ ¼ Pð Þ¼ þ O 2 : ðB8Þ q r 4πr r 1 i Φ¼ σ þ B : ðB1Þ r 2 Hence the Poisson structure, as with the chiral ring, depends only on N (as expected). These quantizations We compute that are distinguished by the coefficients of the subleading terms in the polynomial PðZÞ (computing the commutator 36The fact that the first subleading term in the star product is is simpler than directly computing three-point functions determined by the Coulomb branch also follows from a less because various gauge-fixing ambiguities cancel in the transparent topological descent argument [24]. former).

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The structure constants for the deformation quantizations Using e½2e½−2¼1 gives corresponding to the Higgs branch of the affine A2 3 quivers ; Y Y 2 2 were originally bootstrapped in [24] and later derived from Φ2 M2 2 − ΦM2 2 4 mjNj Φ2 N−1 ð ∞Þ ð ∞Þ ¼ 0 mj ð Þ : localization in [31]. By using the above techniques for the j mj> Coulomb branch of the mirror dual, we obtain these results ðB15Þ and more with very little effort. Then setting 2. SUð2Þ X −1ΦM2 Y − −1M2 2 ¼ C ∞; ¼ iC ∞; Consider SUð Þ SQCD with matter specified by some Y Y ∈ 1 Z 2m N set of spins fjg with multiplicities fNjg, where j 2 >0 Z Φ2 ≡ 2 j j ¼ ;C mj ðB16Þ ∈ Z≥0 and Nj (uncharged matter does not contribute, but j mj>0 we may consider the pure case). In conventions where the weights of SUð2Þ are half-integers and monopole charges yields the equation of a DN singularity, are even integers (b ∈ 2Z), we have X 2 þ ZY2 þ ZN−1 ¼ 0: ðB17Þ Q Q m b ð−1Þð j Þþ 1 Φ Nj j½ m jm bj=2 ð2 þ irmj Þðm bÞ i b j r j j þ −bð ∂σ þ∂ Þ M ¼ e 2 B ; For N ¼ 1 (i.e., N1 2 ¼ 1), one can show that the relevant 1 ðirsgnðbÞΦÞ = rjbj jbj bubbling coefficient vanishes by a polynomiality compu- 2 2 1 i tation at finite r [33], but let us not assume this. We have Φ ¼ σ þ B ; r2 2 c c ðB9Þ M2 2 −2 − ∞ ¼ M∞ þ Φ þ M∞ Φ where mj ∈ f−j; −j þ 1; …;jg. Indicating the commuta- 1 ∞ → ∞ ¼ − ðe½2 − e½−2Þ; ðB18Þ tive limit with an subscript (for r ), we have 2iΦ P Y Y 1 −1 b jbjð2 SjNj Þ jbjmjNj M∞ ¼ð−isgnðbÞΦÞ j m e½b; 2 2 c −2 c j ΦM∞ ¼ Φ M∞ þ − Φ M∞ − j mj>0 Φ Φ 1 ðB10Þ ¼ − ðe½2þe½−2Þ þ 2c; ðB19Þ  2i Xj jðj þ 1Þ if j ∈ Z; S ≡ jm j¼ ðB11Þ so that j j ðj þ 1=2Þ2 if j ∈ Z þ 1 : mj¼−j 2 2 2 2 2 2 P Φ ðM∞Þ − ðΦM∞ − 2cÞ ¼ 1: ðB20Þ Set N ¼ j SjNj. Then, in particular, we see that Equivalently, Δ M2 − 2 Δ ΦM2 − 1 Δ Φ2 2 ð Þ¼N ; ð Þ¼N ; ð Þ¼ : 2 2 X ¼ ΦM∞; Y ¼ −iM∞; ðB12Þ Z ¼ Φ2; ðX − 2cÞ2 þ ZY2 þ 1 ¼ 0: ðB21Þ 2 On dimensional grounds, the bubbling coefficient for M∞ 0 is a monomial in Φ for N ≥ 2, which we can eliminate by a Unless c ¼ , this is a nonsingular deformation of a D1 change of basis. So for N ≥ 2, singularity [as can be seen from the nonvanishing of the partial derivatives at (0, 0, 0)]. For N ¼ 0 (the pure case), 2 2 −2 M∞ ¼ M∞ þ M∞ we have Y Y 2 mjNj Φ N−2 −1 N 2 −2 c c ¼ mj ði Þ ðð Þ e½ þe½ Þ; M2 2 −2 ∞ ¼ M∞ þ 2 þ M∞ þ 2 j mj>0 Φ Φ ðB13Þ 1 2c − e 2 e −2 ; ¼ Φ2 ð ½ þ ½ Þ þ Φ2 ðB22Þ 2 2 −2 ΦM∞ ¼ ΦðM∞ − M∞ Þ Y Y 2 2 c −2 c 2m N −2 ΦM Φ − Φ j j Φ Φ N −1 N 2 − −2 ∞ ¼ M∞ þ 2 M∞ þ 2 ¼ mj ði Þ ðð Þ e½ e½ Þ: Φ Φ 0 j mj> 1 − 2 − −2 ðB14Þ ¼ Φ ðe½ e½ Þ; ðB23Þ

085008-32 NON-ABELIAN MIRROR SYMMETRY BEYOND THE CHIRAL … PHYS. REV. D 101, 085008 (2020) and therefore 1. Affine D4 quiver 2 2 2 2 2 2 The affine D4 quiver contains hypermultiplets ðQ Þ , Φ · M∞ − 2c − Φ ΦM∞ 4: B24 A i ð Þ ð Þ ¼ ð Þ ˜ i ðQAÞ with A ¼ 1; …; 4 and i ¼ 1, 2. The superpotential is Equivalently, X X W ¼ Φ Qi Q˜ j þ ϕ Qi Q˜ j ϵ ; ðC2Þ X ΦM2 Y − M2 Z Φ2 ij A A A A A ij ¼ ∞; ¼ i ∞; ¼ ; A A ZðX 2 þ ZY2 þ 4icYÞþ4ð1 − c2Þ¼0: ðB25Þ where Φ and ϕA are adjoint chirals for the SUð2Þ and Uð1Þ The degree of the relation is reduced when c ¼1 (the sign gauge nodes, respectively. We introduce the notation ≡ ˜ ambiguity is present even when using polynomiality [33]): hABi QAQB; then the F-term relations give X 2 2 X ¼ ΦM∞; Y ¼4M∞; AihA ¼hAAi¼0: ðC3Þ Z ¼ Φ2; X 2 þ ZY2 þ Y ¼ 0; ðB26Þ A where we have slightly redefined the variables. This gives an For fixed A, we have the four relations X alternative way to fix c2. Note that the theory is good for hABihBAi¼0: ðC4Þ N ≥ 3, we expect the D equation to hold for N ≥ 1, and we N B≠A expect it to be modified as above for N ¼ 0. The possibilities for bad theories are simply fjg¼fg(N ¼ 0), fjg¼f1=2g Hence out of the six candidate chiral ring generators with (N ¼ 1), and fjg¼f1=2; 1=2g; f1g (N ¼ 2). Δ ¼ 2, namely hABihBAi with A

pffiffiffi X 2 þ ZY2 þ Z3 ¼ 12 3h23ih31iðh13i2h31ih32iþh32i2h23ih13i − h12i2h23ih31iÞ pffiffiffi ¼ 12 3h23ih31ið−h13ih31ih14ih42i − h23ih32ih14ih42i − h14ih43ih34ih42iÞ pffiffiffi ¼ 12 3h23ih31ih14ih42ið−h31ih13i − h32ih23i − h34ih43iÞ ¼ 0: ðC11Þ

37While (C1) 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 (C1) receives contributions from FI parameters, which we have set to zero.

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Moreover, the S3 generators (4.32) act as ( h12ih23ih31i ↦ h21ih13ih32i¼−h24ih43ih32i¼h24ih41ih12i¼−h23ih31ih12i; rZ ∶ ðC12Þ 2 h13ih31i ↦ h23ih32i and 8 > <> h12ih23ih31i ↦ h23ih31ih12i; sZ ∶ 13 31 ↦ 21 12 34 43 − 13 31 − 23 32 ; C13 3 > h ih i h ih i¼h ih i¼ h ih i h ih i ð Þ :> h23ih32i ↦ h31ih13i; giving the expected (4.31).

2. Affine Dn>4 quiver

For the affine Dn quiver, we have adjoint chirals ϕA and ΦI (I ¼ 1; …;n− 3) for the Uð1Þ and Uð2Þ nodes, respectively. The superpotential is ˜ i Φ j − ϕ ˜ i ˜ i Φ j − ϕ ˜ i W ¼ððQ1Þ ð 1Þi ðQ1Þj 1ðQ1Þ ðQ1ÞiÞþððQ3Þ ð 1Þi ðQ3Þj 3ðQ3Þ ðQ3ÞiÞ ˜ i Φ j − ϕ ˜ i ˜ i Φ j − ϕ ˜ i þððQ2Þ ð n−3Þi ðQ2Þj 2ðQ2Þ ðQ2ÞiÞþððQ4Þ ð n−3Þi ðQ4Þj 4ðQ4Þ ðQ4ÞiÞ Xn−4 ˜ i Φ j k − i Φ j ˜ k þ ððKIÞk ð IÞi ðKIÞj ðKIÞk ð Iþ1Þi ðKIÞj Þ: ðC14Þ I¼1 The signs keep track of orientation in the N ¼ 2 sense (the legs are unoriented in the N ¼ 4 sense). The F-term relations are ˜ i 0 1 2 3 4 ðQAÞ ðQAÞi ¼ ðA ¼ ; ; ; Þ; ðC15Þ ˜ j ˜ j k ˜ j 0 ðQ1ÞiðQ1Þ þðQ3ÞiðQ3Þ þðK1Þi ðK1Þk ¼ ; ðC16Þ

˜ j ˜ j − ˜ k j 0 ðQ2ÞiðQ2Þ þðQ4ÞiðQ4Þ ðKn−4Þi ðKn−4Þk ¼ ; ðC17Þ

˜ k j − k ˜ j 0 1 … − 5 ðKIÞi ðKIÞk ðKIþ1Þi ðKIþ1Þk ¼ ðI ¼ ; ;n Þ: ðC18Þ

It should be kept in mind that the Uð2Þ indices are associated with different nodes. Below, gauge indices are appropriately contracted between pairs of hypers when suppressed. To justify our description of the Higgs branch chiral ring in (4.40)–(4.45), we first list some useful equivalences between chiral ring elements, which are reflected in correlation functions.38 From the F-term relations, we derive

˜ ˜ a ˜ ˜ a Q2ðKn−4Kn−4Þ Q2 ¼ Q4ðKn−4Kn−4Þ Q4 ˜ ˜ ˜ a ˜ ˜ ˜ a ¼ Q2ðQ2Q2 þ Q4Q4Þ Q2 ¼ Q4ðQ2Q2 þ Q4Q4Þ Q4  0 a ∈ 2Z; C19 ¼ ˜ ˜ ðaþ1Þ=2 ð Þ ðQ2Q4Q4Q2Þ a ∈ 2Z þ 1 and

˜ ˜ a ˜ ˜ a Q1ðK1K1Þ Q1 ¼ Q3ðK1K1Þ Q3 a ˜ ˜ ˜ a a ˜ ˜ ˜ a ¼ð−1Þ Q1ðQ1Q1 þ Q3Q3Þ Q1 ¼ð−1Þ Q3ðQ1Q1 þ Q3Q3Þ Q3  0 a ∈ 2Z; C20 ¼ ˜ ˜ ðaþ1Þ=2 ð Þ −ðQ1Q3Q3Q1Þ a ∈ 2Z þ 1:

38In the process, we fix several mistakes in (A. 3) of [46].

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Similarly, we derive that where A ∈ f1; 3g and A0 ∈ f2; 4g. From (C24) and (C20), we have ˜ ˜ a ˜ ˜ ˜ a ˜ Q2ðKn−4Kn−4Þ Q4Q4Q2 ¼Q4ðKn−4Kn−4Þ Q2Q2Q4  ˜ ˜ ˜ ˜ ˜ ˜ ˜ a=2þ1 Q3K1 K −4ðQ2Q2 þ Q4Q4ÞK −4 K1Q3 ðQ2Q4Q4Q2Þ a∈2Z;  n n ¼ 0 ∈ 2Z 1 0 a∈2Zþ1; n þ ; ¼ ˜ ˜ n=2−1 ðC27Þ − Q1Q3Q3Q1 n ∈ 2Z: ðC21Þ ð Þ

˜ ˜ a ˜ ˜ ˜ a ˜ From (C24) and (C22), we also have Q1ðK1K1Þ Q3Q3Q1 ¼ Q3ðK1K1Þ Q1Q1Q3  ˜ ˜ a=2þ1 ˜ ˜ ˜ ˜ ˜ ˜ ðQ1Q3Q3Q1Þ a ∈ 2Z; Q1K1 Kn−4ðQ2Q2 þ Q4Q4ÞKn−4 K1Q3Q3Q1 ¼  0 ∈ 2Z 1 ˜ ˜ ðn−1Þ=2 a þ : ðQ1Q3Q3Q1Þ n ∈ 2Z þ 1; ¼ ðC28Þ ðC22Þ 0 n ∈ 2Z:

Moreover, we see that So we see that the Z2 symmetry that takes 2 ↔ 4 [i.e., ˜ ˜ ðQ2; Q2Þ ↔ ðQ4; Q4Þ] acts as ˜ ˜ ˜ ˜ a QAKn−4 K1ðK1K1Þ K1 Kn−4QA0 ˜ ˜ nþa−4 Z2∶ ðX; Y; ZÞ ↦ ð−X; −Y; ZÞðC29Þ ¼ QAðKn−4Kn−4Þ QA0 ðC23Þ ∈ 2Z ∈ 2Z 1 for A; A0 ∈ f2; 4g and regardless of whether n or n þ . Equivalently, the Z2 symmetry can be implemented by swapping 1 ↔ 3 ˜ ˜ a ˜ ˜ ˜ ↔ ˜ QAK1 Kn−4ðKn−4Kn−4Þ Kn−4 K1QA0 [i.e., ðQ1; Q1Þ ðQ3; Q3Þ]. To see this, note that (C19), (C23), and (C25) imply that ˜ ˜ nþa−4 ¼ QAðK1K1Þ QA0 ðC24Þ ˜ ˜ ˜ ˜ ˜ Q2K −4 K1ðQ1Q1 þ Q3Q3ÞK1 K −4Q2 for A; A0 ∈ f1; 3g (these operators by themselves are not n n gauge invariant unless A ¼ A0). Finally, rearranging and 0 n ∈ 2Z þ 1; ¼ ˜ ˜ n=2−1 ðC30Þ squaring both sides of the F-term equations for the trivalent −ðQ1Q3Q3Q1Þ n ∈ 2Z: Uð2Þ nodes gives Moreover, combining ˜ j ˜ j − k ˜ j ðQ1ÞiðQ1Þ þðQ3ÞiðQ3Þ ¼ ðK1Þi ðK1Þk ˜ k ˜ j ˜ k ˜ j ˜ ˜ ˜ 2 ðQ1Þ ðQ1Þ ðQ3Þ ðQ3Þ þðQ3Þ ðQ3Þ ðQ1Þ ðQ1Þ ⇒ 2Q1Q3Q3Q1 ¼ TrððK1K1Þ Þ; i k i k k ˜ l m ˜ j ˜ j ˜ j ˜ k j ¼ðK1Þi ðK1Þk ðK1Þl ðK1Þm ðC31Þ ðQ2ÞiðQ2Þ þðQ4ÞiðQ4Þ ¼ðKn−4Þi ðKn−4Þk ⇒ 2 ˜ ˜ ˜ 2 Q2Q4Q4Q2 ¼ TrððKn−4Kn−4Þ Þ: with (C19), (C23), and (C25) gives

K K˜ 2 K˜ K 2 K˜ K 2 ˜ ˜ ˜ ˜ ˜ But Trðð I IÞ Þ¼Trðð I IÞ Þ and Trðð I IÞ Þ¼ Q2Kn−4 K1Q1Q3K1 Kn−4Q2Q1Q3 þð1 ↔ 3Þ ˜ 2 1 … − 5  TrððKIþ1KIþ1Þ Þ (the latter for I ¼ ; ;n ), implying ˜ ˜ ðn−1Þ=2 ðQ1Q3Q3Q1Þ n ∈ 2Z þ 1; that ¼ ðC32Þ 0 n ∈ 2Z: ˜ ˜ ˜ ˜ Q1Q3Q3Q1 ¼ Q2Q4Q4Q2 ðC25Þ Hence the Z2 symmetry that takes 1 ↔ 3 acts in exactly the in the chiral ring.39 same way as that which takes 2 ↔ 4. We use 1 ↔ 3 by Now consider the Z2 action on (4.40)–(4.45). The “Uð1Þ convention. Schouten identity” implies that Next, consider the chiral ring relation. First let n ∈ 2Z and set ˜ ˜ ˜ ˜ QAK1 Kn−4QA0 QA0 Kn−4 K1QA 0 n=2−1 ˜ ˜ ˜ ˜ Y ≡ Y þð−ZÞ : ðC33Þ ¼ QA0 Kn−4 K1QAQAK1 Kn−4QA0 ; ðC26Þ Defining the orientation-reversed operator 39This conclusion also holds for n ¼ 4, from squaring both ¯ 0 0 ˜ ˜ ˜ ˜ sides of Y ≡Y j1↔3 ¼2Q1K1 Kn−4Q2Q2Kn−4 K1Q1; ðC34Þ

˜ j ˜ j − ˜ j − ˜ j ðQ1ÞiðQ1Þ þðQ3ÞiðQ3Þ ¼ ðQ2ÞiðQ2Þ ðQ4ÞiðQ4Þ : we see that

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¯ 2−1 ¯ 2 Y0 þ Y0 ¼ −2ð−ZÞn= ; Y0Y0Z ¼ X : ðC35Þ same reasoning as in Sec. VB1, the enhanced S3 symmetry requires that40 Thus we get 1 c 3 c 2 2−1 ¯ Z Y X − 2 Φ2 M2 2 Φ2 − M2 ZY0 ¼ ZY0ð−2ð−ZÞn= − Y0Þ ð C; C; CÞ¼ C 8 þ ;iC 8 ; 2 2 2 ⇔ X þ ZY0 − 2Y0ð−ZÞn= ¼ 0 i C3 ΦM2 − M2 ðC44Þ ⇔ X 2 þ ZY2 ¼ Zn−1; ðC36Þ r as desired. Now let n ∈ 2Z 1 and set c2 2 þ at the quantum level, where we have defined Φ ≡ Φ − 2 c2 X ≡ X 0 − ð−ZÞðn−1Þ=2: ðC37Þ 1=3r (which satisfies hΦ i¼0). In the alternate basis, the Z2 symmetry therefore acts as Defining the orientation-reversed operator c2 2 2 1 c2 2 1 2 3 c2 ¯ 0 0 ðΦ ;M ;ΦM Þ ↦ − Φ þ 4M ; M þ Φ ; X ≡ X j1↔3 2 2 16 2 ˜ ˜ ˜ ˜ ˜ ¼ Q3K1 Kn−4Q2Q2Kn−4 K1Q1Q1Q3; ðC38Þ 3i 1 c − ΦM2 þ M2 þ Φ2 : ðC45Þ 2r 8 we see that

This should be contrasted with the Z2 symmetry acting on X 0 X¯ 0 2 −Z ðn−1Þ=2; X 0X¯ 0 ZY2: þ ¼ ð Þ ¼ ðC39Þ (4.33), which is more natural from the Coulomb branch point of view in that it simply flips the signs of monopoles. The Z2 Thus we get is only ambiguous when n ¼ 4 because it can be conjugated by elements of S3: otherwise, it is unique. X 02 ¼ X 0ð2ð−ZÞðn−1Þ=2 − X¯ 0Þ ⇔ X 02 − 2X 0ð−ZÞðn−1Þ=2 þ ZY2 ¼ 0 3. Affine En quivers ⇔ X 2 ZY2 Zn−1 þ ¼ ; ðC40Þ We now turn to the E-type quiver theories. In all cases, l the fundamental “meson” operators satisfy M I ¼ 0 where as desired. ðIÞ l To conclude, we remark that the basis (4.40)–(4.45) I is the length of leg I. To derive the chiral ring relation for 1 (which we refer to as the “alternate basis”) differs from the E6, we need only the Uð Þ Schouten identity: following earlier one that we used when n ¼ 4, namely (4.33), and [46], the trick is to write the generators containing squares 1 1 that the Z2 acts differently in the two cases. When n ¼ 4, of mesons as Uð Þ × Uð Þ bifundamentals. For E7;8,we we have in the alternate basis that instead employ the Uð2Þ Schouten identity: following [50], 2 ffiffiffi we define auxiliary operators with only Uð Þ indices p ˜ ˜ Z ¼ − 3Q1Q3Q3Q1; uncontracted. We present the derivations for E6;7;8 in ffiffiffi p ˜ ˜ ˜ ˜ decreasing amounts of detail. Y ¼ 3ið2Q2Q3Q3Q2 þ Q1Q3Q3Q1Þ; Deriving a Schouten identity for tensors of given rank 3=4 ˜ ˜ ˜ X ¼ 2 · 3 iQ1Q2Q2Q3Q3Q1; ðC41Þ involves antisymmetrizing over an appropriate number of indices and then contracting a subset of these indices. For where we have rescaled the generators so that they satisfy instance, the Schouten identity for two-component vectors 2 2 3 ½i j k 0 X þ ZY þ Z ¼ 0 and so that X is the same as in (4.33). follows from contracting any two indices in x y z ¼ . At the level of the chiral ring, this alternate basis maps to A Schouten identity for matrices [50] following from ½i1 i2 i3 0 Mk1 Nk2 Kk3 ¼ is ðZ ; Y ; X Þ C C C 1 3 40At the level of the quantized chiral ring, we know that − 2 Φ2 M2 2 Φ2 − M2 3ΦM2 ¼ C 8 þ ∞ ;iC 8 ∞ ;C ∞ 1 1 ˜ ˜ ˜ 2 2 Q1Q2Q2Q3Q3Q1 ↔ − iΦM þ M ; ðC43Þ ðC42Þ 128π3 r on the Coulomb branch of SUð2Þ SQCD with N ¼ 4, where as well as (5.61) and (5.62). These correspondences are consistent f X Y Z ≡ 31=4 4π −1 with (C41) if we define the Higgs branch variables , , at the we have set C ð Þ . A short calculation with the quantum level simply by subtracting their one-point functions. corresponding commutative shift operators shows that these These 1=r corrections ensure that the one-point functions of X 2 Z Y2 Z3 0 Z operators likewise satisfy C þ C C þ C ¼ .Bythe 2-odd operators are zero, in the alternate basis.

085008-36 NON-ABELIAN MIRROR SYMMETRY BEYOND THE CHIRAL … PHYS. REV. D 101, 085008 (2020) X TrðfM; NgKÞ¼ TrðMNÞTrðKÞ − TrðMÞTrðNÞTrðKÞ; ruling out chiral ring elements at Δ ¼ 2.AtΔ ¼ 3, 2 cyc trðMðIÞMðJÞÞ is nontrivial while ðC46Þ − 2 2 trðMðIÞMðJÞMðKÞÞ¼ trðMðJÞMðKÞ þ MðJÞMðKÞÞ where the indices range over f1; 2g. ¼ 0; ðC54Þ

a. E6 giving a single candidate for the chiral ring generator Z (up Our conventions are as in Sec. VI B 2. In this case, to normalization): the symmetry acts as Z2∶ðX; Y; ZÞ ↦ ð−X; Y; −ZÞ. The 2 2 2 Uð2Þ and Uð1Þ D-term relations imply that, for each trðMð1ÞMð2ÞÞ¼trðMð2ÞMð3ÞÞ¼trðMð3ÞMð1ÞÞ T½SUð3Þ leg, − 2 − 2 ¼ trðMð1ÞMð3ÞÞ¼ trðMð2ÞMð1ÞÞ C ˜ i ˜ i 0 ˜ i 0 2 Q Q þ qjq ¼ ;qiq ¼ : ðC47Þ − j C ¼ trðMð3ÞMð2ÞÞ: ðC55Þ For a given leg, one can verify using these relations that, for At Δ ¼ 4, Eq. (C51) implies that example, the Δ ¼ 2 CPOs 1 trðM2 M2 Þ¼trðM2 M2 Þ¼trðM2 M2 Þ; ðC56Þ ˜ j A ˜ i − ˜ j C ˜ i δA ð1Þ ð2Þ ð1Þ ð3Þ ð2Þ ð3Þ qiq Qj QB 3 qiq Qj QC B ðC48Þ giving a single candidate for the chiral ring generator Y. − A C are equivalent to M CM B in the chiral ring. Since the This is the only candidate because trace part vanishes in the chiral ring, we may simply A A ˜ i 2 − write M B ¼ Qi QB. trððMðIÞMðJ=KÞÞ Þ¼ trðMðIÞMðJÞMðIÞMðKÞÞ We first summarize some useful relations. Writing the − 2 2 2 2 A ¼ trðMðIÞMðJÞ þ MðIÞMðKÞÞ; ðC57Þ MðIÞ B as matrices, we have 0 2 2 2 2 Mð1Þ þ Mð2Þ þ Mð3Þ ¼ ðC49Þ trðMðIÞMðJÞMðKÞÞ¼trððMðJÞMðKÞÞ þ MðJÞMðKÞÞ: ðC58Þ from the SUð3Þ D-term relation. We see from the D-term At Δ ¼ 5, there are no nontrivial chiral ring elements. relations for each leg that Indeed, with two types of MðIÞ, there is only one pattern of p 0 contraction: trðMðIÞÞ¼ ðC50Þ trðM2 M M M Þ: ðC59Þ for integers p ≥ 1 and I ¼ 1, 2, 3. Moreover, we have ðIÞ ðJÞ ðIÞ ðJÞ 3 0 With three types, we have the possibilities MðIÞ ¼ : ðC51Þ 2 2 2 Indeed, trðMðIÞMðJÞMðIÞMðKÞÞ; trðMðIÞMðJÞMðKÞÞ; 2 A B C trðMðIÞMðJÞMðKÞMðJÞÞ; trðMðIÞMðJÞMðIÞMðJÞMðKÞÞ: MðIÞ BMðIÞ CMðIÞ D A ˜ i B ˜ j C ˜ k ðC60Þ ¼ðQðIÞÞi ðQðIÞÞBðQðIÞÞj ðQðIÞÞCðQðIÞÞk ðQðIÞÞD A ˜ i ˜ j ˜ k ¼ðQðIÞÞi ðqðIÞÞ ðqðIÞÞjðqðIÞÞ ðqðIÞÞkðQðIÞÞD But we have ¼ 0; ðC52Þ 2 − 2 trðMðIÞMðJÞMðIÞMðKÞÞ¼ trðMðIÞMðJÞMðIÞMðJÞÞ ˜ j 0 2 since ðqðIÞÞjðqðIÞÞ ¼ . ¼ trðMðIÞMðJÞMðKÞMðJÞÞ Let us now enumerate the nontrivial chiral ring elements 2 2 ¼ −trðM M M Þ¼0; ðC61Þ of small dimension (compare to [46]). The p ¼ 1 case of ðIÞ ðKÞ ðJÞ Δ 1 (C50) rules out chiral ring elements at ¼ . From (C49) trðM M M M M Þ and (C50), we also have ðIÞ ðJÞ ðIÞ ðJÞ ðKÞ − 2 − 2 ¼ trðMðIÞMðJÞMðIÞMðJÞÞ trðMðJÞMðIÞMðJÞMðIÞÞ trðM M Þ¼−trðM M Þ¼trðM M Þ ðIÞ ðJÞ ðIÞ ðKÞ ðJÞ ðKÞ ¼ −0 − 0 ¼ 0: ðC62Þ − ¼ trðMðJÞMðIÞÞ At Δ ¼ 6, the possible contractions involving two types of ⇒ 0 trðMðIÞMðJÞÞ¼ ; ðC53Þ MðIÞ are

085008-37 YALE FAN and YIFAN WANG PHYS. REV. D 101, 085008 (2020)

2 2 2 2 2 2 2 2 2 2 trðMðIÞMðJÞMðIÞMðJÞÞ; trððMðIÞMðJÞÞ Þ; trðMð1ÞMð2ÞMð3ÞÞtrðMð1ÞMð3ÞMð2ÞÞ 3 12 21 13 31 23 32 trððMðIÞMðJÞÞ Þ; ðC63Þ ¼ð Þð Þð Þð Þð Þð Þ ¼ trðM2 M2 ÞtrðM2 M2 ÞtrðM2 M2 Þ; ðC75Þ and those involving three types can all be written as linear ð1Þ ð2Þ ð1Þ ð3Þ ð2Þ ð3Þ combinations of those involving two types using (C49). Restricting our attention to two types, we derive that meaning [by virtue of (C56)]

2 2 2 2 UU¯ V3 trðMðIÞMðJÞMðIÞMðJÞÞþtrðMðIÞMðKÞMðIÞMðKÞÞ ¼ : ðC76Þ ¼ −trððM2 M Þ2ÞðC64Þ ðIÞ ðJ=KÞ Now let and ≡ ˜ i A ˜ j B ˜ k ðIJKÞ ðqðIÞÞ ðQðIÞÞi ðQðJÞÞAðQðJÞÞj ðQðKÞÞBðqðKÞÞk: 3 − 3 3 trððMðIÞMðJÞÞ Þ¼ trððMðIÞMðKÞÞ Þ¼trððMðJÞMðKÞÞ Þ ðC77Þ − 3 0 ¼ trððMðIÞMðJÞÞ Þ¼ ; ðC65Þ Recall (C64), which is equivalent to so it suffices to consider contractions of the form 2 2 2 2 2 2 2 2 trðM M MðIÞMðJÞÞ.But ðIÞ ðJÞ trðMðIÞMðJÞMðKÞÞþtrðMðIÞMðKÞMðJÞÞ 2 2 2 2 2 2 2 2 2 − − ¼ trððMðIÞMðJÞÞ Þ¼ trððMðIÞMðKÞÞ Þ trðMðIÞMðJÞMðIÞMðJÞÞ¼trðMðIÞMðJÞMðKÞÞ; ðC66Þ − 2 2 − 2 2 ¼ trððMðJÞMðIÞÞ Þ¼ trððMðJÞMðKÞÞ Þ so we are left with only two independent chiral ring −tr M2 M 2 −tr M2 M 2 ; C78 elements at Δ ¼ 6: ¼ ðð ðKÞ ðIÞÞ Þ¼ ðð ðKÞ ðJÞÞ Þ ð Þ

2 2 2 2 2 2 trðMð1ÞMð2ÞMð3ÞÞ; trðMð1ÞMð3ÞMð2ÞÞ: ðC67Þ and note that

One linear combination of them should give the square of 2 − 121 2 2 121 2 trðMð1ÞMð2ÞÞ¼ ð Þ; trððMð1ÞMð2ÞÞ Þ¼ð Þ : the generator at Δ ¼ 3, and the other should give the new generator X at Δ ¼ 6. ðC79Þ To derive the chiral ring relation (compare to [50],but without FI parameters), set So we get

2 2 2 2 2 W ≡ trðM M 2 Þ; ðC68Þ ð1Þ ð Þ trððMð1ÞMð2ÞÞ Þ¼trðMð1ÞMð2ÞÞ ; ðC80Þ V ≡ 2 2 trðMð1ÞMð2ÞÞ; ðC69Þ which implies that

U ≡ 2 2 2 ¯ 2 trðMð1ÞMð2ÞMð3ÞÞ; ðC70Þ U þ U ¼ −W : ðC81Þ

U¯ ≡ 2 2 2 trðMð1ÞMð3ÞMð2ÞÞ: ðC71Þ Combining (C76) and (C81) gives

Let U2 þ UW2 þ V3 ¼ 0; ðC82Þ

IJ ≡ q˜ i Q A Q˜ j q : ð Þ ð ðIÞÞ ð ðIÞÞi ð ðJÞÞAð ðJÞÞj ðC72Þ and making the change of variables Using ðM2 ÞA ¼ðq Þ ðq˜ ÞiðQ ÞAðQ˜ Þj and rearrang- pffiffiffi ðIÞ B ðIÞ j ðIÞ ðIÞ i ðIÞ B U X − Z2 V −Y W 2Z ing, we have ¼ i ; ¼ ; ¼ ðC83Þ

2 2 gives trðMðIÞMðJÞÞ¼ðIJÞðJIÞ; ðC73Þ

2 2 2 − X 2 þ Y3 þ Z4 ¼ 0; ðC84Þ trðMðIÞMðJÞMðKÞÞ¼ ðIKÞðKJÞðJIÞ: ðC74Þ

Hence we derive that where

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1 i i 2 2 2 M ≡ ðq˜ 12 1 Þ ðq12 1 Þ ; ðC94Þ Z ¼ pffiffiffi trðM M 2 Þ; Y ¼ −trðM M Þ; j ð Þ ð Þ j 2 ð1Þ ð Þ ð1Þ ð2Þ 2 2 2 2 N i ≡ ˜ i B ˜ A N M X − Z j ðq23ð1ÞÞAðq23ð1ÞÞj ðq34ð1ÞÞMðq34ð1ÞÞB ðMð3ÞÞ N; ¼ i½trðMð1ÞMð2ÞMð3ÞÞþ : ðC85Þ ðC95Þ In this presentation, the Z2 symmetry acts as ð1Þ ↔ ð2Þ. Ki ≡ − ˜ i B ˜ A N 3 M j ðq23ð1ÞÞAðq23ð1ÞÞj ðq34ð1ÞÞMðq34ð1ÞÞB ðMð2ÞÞ N: b. E7 ðC96Þ For each T½SUð4Þ leg, we denote the bifundamental hypers by Then we have by construction that

˜ i A ˜ i N ˜ A ððq12Þi; ðq12Þ Þ; ððq23Þi ; ðq23ÞAÞ; ððq34ÞA ; ðq34ÞNÞ; trðMN Þ¼Z; trðMKÞ¼−Y; ðC97Þ ðC86Þ and we compute using the D=F-term relations that where i ¼ 1,2;A ¼ 1, 2, 3; and N ¼ 1,2,3,4.The 2 2 subscripts indicate the ranks of the gauge nodes. trðN Þ¼−2Y; trðNKÞ¼−Z ; Accounting for orientation, the D=F-term equations are YtrðMN KN Þ¼X 2: ðC98Þ (in our conventions)

˜ i 0 ˜ j A ˜ j 0 We now write ðq12Þiðq12Þ ¼ ; ðq12Þiðq12Þ þðq23Þi ðq23ÞA ¼ ; A ˜ i − ˜ A N 0 MN KN M N KN ðq23Þi ðq23ÞB ðq34ÞNðq34ÞB ¼ ; ðC87Þ trð Þ¼trðf ; g Þ 1 and we have the mesons − MK N N 2 trð f ; gÞ ðC99Þ M ≡ M ˜ A M N ðq34ÞA ðq34ÞN; ðC88Þ and use the 2 × 2 Schouten identity (C46) as well as trðMÞ¼trðN Þ¼trðKÞ¼0 to get which are traceless in the chiral ring by the D=F-term relations. We can also write trðMN KN Þ¼trðMN ÞtrðKN Þ 2 M A ˜ i M ˜ B 1 ðM Þ N ¼ðq23Þi ðq23ÞBðq34ÞA ðq34ÞN; ðC89Þ − MK N 2 2 trð Þtrð Þ; ðC100Þ 3 M − ˜ j A ˜ i M ˜ B ðM Þ N ¼ ðq12Þiðq12Þ ðq23Þj ðq23ÞBðq34ÞA ðq34ÞN: which implies that ðC90Þ X 2 þ Y3 þ YZ3 ¼ 0; ðC101Þ Higher powers vanish, as do all traces of powers: p 0 ≥ 1 trðM Þ¼ for p . as desired. For the leg of length two, we have the D=F-term relation

M ˜ j 0 c. E8 qi qM ¼ ðC91Þ In this case, we define the traceless Uð2Þ matrices and the meson Ai ≡ M ˜ A N ˜ i B j ðMð1ÞÞ Nðq46ð2ÞÞMðq46ð2ÞÞB ðq24ð2ÞÞAðq24ð2ÞÞj ; MM ≡ qMq˜ i ; ðC92Þ N i N ðC102Þ whose trace and higher powers vanish. Bi ≡ 3 M ˜ A N ˜ i B For the quiver as a whole, we have j ðMð1ÞÞ Nðq46ð2ÞÞMðq46ð2ÞÞB ðq24ð2ÞÞAðq24ð2ÞÞj ; 0 ðC103Þ Mð1Þ þ Mð2Þ þ Mð3Þ ¼ ðC93Þ i 5 M A N i B C ≡ −ðM Þ ðq˜ 46 2 Þ ðq46 2 Þ ðq˜ 24 2 Þ ðq24 2 Þ ; by the SUð4Þ D-term relation. j ð1Þ N ð Þ M ð Þ B ð Þ A ð Þ j To proceed, define (as in [50]) the traceless Uð2Þ ðC104Þ matrices41 with the minus sign due to our conventions for Uð1Þ 41The minus sign in K is a consequence of our conventions nodes; then by construction, we have Y ¼trðACÞ and (C87). X ¼trðABCÞ. We also compute that

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2 1 trðABÞ¼0; trðBCÞ¼Z ; ABCB AB B C − A B B C trð Þ¼trð f ; gÞ 2 trð f ; g Þ A2 −2Z B2 −2Y trð Þ¼ ; trð Þ¼ : ðC105Þ 1 ¼ trðABÞtrðBCÞ − trðACÞtrðB2Þ; ðC108Þ Now consider the following expression, which we simplify 2 2 2 by writing in terms of anticommutators, using the × AB 0 Schouten identity (C46), and using that individual traces of which, in combination with trð Þ¼ , implies that A, B, C vanish: 1 ABCABC AC 2 B2 trð Þ¼2 trð Þ trð Þ trðABCABCÞ¼trðfA; BgCABCÞ − trðBfA; CgABCÞ 1 2 2 1 þ trðA ÞtrððBCÞ Þ: ðC109Þ BC A A BC 2 þ 2 trð f ; g ÞðC106Þ By the 1D Schouten identity, we have trððABCÞ2Þ¼ 2 2 2 2 4 ¼ trðABÞtrðABC2Þ − trðACÞtrðABCBÞ trðABCÞ ¼ X as well as trððBCÞ Þ¼trðBCÞ ¼ Z ,so 1 we arrive at þ trðA2ÞtrððBCÞ2Þ: ðC107Þ 2 X 2 þ Y3 þ Z5 ¼ 0; ðC110Þ We also have that as desired.

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