Coulomb Branch Quantization and Abelianized Monopole Bubbling
CCALT-TH 2018-056PUPT-2579
Coulomb Branch Quantization andAbelianized Monopole Bubbling
Mykola Dedushenko, Yale Fan, Silviu S. Pufu, and Ran Yacoby Walter Burke Institute for Theoretical Physics, California Institute of Technology,Pasadena, CA 91125, USA Department of Physics, Princeton University, Princeton, NJ 08544, USA Department of Particle Physics and Astrophysics, Weizmann Institute of Science,Rehovot 76100, Israel
Abstract
We develop an approach to the study of Coulomb branch operators in 3D N = 4 gaugetheories and the associated quantization structure of their Coulomb branches. This structureis encoded in a one-dimensional TQFT subsector of the full 3D theory, which we describeby combining several techniques and ideas. The answer takes the form of an associativeand noncommutative star product algebra on the Coulomb branch. For “good” and “ugly”theories (according to the Gaiotto-Witten classification), we also exhibit a trace map onthis algebra, which allows for the computation of correlation functions and, in particular,guarantees that the star product satisfies a truncation condition. This work extends previouswork on abelian theories to the non-abelian case by quantifying the monopole bubbling thatdescribes screening of GNO boundary conditions. In our approach, monopole bubbling isdetermined from the algebraic consistency of the OPE. This also yields a physical proof ofthe Bullimore-Dimofte-Gaiotto abelianization description of the Coulomb branch. a r X i v : . [ h e p - t h ] O c t ontents A Theories . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 444.4.2 B ∼ = C Theories . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 494.4.3 G Theories . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 544.5 General Case . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57 SU (2) with N f Fundamentals and N a Adjoints . . . . . . . . . . . . . 595.1.2 Pure SU (2) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 605.1.3 G with N f Fundamentals . . . . . . . . . . . . . . . . . . . . . . . . 615.2 Quantized Chiral Rings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 635.2.1 SU (2) with N f Fundamentals and N a Adjoints . . . . . . . . . . . . . 641.2.2 G with N f Fundamentals . . . . . . . . . . . . . . . . . . . . . . . . 655.3 Correlation Functions and Mirror Symmetry . . . . . . . . . . . . . . . . . . 665.3.1 Matrix Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 665.3.2 An N = 8 Example . . . . . . . . . . . . . . . . . . . . . . . . . . . . 675.3.3 U (2) with N a = N f = 1 . . . . . . . . . . . . . . . . . . . . . . . . . 685.3.4 U ( N c ) with N a = N f = 1 . . . . . . . . . . . . . . . . . . . . . . . . . 70 A Conventions 74B Twisted-Translated Operators 75C Matrix Nondegeneracy and Abelianized Bubbling 77D Bubbling Coefficients from 4D 79
D.1 The IOT Algorithm . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 79D.2 Bubbling Patterns . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81D.3 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82D.3.1 U (2) with N a = N f = 1 . . . . . . . . . . . . . . . . . . . . . . . . . 82D.3.2 U (2) with N a = 0 and N f ≥ U (3) with N a = 0 and N f ≥ E More (Quantized) Chiral Rings 91
E.1 SQED N versus U (1) with One Hyper of Charge N . . . . . . . . . . . . . . . 92E.2 Theories on D2-Branes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93E.2.1 U (2) with N a = N f = 1 . . . . . . . . . . . . . . . . . . . . . . . . . 93E.2.2 U (2) with N a = 1 and N f ≥ U ( N c ) with N a = 1 and N f ≥ SU (2) with N f ≥ SU (3) with N f ≥ F Correlation Functions 105
F.1 Mirror Symmetry Check for an N = 8 SCFT . . . . . . . . . . . . . . . . . . 1052.1.1 U (2) with N a = N f = 1 . . . . . . . . . . . . . . . . . . . . . . . . . 106F.1.2 U ( N c ) with N a = N f = 1 . . . . . . . . . . . . . . . . . . . . . . . . . 110F.2 An Abelian/Non-Abelian Mirror Symmetry Example . . . . . . . . . . . . . 1123 Introduction
Gauge theories in three dimensions contain special local defect operators called monopoleoperators, which are defined by requiring certain singular behavior of the gauge field closeto the insertion point [1]. These operators play important roles in the dynamics of thesetheories, and in particular in establishing various interesting properties such as infrared (IR)dualities between theories with different ultraviolet (UV) descriptions (see [2–15] for somerecent examples). Because these operators are not polynomial in the Lagrangian fields,they are notoriously difficult to study, and most studies so far have focused on determiningonly their quantum numbers [1, 16–29]. The goal of this paper is to present the first directcomputations of operator product expansion (OPE) coefficients and correlation functions ofmonopole operators in 3D non-abelian gauge theories.We focus on a class of 3D gauge theories with N = 4 supersymmetry (eight Poincar´esupercharges) constructed by coupling a vector multiplet with gauge group G to a matterhypermultiplet that transforms in some representation of G . For a matter representation ofsufficiently large dimension, these theories flow in the IR to interacting superconformal fieldtheories (SCFTs), whose correlation functions are generally intractable. However, as shownin [35,36], these theories also contain one-dimensional protected subsectors whose correlationfunctions are topological, and one may hope that computations in these protected subsectorsbecome tractable. This is indeed the case, as was shown in [37, 38] and as will be exploredin further detail here. While 3D N = 4 SCFTs have in general two inequivalent protectedtopological sectors, one associated with the Higgs branch and one with the Coulomb branch,it is the Coulomb branch sector that contains monopole operators and that will therefore bethe focus of our work. (The Higgs branch sector was studied in [37].) From the 3D SCFTpoint of view, the information contained in either of the two protected sectors is equivalentto that contained in the ( n ≤ Q C that is a linear combination of a Poincar´e and a conformal supercharge. As such, one may think that the protected sector mentioned above is emergent at the IRfixed point, and therefore inaccessible in the UV description. This is indeed true for SCFTsdefined on R . However, as was shown in [37, 38], if one defines the QFT on a round S These theories do not allow the presence of Chern-Simons terms. While it is possible to construct N = 4Chern-Simons-matter theories [30–34], we do not study them here. Similar statements hold about the Higgs branch protected sector if one replaces Q C with another super-charge Q H . R , then the protected sector becomes accessible in the UV because on S , thesquare of Q C does not contain special conformal generators. Indeed, Poincar´e and specialconformal generators are mixed together when mapping a CFT from R to S . As we willexplain, the square of Q C includes an isometry of S that fixes a great circle, and this is thecircle where the 1D topological quantum field theory (TQFT) lives.Previous work [37] used the idea of defining the QFT on S together with supersymmetriclocalization to solve the 1D Higgs branch theory by describing a method for computing itsstructure constants. The Coulomb branch case is much more complicated because it involvesmonopole operators. A complete solution of the 1D Coulomb branch theory was obtained forabelian gauge theories in [38]. Building on the machinery developed in [38], we describe howto compute all observables within the 1D Coulomb branch topological sector of an arbitrarynon-abelian 3D N = 4 gauge theory by constructing “shift operators” whose algebra is arepresentation of the OPE of the 1D TQFT operators (also known as twisted(-translated)Coulomb branch operators, for reasons that will become clear).The mathematical physics motivation for studying the 1D TQFT is that it provides a“quantization” of the ring of holomorphic functions defined on the Coulomb branch M C .This can be explained as follows. The 3D theories that we study have two distinguishedbranches of the moduli space of vacua: the Higgs branch and the Coulomb branch. These areeach parametrized, redundantly, by VEVs of gauge-invariant chiral operators whose chiralring relations determine the branches as generically singular complex algebraic varieties.While the Higgs branch chiral ring relations follow from the classical Lagrangian, those forthe Coulomb branch receive quantum corrections. The Coulomb branch is constrained byextended SUSY to be a generically singular hyperk¨ahler manifold of quaternionic dimensionequal to the rank of G , which, with respect to a choice of complex structure, can be viewed asa complex symplectic manifold. The half-BPS operators that acquire VEVs on the Coulombbranch, to be referred as Coulomb branch operators (CBOs), consist of monopole operators,their dressings by vector multiplet scalars, and operators built from the vector multipletscalars themselves. (Monopole operator VEVs encode those of additional scalar moduli, thedual photons.) All of the holomorphic functions on M C are given by VEVs of the subset ofCBOs that are chiral with respect to an N = 2 subalgebra. Under the OPE, these operatorsform a ring, which is well-known to be isomorphic to the ring C [ M C ] of holomorphic functionson M C . It was argued in [36] that because the operators in the 1D TQFT are in one-to-one correspondence with chiral ring CBOs, the 1D TQFT is a deformation quantization of C [ M C ]. Indeed, the 1D OPE induces an associative but noncommutative product on C [ M C ]5eferred to as a star product, which in the limit r → ∞ ( r being the radius of S ) reducesto the ordinary product of the corresponding holomorphic functions, and that at order 1 /r gives the Poisson bracket of the corresponding holomorphic functions.Note that both the quantization of [36] in the “ Q + S ” cohomology and our quantizationon a sphere are realizations of the older idea of obtaining a lower-dimensional theory bypassing to the equivariant cohomology of a supercharge, which originally appeared in thecontext of the Ω-deformation in 4D theories [39–41] and was also applied to 3D theoriesin [42–44].Our procedure for solving the 1D Coulomb branch theory uses a combination of thecutting and gluing axioms, supersymmetric localization, and a consistency requirement thatwe refer to as polynomiality. We first cut S into two hemispheres HS ± along an equatorial S = ∂HS ± orthogonal to the circle along which the 1D operators live (see Figure 1).Correlators are then represented by an inner product of wavefunctions generated by thepath integral on HS ± with insertions of twisted CBOs. In [38, 45], it was shown that itsuffices to consider such wavefunctions Ψ ± ( B BPS ) with operator insertions only at the tip of HS ± , and evaluated on a certain class of half-BPS boundary conditions B BPS . Insertions oftwisted CBOs anywhere on the great semicircles of HS ± can then be realized, up to irrelevant Q C -exact terms, as simple shift operators acting on this restricted class of wavefunctions. Itwas shown in [38] that these shift operators can be fully reconstructed from general principlesand knowledge of Ψ ± ( B BPS ). Moreover, their algebra provides a faithful representation of thestar product. Finally, one can determine expectation values (which is to say, more abstractly,that one can define an evaluation map on C [ M C ], known as the trace map in deformationquantization) by gluing Ψ + ( B BPS ) and Ψ − ( B BPS ) with an appropriate measure, as will bereviewed in Section 2.2.The fact that the star product algebra can be determined independently of evaluatingcorrelators is very useful. First, calculating correlators using the above procedure involvessolving matrix integrals, which can be complicated for gauge groups of high rank. On theother hand, the star product can be inferred from the comparatively simple calculation ofthe wavefunctions Ψ ± ( B BPS ). Second, the matrix models representing correlators divergefor “bad” theories in the sense of Gaiotto and Witten [30]. Nevertheless, as we will see,the HS wavefunctions and the star product extracted from them are well-defined evenin those cases. Therefore, we emphasize that our formalism works perfectly well even forbad theories, as far as the Coulomb branch and its deformation quantization are concerned. In bad theories [46–49], the IR superconformal R-symmetry is not visible in the UV, which invalidatesthe usual localization logic [50, 51]. S HS + S S N HS – HS S N S N φ φ = ±π φ = 0 φ = π /2 φ = –π /2 ∂ HS = S S Figure 1: A schematic depiction of S , obtained by gluing two hemispheres HS ± ∼ = B . The1D TQFT lives on the S parametrized by the angle ϕ (thick orange line). The 1D TQFTcircle intersects the equatorial S = ∂HS ± at two points identified with its North ( N ) andSouth ( S ) poles.However, correlation functions cannot be computed for such theories, and the star productsmight not satisfy the truncation property introduced in [36].On a more technical note, we provide a new way of analyzing “monopole bubbling” [52].Monopole bubbling is a phenomenon whereby the charge of a singular monopole is screenedto a lower one by small ’t Hooft-Polyakov monopoles. In our setup, this phenomenon mani-fests itself through the fact that our shift operators for a monopole of given charge containcontributions proportional to those of monopoles of smaller charge, with coefficients that werefer to as bubbling coefficients. While we do not know of a localization-based algorithm forobtaining these coefficients in general, we propose that the requirement that the OPE of anytwo 1D TQFT operators should be a polynomial in the 1D operators uniquely determinesthe bubbling coefficients, up to operator mixing ambiguities. In Section 4, we provide manyexamples of gauge theories of small rank where we explicitly carry out our algorithm todetermine the shift operators and bubbling coefficients. These results are also interesting forthe purpose of comparison with the literature on direct localization computations of bubblingin 4D, e.g., [53–55], which were subsequently refined by [56–58].The main mathematical content of this work is a construction of deformation quantiza-7ions of Coulomb branches of 3D N = 4 theories that also satisfy the truncation conditionof [36] in the case of good or ugly theories, as a consequence of the existence of the natu-ral trace map (the one-point function). By taking the commutative limit, we recover theordinary Coulomb branch of the theory in the form of the “abelianization map” proposedby [59]. Therefore, our approach also provides a way to prove the abelianization proposalof [59] starting from basic physical principles. Moreover, the knowledge of bubbling coeffi-cients that our approach provides vastly expands the domain of applicability of abelianizationto all Lagrangian 3D N = 4 theories of cotangent type. Finally, we expect that translat-ing our approach into a language that uses the mathematical definition of the Coulombbranch [60–63] might be of independent interest in the study of deformation quantization.The rest of this paper is organized as follows. Section 2 contains a review of the setupof our problem as well as a derivation of the shift operators without taking bubbling intoaccount. Section 3 discusses the dressing of monopole operators with vector multiplet scalarsand sets up the computation of the bubbling coefficients. In Section 4, we provide explicitexamples of shift operators and bubbling coefficients in theories of small rank. In Section 5,we discuss several applications of our formalism: to determining chiral rings, to chiral ringquantization, and to computing correlation functions of monopole operators and perform-ing checks of non-abelian mirror symmetry. Many technical details, further examples, andcomments on connections between our approach and existing ones can be found in the ap-pendices. We study 3D N = 4 gauge theories of cotangent type, which are the same theories whosequantized Higgs branches were the subject of [37]. Coulomb branches of abelian gauge theo-ries were scrutinized in [38] using different techniques, and here we extend those techniquesto the case of general gauge groups G ∼ = (cid:81) i G i , where each G i is either simple or abelian.As the construction of such theories was detailed in [37, 38], we only briefly describe it here.These theories are built from a 3D N = 4 vector multiplet V taking values in the Liealgebra g = Lie( G ) and from a 3D N = 4 hypermultiplet H valued in a (generally reducible) Such star products are also called “short” in ongoing mathematical work on their classification, as welearned from P. Etingof. R of G . H can be written in terms of half-hypermultiplets taking values in R ⊕ R , which is the meaning of the term “cotangent type.” More general representations ofhalf-hypermultiplets should also be possible to address using our techniques, but we do notconsider them in the present work.Our focus is on such theories supersymmetrically placed on the round S of radius r .There are several good reasons for choosing this background. One is that compactnessmakes the application of supersymmetric localization techniques more straightforward. Butthe most important reason, as should be clear to readers familiar with [37, 38], is that thesphere is a natural setting for deformation quantization of moduli spaces: the Coulomb andHiggs branches in such a background can be viewed as noncommutative, with 1 /r playingthe role of a quantization parameter. As with the 2D Ω-background in flat space [59], theresult is an effective compactification of spacetime to a line.Furthermore, quantized Coulomb and Higgs branch chiral rings are directly related tophysical correlation functions, and in particular encode the OPE data of the BPS operatorsin the IR superconformal theory, whenever it exists. This relation equips the noncommu-tative star product algebra of observables with a natural choice of “trace” operation — theone-point function of the QFT — as well as natural choices of basis corresponding to op-erators that are orthogonal with respect to the two-point function and have well-definedconformal dimensions at the SCFT point. These extra structures are a significant advantageof quantization using the spherical background, and they are responsible for much of theprogress that we make in this paper.The N = 4 supersymmetric background on S is based on the supersymmetry algebra s = su (2 | (cid:96) ⊕ su (2 | r , which also admits a central extension (cid:101) s = (cid:94) su (2 | (cid:96) ⊕ (cid:94) su (2 | r , withcentral charges corresponding to supersymmetric mass and FI deformations of the theory. Inthe flat-space limit r → ∞ , this algebra becomes the usual N = 4 super-Poincar´e algebra,implying that all results of this paper should have a good r → ∞ limit. All of the necessarydetails on the SUSY algebra s , and how the vector and hypermultiplets transform underit, can be found either in Section 2 of [37] or in Section 2.1 and Appendix A.2 of [38].Supersymmetric actions for V , H , and their deformations by mass and FI terms can also befound in those sections.The SUSY algebra (cid:101) s contains two interesting choices of supercharge, Q H and Q C . Theysatisfy the following relations:( Q H ) = 4 ir ( P τ + R C + ir (cid:98) ζ ) , ( Q C ) = 4 ir ( P τ + R H + ir (cid:98) m ) , (2.1)9here P τ denotes a U (1) isometry of S whose fixed-point locus is a great circle parametrizedby ϕ ∈ ( − π, π ): call it S ϕ ⊂ S . Here, R C and R H are the Cartan generators of the usual SU (2) C × SU (2) H R-symmetry of N = 4 SUSY, which in terms of the inner U (1) (cid:96) × U (1) r R-symmetry of s are identified as: R H = 12 ( R (cid:96) + R r ) , R C = 12 ( R (cid:96) − R r ) . (2.2)The notations (cid:98) ζ and (cid:98) m stand for the FI and mass deformations, i.e., central charges of (cid:101) s .The most important features of Q H and Q C are that if we consider their actions on thespace of local operators and compute their equivariant cohomologies, the answers have veryinteresting structures. The operators annihilated by Q H are the so-called twisted-translatedHiggs branch operators (HBOs), whose OPE encodes a quantization of the Higgs branch;such operators for the theories of interest were fully studied in [37]. Correspondingly, thecohomology of Q C contains twisted-translated Coulomb branch operators, whose structurehas so far been explored only for abelian theories [38]. Such operators must be inserted alongthe great circle S ϕ fixed by ( Q C ) , and their OPE encodes a quantization of the Coulombbranch. More details on twisted-translated operators are given in Appendix B. The purpose of this work is to study the cohomology of Q C and associated structures forgeneral non-abelian gauge theories of cotangent type. The operators annihilated by Q C areconstructed from monopole operators and a certain linear combination of scalars in the vectormultiplet. Recall that the vector multiplet contains an SU (2) C triplet of scalars Φ ˙ a ˙ b = Φ ˙ b ˙ a .Using the notation of [38], the following linear combination is annihilated by Q C :Φ( ϕ ) = Φ ˙ a ˙ b ( ϕ ) v ˙ a v ˙ b , v = 1 √ (cid:32) e iϕ/ e − iϕ/ (cid:33) , (2.3)whenever this operator is inserted along S ϕ ⊂ S . On the other hand, (bare) BPS monopoleoperators are defined as defects imposing special boundary conditions on the gauge field andon Φ ˙ a ˙ b . They were first defined for 3D N = 4 theories in [64], while the non-supersymmetricversion was introduced earlier in [1]. The twisted-translated monopole operators that westudy — which are essentially those of [64] undergoing an additional SU (2) C rotation as we Concretely, τ is the fiber coordinate in an S fibration over the disk D , i.e., S τ → S → D . Afterconformally mapping to flat space, P τ would be a rotation that fixes the image of S ϕ , which is a line. S ϕ — were described in detail in [38]. Their definition is rather intricate, so itwill be helpful to review it, with an eye toward the additional complications that arise innon-abelian gauge theories.First recall that in a U (1) gauge theory, a (bare) non-supersymmetric monopole operatoris a local defect operator that sources magnetic flux at a point in 3D spacetime. In a non-abelian gauge theory, the quantized charge b is promoted to a matrix, or more precisely,a cocharacter of G (referred to as the GNO charge [65]). A cocharacter is an element ofHom( U (1) , G ) /G ∼ = Hom( U (1) , T ) / W . Passing from the element of Hom( U (1) , T ) / W to themap of algebras R → t , we see that cocharacters can also be identified with Weyl orbits inthe coweight lattice Λ ∨ w ⊂ t of G , i.e., in the weight lattice of the Langlands dual group L G .Since every Weyl orbit contains exactly one dominant weight (lying in the fundamental Weylchamber), it is conventional to label monopole charges by dominant weights of L G [59]. Let b ∈ t be such a dominant weight of L G . Then a bare monopole operator is defined by a sumover W b , the Weyl orbit of b , of path integrals with singular boundary conditions definedby elements of W b . Specifically, the insertion of a twisted-translated monopole operator ata point ϕ ∈ S ϕ is defined by the following singular boundary conditions for F µν and Φ ˙ a ˙ b : ∗ F ∼ b y µ d y µ | y | , Φ ˙1˙1 = − (Φ ˙2˙2 ) † ∼ − b | y | e − iϕ , Φ ˙1˙2 ∼ , (2.4)where it is understood that one must compute not a single path integral, but rather a sumof path integrals over field configurations satisfying (2.4) with b ranging over the full Weylorbit of a given dominant weight. Here, “ ∼ ” means “equal up to regular terms” and y µ are Riemann normal coordinates centered at the monopole insertion point. The origin of(2.4) is that twisted-translated monopoles are chiral with respect to the N = 2 subalgebradefined by the polarization vector in (2.3) at any given ϕ . This requires that the real scalarin the N = 2 vector multiplet diverge as b | y | near the monopole [64] and results in nontrivialprofiles for the N = 4 vector multiplet scalars near the insertion point.We denote such twisted-translated monopole operators by M b ( ϕ ), or simply M b . The Q C -cohomology, in addition to M b ( ϕ ) and gauge-invariant polynomials in Φ( ϕ ), containsmonopole operators dressed by polynomials P (Φ), or dressed monopoles, which we denoteby [ P (Φ) M b ]. Note that because monopoles are really given by sums over Weyl orbits, the This is a more refined notion than the topological charge labeled by π ( G ) (when it exists): such chargescorrespond to global symmetries of the Coulomb branch whose conserved currents in the UV are the abelianfield strengths and which may be enhanced in the IR. P (Φ) M b ] is not merely a product of P (Φ) and M b , but rather: (cid:2) P (Φ) M b (cid:3) = 1 |W b | (cid:88) w ∈W P (Φ w ) × (insertion of a charge-(w · b ) monopole singularity) , (2.5)where Φ w means that as we sum over the Weyl orbit, we act on the P (Φ) insertion as well.Because M b breaks the gauge group at the insertion point down to the subgroup G b ⊂ G that preserves b , P (Φ) must be invariant under the G b action. Also, to avoid overcounting,we must divide by the order of the stabilizer of b in W . At this point, we pause to discuss a few subtleties inherent to the above definition.They are important for precise understanding, and ultimately for performing computationscorrectly, but may be skipped on first reading.First, let us ask ourselves what exactly Φ w is. After all, the Weyl group acts canonicallyon the Cartan subalgebra t , but it does not have a natural action on the full Lie algebra g where Φ is valued. Indeed, from the identification W = N ( T ) /Z ( T ), a Weyl group elementis interpreted as an element of the normalizer N ( T ) ⊂ G of the maximal torus T , up to anelement of the centralizer Z ( T ) ⊂ G . On t , the centralizer Z ( T ) acts trivially, but it certainlyacts nontrivially on the full algebra, making the action of W on g ambiguous. However, theaction of W on a G b -invariant polynomial P (Φ) is nevertheless unambiguous. To understandthis, note that the magnetic charge b ∈ t is obviously preserved by Z ( T ), so the group G b includes Z ( T ) as a subgroup. In particular, it means that P (Φ) is Z ( T )-invariant, and hencethe action of w ∈ N ( T ) /Z ( T ) on P (Φ) is unambiguous — this is the action that appears in(2.5).This is not the only subtlety to take care of. It is also worth noting that the action of theWeyl group on the dressing factor is different from its action on b . The fundamental reasonfor this is that Φ represents a “non-defect” observable (given by an insertion of fundamentalfields in the path integral), while b characterizes the defect: namely, it describes the strengthof the monopole singularity that plays the role of a boundary condition for the fundamentalfields. In Appendix B.1 of [38], it was explained how symmetries act on observables of thesetwo types (it was emphasized for global symmetries there, but the argument is exactly the We will see that after localization, Φ takes values in the Cartan subalgebra t C , in which case the G b -invariance of P (Φ) boils down to W b -invariance, where W b is the Weyl group of G b . But then, because (2.5)includes summation over the Weyl orbit of W , there is no real need to require W b -invariance of P (Φ), as itwill be automatically averaged over the subgroup W b ⊂ W upon this summation. Therefore, later on, whenwe write formulas in terms of Φ ∈ t C , we can insert arbitrary polynomials P (Φ) in [ P (Φ) M b ]. The factor |W b | − only appears when we sum over elements of W , while equations written directly interms of a sum over the Weyl orbit do not need such a factor. b and Φ are elements of g , and if the gauge symmetry acts on Φ by U (Φ (cid:55)→ U Φ U − ),then it acts on b by U − ( b (cid:55)→ U − bU ). The monopole singularity is labeled by b ∈ t , andthe Weyl group has a natural action on it coming from the identification W = N ( T ) /Z ( T ).The dressing factor is a G b -invariant polynomial P (Φ) that is also acted on by W , as wejust explained. If we act on b by w ∈ W (that is, b (cid:55)→ w · b ), then we should act on P (Φ)by w − : P (Φ w ) = w − · P (Φ). After restricting Φ to take values in t C (which happensafter localization and gauge fixing), it is convenient to note that W acts on t by orthogonalmatrices, and hence the left action by w − is the same as the right action by w. This providesa convenient way to perform actual calculations: the Weyl group acts from the left on b ∈ t and from the right on Φ ∈ t once we represent them as a column vector ( b i ) and a row vector(Φ i ), respectively, in some orthonormal basis of t .Another convention that we choose to follow is that by b in [ P (Φ) M b ], we mean someweight of L G within the given Weyl orbit, though not necessarily the dominant one. When-ever we label monopoles by dominant weights, we explicitly say so. The polynomial P (Φ)appearing inside the square brackets is always the one attached to the charge- b singularity(whether or not b is dominant), while the Weyl-transformed singularities w · b are multipliedby Weyl-transformed polynomials, as in (2.5).In this paper, we develop methods for computing correlation functions of dressed mono-pole operators of the form (2.5). There are several techniques that we combine in order toachieve our results: cutting and gluing techniques [45, 66], localization, and algebraic consis-tency of the resulting OPE. In what follows, we describe each of them and what role theyplay in the derivation. The cutting and gluing property [45, 66] holds in any local quantum field theory, and it hasalready been applied to the abelian version of our problem in [38]. This is also one of thekey ingredients in the non-abelian generalization here. We can motivate its application asfollows. As explained in [38], only a very restricted class of configurations of twisted CBOson S is amenable to a direct localization computation. A less direct approach is to endowthe path integral on S with extra structure by dividing it into path integrals on two openhalves. These path integrals individually prepare states in the Hilbert space of the theoryon S . The advantage of this procedure is that it allows for operator insertions within S to be implemented by acting on these boundary states with operators on their associated13ilbert spaces.Specifically, the round S is glued from two hemispheres, HS and HS − , and we needto know how this procedure is represented at the level of quantum field theories living onthem. Recall that gluing corresponds to taking (cid:104) Ψ − | Ψ + (cid:105) , where | Ψ + (cid:105) ∈ H S and (cid:104) Ψ − | ∈ H ∨ S are states generated at the boundaries of the two hemispheres. Furthermore, in Lagrangiantheories with no more than two derivatives, this operation is represented by an integral overthe space of polarized boundary conditions [45, 66] for a choice of polarization on P ( S ), thephase space associated with S = ∂HS . For a special choice of supersymmetry-preservingpolarization, this integral can be localized to the finite-dimensional subspace of half-BPSboundary conditions of a certain type, which results in a simple gluing formula [38, 45]: (cid:104) Ψ − | Ψ + (cid:105) = 1 |W| (cid:88) B ∈ Λ ∨ w (cid:90) t d r σ µ ( σ, B ) (cid:104) Ψ − | σ, B (cid:105)(cid:104) σ, B | Ψ + (cid:105) . (2.6)Here, the integration goes over the Cartan t ⊂ g , Λ ∨ w ⊂ t is the coweight lattice, µ ( σ, B ) isthe gluing measure given by the one-loop determinant on S , µ ( σ, B ) = Z c . m . one-loop ( σ, B ) Z v . m . one-loop ( σ, B ) J ( σ, B ) ,Z v . m . one-loop ( σ, B ) J ( σ, B ) = (cid:89) α ∈ ∆ + ( − α · B (cid:34)(cid:16) α · σr (cid:17) + (cid:18) α · B r (cid:19) (cid:35) ,Z c . m . one-loop ( σ, B ) = (cid:89) w ∈R ( − | w · B |− w · B Γ (cid:16) + iw · σ + | w · B | (cid:17) Γ (cid:16) − iw · σ + | w · B | (cid:17) , (2.7)and (cid:104) Ψ − | σ, B (cid:105) , (cid:104) σ, B | Ψ + (cid:105) are the hemisphere partition functions with prescribed boundaryconditions determined by σ ∈ t and B ∈ Λ ∨ w ⊂ t . We think of (cid:104) Ψ − | σ, B (cid:105) , (cid:104) σ, B | Ψ + (cid:105) aswavefunctions on t × Λ ∨ w : they are elements of an appropriate functional space, such as L ( t × Λ ∨ w ), a precise identification of which is not important. The boundary conditionsparametrized by σ, B are half-BPS boundary conditions on bulk fields preserving 2D (2 , S , namely an su (2 |
1) subalgebra of s containing Q C . In terms of the on-shellcomponents of the multiplets H = ( q a , (cid:101) q a , ψ α ˙ a , (cid:101) ψ α ˙ a ) and V = ( A µ , Φ ˙ a ˙ b , λ αa ˙ a ), as well as thevariables q ± ≡ q ± iq and (cid:101) q ± ≡ (cid:101) q ± i (cid:101) q , these boundary conditions are given by: q + (cid:12)(cid:12) = (cid:101) q − (cid:12)(cid:12) = (cid:18) D ⊥ q − + Φ ˙1˙1 − Φ ˙2˙2 q − (cid:19) (cid:12)(cid:12)(cid:12)(cid:12) = (cid:18) D ⊥ (cid:101) q + + Φ ˙1˙1 − Φ ˙2˙2 (cid:101) q + (cid:19) (cid:12)(cid:12)(cid:12)(cid:12) = 0 , In (2.7), J is a standard Vandermonde determinant and we have omitted an overall power of r from thelogarithmic running of the 2D FI parameters. ψ ˙1 − σ ψ ˙2 ) (cid:12)(cid:12) = ( (cid:101) ψ ˙1 + σ (cid:101) ψ ˙2 ) (cid:12)(cid:12) = 0 ,A (cid:107) (cid:12)(cid:12) = ± B θ − τ, Φ ˙1˙1 + Φ ˙2˙2 i (cid:12)(cid:12)(cid:12)(cid:12) = B r , Φ ˙1˙2 (cid:12)(cid:12) = σr , ( λ − iλ + σ ( λ − iλ )) (cid:12)(cid:12) = ( λ + iλ − σ ( λ + iλ )) (cid:12)(cid:12) = 0 . (2.8)Note that such boundary conditions specify the magnetic flux B ∈ Λ ∨ w through the boundary S . Thus we could alternatively think of B as the corresponding cocharacter, i.e., the fullWeyl orbit W B , in which case the sum in (2.6) would run over the set of cocharacters(allowed magnetic charges) Γ m = Λ ∨ w / W . In such a case, the boundary conditions abovewould have to be understood in the same way as the definition of the monopole operator:one would have to evaluate the hemisphere partition function for every element of the Weylorbit W B ⊂ Λ ∨ w and sum the results. We find it more convenient to treat B as an elementof Λ ∨ w , in which case we simply sum over B ∈ Λ ∨ w in the gluing formula and there is no needfor a separate sum over Weyl reflections.The gluing formula (2.6) holds as long as the states Ψ ± are supersymmetric, i.e., anni-hilated by Q C [38, 45]. This is true for the state generated at the boundary of the emptyhemisphere, and remains valid if we start inserting Q C -closed observables inside. Such in-sertions will modify the hemisphere partition function, and can be represented as certainoperators acting on the empty hemisphere partition function. In this paper, we are onlyconcerned with local observables, described above as gauge-invariant polynomials in Φ( ϕ )and dressed monopole operators. Such local observables form an OPE algebra A C , whichwill turn out to be a quantization of the Coulomb branch. Therefore, all we need to do isfind how Φ( ϕ ) and dressed monopoles act on the hemisphere partition function. An important step is to compute the hemisphere partition function with insertions of local Q C -closed observables. As explained in [38], because correlation functions do not dependon the positions of the insertions, we can move them all to the tip of the hemisphere andreplace them by an equivalent composite operator located there. In the abelian case, theGNO charge of the twisted CBO at the tip is equal to the sum of the GNO charges ofall insertions, while in the non-abelian case, it is determined by taking tensor products ofrepresentations of L G . It suffices to consider a bare monopole at the tip, as it is trivial toinclude insertions of (gauge-invariant monomials in) the scalar Φ( ϕ ) anywhere along S ϕ .The hemisphere partition function can be computed using supersymmetric localization.15n fact, half of the computation that we need has already been performed in [38], whoseconventions we closely follow. Recall that the round sphere is parametrized by 0 ≤ θ ≤ π/ ≤ ϕ ≤ π , and − π ≤ τ ≤ π , and S ϕ is located at θ = π/
2, where the τ -circle shrinks.The sphere is cut into two hemispheres along the S located at ϕ = 0 and ϕ = ± π . Itis also sometimes convenient to use spherical coordinates ( η, ψ, τ ), which are related to the“fibration” coordinates ( θ, ϕ, τ ) by(cos θ, sin θ cos ϕ, sin θ sin ϕ ) = (sin η sin ψ, − sin η cos ψ, cos η ) , τ = τ, (2.9)where η, ψ ∈ [0 , π ]. In terms of such coordinates, the cut is located at η = π/ b at ( θ, ϕ ) = ( π/ , π/ η = 0. We also impose the conditions (2.8) at the boundary of the hemisphere. The BPSequations that follow from Q C can be conveniently written in terms of R ≡ sin θ, Φ r ≡ Re( Re iϕ Φ ˙1˙1 ) , Φ i ≡ Im( Re iϕ Φ ˙1˙1 ) , (2.10)and they take the form: [Φ ˙1˙2 , Φ i ] = [Φ ˙1˙2 , Φ r ] = 0 ,D = Re( D ) = 0 , Im( D ) = − r Φ ˙1˙2 , D µ Φ ˙1˙2 = D τ Φ i = 0 , D τ Φ r = ir [Φ r , Φ i ] ,R D R Φ i + D ϕ Φ r = 0 , R (1 − R ) D R Φ r − D ϕ Φ i = 0 ,F µν = √ g(cid:15) µνρ κ ρ D ρ Φ r , where κ ρ = (cid:0) , , θ (cid:1) ρ . (2.11)In the last equation, the index ρ is summed over, and indices are raised and lowered usingthe metric in [38]. These equations have a straightforward (non-bubbling) solution that onlyexists if the boundary (flux) coweight B matches one of the coweights in the Weyl orbitcorresponding to the monopole charge. In other words, if the monopole’s dominant coweightis b , then the straightforward solution exists iff B = wb for some w ∈ W . This solution hasvanishing fields in the hypermultiplet as well as vanishing fermions in the vector multiplet,while the bosons in the vector multiplet take the form: D = 0 , Φ ˙1˙2 = irD = irD = σr ∈ t , ˙1˙1 = Φ ˙2˙2 = iB r sin η = iB r (cid:112) cos θ + sin θ cos ϕ ,A ± = − B ψ ∓
1) d τ = B (cid:32) sin θ cos ϕ (cid:112) − sin θ sin ϕ ± (cid:33) d τ, (2.12)where A − is defined everywhere on the hemisphere except the interval π/ ≤ ϕ ≤ π at θ = π/
2; similarly, A + is defined everywhere except on 0 ≤ ϕ ≤ π/ , θ = π/
2. Here, D ab are the auxiliary fields in the vector multiplet.The above straightforward solution is a direct generalization of the abelian one from [38].Therefore, (2.12) can be called the “abelian solution.” Indeed, since B ∈ t , we see thatonly components valued in the maximal torus of the gauge group have VEVs. It is known,however, that in the non-abelian case, the equations (2.11) might have additional loci ofsolutions. They correspond to screening effects that go by the name of “monopole bubbling”[52]. In particular, one notices that close to the special circle θ = π/
2, the last equationin (2.11) becomes the Bogomolny equation, and the bubbling loci in the moduli spaces ofBogomolny equations have been an active area of study. We will discuss bubbling in moredetail soon, but for now let us focus on (2.12).The abelian solution (2.12) has the feature that all fields with nontrivial VEVs on thelocalization locus are vector multiplet fields valued in t . In other words, the VEVs look asthough the gauge group were actually T , the maximal torus of G . This is essentially howthe “abelianization” of [59] makes an appearance in our approach.Note that since the Yang-Mills action is Q C -exact [37, 38], one can use it for localizationand to compute the relevant determinants in the weak-coupling limit g YM →
0. The action(with boundary terms included such that the sum of the bulk and boundary pieces is Q C -exact [38]) vanishes on the localization locus, and it remains only to compute the one-loopdeterminants in the background of (2.12).The action for hypermultiplets in the background of (2.12) becomes quadratic, so thereis no need to localize them separately: one can directly integrate them out. Furthermore,this action is simply that of free hypermultiplets coupled to the T -valued gauge background.Each representation R of G gives a set of abelian charges under T given by the weights w ∈ R . Therefore, we can borrow the corresponding one-loop determinant from the previouswork [38], where the abelian case was studied: Z hyper1-loop = (cid:89) w ∈R r | w · B | Γ (cid:16) | w · B | − iw · σ (cid:17) √ π . (2.13)17he only novelty in the computation of non-abelian one-loop determinants is that vectormultiplets contribute: we need to include the contributions of W-bosons and gaugini. Anindirect derivation of these determinants will be presented in Section 2.6. The answer isgiven by Z vec1-loop = (cid:89) α ∈ ∆ r | α · B | √ π Γ (cid:16) | α · B | − iα · σ (cid:17) . (2.14)Therefore, the contribution from the abelian solution to the hemisphere partition functionwith a monopole labeled by a coweight b ∈ Λ ∨ w ⊂ t inserted at the tip is given by Z ( b ; σ, B ) = (cid:88) b (cid:48) ∈W b δ B,b (cid:48) (cid:81) w ∈R √ πr | w · b (cid:48)| Γ (cid:16) | w · b (cid:48) | − iw · σ (cid:17)(cid:81) α ∈ ∆ 1 √ πr | α · b (cid:48)| Γ (cid:16) | α · b (cid:48) | − iα · σ (cid:17) ≡ (cid:88) b (cid:48) ∈W b Z ( b (cid:48) ; σ, B ) , (2.15)where the δ B,b (cid:48) enforces flux conservation: the flux sourced by the monopole equals theflux exiting through S . We have introduced the notation Z for an “incomplete” partitionfunction that does not include a sum over the Weyl orbit of b . Such a quantity does notrepresent a physical monopole operator, but it will prove to be convenient in the followingsections. In general, Z as given above is not the full answer, because there are contributions fromadditional loci in the localization computation. We now discuss them. Close to the monopole insertions, our BPS equations behave as Bogomolny equations on R with a monopole singularity at the origin. Such equations are known to have “screeningsolutions” in addition to the simple abelian “Dirac monopole” solution described in theprevious subsection. The main property of such solutions is that while at the origin of R they have a monopole singularity characterized by b ∈ Λ ∨ w , at infinity they behave as Diracmonopoles of different charges v ∈ Λ ∨ w . It is also known that such solutions only exist when v is a weight in the representation determined by the highest weight b such that | v | < | b | (in which case v is said to be “associated to” b , sometimes written simply as v < b ). Let M ( b, v ) denote the moduli space of such screening solutions. For given b and v , let ρ be thelength scale over which the screening takes place. It is one of the moduli for solutions of the We do not keep careful track of the overall sign of the hemisphere wavefunction, as it cancels in thegluing formula. ρ → M ( b, v ).In this limit, the solution approaches a Dirac monopole of charge v everywhere on R exceptfor an infinitesimal neighborhood of the origin where the non-abelian screening takes place.This solution can be thought of as a singular (Dirac) monopole screened by coincident andinfinitesimally small smooth (’t Hooft-Polyakov) monopoles; the latter have GNO chargeslabeled by coroots. It is natural to suppose that such solutions also exist on S : while atfinite ρ they are expected to receive 1 /r corrections compared to the flat-space case, in the ρ → R .Notice that our general BPS equations require solutions to be abelian away from theinsertion point. However, within the radius ρ , the screening solutions to the Bogomolnyequations are essentially non-abelian. Therefore, smooth screening solutions correspondingto generic points of M ( b, v ) cannot give new solutions to the BPS equations. However,boundary components of M ( b, v ) where ρ → v everywhere except at the insertionpoint, it is convenient to factor out Z ( v ; σ, B ) computed in the previous subsection, and tosay that the full contribution from the “ b → v ” bubbling locus is given roughly by Z mono ( b, v ; σ, B ) Z ( v ; σ, B ) , (2.16)where Z mono characterizes the effect of monopole bubbling. We call it the bubbling factor.In fact, such a simple presentation is not quite correct, and we need to be more precisehere. Recall that the monopole insertion is not just defined by a single singular boundarycondition (2.4): rather, one sums over the Weyl orbit of such singular boundary conditions.Therefore, the above expression is expected to have sums over such orbits for both b and v .A more general expectation, which turns out to be correct, is that the contribution of thebubbling locus takes the form (cid:88) b (cid:48) ∈W bv (cid:48) ∈W v Z abmono ( b (cid:48) , v (cid:48) ; σ, B ) Z ( v (cid:48) ; σ, B ) (2.17)where, as before, b and v are understood to be (dominant) coweights representing magnetic19harges, and we sum over their Weyl orbits. The new quantity appearing in this equation, Z abmono ( b (cid:48) , v (cid:48) ; σ, B ) , (2.18)is called the “abelianized bubbling factor.” It depends on coweights b (cid:48) , v (cid:48) ∈ Λ ∨ w ⊂ t ratherthan on cocharacters, while physical answers in the full non-abelian theory depend on cochar-acters and thus always include sums over Weyl orbits. The abelianized bubbling factorsintroduced here prove to be of great importance for the formalism of this paper. Later on,we will provide more rigorous evidence for their relevance based purely on group theoryarguments that are independent of the heuristic path integral–inspired explanation of thissection. Note also that the abelianized bubbling factors are expected to behave under Weylreflections in the following way: Z abmono (w · b, w · v ; w · σ, w · B ) = Z abmono ( b, v ; σ, B ) , w ∈ W . (2.19)Now we can write the complete answer for the hemisphere partition function: (cid:104) σ, B | Ψ b (cid:105) = Z ( b ; σ, B ) + (cid:88) | v | < | b | (cid:88) b (cid:48) ∈W bv (cid:48) ∈W v Z abmono ( b (cid:48) , v (cid:48) ; σ, B ) Z ( v (cid:48) ; σ, B ) , (2.20)where Ψ b represents the state generated at the boundary of the hemisphere with a physicalmonopole of charge b inserted at the tip. Here, the first sum goes over dominant coweights v satisfying | v | < | b | , while the second sum goes over the corresponding Weyl orbits.The localization approach to the computation of Z mono ( b, v ; σ, B ) is quite technical, hav-ing been a subject of several works in the past [53–55], and more recently [56, 58]. In thecurrent paper, we do not attempt a direct computation of Z mono ( b, v ; σ, B ) or Z abmono ( b, v ; σ, B ).Instead, we describe a roundabout way to find them from the algebraic consistency of ourformalism. We find that the Z abmono ( b, v ; σ, B ) are always given by certain rational functions,but we do not need to assume anything about their form. In this section, we derive how insertions of local Q C -closed observables are representedby operators acting on the hemisphere wavefunction (up to the so-far unknown bubblingfactors). The easiest ones are polynomials in Φ( ϕ ). Just like in [38], we can think of them as Here, all variables take values in t , so the action of W is unambiguous. ϕ, θ ) = (0 , π/
2) or through the Southpole ( ϕ, θ ) = ( π, π/ ϕ ) either for 0 < ϕ < π/ π/ < ϕ < π . We find that for the North pole,Φ( ϕ = 0) = 12 (Φ ˙1˙1 + 2Φ ˙1˙2 + Φ ˙2˙2 ) = 1 r (cid:18) σ + i B (cid:19) . (2.21)Similarly, for the South pole,Φ( ϕ = π ) = 12 ( − Φ ˙1˙1 + 2Φ ˙1˙2 − Φ ˙2˙2 ) = 1 r (cid:18) σ − i B (cid:19) . (2.22)This operator simply measures the values of σ and B away from the monopole insertion, andbubbling is accounted for trivially. In particular, on the unbubbled locus, B evaluates to b ,while for the bubbling loci it evaluates to the corresponding B = v . Thus we conclude thatΦ( ϕ ) is represented by the following North and South pole operators:Φ N = 1 r (cid:18) σ + i B (cid:19) ∈ t C , Φ S = 1 r (cid:18) σ − i B (cid:19) ∈ t C , (2.23)where B should be thought of as measuring B ∈ Λ ∨ w at the boundary S , i.e., it multipliesthe wavefunction Ψ( σ, B ) by B , and thus is simply a diagonal multiplication operator.It is also not too hard to obtain the generalizations of the abelian shift operators from [38]that represent insertions of non-abelian monopoles. From the structure of the partitionfunctions above, it is clear that they take the following form: M b = (cid:88) b (cid:48) ∈W b M b (cid:48) + (cid:88) | v | < | b | (cid:88) b (cid:48) ∈W bv (cid:48) ∈W v Z abmono ( b (cid:48) , v (cid:48) ; σ, B ) M v (cid:48) . (2.24)Here, M b is an abelianized (non-Weyl-averaged) shift operator which represents the insertionof a bare monopole singularity characterized by the coweight (not cocharacter!) b , and whosedefinition ignores bubbling phenomena. The inclusion of abelianized bubbling coefficients Z abmono takes care of screening effects, and summing over Weyl orbits corresponds to passingto cocharacters, i.e., true physical magnetic charges.The expression (2.24) is evident from the structure of the hemisphere partition functionwith a monopole inserted, as described in the previous subsections. Indeed, away from themonopole insertion, its effect must be represented by a sum over bubbling sectors, and withineach bubbling sector, the contribution must take the form of a sum over the Weyl reflections21f the basic contribution. The expression (2.24), in fact, represents nothing else but theabelianization map proposed in [59]: the full non-abelian operator M b is written in terms ofthe abelianized monopoles M b acting on Ψ( σ, B ), wavefunctions on t × Λ ∨ w .It remains to determine the expressions for M b acting on wavefunctions Ψ( σ, B ). Justlike in [38], there are separate sets of operators that implement insertions through the Northand South poles. These generate isomorphic algebras, and they are uniquely determined bythe following set of consistency conditions:1) They should shift the magnetic flux at which Ψ( σ, B ) is supported by b ∈ Λ ∨ w .2) They should commute with Φ at the opposite pole, i.e., [ M bN , Φ S ] = [ M bS , Φ N ] = 0.3) They should commute with another monopole at the opposite pole, i.e., [ M bN , M b (cid:48) S ] = 0.4) When acting on the vacuum (empty hemisphere) wavefunction, the result should agreewith (2.15).This set of conditions determines the North shift operator to be M bN = (cid:81) w ∈R (cid:104) ( − ( w · b )+ r | w · b | / (cid:0) + irw · Φ N (cid:1) ( w · b ) + (cid:105)(cid:81) α ∈ ∆ (cid:104) ( − ( α · b )+ r | α · b | / ( irα · Φ N ) ( α · b ) + (cid:105) e − b · ( i ∂ σ + ∂ B ) , (2.25)where ( a ) + ≡ a if a ≥ a ) + ≡ x ) n stands for the Pochhammer symbolΓ( x + n ) / Γ( x ), and x · y represents the canonical pairing t ∗ × t → R . The analogous Southpole operator is M bS = (cid:81) w ∈R (cid:104) ( − ( − w · b )+ r | w · b | / (cid:0) + irw · Φ S (cid:1) ( − w · b ) + (cid:105)(cid:81) α ∈ ∆ (cid:104) ( − ( − α · b )+ r | α · b | / ( irα · Φ S ) ( − α · b ) + (cid:105) e b · ( i ∂ σ − ∂ B ) . (2.26)By counting powers of r − in the general expression (2.25), we find that the dimension of acharge- b monopole is ∆ b = 12 (cid:32)(cid:88) w ∈R | w · b | − (cid:88) α ∈ ∆ | α · b | (cid:33) . (2.27)This dimension formula will come in handy later.The shift operators satisfy an important multiplication property, which later on will allowus to generate monopoles of arbitrary charge starting from a few low-charge monopoles: M b N (cid:63) M b N = P b ,b (Φ) M b + b N for dominant b and b , (2.28)22nd similarly for the South pole operators, where P b ,b (Φ) is some polynomial in Φ. Weuse (cid:63) to denote products as operators (in particular, shift operators act on the Φ-dependentprefactors in M N,S ), emphasizing that they form an associative noncommutative algebra. Infact, (2.28) holds slightly more generally than for dominant weights: if ∆ + is some choice ofpositive roots (determined by a hyperplane in t ∗ ), then (2.28) holds whenever the condition( b · α )( b · α ) ≥ α ∈ ∆ + . The property (2.28) ensures that in the productof two physical bare monopoles, the highest-charge monopole appears without denominators.If in addition, b and b satisfy the property that ( b · w )( b · w ) ≥ w ∈ R , then a stronger equality holds: M b N (cid:63) M b N = M b + b N . (2.29)Finally, for general b and b , we have: M b N (cid:63) M b N = (cid:81) w ∈R ( − iw · Φ N ) ( w · b ) + +( w · b ) + − ( w · ( b + b )) + (cid:81) w ∈ ∆ ( − iα · Φ N ) ( α · b ) + +( α · b ) + − ( α · ( b + b )) + M b + b N + O (1 /r ) . (2.30)These are precisely the abelian chiral ring relations of [59].In addition to defining M N,S , we also need to properly define dressed monopoles. Thisis an important and somewhat subtle consideration, especially due to the interplay withbubbling. We discuss it in Section 3. Before doing so, let us first fill a gap in the abovediscussion by comparing our results to supersymmetric indices in four dimensions, whichprovide a way to derive the vector multiplet one-loop determinant.
Our setup has a natural uplift to a supersymmetric index of 4D N = 2 theories on S × S .The operators constructed from Φ( ϕ ) lift to supersymmetric Wilson loops wrapping the S , while monopole operators correspond to supersymmetric ’t Hooft loops on S . Certainquestions relevant to this 4D setup have been studied in the literature in great detail, andwe can use the answers to determine the unbubbled partition functions. By shrinking the S factor, the 4D results allow us to infer the unbubbled one-loop determinants mentionedin the previous subsections. Doing this for the bubbling contributions is more subtle and isdiscussed in Appendix D, where we find agreement with our method of deriving bubblingterms in cases where the 4D results are known. For simplicity, let us first set the radius r of S to 1, and let us denote the circumference of S by β . To restore r , we simply send23 → β/r in all formulas.Since the one-loop determinant for hypermultiplets is already known, we concern our-selves only with determining the vector multiplet contribution. This can be done in a theorywith any conveniently chosen matter content. We can always choose the matter contentin such a way that both the 4D N = 2 and the 3D N = 4 theories are conformal. Thecorresponding 4D index is known as the Schur index. The Schur index is defined as I ( p ) = Tr H S ( − F p E − R (2.31)where the trace is taken over the Hilbert space of the 4D theory on S and R is the Cartangenerator of the su (2) R-symmetry, normalized so that the allowed charges are quantized inhalf-integer units. In the path integral description, I ( p ) evaluated when p = e − β is given byan S × S partition function, with S of circumference β and with an R-symmetry twistby e βR as we go once around the S . This S × S partition function is invariant underall 4D superconformal generators that commute with E − R , or in other words, that have E = R . One can easily list these generators and check that they form an su (2 | ⊕ su (2 | su (2 | ⊕ su (2 | g YM and can be computed at weak coupling.One can additionally insert an ’t Hooft loop of GNO charge b (taken to be a dominantcoweight) wrapping S at one pole of S and the oppositely charged loop at the oppositepole of S . The answer for this modified index in a general 4D N = 2 gauge theory, up to asign and ignoring the bubbling effect, can be found in [55]: I b ( p ) = 1 |W b | (cid:90) rank( G ) (cid:89) i =1 dλ i π (cid:34) (cid:89) α ∈ ∆ (cid:16) − e iα · λ p | α · b | (cid:17)(cid:35) P.E.[ I v ( e iλ i , p )] P.E.[ I h ( e iλ i , p )] , (2.32)where ∆ is the set of all roots, P.E. is the plethystic exponential defined as P.E.[ f ( x )] ≡ exp (cid:104)(cid:80) ∞ n =1 f ( x n ) n (cid:105) , I v is the contribution from the N = 2 vector, and I h is the contributionfrom the N = 2 hyper in the representation R : I v ( e iλ i , p ) = − (cid:88) α ∈ adj p | α · b | − p e iα · λ , We set η a = 1 and x = √ p in Equation (3.44) of [55]. h ( e iλ i , p ) = (cid:88) w ∈R p + | w · b | − p ( e iw · λ + e − iw · λ ) . (2.33)Using the identity exp (cid:104) − (cid:80) ∞ n =1 a n n (1 − q n ) (cid:105) = ( a ; q ) where ( a ; q ) ≡ (cid:81) ∞ n =0 (1 − aq n ) is the q -Pochhammer symbol, we can rewrite I b ( p ) as I b ( p ) = ( p ; p ) G ) |W b | (cid:90) π − π rank( G ) (cid:89) i =1 dλ i π (cid:81) α ∈ ∆ (cid:104)(cid:16) − e iα · λ p | α · b | (cid:17) ( e iα · λ p | α · b | ; p ) (cid:105)(cid:81) w ∈R ( e iw · λ p + | w · b | ; p )( e − iw · λ p + | w · b | ; p ) . (2.34)We would like to determine the 3D hemisphere partition function. One way to do so is touse the results of [45] to first extract the 4D half-index from (2.34), and then dimensionallyreduce it. Alternatively (and this is how we proceed), we first reduce the index (2.34) to 3Dto find the S partition function, and then use the gluing formula from Section 2.2 to recoverthe hemisphere partition function as the square root of the absolute value of the integrand.One can then fix signs by consistency with gluing.To reduce (2.34) down to three dimensions, we take the β → p = e − β , we scale the integration variable accordingly: λ = βσ ∈ t . (2.35)The angular variable λ (parametrizing the maximal torus T ⊂ G ) then “opens up” into anaffine variable σ ∈ t . To take the limit, one needs the following identities (see [38]):1( p x ; p ) = e π β β x − √ π Γ( x )(1 + O ( β )) , ( p ; p ) = (cid:114) πβ e − π β (1 + O ( β )) , (2.36)which give I b ≈ e − π r β (cid:18) dim G − (cid:80) NfI =1 dim R I (cid:19) |W b | (cid:90) ∞−∞ rank( G ) (cid:89) i =1 dσ i (cid:89) α ∈ ∆ + (cid:18) ( α · σ ) + | α · b | (cid:19) × (cid:81) w ∈R (cid:12)(cid:12)(cid:12)(cid:12) β | w · b | √ πr | w · b | Γ (cid:16) | w · b | − iw · σ (cid:17)(cid:12)(cid:12)(cid:12)(cid:12) (cid:81) α ∈ ∆ (cid:12)(cid:12)(cid:12)(cid:12) β | α · b | √ πr | α · b | Γ (cid:16) | α · b | − iα · σ (cid:17)(cid:12)(cid:12)(cid:12)(cid:12) (2.37)as β →
0. In (2.37), we restored the radius r of S by dimensional analysis.The exponential prefactor in (2.37) is precisely the Cardy behavior of [67]. In the in-25egrand, we recognize the one-loop contribution of the hypermultiplet to the S partitionfunction, Z hyper1-loop, S ( σ ) = (cid:89) w ∈R (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) √ πr | w · b | Γ (cid:18) | w · b | − iw · σ (cid:19)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) , (2.38)multiplied by β | w · b | . The remaining factor in the integrand must be proportional to theone-loop contribution of the vector multiplet to the S partition function. Assuming thatthe one-loop vector multiplet contribution comes multiplied by β − | α · b | (by analogy with thehypermultiplet factor), we conclude that it is equal to Z vec1-loop, S ( σ ) = (cid:81) α ∈ ∆ + (cid:16) ( α · σ ) + | α · b | (cid:17)(cid:81) α ∈ ∆ (cid:12)(cid:12)(cid:12)(cid:12) √ πr | α · b | Γ (cid:16) | α · b | − iα · σ (cid:17)(cid:12)(cid:12)(cid:12)(cid:12) . (2.39)The S partition function is then given by the expression Z b = 1 |W b | (cid:90) ∞−∞ rank( G ) (cid:89) i =1 dσ i Z vec1-loop, S ( σ ) Z hyper1-loop, S ( σ ) . (2.40)Note that using this method, the overall normalization of Z b is ambiguous, but we proposethat the correct expression is given by (2.40). This expression passes the check that when b = 0, it reduces to the S partition function derived in [29], namely Z = Z = 1 |W| (cid:90) ∞−∞ rank( G ) (cid:89) i =1 dσ i (cid:81) α ∈ ∆ + ( πα · σ ) (cid:81) w ∈R πw · σ ) . (2.41)What remains to be done is to use (2.38) and (2.39) to verify the hemisphere one-loopdeterminants given in (2.13) and (2.14). To do so, we use the gluing formula (2.6) as wellas the explicit expression for the gluing measure in (2.7). It immediately follows that thehypermultiplet and vector multiplet contribute (2.13) and (2.14) to the hemisphere partitionfunction, respectively. The hypermultiplet contribution (2.13) was previously determinedby an explicit computation of the one-loop determinant on the hemisphere [38]. It wouldbe interesting to carry out the analogous computation for the non-abelian vector multiplet,which we have bypassed by means of the above argument. Note that the hemisphere and the Q C -invariant background (2.12) with a monopole at the tip η = 0 Dressing and Abelianized Bubbling
We have now derived the structure of bare monopoles, up to the bubbling factors. In thissection, we extend this construction to more general dressed monopoles. Recall that themagnetic charge b breaks the gauge group at the insertion point down to G b , the centralizerof b . As is well-known in the literature [59] and as reviewed in Section 2.1.2, one may dressthe monopole operator by some G b -invariant polynomial P (Φ) in the variable Φ( ϕ ).If P (Φ) is invariant under the full gauge group G , then it is a valid Q C -closed observableon its own. This makes the definition of the corresponding dressed monopole essentially triv-ial: we simply “collide” two separate observables P (Φ) and M b , which within our formalismmeans multiplying them as operators acting on the hemisphere wavefunction. Using the starproduct notation for such multiplication, we thus have: (cid:2) P (Φ) M b (cid:3) := P (Φ) (cid:63) M b . (3.1)When the polynomial P (Φ) is only invariant under a subgroup G b rather than the full gaugegroup G , we must proceed differently because P (Φ) only makes sense as part of [ P (Φ) M b ],not as a separate observable. Had bubbling not been an issue, the solution would again be preserve an N = 2 subalgebra su (2 | Q ± and Q ± in [38]. The suggestiveform of (2.14) then leads one to wonder whether it can be explained by a Higgsing argument familiar inthe study of theories with four supercharges (see, e.g., [68]). Namely, with respect to the aforementioned N = 2 subalgebra, the hypermultiplet decomposes into N = 2 chiral multiplets of R-charge 1 / N = 2 vector multiplet and an adjoint chiral multiplet of R-charge 1.Suppose that one could deform the action in such a way as to accommodate arbitrary R-charge q for thechiral multiplets transforming in representations R , R of G (as in, e.g., [50, 51]) while preserving the N = 2superpotential coupling that descends from the Φ ˙ a ˙ b Φ ˙ a ˙ b term in the N = 4 Lagrangian. Then one mightexpect the corresponding chiral multiplet one-loop determinant on the hemisphere to take the form of aproduct over weights w ∈ R of Z q chiral ( w · σ ) ∼ Γ (cid:18) − q + | w · B | − iw · σ (cid:19) , (2.42)so that the numerator of Z ( b (cid:48) ; σ, B ) in (2.15) comes from Z / ( w · σ ) and the denominator from Z vector ( α · σ ) = 1 Z ( − α · σ ) , (2.43)by reflection symmetry of the roots α . Here, (2.43) follows from the Higgs mechanism and Z vector ( α · σ )denotes the contribution to the vector multiplet one-loop determinant from a mode in the α -direction.It would be interesting to make this intuition precise. However, due to our choice of N = 2 superalgebraon S , ours is not the standard N = 2 Coulomb branch localization, where chiral multiplet fields vanishon the localization locus. Indeed, (2.12) implies a nontrivial background for the scalar in the adjoint chiralmultiplet (i.e., σ/r ) as well as for the scalar in the N = 2 vector multiplet (i.e., − B/r sin η ). In particular, σ ∈ t is not the standard Coulomb branch parameter: it labels the scalar zero mode of the adjoint chiraland not of the vector. P (Φ) M b ] = |W b | − (cid:80) w ∈W P (Φ w ) M w · b . In general,however, the presence of bubbling makes such a simple definition incomplete.For the remainder of this section, we focus on the case of a simple gauge group G . Thegeneralization to the situation where G is a simple factor of a larger gauge group is straight-forward: different simple factors couple to each other only through the matter multiplets,and representation-theoretic issues can be addressed for each simple factor separately. Thefinal conclusion of this section, Theorem 1, holds for general G with the understanding thatfor non-simple gauge groups, bubbling terms for a monopole operator magnetically chargedunder one simple factor might also depend on scalars Φ from other simple factors. From thepoint of view of a given simple factor G , Φ’s valued in other simple factors G (cid:48) act as twistedmasses for G (cid:48) . A general dressed monopole operator takes the form (cid:2) P (Φ) M b (cid:3) = 1 |W b | (cid:88) w ∈W P (Φ w ) M w · b + · · · (3.2)where the ellipses stand for bubbling contributions. It is intuitively clear that such observ-ables, constructed for all possible choices of P (Φ), cannot all be independent: there shouldexist a minimal set of dressed monopoles, a basis in some sense, from which all other dressedmonopoles follow. In this subsection, we make this intuition precise by rigorously provingthat for a given magnetic charge b , there exists a set of primitive dressed monopoles thataccomplish this. Definition 1.
Dressed monopoles [ P (Φ) M b ] , [ P (Φ) M b ] , . . . , [ P p (Φ) M b ] are called primi-tive (of magnetic charge b ) if they form a basis for the (free) module of dressed charge- b monopoles over the ring of G -invariant polynomials. This means that by taking linear com-binations p (cid:88) i =1 Q i (Φ) (cid:63) (cid:2) P i (Φ) M b (cid:3) (3.3)where the Q i (Φ) are G -invariant polynomials, we obtain dressed monopoles with all possibleleading terms of the form (3.2), and furthermore, that p is the minimum number that makesthis possible. We will always assume P (Φ) = 1, so that the first primitive monopole is thebare monopole itself. 28 xample. In SU (2) gauge theory, the Weyl group is Z , which simply flips b → − b and Φ →− Φ ∈ t C . A dressed monopole of charge b takes the form P (Φ) M b + P ( − Φ) M − b + bubbling.In this case, there are only two primitive dressed monopoles for each b : M b = M b + M − b + bubbling , (cid:2) Φ M b (cid:3) = Φ( M b − M − b ) + bubbling . (3.4) By writing P (Φ) = P (Φ) + P ( − Φ)2 + P (Φ) − P ( − Φ)2Φ Φ , (3.5) it becomes obvious that any other dressed monopole can be defined as: (cid:2) P (Φ) M b (cid:3) := P (Φ) + P ( − Φ)2 (cid:63) M b + P (Φ) − P ( − Φ)2Φ (cid:63) (cid:2) Φ M b (cid:3) . (3.6)To describe primitive monopoles for general gauge groups, it is enough to focus on theleading term of (3.2), as we do in this subsection. Bubbling contributions will be analyzedfrom this point of view in the next subsection.The leading term in (3.2) is constructed to be invariant under the Weyl group action.Therefore, we can classify such leading terms by identifying invariants of the Weyl group inthe corresponding (reducible) representations of W . Alternatively, we could achieve this byfocusing on the dressing factors and classifying polynomials P (Φ) invariant under G b . SinceΦ ∈ t C after localization and gauge fixing, it is enough to impose invariance under W b (theWeyl group of G b ). Thus dressed monopoles can be classified by W b -invariant polynomialsin Φ. Nevertheless, we find it more convenient to study the invariants of W directly. Proposition 1.
Let G be a simple Lie group, W its Weyl group, and b a dominant coweight(a magnetic charge). Then there exists a set of primitive monopoles (of magnetic charge b ) [ P (Φ) M b ] , [ P (Φ) M b ] , . . . , [ P p (Φ) M b ] , where p = |W b | is the size of the Weyl orbit of b . The remainder of this subsection is devoted to proving this proposition using classicalfacts from invariant theory. Less mathematically inclined readers are free to skip it.
Proof.
Consider ρ b , a representation of W spanned as a C -linear space by the Weyl orbit ofthe coweight b . We write it in terms of shift operators M w · b , w ∈ W , as ρ b := Span C { M w · b | w ∈ W} . (3.7) W b -invariant polynomials on t can be uniquely extended to G b -invariant polynomials on g . (cid:80) b (cid:48) ∈W b M b (cid:48) , which is the simplest invariant corresponding to the bare monopole operator.The Cartan subalgebra t itself is also a W -module: W acts on it in an irreducible r -dimensional representation, where r = rank( G ). We will denote such a representation simplyby t . The variable Φ = (cid:80) ra =1 Φ a h a clearly takes values in this representation.However, recall from the discussion after (2.5) that when w ∈ W acts on a dressedmonopole operator by transforming the weight according to b (cid:55)→ w · b , physics tells us thatthe dressing factor should be acted on by w − : Φ (cid:55)→ Φ w = w − · Φ. Since w is representedby an orthogonal matrix on t , this is the same as acting with w T from the left or with wfrom the right. This is how one acts in a dual representation. Thus in a dressed monopoleoperator, we think of Φ as transforming in the dual representation t ∗ . The dressing factor P (Φ) entering (3.2), being a polynomial in Φ, transforms in S t ∗ , the symmetric algebra of t ∗ ,or equivalently, the algebra C [ t ] of polynomial functions on t . This implies that any dressedmonopole operator is determined by an invariant vector inside the following W -module: R b := S t ∗ ⊗ ρ b ∼ = C [ t ] ⊗ ρ b . (3.8)Thus the leading terms in dressed monopoles of charge b are classified by invariants R W b .Questions of this sort have been studied extensively in the ancient subject of invarianttheory (see, for example, [69]). To begin, let us understand the structure of S t ∗ ∼ = C [ t ] as arepresentation of W , in particular its isotypic decomposition. It is well-known that the ringof invariants for a reflection group (such as the Weyl group) has the structure of anotherpolynomial ring (see [69, Part V], in particular [69, Section 18-1]): C [ t ] W ∼ = C [ f , . . . , f r ] , where r = dim t = rank( G ) . (3.9)Here, the f i are invariant homogeneous polynomials whose degrees d i satisfy (cid:81) ri =1 d i = |W| .Another well-known object is the ring of covariants [69, Part VII], which is defined as follows.Consider an ideal in C [ t ] generated by non-constant invariant polynomials: I = (cid:0) C [ t ] W deg > (cid:1) = ( f , . . . , f r ) . (3.10)The ring of covariants is defined as C [ t ] W = C [ t ] /I. (3.11)30t is again well-known [69, Section 24-1] that C [ t ] W ∼ = C [ W ] as a W -module, where C [ W ] is theregular representation. Since W maps I to itself, Maschke’s theorem implies that one can finda W -invariant subspace M W ⊂ C [ t ] such that C [ t ] ∼ = I ⊕ M W , and this M W ∼ = C [ t ] W ∼ = C [ W ].Finally, [69, Section 18-3] implies that C [ t ] is a free C [ t ] W -module generated by the basis of M W : C [ t ] ∼ = C [ t ] W ⊗ C M W . To summarize, the structure of S t ∗ ∼ = C [ t ] as a W -module is C [ t ] ∼ = C [ t ] W ⊗ C C [ W ] , (3.12)where C [ W ] is realized on polynomials from M W ⊂ C [ t ]. Equation (3.12) also encodes theisotypic decomposition since every m -dimensional irrep of W appears in C [ W ] precisely m times.With this knowledge, our representation of interest becomes R b ∼ = C [ t ] W ⊗ C C [ W ] ⊗ C ρ b . (3.13)Now the problem of identifying R W b simplifies substantially: R W b ∼ = C [ t ] W ⊗ C (cid:0) C [ W ] ⊗ C ρ b (cid:1) W . (3.14)Namely, we must find an invariant subspace in C [ W ] ⊗ C ρ b , which is a product of two finite-dimensional representations of W . Any other element of R W b is obtained by multiplicationwith invariant polynomials from C [ t ] W = C [ f , . . . , f r ].What we have proven so far is the following: R W b is a free C [ t ] W -module, and any C -basisof (cid:0) C [ W ] ⊗ C ρ b (cid:1) W gives a C [ t ] W -basis of R W b , i.e., a set of primitive dressed monopoles ofmagnetic charge b .To compute (cid:0) C [ W ] ⊗ C ρ b (cid:1) W , we simply decompose each of the two representations intoirreducible components and pair up dual representations. Indeed, by Schur’s lemma, onlytensor products like V ⊗ V ∗ , where V is some irrep and V ∗ is its dual, can contain invariantsubspaces. We can also easily find the dimension of (cid:0) C [ W ] ⊗ C ρ b (cid:1) W . Since C [ W ] containseach irreducible representation ρ i of W exactly dim( ρ i ) times,( C [ W ] ⊗ C ρ i ) W ∼ = C dim( ρ i ) . (3.15)Decomposing ρ b into irreducible components as ρ b ∼ = ⊕ i ∈ I ( b ) ρ i , this obviously gives: (cid:0) C [ W ] ⊗ C ρ b (cid:1) W ∼ = ⊕ i ∈ I ( b ) ( C [ W ] ⊗ C ρ i ) W ∼ = C dim( ρ b ) . (3.16)31ence there are exactly dim( ρ b ) = |W b | primitive dressed monopoles of charge b .We have now classified the leading terms in dressed monopoles. Any such leading termmust be extended by the appropriate bubbling contributions to give a complete physicaldressed monopole, and primitive monopoles are no exception: (cid:2) P i (Φ) M b (cid:3) = 1 |W b | (cid:88) w ∈W P i (Φ w ) M w · b + bubbling , i = 1 , . . . , |W b | . (3.17)We now turn to the study of these bubbling contributions. Suppose we have found a set of polynomials P , . . . , P |W b | such that the dressed monopoles[ P i (Φ) M b ] form the primitive set for a given magnetic charge b , in the sense explained inthe previous subsection. That is, |W b | − (cid:80) w ∈W P i (Φ w ) M w · b for i = 1 , . . . , |W b | form a basisfor R W b (the space of dressed charge- b monopoles) over C [ t ] W (the algebra of gauge-invariantpolynomials in Φ). In this subsection, we will show that there exists a special bubbled andabelianized monopole shift operator (cid:102) M b = M b + · · · such that (cid:2) P i (Φ) M b (cid:3) = 1 |W b | (cid:88) w ∈W P i (Φ w ) (cid:102) M w · b . (3.18)The left-hand side has the following structure: for each i , (cid:2) P i (Φ) M b (cid:3) = 1 |W b | (cid:88) w ∈W P i (Φ w ) M w · b + 1 |W b | (cid:88) | v | < | b | (cid:88) w ∈W V b → vi (Φ w ) M w · v . (3.19)Here, the first sum corresponds to the sector with no screening effects, and the remainingterms describe monopole bubbling, with V b → vi given by some rational functions of Φ ∈ t C that encode the bubbling data (because the V i are not yet known, the |W b | − in the secondterm is optional). By equating the right-hand sides of (3.18) and (3.19), we obtain a systemof linear equations for (cid:102) M w · b , w ∈ W : (cid:88) w ∈W P i (Φ w ) (cid:102) M w · b = (cid:88) w ∈W P i (Φ w ) M w · b + (cid:88) | v | < | b | (cid:88) w ∈W V b → vi (Φ w ) M w · v . (3.20)Its solution provides the definition of (cid:102) M b , but first we need to show that such a solutionexists, i.e., that the matrix of coefficients P i (Φ w ) is nondegenerate. This essentially follows32rom the primitivity of [ P i (Φ) M b ], and the proof is given in Appendix C.The solution to (3.20) takes the form (cid:102) M b = M b + (cid:88) | v | < | b | Z ab b → v (Φ) M v , (3.21)where the first term has an obvious origin: it is the shift operator that describes the sectorwithout bubbling. Here b is a fixed coweight, whereas the sum in the second term is takenover all coweights whose length is less than that of | b | .The functions Z ab b → v (Φ) are some rational functions of Φ ∈ t C that encode the bubblingphenomena. They do not have any invariance property under the action of W . We mayextend them to non-dominant b by postulating the following transformation property: Z abw · b → w · v (Φ) = Z ab b → v (Φ w ) , (3.22)consistent with (2.19). These functions are what we refer to as abelianized bubbling factors .Recall that we previously argued for their existence using heuristic path integral reasoning.We have now rigorously proven their existence by relying solely on group theory.As also mentioned in Appendix C, the expression for (cid:102) M w · b can be obtained from theexpression for (cid:102) M b by a Weyl reflection: (cid:102) M w · b = M w · b + (cid:88) | v | < | b | Z ab b → v (Φ w ) M w · v = M w · b + (cid:88) | v | < | b | Z abw · b → w · v (Φ) M w · v . (3.23)Having established the existence of abelianized and bubbled monopoles (cid:102) M b , one can veryeasily construct arbitrary dressed monopoles. In fact, this proves the following theorem. Theorem 1.
A shift operator describing an arbitrary physical dressed monopole of magneticcharge b can be constructed as |W b | (cid:88) w ∈W F (Φ w ) (cid:102) M w · b , (3.24) where F (Φ) is a polynomial in Φ ∈ t C . Such an expression will automatically produce, in the leading term, F averaged over W b ,the stabilizer of b in W , as well as generating the appropriate subleading terms describingbubbling.The abelianized bubbling coefficients prove to be very useful below.33 .3 Relation to the Abelianization Map Before putting the notion of abelianized bubbling to work, let us comment on how it fitsinto the context of previous studies.One approach to understanding the geometry of the Coulomb branch of a 3D N = 4theory was proposed in [59]. Let M abel C ⊂ M C denote the generic points on the Coulombbranch where the gauge group G is broken to its maximal torus T . Using the fact thatthe chiral ring is independent of gauge couplings, it was argued in [59] that the abelianizedchiral ring C [ M abel C ] can be determined by integrating out the massive W-bosons at one loop,ignoring nonperturbative effects. This ring is generated by (VEVs of) dressed chiral mono-pole operators of T , the complex scalars Φ a ( a = 1 , . . . , r ), and the inverses of the W-bosoncomplex masses α (Φ) for all roots α ∈ ∆. At points on M C where a non-abelian subgroupof G is restored, some α (Φ) → C [ M abel C ] becomes ill-defined. Nonperturbativeeffects cannot be ignored at such points.These nonperturbative effects are encoded in the so-called abelianization map, whichexpresses a chiral monopole operator M in the non-abelian theory as a linear combinationof monopole operators M in the low-energy abelian gauge theory, with coefficients beingmeromorphic functions of the complex abelian vector multiplet scalars. In our notation, thismap takes precisely the form (2.24) or, before Weyl-averaging, (3.21). The abelianizationmap realizes C [ M C ] as the subring of C [ M abel C ] generated by the operators on the RHS of(2.24). To obtain C [ M C ], all we need are the abelianized bubbling coefficients Z ab . Thesebubbling coefficients ensure that C [ M C ] closes without needing to include α (Φ) − , so thatit is well-defined everywhere on M C .The shift operators that we construct allow us to directly compute the OPE of chiralmonopole operators within a cohomological truncation of a given 3D N = 4 gauge theory.As explained in the introduction, this OPE encodes information about the geometry ofthe Coulomb branch beyond the chiral ring data. In particular, our shift operators give aconcrete realization of the abelianization map of [59], allowing us to determine the bubblingcoefficients from the bottom up. In our approach, the bubbling coefficients so obtained canfurther be used as input to calculate SCFT correlators via the gluing formula of Section 2.2.In fact, as explained in Appendix D.1, previous formulations of the abelianization map donot distinguish between the abelianized bubbling coefficients Z ab and certain coarser, Weyl-averaged counterparts thereof (denoted by Z mono ), as written in (A.8). To our knowledge,even the Z mono remain inaccessible to direct localization computations except in a few classesof examples, namely G = U ( N ) with fundamental and adjoint hypermultiplets [54]. The fact34hat the previously considered Weyl-averaged bubbling coefficients Z mono can be written interms of the more basic Z ab is one of the key observations of our work, and the computabilityof Z ab is one of our main results. For a bare monopole, decomposing Z mono into Z ab is merely a rewriting of the Weyl sum.However, the refinement of bubbling by abelianized bubbling turns out to be crucial for con-structing dressed monopoles. Given a bare monopole, its abelianized bubbling coefficientsallow us to construct all of its dressings in a way that guarantees closure of the star algebra.As we discuss next, our claim is that the closure of this algebra, or “polynomiality,” deter-mines Z ab uniquely up to operator mixing, in a sense to be made precise. By taking starproducts of (dressed) monopoles whose bubbling coefficients are known, one can inductivelyextract Z mono for all pairs of monopole charges ( b, v ) with v < b . The algebra of quantum Coulomb branch operators, A C , is believed to consist of gauge-invariant polynomials P (Φ) in the Q C -closed variable Φ( ϕ ) and dressed monopole operators[ F (Φ) M b ], where the dressing factor F (Φ) is a G b -invariant polynomial in Φ( ϕ ). Note thatthe subleading (bubbling) terms in [ F (Φ) M b ] can involve rational functions of Φ, but theleading term must be built solely from the polynomial F (Φ). Such an assumption has alsobeen made in the recent literature on 3D N = 4 Coulomb branches [59, 70]. One of thereasons that we expect this to be true is that VEVs of such operators should be algebraicfunctions on the Coulomb branch. Thus it would be unnecessary (and problematic) tointroduce poles by choosing P (Φ) or F (Φ) rational. The appearance of rational functions inthe OPE can be ruled out using similar reasoning.In good or ugly theories, we can make this argument slightly more explicit. The Coulombbranch in such theories is expected to be a hyperk¨ahler cone. Furthermore, because conformaldimensions are bounded from below, there are only finitely many operators below any fixedconformal dimension, and because 1 /r has dimension one, only finitely many operators can Some hints as to the necessity of the refined quantities Z ab were obtained for U Sp (2 N ) theories withfundamental matter in [49]. There, a proposal was made for extending the abelianized chiral ring relationsto the full Coulomb branch, relying on a subtle change of variables for abelian monopole operators ((2.3)in [49]). While not phrased in the language of monopole bubbling, this proposal should really be understoodas a conjecture for the abelianized bubbling coefficients of this class of theories (in the commutative limit).Indeed, the prediction of [49] agrees with what we explicitly derive in Section 4.4 for the case N = 2: compareto the r → ∞ limit of our (4.65). To match conventions, note that our abelianized monopoles whose chargesare the four Weyl images of ω ∨ in U Sp (4) gauge theory correspond to what are called u ± a ( a = 1 ,
2) in [49].Our perspective puts the proposal of [49] into the more general context of abelianized monopole bubbling. A C ,which is simply a sector of the OPE algebra in the IR CFT. This excludes denominatorsof the form (1 /r + P (Φ)) − where P (0) = 0, as such denominators, when expanded in 1 /r ,give infinitely many terms. The remaining possibility is to have denominators of the form1 /P (Φ) where P (0) = 0. But such operators blow up at the origin of the cone: they are notpart of the coordinate ring and thus should not appear in the algebra.In general, it is hard to give a more rigorous argument for polynomiality of observablesdue to the absence of a mathematical definition of the QFTs that concern us here. Neverthe-less, we proceed under the assumption that polynomiality holds, using the above heuristicreasoning and support from the existing literature as good evidence for it. Furthermore, theresults that we describe are in complete agreement with this assumption, implying that thealgebra A C constructed to satisfy polynomiality is self-consistent.One important observation is that if we neglect to include bubbling terms in the defini-tion of [ P (Φ) M b ], then polynomiality in general fails: operator products of such observablesproduce denominators that do not cancel. Therefore, one role of the bubbling terms is toguarantee polynomiality. In this section, we argue that polynomiality actually fully deter-mines the algebra A C , up to the natural ambiguity of operator mixing. In quantum field theories, an arbitrarily chosen basis of observables need not be diagonalwith respect to the two-point function, nor does it need to diagonalize the dilatation operatorin the case of a CFT. Observables can mix with others of the same dimension, and on curvedspaces, they can also mix with those of lower dimension, the difference being compensated forby powers of background (super)gravity fields. The mixing patterns often depend on short-distance effects, in particular how we define composite observables, creating ambiguities thatmust be resolved in the end by diagonalizing the two-point function.For our theories on S , the Riemann curvature is proportional to 1 /r . Mixing with oddpowers of 1 /r might not necessarily be generated by coupling to background SUGRA, but weinclude it in the formalism because it helps with the polynomiality argument in the followingsections. It could be that imposing some other requirement along with polynomiality wouldallow us to determine bubbling coefficients uniquely up to mixing with only even powersof 1 /r . However, we will not need to do so in this paper: mixing ambiguities can still beresolved in the end. The presence of operator mixing implies that in our problem, it is36atural to make r -dependent basis changes of the form O (cid:55)→ O + (cid:88) n ≥ r n O n (4.1)where if O has dimension ∆, then O n has dimension ∆ − n . Other quantum numbers, ifpresent, should also be preserved by such transformations.One might recognize redefinitions of the form (4.1) as typical “gauge” transformationsconsidered in (equivariant) deformation quantization. In the present context, they werediscussed in [36], where the problem was posed for SCFTs in flat space and transformationsof the form (4.1) were less relevant due to the absence of a natural “mixing” parameter like1 /r . In the S setup, however, (4.1) does naturally arise due to mixing. Such transformationsfirst appeared in the deformation quantization literature [71–77], where the classification ofquantizations often drastically simplifies once the problem is studied modulo (4.1). It istherefore reasonable to first solve our problem of constructing A C modulo transformationsof the form (4.1) (or rather, similar ones defined in the next paragraph). After that, themixing ambiguities can be resolved. In an SCFT, this can be achieved by diagonalizing thetwo-point function: the diagonalization determines a preferred basis of “SCFT operators”in the algebra A C . Alternatively, it might sometimes be enough to have an answer given in some basis, not necessarily the diagonal one (especially in bad theories, where one cannotstraightforwardly compute correlators).For the study of Coulomb branch operators, transformations of the general form (4.1)might not be the most adequate choice. We wish to think of the leading (i.e., no-bubbling)term of a dressed monopole [ P (Φ) M b ] as canonically defined, while the subleading bubblingterms might be ambiguous. If P (Φ) has large enough degree, one can find other monopoleoperators in the theory that have higher magnetic charge but lower dimension. Accordingto (4.1), they can mix with [ P (Φ) M b ]. This can indeed happen in physical operator mixing.However, for studying the structure of monopole operators, such a mixing is too crude, asit would alter the leading term of [ P (Φ) M b ]. We therefore define a class of less generaltransformations that respect the GNO charge. Namely, if O is a monopole operator of GNOcharge b , we only consider mixing with operators corresponding to GNO charges v (includingzero) such that b can bubble into v . Recall that this means | v | < | b | and that v belongs to the L G -representation of highest weight b . Such a relation determines a partial order on the setof monopole operators, and we denote by |O n | < |O| the situation where the GNO chargeof O “can bubble” into the GNO charge of O n . Then we may consider more restrictive37ransformations of the form O (cid:55)→ O + (cid:88) n ≥ |O n | < |O| r n O n , (4.2)where as before, the dimension of O n is n units smaller than that of O .We wish to first study monopole operators modulo such transformations. This meansthat for a given monopole [ P (Φ) M b ], the bubbling terms are not uniquely determined. Wecan shift [ P (Φ) M b ] by a linear combination of dressed monopoles of lower magnetic chargeand lower dimension (differences in dimension being compensated for by powers of 1 /r ), inthis way obtaining a valid, though different, definition of a dressed monopole operator. Werefer to (4.2) as the mixing ambiguity later on in this paper. The more general mixing (4.1)would only be relevant in an SCFT if we were to look for an orthonormal basis of observablesin the end.Such shifts significantly alter the bubbling coefficients V b → vi (Φ) appearing in the def-inition of [ P (Φ) M b ]: they can be shifted by polynomials or even by multiples of otherbubbling terms, which translates into complicated rational ambiguities of abelianized bub-bling coefficients Z ab b → v (Φ). Any concrete expressions for bubbling coefficients available in theliterature always implicitly refer to some choice of basis, thus resolving the mixing ambiguityin the algebra of observables. The presence of such ambiguities inherent to A C means thatthere is no chance of determining A C simply from polynomiality. In particular, this givesa negative answer to a question raised, e.g., in [70] on whether structural properties of A C (polynomiality and gauge invariance) determine it uniquely. We argue, however, that thenext simplest possibility holds: A C is uniquely determined by polynomiality precisely up tomixing ambiguities of the form (4.2). We start by proving this claim in the simplest cases. The simplest case is that in which the algebra A C is fully generated by monopole operatorsin minuscule representations of L G . Such monopoles cannot bubble because for minusculecoweights b , there are no v such that both | v | < | b | holds and b − v is a coroot. For suchmonopole operators, we have the following simple expressions: (cid:2) P (Φ) M b (cid:3) = 1 |W b | (cid:88) w ∈W P (Φ w ) M w · b . (4.3)Higher-charge monopole operators might contain bubbling terms, but they are easily deter-mined by taking products of lower-charge monopoles. Such cases were previously addressed38n the literature using different methods, and essentially comprise the main examples in [59]because abelianization has a simpler structure in these cases.Theories with minuscule generators include those with the gauge group P SU ( N ) = SU ( N ) / Z N , whose Langlands dual is SU ( N ): the fundamental weights of SU ( N ) are mi-nuscule and thus cannot bubble. Another example is U ( N ) gauge theory, since U ( N ) isself-dual and its fundamental weights are also minuscule. We discuss further aspects ofthese theories in Section 5 and Appendix E. We now move on to the more interesting (andnovel) case of theories with no minuscule generators, starting from the lower-rank gaugegroups.
The only rank-one gauge theory with no minuscule generators is SU (2) gauge theory. Thedual group is SO (3), so the lowest monopole operator corresponds to a root, i.e., the vectorrepresentation of SO (3). In a normalization where the weights of SU (2) are half-integersand products of weights with monopole charges (cocharacters, or dominant coweights) areintegers, the minimal monopole has b = 2. It can bubble to the zero-charge sector, because0 < | b | and b − (cid:102) M = M + Z (Φ) (4.4)with a single abelianized bubbling term, a function Z (Φ). Knowledge of Z (Φ) allows one toconstruct arbitrary dressed monopole operators of charge 2, and ultimately, by taking starproducts of the latter, monopoles of arbitrary charge.For the sake of generality, we may suppose that SU (2) is a simple factor in a larger gaugegroup G = SU (2) × · · · . Therefore, we implicitly assume that Z (Φ) might also depend onscalars Φ valued in other simple factors of G , which from the point of view of a given SU (2)factor play the role of masses.The basic shift operator of charge b ∈ Z is (we work in the North picture from now on,so we drop the “ N ” subscripts): M b = (cid:81) w ∈R (cid:104) ( − ( w · b )+ r | w · b | / (cid:0) + irw · Φ (cid:1) ( w · b ) + (cid:105)(cid:81) α ∈{ +1 , − } (cid:104) ( − ( α · b )+ r | α · b | / ( irα · Φ) ( α · b ) + (cid:105) e − b · ( i ∂ σ + ∂ B ) . (4.5) One can also use the U ( N ) results to solve the SU ( N ) theory, even though the latter has no minusculemonopoles. This point will be discussed later.
39y counting powers of r − , we read off the dimension of a bare monopole of charge b :∆ b = (cid:88) w ∈R | w · b | − | b | . (4.6)The dressed monopole is constructed as (cid:2) P (Φ) M (cid:3) = P (Φ) (cid:102) M + P ( − Φ) (cid:102) M − = P (Φ) M + P ( − Φ) M − + P (Φ) Z (Φ) + P ( − Φ) Z ( − Φ) . (4.7)Since an arbitrary P (Φ) can be written as (3.5), we clearly see that the primitive dressedmonopoles in this case are: M = M + M − + Z (Φ) + Z ( − Φ) , (cid:2) Φ M (cid:3) = Φ( M − M − ) + Φ( Z (Φ) − Z ( − Φ)) . (4.8)We then compute the following star products of these primitive monopoles with the Weyl-invariant polynomial Φ : M (cid:63) Φ = (cid:34)(cid:18) Φ − ir (cid:19) M (cid:35) + 4 r ( Z (Φ) + Z ( − Φ)) + 4 ir Φ ( Z (Φ) − Z ( − Φ)) , (cid:2) Φ M (cid:3) (cid:63) Φ = (cid:34)(cid:18) Φ − ir (cid:19) Φ M (cid:35) + 4 r Φ ( Z (Φ) − Z ( − Φ)) + 4 ir Φ ( Z (Φ) + Z ( − Φ)) , (4.9)with the first terms on the right being dressed monopoles with dressing factors (cid:0) Φ − ir (cid:1) and (cid:0) Φ − ir (cid:1) Φ, respectively. The polynomiality condition implies that the remaining termsmust be Weyl-invariant polynomials in Φ ∈ su (2) (and possibly other simple factors):4 r ( Z (Φ) + Z ( − Φ)) + 4 ir Φ ( Z (Φ) − Z ( − Φ)) ≡ r A (Φ ) ∈ C [Φ ] , r Φ ( Z (Φ) − Z ( − Φ)) + 4 ir Φ ( Z (Φ) + Z ( − Φ)) ≡ r A (Φ ) ∈ C [Φ ] . (4.10)Recall that the operator mixing ambiguity allows one to shift the bubbling factors Z (Φ) + Z ( − Φ) and Φ ( Z (Φ) − Z ( − Φ)) by arbitrary Weyl-invariant polynomials whose degrees are If Z (Φ) and/or P (Φ) depend on Φ’s valued in other simple factors, we only reverse the sign of Φ valuedin SU (2), as we are only concerned with the action of the Weyl group of SU (2) here. Z (Φ) − Z ( − Φ)), we can make A (Φ ) vanish. After doing so, we solve Equation (4.10) for Z (Φ): Z (Φ) = − iA (Φ )8Φ(Φ − ir ) . (4.11)We have not yet used the ambiguity to shift Z (Φ) + Z ( − Φ) by a Weyl-invariant polynomial.Such shifts that leave r ( Z (Φ) + Z ( − Φ)) + i Φ ( Z (Φ) − Z ( − Φ)) invariant (because we havefixed the latter expression by demanding A (Φ) = 0) give one the freedom to shift Z (Φ) by∆ Z (Φ) = Φ + ir V (Φ ) , (4.12)with V (Φ ) an arbitrary Weyl-invariant polynomial. Adding this ambiguity to (4.11) gives: Z (Φ) = − i A (Φ ) + 4 i (Φ + r ) V (Φ )8Φ(Φ − ir ) . (4.13)For any A (Φ ), there exists a unique polynomial V (Φ ) such that the numerator A (Φ ) +4 i (Φ + r ) V (Φ ) ≡ ic does not depend on Φ ∈ su (2), where c is a dimensionful constant. Therefore, by completely fixing the mixing ambiguity, we find that: Z (Φ) = c Φ(Φ − ir ) . (4.14)It remains to determine c . To this end, we compute the following expression: M (cid:63) (cid:2) Φ M (cid:3) − (cid:2) (Φ − i/r ) M (cid:3) (cid:63) M , (4.15)which must satisfy the polynomiality constraint. This is where the answer starts to dependon the precise matter content of the theory (all previous steps apply equally well to allmatter representations R ). Assume that the gauge group is precisely SU (2) (with no othersimple factors), and that the theory has N f fundamental and N a adjoint hypermultiplets.The dimension of a charge- b monopole is hence∆ b = | b | N f + | b | ( N a − . (4.16) However, it can still depend on Φ valued in other simple factors.
41 straightforward computation with shift operators gives(4.15) = 8 ic r (1 + r Φ ) + (cid:34) (cid:18) i r + Φ2 (cid:19) N f − (cid:18) i r + Φ (cid:19) N a (cid:18) i r + Φ (cid:19) N a + (Φ ↔ − Φ) (cid:35) . (4.17)At this point, we see that the precise answer for c depends on whether N f ≥ N f = 0.If N f ≥
1, then the second term on the right is a Weyl-invariant polynomial and the onlynon-polynomial piece is ic r (1+ r Φ ) , implying that only c = 0 is consistent with polynomiality.However, if N f = 0, then one finds that the poles at Φ = ± i/r (whose presence would violatepolynomiality) vanish when c = (2 r ) − N a = ⇒ c = ± (2 r ) − N a . (4.18)The sign of c remains undetermined, and indeed, the algebra is consistent for both signs of c . In fact, it is not hard to see that flipping the sign of c has the same effect on the algebra A C as flipping the overall sign of M , which is simply a change of basis. This, in particular,shows that after performing Gram-Schmidt orthogonalization, the algebra is unaffected, andthe physical correlation functions do not depend on the sign of c .We will soon see that, quite curiously, such a sign ambiguity is not present in higher-rankcases. In the present case, there exists a convenient way to fix the sign. Notice that a theorywith only adjoint matter admits two possible global forms of the gauge group: either SU (2)or SO (3). They differ by the spectrum of allowed monopole operators. While M is thelowest monopole in the SU (2) case, the SO (3) gauge theory also admits M . Indeed, theLanglands dual of SO (3) is SU (2), and M is in the fundamental representation. Because M is minuscule, it contains no bubbling term: M = M + M − , (cid:2) Φ M (cid:3) = Φ( M − M − ) . (4.19)We can then define M = M (cid:63) M and [Φ M ] = [Φ M ] (cid:63) M , and calculate the bubblingterm generated in this way. This gives the following value of c for the SO (3) gauge theory: c = ( − r ) − N a . (4.20)One could wonder whether the SU (2) global form corresponds to a different sign, but this42s not the case. There exists another trick to access bubbling terms in SU (2) (and moregenerally, in SU ( N )) gauge theory. It consists of studying the U (2) theory first, and thengauging the U (1) top symmetry that rotates the dual photon in the diagonal U (1) gaugegroup (this approach was also used in [48, 49]). In a U (2) gauge theory, monopole chargesare labeled by two integers ( n, m ) ∈ Z , and some of them are minuscule. In particular, M (1 , and M ( − , are minuscule, and their product can be used to determine the non-minuscule M (1 , − . After gauging U (1) top , the latter becomes M of the SU (2) gauge theory.Proceeding along these lines gives the same value for c as in (4.20).So in the end, we find that the bubbling coefficient in SU (2) (or SO (3), when possible)gauge theory with N f fundamentals and N a adjoints, up to the operator mixing ambiguity,takes the form Z (Φ) = N f > , ( − r ) − Na Φ(Φ − ir ) if N f = 0 , (4.21)which then determines (cid:103) M = M + Z (Φ).Let us now generalize to the case where SU (2) is a simple factor in a larger gauge group,that is, G = SU (2) × G (cid:48) . Then the N f fundamentals of SU (2) form some generally reduciblerepresentation R (cid:48) f of G (cid:48) , while the N a adjoints of SU (2) form another representation R (cid:48) a of G (cid:48) . This modifies the computation of (4.15) as follows:(4.15) = 8 ic r (1 + r Φ ) + (cid:34) (cid:0) i r + Φ2 (cid:1) (cid:89) w ∈R (cid:48) f (cid:32)(cid:18) i r + Φ2 (cid:19) − ( w · Φ (cid:48) ) (cid:33) × (cid:89) w ∈R (cid:48) a (cid:32)(cid:18) i r + Φ (cid:19) − ( w · Φ (cid:48) ) (cid:33) (cid:32)(cid:18) i r + Φ (cid:19) − ( w · Φ (cid:48) ) (cid:33) + (Φ ↔ − Φ) (cid:35) , (4.22)where Φ ∈ t ⊂ su (2) and Φ (cid:48) ∈ t (cid:48) ⊂ Lie( G (cid:48) ). The cancellation of poles determines c : c = (cid:89) w ∈R (cid:48) a (cid:18) − r − ( w · Φ (cid:48) ) (cid:19) (cid:89) w ∈R (cid:48) f ( − iw · Φ (cid:48) ) , (4.23)where the sign was fixed by passing to the U (2) theory and applying the “gauging U (1) top ”trick. This shows that in a general gauge theory with gauge group G = SU (2) × G (cid:48) , theabelianized bubbling term for monopoles magnetically charged under the SU (2) factor takesthe same form c Φ(Φ − ir ) where Φ ∈ t ⊂ su (2), while c is no longer a constant, but rather a43ontrivial function of Φ (cid:48) from the G (cid:48) vector multiplets. This last result is enough to studythe algebra A C and corresponding correlators for arbitrary quivers of SU (2) gauge groups. In this subsection, we repeat the above analysis for rank-two gauge groups, namely SU (3), P SU (3),
U Sp (4) ∼ = Spin (5), SO (5), and G , demonstrating how polynomiality determinesbubbling coefficients. This will further clarify the general procedure, which was applied torank-one theories in the previous subsection. A Theories
Consider the A gauge theories, i.e., those based on either SU (3) or P SU (3) = SU (3) / Z gauge group. The P SU (3) case is trivial, as the theory admits monopoles in fundamentalrepresentations of the dual group SU (3). Such monopoles are minuscule, so they do notbubble, and being the generators, they fully determine the algebra. In the SU (3) gaugetheory, however, the monopole charges take values in the weight lattice of P SU (3), whichcoincides with its root lattice. Letting α and α denote simple roots of SU (3), the coroots α ∨ = 2 α / ( α , α ) and α ∨ = 2 α / ( α , α ) generate the root lattice of P SU (3), and physicalmonopole charges (cocharacters) correspond to Weyl orbits in this lattice.The minimal monopole operator corresponds to the Weyl orbit of α ∨ , which coincideswith the root system of P SU (3). In standard conventions, α ∨ + α ∨ is the dominant coroot, sowe could use it to label the minimal-charge monopole operator [ P (Φ) M α ∨ + α ∨ ]. In practice,we find it slightly more convenient to label it by α ∨ . Such a monopole can only bubbleto a trivial representation, since the only weight shorter than | α ∨ | is a zero weight, and itbelongs to the highest-weight representation generated by α ∨ + α ∨ . Therefore, there existsonly one bubbling coefficient in this case, Z (Φ), which determines the abelianized version ofthe minimal monopole and its dressings: (cid:102) M α ∨ = M α ∨ + Z (Φ) , (cid:104) P (Φ) M α ∨ (cid:105) = (cid:88) w ∈W P (Φ w ) (cid:102) M w · α ∨ . (4.24)In the A case, Φ = (Φ , Φ ) and W = S ; the ring of invariants can be described as C [Φ , Φ ] W = C [ f , f ] where f = Φ + Φ , f = Φ (Φ − ) . (4.25)44here are six primitive dressed monopoles of minimal charge that generate the space ofdressed monopoles (of minimal charge) as a C [Φ , Φ ] W -module. They can be chosen as: M α ∨ , (cid:104) Φ M α ∨ (cid:105) , (cid:104) Φ M α ∨ (cid:105) , (cid:104) Φ M α ∨ (cid:105) , (cid:104) Φ Φ M α ∨ (cid:105) , (cid:104) Φ M α ∨ (cid:105) . (4.26)The next step, just like in the rank-one case, is to compute star products of these with thelowest invariant polynomial Φ + Φ (often referred to as the quadratic Casimir in physicsliterature). A straightforward computation for general dressed [ P (Φ) M α ∨ ] gives: (cid:104) P (Φ) M α ∨ (cid:105) (cid:63) (Φ + Φ ) = (cid:104) ((Φ − i/r ) + Φ ) P (Φ) M α ∨ (cid:105) + (cid:88) w ∈W (cid:18) r + 4 ir Φ w1 (cid:19) P (Φ w ) Z (Φ w ) . (4.28)The last term above must be a Weyl-invariant polynomial for all possible polynomials P . Itis enough to impose this requirement for P = 1 , Φ , Φ , Φ , Φ Φ , Φ . Recall that (cid:104) Φ j Φ k M α ∨ (cid:105) = (cid:88) w ∈W (Φ w1 ) j (Φ w2 ) k M w · α ∨ + V jk (Φ) , V jk (Φ) ≡ (cid:88) w ∈W (Φ w1 ) j (Φ w2 ) k Z (Φ w ) . (4.29)We see that the last term in (4.28) for P = 1 , Φ , Φ , Φ , Φ Φ , Φ is simply: (cid:88) w ∈W (cid:18) r + 4 ir Φ w1 (cid:19) Z (Φ w ) = 4 r V (Φ) + 4 ir V (Φ) , (cid:88) w ∈W (cid:18) r + 4 ir Φ w1 (cid:19) Φ w1 Z (Φ w ) = 4 r V (Φ) + 4 ir V (Φ) , (cid:88) w ∈W (cid:18) r + 4 ir Φ w1 (cid:19) Φ w2 Z (Φ w ) = 4 r V (Φ) + 4 ir V (Φ) , (cid:88) w ∈W (cid:18) r + 4 ir Φ w1 (cid:19) (Φ w1 ) Z (Φ w ) = 4 r V (Φ) + 4 ir V (Φ) , (4.30) (cid:88) w ∈W (cid:18) r + 4 ir Φ w1 (cid:19) Φ w1 Φ w2 Z (Φ w ) = 4 r V (Φ) + 4 ir V (Φ) , Our conventions are that W = { , w a , w a , w b , w b w a , w b w a } wherew a = (cid:18) − / −√ / √ / − / (cid:19) , w b = (cid:18) − (cid:19) , (4.27)and w : (cid:0) Φ Φ (cid:1) (cid:55)→ (cid:0) Φ Φ (cid:1) w for w ∈ W . w ∈W (cid:18) r + 4 ir Φ w1 (cid:19) (Φ w1 ) Z (Φ w ) = 4 r V (Φ) + 4 ir V (Φ) . The right-hand side of each of these equations should be a Weyl-invariant polynomial. Thislinear system can be solved for Z (Φ), but we will do better if we first use the operator mixingfreedom. Recall that V , V , V , V , V , V , being the bubbling terms in (4.26), can beshifted by Weyl-invariant polynomials in Φ , Φ (and r − ). Using such shifts of V , V , V , V , we can make the right-hand sides of the first four equations in (4.30) vanish, while thoseof the fifth and sixth ones should be Weyl-invariant polynomials. In other words, we obtain:4 r V (Φ) + 4 ir V (Φ) = 0 , r V (Φ) + 4 ir V (Φ) = 0 , r V (Φ) + 4 ir V (Φ) = 0 , r V (Φ) + 4 ir V (Φ) = 0 , (4.31)4 r V (Φ) + 4 ir V (Φ) = 1 r A (Φ) , r V (Φ) + 4 ir V (Φ) = 1 r B (Φ) , where A and B are Weyl-invariant polynomials (hence polynomials in f and f ). Insertingthis into (4.30), we solve the resulting linear system for Z (Φ) to find that Z (Φ) = i (Φ A (Φ) − B (Φ))6Φ (Φ − i/r )(Φ − ) . (4.32)So far, we have not used the freedom to shift V and V by Weyl-invariant polynomials F (Φ) = F ( f , f ) and F (Φ) = F ( f , f ). To preserve the first four equations in (4.31),such shifts should be accompanied by V k → V k + (cid:18) ir (cid:19) k F ( k = 1 , , , V → V + ir F . (4.33)Solving another linear system, namely (cid:88) w ∈W ∆ Z (Φ w ) = F (Φ) , (cid:88) w ∈W Φ w2 ∆ Z (Φ w ) = F (Φ) , (cid:88) w ∈W (Φ w1 ) k ∆ Z (Φ w ) = (cid:18) ir (cid:19) k F (Φ) ( k = 1 , , , (cid:88) w ∈W Φ w1 Φ w2 ∆ Z (Φ w ) = ir F (Φ) , (4.34)46e find that such shifts trace back to the following shift in Z (Φ):∆ Z (Φ) = − Φ [( f + 4 /r ) F (Φ) − f F (Φ)] + ( f + 1 /r )( f + 4 /r ) F (Φ) − f F (Φ)6Φ (Φ − i/r )(Φ − ) . (4.35)Shifting Z by such an expression is equivalent to shifting A (Φ) = A ( f , f ) and B (Φ) = B ( f , f ) by ∆ A ( f , f ) = i ( f + 4 /r ) F ( f , f ) − if F ( f , f ) , ∆ B ( f , f ) = if F ( f , f ) − i ( f + 1 /r )( f + 4 /r ) F ( f , f ) . (4.36)We can use such shifts to eliminate the f -dependence of A and B . Indeed, we first choose F to eliminate the f -dependence of A . We then choose F ( f , f ) = F ( f ) to preserve thecondition that A ( f , f ) = A ( f ). Noting that shifts of the form∆ F ( f , f ) = ( f + 4 /r ) P ( f , f ) , ∆ F ( f , f ) = f P ( f , f ) (4.37)leave ∆ A ( f , f ) invariant, we still have the freedom to shift B by∆ B ( f , f ) = if F ( f ) + i ( f − ( f + 1 /r )( f + 4 /r ) ) P ( f , f ) . (4.38)The P -dependent part of this shift can be used to make B at most linear in f (by polynomiallong division), whereupon F can be chosen to eliminate the remaining f -dependence of B .Having completely used the mixing freedom in this way, we find that the abelianized bubblingterm takes the form: Z (Φ) = i (Φ A ( f ) − B ( f ))Φ (Φ − i/r )(Φ − ) , f = Φ + Φ , f = Φ (Φ − ) . (4.39)We have now reached the limits of what can be done based on the gauge group only. Theconcrete expressions for the polynomials A ( f ) and B ( f ) depend on the matter content,as in the rank-one case. For simplicity, let us consider only the case of an SU (3) vectormultiplet coupled to N f fundamental flavors. We then compute the following star product: M α ∨ (cid:63) (cid:104) Φ M α ∨ (cid:105) − (cid:104) (Φ − i/r ) M α ∨ (cid:105) (cid:63) M α ∨ = (cid:104) P (Φ) M α ∨ (cid:105) + R (Φ) . (4.40)The combination above is devised in such a way that M α ∨ does not show up on the right.47he monopole M α ∨ +2 α ∨ would be present for more general matter representations (e.g., ifwe included adjoint matter), but it does not show up in our case either, which is why thetheory with only fundamental matter is somewhat simpler. Here, P (Φ) is some polynomialdressing factor, while R (Φ) is a Weyl-invariant polynomial.The expressions for P and R are lengthy, so we do not provide them here for brevity.Polynomiality of P (Φ) — that is, cancellation of poles — determines the unknown terms A ( f ) and B ( f ). We find thatfor even N f : A ( f ) = 0 , B ( f ) = − i (cid:16) − i √ (cid:17) N f ( f + 1 /r ) N f / , for odd N f : B ( f ) = 0 , A ( f ) = 4 i (cid:16) − i √ (cid:17) N f ( f + 1 /r ) ( N f − / . (4.41)We have thus determined (4.39).As in the rank-one case, this result can be generalized to a gauge group G = SU (3) × G (cid:48) and SU (3)-valued monopoles. If the N f fundamentals of SU (3) form a representation R (cid:48) of G (cid:48) , then the no-pole condition encodes the polynomials A ( f ) and B ( f ) as follows: xA ( x − r − ) − B ( x − r − ) = 4 i (cid:89) w ∈R (cid:48) (cid:18) − ix √ − iw · Φ (cid:48) (cid:19) . (4.42)Here, Φ (cid:48) corresponds to scalars from the G (cid:48) vector multiplet. This formula reproduces (4.41)if we take R (cid:48) = C N f to be a trivial representation of G (cid:48) , that is, all weights w to be zero.This final result allows one to study quivers of SU (3) groups in which every gauge nodeonly couples to fundamental matter; it also allows for the inclusion of masses by treating Φ (cid:48) as a background. Higher magnetic charges.
Finally, we would like to explain how to construct monopolesof other magnetic charges. This is straightforward in the theory with
P SU (3) gauge group:the dual group is SU (3), so both fundamental representations of SU (3) give allowed mono-pole charges. Their tensor products generate arbitrary representations of SU (3). In the caseof SU (3) gauge theory, things are slightly more involved, but still tractable.We have derived an expression for Z (Φ), which is enough to build a dressed monopole[ P (Φ , Φ ) M α ∨ ] corresponding to the Weyl orbit of α ∨ , with arbitrary polynomial P . Is itenough to construct all allowed monopoles in the theory? After all, the coroots take values ina two-dimensional lattice spanned by α ∨ , α ∨ , and merely on representation-theoretic grounds,one cannot construct all representations labeled by dominant weights in this lattice just from48ensor products of the adjoint representation. However, by taking star products of dressedmonopoles [ P (Φ , Φ ) M α ∨ ], one can actually generate everything else.From (2.27), we see that in an SU (3) theory with N f fundamentals, the dimensions ofthree lowest bare monopoles are∆ α ∨ = N f − , ∆ α ∨ = 2 N f − , ∆ α ∨ +2 α ∨ = 2 N f − . (4.43)Since 2∆ α ∨ < ∆ α ∨ +2 α ∨ , the product of two bare monopoles M α ∨ cannot generate M α ∨ +2 α ∨ .However, the latter can appear if we compensate for the mismatch in dimensions by dressingthe monopoles with an appropriate number of Φ’s. For example, the star product (cid:104) Φ (Φ − i/r ) M α ∨ (cid:105) (cid:63) M α ∨ − (cid:104) Φ M α ∨ (cid:105) (cid:63) (cid:104) Φ M α ∨ (cid:105) (4.44)has M α ∨ +2 α ∨ as a leading term, and can therefore serve as a definition of M α ∨ +2 α ∨ . Similarlytaking products of monopoles dressed by higher-degree polynomials, we can obtain dressedversions of M α ∨ +2 α ∨ . Having constructed in this way both M α ∨ , M α ∨ +2 α ∨ and their dressedversions, we can generate all other allowed monopoles. B ∼ = C Theories
There are two compact rank-two gauge groups that correspond to the B ∼ = C Lie algebra:
U Sp (4) ∼ = Spin (5) and SO (5) ∼ = U Sp (4) / Z , which are Langlands dual to each other. Thegroup U Sp (4) is often called Sp (2), but we will use the former notation. The root latticeis generated by the short simple root α and the long simple root β . In our conventions,we write them in Cartesian coordinates as α = (1 ,
0) and β = ( − , α ∨ = 2 α = (2 ,
0) and β ∨ = β = ( − , α ∨ is a long coroot. SO (5) gauge theory. First consider the SO (5) gauge theory. The monopoles are labeledby (Weyl orbits of) the dominant weights of the dual group U Sp (4), whose root latticeis generated by α ∨ and β ∨ . The group U Sp (4) has two fundamental representations: thefour-dimensional defining representation and the five-dimensional vector representation of SO (5) = U Sp (4) / Z . The four-dimensional representation has weights ω ∨ = 12 α ∨ = (1 , , ω ∨ + β ∨ = (0 , ,ω ∨ − α ∨ = ( − , , ω ∨ − α ∨ − β ∨ = (0 , − . (4.45)49his representation is minuscule, so the smallest monopole of the model does not bubble: (cid:104) P (Φ) M ω ∨ (cid:105) = (cid:88) w ∈W P (Φ w ) M w · ω ∨ . (4.46)The five-dimensional representation of U Sp (4) is not minuscule. Its weights are ω ∨ = α ∨ + β ∨ = (1 , , ω ∨ − α ∨ = β ∨ = ( − , , − β ∨ = (1 , − , − β ∨ − α ∨ = ( − , − , (0 , . (4.47)We see that the charge- ω ∨ monopole can bubble to zero magnetic charge. Therefore, theabelianized monopole takes the form (cid:102) M ω ∨ = M ω ∨ + Z (Φ) . (4.48)This Z (Φ) can be deduced by computing star products involving only the minimal monopole[ P (Φ) M ω ∨ ]. On the other hand, it can also be found using our algorithmic polynomialityapproach (which is really a different application of the same idea, namely consistency of theOPE algebra). Let us determine it using such an approach — both for practice, and becauseit will soon be useful for the study of the U Sp (4) gauge theory.The charge- ω ∨ dressed monopoles are constructed as (cid:104) P (Φ) M ω ∨ (cid:105) = 12 (cid:88) w ∈W P (Φ w ) (cid:102) M w · ω ∨ . (4.49)As before, Φ = (Φ , Φ ) ∈ t C . The Weyl group is D = Z (cid:111) Z , and the ring of invariants is C [Φ , Φ ] W = C [ f , f ] where f = Φ + Φ , f = Φ Φ . (4.50)Notice that ω ∨ is preserved by the subgroup of W that switches Φ ↔ Φ , which explainsthe in (4.49). Therefore, such monopoles can only be dressed by polynomials symmetricunder Φ ↔ Φ (this happens automatically once we apply (4.49)). Dressing by a symmetricpolynomial only depends on the symmetric part of Z (Φ , Φ ). Therefore, we may assumethat Z (Φ , Φ ) = Z (Φ , Φ ).Since the Weyl orbit of ω ∨ has four elements, there are four primitive dressed monopoles50hat generate all dressed charge- ω ∨ monopoles as a C [Φ , Φ ] W -module. Choose them to be: M ω ∨ , (cid:104) (Φ + Φ ) M ω ∨ (cid:105) , (cid:104) (Φ + Φ ) M ω ∨ (cid:105) , (cid:104) (Φ + Φ ) M ω ∨ (cid:105) . (4.51)The next step is to compute their star products with f . For arbitrary P (Φ), we find: (cid:104) P (Φ) M ω ∨ (cid:105) (cid:63) (Φ + Φ ) − (cid:34)(cid:32)(cid:18) Φ − ir (cid:19) + (cid:18) Φ − ir (cid:19) (cid:33) P (Φ) M ω ∨ (cid:35) = (cid:88) w ∈W (cid:18) r + ir (Φ w1 + Φ w2 ) (cid:19) P (Φ w ) Z (Φ w ) . (4.52)We require that the second line be a polynomial, in particular for P = 1, Φ + Φ , (Φ + Φ ) ,and (Φ + Φ ) . Using notation similar to that in the SU (3) case, (cid:104) (Φ + Φ ) k M ω ∨ (cid:105) = 12 (cid:88) w ∈W (Φ w1 + Φ w2 ) k (cid:102) M w · ω ∨ + V k (Φ) , V k (Φ) ≡ (cid:88) w ∈W (Φ w1 + Φ w2 ) k Z (Φ w ) , (4.53)we identify the last term in (4.52) for P (Φ) = (Φ + Φ ) k as r V k (Φ) + ir V k +1 (Φ), which wedemand to be a Weyl-invariant polynomial. Using the operator mixing freedom to shift V , V , and V , we can make the first three of these polynomials vanish, but not the last:2 r V k (Φ) + 2 ir V k +1 (Φ) = 0 ( k = 0 , , , r V (Φ) + 2 ir V (Φ) = 1 r A (Φ + Φ , Φ Φ ) ∈ C [Φ + Φ , Φ Φ ] . (4.54)Using the expressions V k (Φ) = (cid:80) w ∈W (Φ w1 + Φ w2 ) k Z (Φ w ), we solve this system of four equa-tions under the assumption that Z (Φ , Φ ) = Z (Φ , Φ ) to find: Z (Φ) = − iA (Φ + Φ , Φ Φ )32Φ Φ (Φ + Φ )(Φ + Φ − i/r ) . (4.55)The next step is to fix the remaining mixing freedom, which allows for shifts of V by Weyl-invariant polynomials F (Φ + Φ , Φ Φ ), namely V → V + F . To preserve the form of theequations (4.54), we also shift V k → V k + (cid:0) ir (cid:1) k F for k = 1 , ,
3. This can be solved for thecorresponding shift ∆ Z (Φ) of Z (Φ):∆ Z (Φ) = − (Φ + Φ + i/r )(Φ − Φ + i/r )(Φ − Φ − i/r ) F (Φ + Φ , Φ Φ )16Φ Φ (Φ + Φ ) . (4.56)51omparing with (4.55), we see that such shifts are equivalent to shifting A by∆ A = (cid:18) r + 4 f r + f − f (cid:19) F ( f , f ) , (4.57)where f = Φ + Φ and f = Φ Φ . Because the expression in parentheses is no more thanlinear in f , such shifts can completely eliminate the f -dependence from A . Indeed, for anarbitrary polynomial A ( f , f ), there is a unique F ( f , f ) such that A ( f , f ) + ∆ A ( f , f )depends only on f . This fully fixes the mixing freedom, so that in the end, we have Z (Φ) = a (Φ + Φ )Φ Φ (Φ + Φ )(Φ + Φ − i/r ) . (4.58)Determining a requires computing an appropriate star product. The answer depends onthe matter content, and in this case it is not too hard to include both N f five-dimensionalflavors of SO (5) and N a adjoint flavors. We consider the following star product of minimalmonopoles, which is enough to generate the next-to-minimal monopole of charge ω ∨ : M ω ∨ (cid:63) (cid:104) Φ M ω ∨ (cid:105) − (cid:104) (Φ − i/r ) M ω ∨ (cid:105) (cid:63) M ω ∨ . (4.59)The reason for including Φ can be seen from the dimensions of the monopoles:∆ ω ∨ = N f + 3( N a − , ∆ ω ∨ = 2 N f + 4( N a − . (4.60)Only with the insertion of (at least) Φ do we find that the dimension of (4.59), given by2∆ ω ∨ + 3 = 2 N f + 6 N a −
3, exceeds ∆ ω ∨ for all values of N a , thus allowing the monopoleof charge ω ∨ to appear on the right. It indeed appears, in bare form for N a = 0 and indressed form for N a (cid:54) = 0. We subtract it from the above star product and look at the free(charge-zero) term, demanding its polynomiality. This determines a (Φ + Φ ). For brevity,we do not present the cumbersome intermediate formulas and only give the final answer: a ( x ) = − (cid:18) x r (cid:19) N f (cid:18) − x r − r (cid:19) N a . (4.61)This expression determines Z (Φ), from which we can construct arbitrary dressed monopolesof charge ω ∨ . With the two monopoles corresponding to fundamental weights of U Sp (4) inhand, we can construct arbitrary monopoles in the SO (5) gauge theory. Notice that it wasclear from the beginning that the charge- ω ∨ monopole suffices to generate the algebra.52ike in all cases so far, it is not hard to generalize to a non-simple gauge group G = SO (5) × G (cid:48) , assuming that the N f fundamentals of SO (5) form a representation R (cid:48) f of G (cid:48) while the N a adjoints transform in R (cid:48) a of G (cid:48) . Modifying the above calculation appropriatelygives a ( x ) = − (cid:89) w ∈R (cid:48) f (cid:18) x r − ( w · Φ (cid:48) ) (cid:19) (cid:89) w ∈R (cid:48) a (cid:20)(cid:18) − r − ( w · Φ (cid:48) ) (cid:19) (cid:18) x r − ( w · Φ (cid:48) ) (cid:19)(cid:21) . (4.62)Here, as before, Φ (cid:48) is valued in the G (cid:48) vector multiplets. This answer for a ( x ) of coursereduces to (4.61) when all weights in R (cid:48) f and R (cid:48) a vanish. As usual, Φ (cid:48) plays the role of amass matrix if we treat G (cid:48) as a global symmetry. U Sp (4) gauge theory.
Consider the
U Sp (4) gauge theory. It has the same simple roots α, β and simple coroots α ∨ , β ∨ as in the SO (5) case. Only the lattice of allowed weights ofmatter representations is different.The dual group is SO (5), which has no minuscule representations. The minimal monopolehas charge ω ∨ , like the next-to-minimal monopole of the SO (5) gauge theory. It is definedby the same equations (4.48) and (4.49). Further steps involving the ring of invariants, theprimitive dressed monopoles, and ultimately the answer (4.58) are applicable to the U Sp (4)case as well — they do not depend on the global form of the gauge group. To proceed, weneed to find the polynomial a entering the abelianized bubbling coefficient (4.58). This stepdepends on the matter content, and hence on the global form of the gauge group.Let us consider for simplicity a theory which only has matter in N copies of the four-dimensional representation of U Sp (4). We compute the star product M ω ∨ (cid:63) (cid:104) (Φ + Φ ) M ω ∨ (cid:105) − (cid:104) (Φ + Φ − i/r ) M ω ∨ (cid:105) (cid:63) M ω ∨ . (4.63)The charges 2 ω ∨ and 2 ω ∨ cancel from the result. Polynomiality of the remainder requires a ( x ) to be a constant, and determines its square. We describe the answer in the more generalcase of G = U Sp (4) × G (cid:48) , assuming that the N fundamentals of U Sp (4) transform in R (cid:48) of G (cid:48) : a = − (cid:89) w ∈R (cid:48) ( − iw · Φ (cid:48) ) , (4.64)53hich determines the bubbling coefficient Z (Φ) = a Φ Φ (Φ + Φ )(Φ + Φ − i/r ) . (4.65)The situation here is reminiscent of the SU (2) case: without an extra group G (cid:48) , all weights w vanish, and we find that a = 0. In other words, in the U Sp (4) gauge theory with N >
0, inthe absence of extra gaugings and masses, the bubbling coefficient of the minimal monopolecan be removed using operator mixing. If N = 0, then a = −
1, where the sign was chosento agree with the SO (5) answer (in the N = 0 case, the absence of matter allows for both U Sp (4) and SO (5) gauge groups, and we can determine the U Sp (4) answer from the SO (5)answer: the theories differ only by a Z gauging). For other values of N , we fixed the signof a arbitrarily, since it only affects the algebra A C up to a change of basis.Finally, let us add that at this point, using some physics intuition, we can easily guessthe answer for a ( x ) in the more general case where we have matter in a representation[ ⊗ R (cid:48) ] ⊕ [ ⊗ R (cid:48) ] ⊕ [ adj ⊗ R (cid:48) a ] of the gauge group G = U Sp (4) × G (cid:48) . Here, , , and adj are the four-dimensional, five-dimensional, and adjoint representations of U Sp (4), and R (cid:48) , R (cid:48) , R (cid:48) a are some representations of G (cid:48) . We have seen before that contributions ofdifferent matter multiplets enter the answer for a ( x ) multiplicatively. This makes sense fromthe localization point of view: bubbling terms are given by one-loop determinants aroundfixed points in the bubbling loci, and one-loop determinants of various matter multipletscontribute multiplicatively. So it is natural to expect that the answer in this general caseshould be given by a ( x ) = − (cid:89) w ∈R (cid:48) ( − iw · Φ (cid:48) ) (cid:89) w ∈R (cid:48) (cid:18) x r − ( w · Φ (cid:48) ) (cid:19) × (cid:89) w ∈R (cid:48) a (cid:20)(cid:18) − r − ( w · Φ (cid:48) ) (cid:19) (cid:18) x r − ( w · Φ (cid:48) ) (cid:19)(cid:21) . (4.66)Above, we borrowed the contributions of and adj from the subsection on the SO (5) case,as the theory with only these types of matter allows for either SO (5) or U Sp (4) gauge group. G Theories
The remaining rank-two simple gauge group is G . It has only one compact form, which is ofcourse centerless and Langlands dual to itself, meaning that we do not have to study variouscases as before. We describe the root system ∆ of G in Cartesian coordinates such that the54hort simple root is α = (1 ,
0) and the long simple root is β = ( − , √ ). The correspondingcoroots are α ∨ = 2 α = (2 ,
0) and β ∨ = β = ( − , √ ), which are now long and short,respectively, and generate the root system ∆ ∨ of the dual G . It is convenient to describe∆ ∨ in terms of another pair of simple coroots, which we define as α mon = α ∨ + 2 β ∨ = (0 , √ )and β mon = − α ∨ − β ∨ = ( − , −√ α mon is short and β mon is long.The smallest irreducible representation is 7-dimensional: its weights are given by a zeroweight (0 ,
0) and the six short roots in ∆, namely α and its Weyl images. Because of the zeroweight, the representation is not minuscule. The next-smallest irreducible representation isthe 14-dimensional adjoint representation. The and are fundamental representations,but we will refer only to as the fundamental, and to as the adjoint.The minimal monopole charge is described by the nonzero weights in the of the dual G , i.e., by the Weyl orbit of the short coroot α mon (or equivalently, by β ∨ , which belongsto the same Weyl orbit). Because is not minuscule, it can bubble into the identity: (cid:102) M α mon = M α mon + Z (Φ) , (4.67)and the physical dressed monopole of minimal charge is defined by[ P (Φ) M α mon ] = 12 (cid:88) w ∈W P (Φ w ) (cid:102) M w · α mon . (4.68)The Weyl group is W = D = Z (cid:111) Z , the group of symmetries of a hexagon. As usual,Φ = (Φ , Φ ) ∈ t C , and the ring of invariants is C [Φ , Φ ] W = C [ f , f ] where f = Φ + Φ , f = Φ (Φ − ) . (4.69)Because the Weyl orbit of α mon has order 6, there are six primitive dressed monopoles: M α mon , [Φ M α mon ] , (cid:2) Φ M α mon (cid:3) , (cid:2) Φ M α mon (cid:3) , (cid:2) Φ M α mon (cid:3) , (cid:2) Φ M α mon (cid:3) . (4.70)The next few steps are exactly the same as before. Namely, we compute the star product[ P (Φ) M α mon ] (cid:63) (Φ + Φ ) − (cid:34)(cid:32) Φ + (cid:18) Φ − ir √ (cid:19) (cid:33) P (Φ) M α mon (cid:35) = 12 (cid:88) w ∈W (cid:18) r + 4 ir √ w2 (cid:19) P (Φ w ) Z (Φ w ) (4.71)55nd demand polynomiality for P = 1 , Φ , . . . , Φ . Because α mon is preserved by the Weylreflection (Φ , Φ ) → ( − Φ , Φ ), it is sufficient to consider only dressings by polynomialsinvariant under such a reflection (which also explains the factor of in the definition of themonopole). Therefore, one can assume from the beginning that Z (Φ , Φ ) = Z ( − Φ , Φ ) . (4.72)Polynomiality of the last term in (4.71) and the operator mixing freedom almost completelydetermine Z (Φ). To avoid repetition, we simply state the final answer: Z (Φ) = A (Φ + Φ )Φ ( √ − ir )(Φ − )(3Φ − Φ ) . (4.73)Finally, to determine the polynomial A , we compute another star product: M α mon (cid:63) [Φ M α mon ] − (cid:20)(cid:18) Φ − ir √ (cid:19) M α mon (cid:21) (cid:63) M α mon . (4.74)At this point, we limit ourselves to the theory with N f seven-dimensional flavors of G .In this case, higher magnetic charges 2 α mon and β mon cancel from the above expression.The monopole of charge 2 α mon cancels because (4.74) is specifically constructed to ensureits cancellation, while the charge β mon cannot appear for dimensional reasons. Indeed, thedimensions of the lowest monopoles are∆ α mon = 2 N f − , ∆ β mon = 4 N f − . (4.75)The dimension of (4.74) is 2∆ α mon + 1 < ∆ β mon , so the monopole of dimension ∆ β mon cannotappear on the right. However, the dressed monopole of charge α mon appears, and demandingpolynomiality of its dressing factor determines A to be A ( x ) = 163 √ (cid:18) x r (cid:19) N f . (4.76)It is also not difficult to generalize to the case of a gauge group G = G × G (cid:48) , assumingthat the N f fundamentals of G transform in a representation R (cid:48) of G (cid:48) : A ( x ) = 163 √ (cid:89) w ∈R (cid:48) (cid:18) x r − ( w · Φ (cid:48) ) (cid:19) . (4.77)56 .5 General Case The detailed exploration of the lower-rank theories in the above subsections should give thereader a sense of what polynomiality-based computations look like. Further, it shows a clearpattern and allows us to formulate a strategy that should work for general gauge groups.To begin, one identifies the set of minimal monopoles that are expected to generate thealgebra. They can either be minuscule or bubble into the charge-zero sector (which we referto as “bubbling into the identity”). They cannot bubble into smaller nonzero charges, asthat would contradict their minimality. If all of them are minuscule, we are done: it onlyremains to make sure that they indeed generate everything, and to determine the relations.If there exists a minimal monopole of charge ω that is not minuscule, then it can bubbleinto the identity, and we should determine the corresponding abelianized bubbling factor Z (Φ). First, we use invariant theory to identify the set of primitive dressed monopoles ofcharge ω . Then we compute their star products with the quadratic Casimir f = (cid:80) ri =1 Φ i .By demanding polynomiality of the answer and using the operator mixing freedom, wealmost completely determine the bubbling factor Z (Φ), up to an unknown Weyl-invariantpolynomial A . These steps clearly work in an arbitrary gauge theory. The next step is themost challenging one: we need to construct a star product that determines the unknownpolynomial A . We have seen that at this step, sometimes A is uniquely determined, andsometimes it is only determined up to a sign, which is a harmless ambiguity that can berelated to a change of basis in A C .We consider the above procedure as strong evidence that polynomiality fully determinesthe algebra A C (if not a proof, at a physical level of rigor). It would still be desirable to finda more elegant and mathematically illuminating way to reach this conclusion. We now demonstrate the applications of our shift operator formalism in a number of simpleexamples. More elaborate examples can be found in the appendices.
In the commutative limit ( r → ∞ ), the quantum algebra A C reduces to the Coulomb branchchiral ring. Because finite- r computations, as shown above, allow one to determine bubblingcoefficients and thus A C in any theory, this provides a simple way to construct Coulomb57ranches even when other approaches face difficulties. However, finite- r computations canbe very hard, so it is convenient to first develop the commutative version of shift operators.This is the subject of this section, and the answer takes the form of abelianization as in [59].We begin by noting that the shift operator M bN from (2.25) has a well-defined r → ∞ limit. First, because the operator e − b · ( i ∂ σ + ∂ B ) acts on Φ by a shiftΦ (cid:55)→ Φ − ir b, (5.1)this shift vanishes in the r → ∞ limit, so that e − b · ( i ∂ σ + ∂ B ) no longer acts on Φ-dependentterms. Instead, it effectively turns into a generator of the group ring C [Λ ∨ w ] associated to thelattice of coweights (considered as an abelian group). Such generators, denoted by e [ b ], aresubject to the relations e [ b ] e [ b ] = e [ b + b ] . (5.2)Next, we observe that the Φ-dependent rational prefactor in the definition (2.25) of M bN alsohas a well-defined r → ∞ limit. Denoting the commutative limit of M bN by v b , we find thatit looks as follows: v b = (cid:81) w ∈R ( − iw · Φ) ( w · b ) + (cid:81) α ∈ ∆ ( − iα · Φ) ( α · b ) + e [ b ] . (5.3)This expression includes the case where some matter multiplets have masses, in which case R is considered to be a representation of both gauge and flavor groups, and some Φ’s areVEVs of the background vector multiplets (that is, masses). Note that (5.3) immediatelyimplies (2.30).This (5.3) is precisely as in [59], showing that we indeed recover their abelianization mapin the r → ∞ limit. A bonus of our formalism is that the abelianized bubbling coefficientsof Sections 3 and 4 are known, and also have a well-defined r → ∞ limit. Introducing thenotation z b → v (Φ) ≡ lim r →∞ Z ab b → v (Φ) (5.4)and the corresponding notation for the commuting abelianized monopole shift operator, (cid:101) v b ≡ lim r →∞ (cid:102) M b = v b + (cid:88) | u | < | b | z b → u (Φ) v u , (5.5) The ( w · b ) + in the exponent is not a typo. It was previously the lower index of a Pochhammer symbol,but in the commutative limit, it turns into a power. Note also that (5.3), as written, holds for semisimplegauge groups, for which the sum of the weights w vanishes. Otherwise, it should include an additional factorof r b Σ / where Σ is the sum of all U (1) charges of hypermultiplets in the theory, as can be seen by writing( w · b ) + = ( | w · b | + w · b ) /
58e conclude that the commuting versions of general physical dressed monopoles are givenby (cid:2) P (Φ) V b (cid:3) = 1 |W b | (cid:88) w ∈W P i (Φ w ) (cid:101) v w · b . (5.6)Let us consider a few examples of Coulomb branches determined using this technique. SU (2) with N f Fundamentals and N a Adjoints
In Section 4.3, we showed that in the SU (2) gauge theory with N f > N a of adjoints, the abelianized bubbling coefficient Z ab2 → (Φ) is a polynomial.Hence, up to operator mixing, we can take Z ab2 → (Φ) = 0. The same is then true for its r → ∞ limit, z → (Φ) = 0.When N f = 0, the bubbling term is a nontrivial rational function, Z → (Φ) = ( − r ) − N a Φ(Φ − ir ) . (5.7)However, we see that its r → ∞ limit is zero unless N a = 0. Hence we can again take z → (Φ) = 0, except in a pure gauge theory, which will be treated separately.Since the Cartan is one-dimensional, we write the Cartan-valued Φ simply as a complexnumber. The two primitive monopoles of minimal charge b = 2 in the commuting limit takethe form: v + v − = (cid:18) − i Φ2 (cid:19) N f ( i Φ) N a − ( e [2] + ( − N f e [ − , Φ( v − v − ) = Φ (cid:18) − i Φ2 (cid:19) N f ( i Φ) N a − ( e [2] − ( − N f e [ − . (5.8)In addition, we have the variable Φ . Define: U = 2 N f − ( v + v − ) , V = − i N f − Φ( v − v − ) , W = Φ . (5.9)The only relation between these variables follows from e [2] e [ −
2] = 1 and takes the form V + U W = W N f +2 N a − , (5.10)which is the defining equation of a D N f +2 N a singularity. According to Equation (4.16), thedimension of the lowest monopole operator is ∆ = N f + 2 N a −
2. We see that the theory59s good whenever N f + 2 N a >
2. Precisely for such values, (5.10) determines a cone. For N f + 2 N a = 2, U has dimension (or rather R-charge) zero, while for N f + 2 N a = 1, thatis, N a = 0 and N f = 1, the monopole has negative R-charge — in both of these cases, thetheory is bad and (5.10) is not a cone.It is also straightforward to include masses by turning on background VEVs for flavorsymmetries. In such a case, the bubbling term remains nontrivial in the r → ∞ limit, as weknow from (4.23), and is given by z → (Φ) = (cid:81) N a a =1 ( − M a ) (cid:81) N f i =1 ( − iM i )Φ (5.11)where M a and M i are the masses of the adjoint and fundamental hypers, respectively. Theexpressions for the commuting shift operators are also modified (as follows from coupling tothe background multiplet): v = (cid:81) N f i =1 (cid:0) − i Φ2 − iM i (cid:1) (cid:81) N a a =1 ( i Φ + iM a ) ( i Φ) e [2] ,v − = (cid:81) N f i =1 (cid:0) i Φ2 − iM i (cid:1) (cid:81) N a a =1 ( − i Φ + iM a ) ( i Φ) e [ − . (5.12)Using the variables U = 2 N f − ( v + v − + 2 z → (Φ)) , V = − i N f − Φ( v − v − ) , W = Φ (5.13)and the relation e [2] e [ −
2] = 1, we find: V W + U W − N a (cid:89) a =1 (cid:0) − M a (cid:1) N f (cid:89) i =1 ( − iM i ) = N f (cid:89) i =1 (cid:0) W − M i (cid:1) N a (cid:89) a =1 (cid:0) W − M a (cid:1) , (5.14)which at N a = 0 agrees with the result in [49] found by gauging U (1) top of the U (2) theory. SU (2)For a pure SU (2) gauge theory, the bubbling term in the commutative limit is z → (Φ) = 1Φ , (5.15)60o the abelianized shift operators are (cid:101) v ± = v ± + 1Φ . (5.16)The primitive monopoles take the form (cid:101) v + (cid:101) v − = − ( e [2] + e [ − , Φ( (cid:101) v − (cid:101) v − ) = −
1Φ ( e [2] − e [ − . (5.17)If we define U = 12 ( (cid:101) v + (cid:101) v − ) , V = 12 Φ( (cid:101) v − (cid:101) v − ) , W = Φ , (5.18)then we find that e [2] e [ −
2] = 1 implies the relation V = U W − U , (5.19)which does not belong to the series (5.10) and agrees with (A.9) in [49]. G with N f Fundamentals
To demonstrate the effectiveness of our formalism, we now discuss the theory with gaugegroup G and N f hypermultiplets in the seven-dimensional fundamental representation of G . Recall from Section 4.4.3 that the lattice of coweights is generated by a short coroot α mon and a long coroot β mon . At zero magnetic charge, there are two Casimir invariants f = Φ + Φ , f = Φ (Φ − ) , (5.20)and at magnetic charge α mon , there are six primitive dressed monopoles, which in the com-mutative limit give six primitive commutative monopoles: m = V α mon , m = [Φ V α mon ] , m = [Φ V α mon ] ,m = [Φ V α mon ] , m = [Φ V α mon ] , m = [Φ V α mon ] . (5.21)In Section 4.4.3, we found the abelianized bubbling factor Z (Φ) for “ α mon → r → ∞ limit is z (Φ) = 4 − N f (Φ + Φ ) N f (Φ − )(3Φ − Φ ) . (5.22)61e now have the ingredients in place to determine the chiral ring.We first observe from (4.75) that by taking the products m − m m , m m − m m , m m − m m ,m m − m m , m m − m m , m m − m m , (5.23)we can obtain all six primitive dressed monopoles of magnetic charge β mon . Because α mon and β mon are fundamental coweights, they obviously generate the rest of the charges. Fur-thermore, (5.23) implies that monopoles of charge β mon are generated from those of charge α mon . Therefore, the six monopoles m , . . . , m and two Casimirs f and f generate the fullchiral ring.It remains to determine their relations. They follow from the relations in C [Λ ∨ w ]: e [ α mon ] e [ − α mon ] = 1 ,e [ β mon + α mon ] e [ − β mon − α mon ] = 1 ,e [ β mon + 2 α mon ] e [ − β mon − α mon ] = 1 ,e [ β mon + α mon ] e [ α mon ] − e [ β mon + 2 α mon ] = 0 . (5.24)These relations can easily be seen to follow, in turn, from linear dependences between theshort (co)roots of G . Moreover, they generate a complete set of relations in C [Λ ∨ w ]: short(co)roots generate the full (co)weight lattice, and relations between the short (co)roots de-termine everything.Using the definition (5.3) of commuting shift operators, incorporating the abelianizedbubbling factor (5.22) according to (5.5) and (5.6), and using the relations (5.24), one canderive the relations L i = [ P i (Φ) V α mon ] + F i (Φ) ( i = 1 , , ,
4) (5.25)between the chiral ring generators, where L ≡ m + 12 m m − m m + 38 ( m − m m ) f ,L ≡ m m + m m − m m + 34 ( m m − m m ) f , (5.26) L ≡ m + 12 m m − m m + 316 ( m m − m m ) f −
964 ( m − m m ) f ,L ≡ m m − m m − m m f −
34 (3 m m − m m ) f + 916 (2 m m − m m ) f , P i and F i are N f -dependent polynomials in Φ that can be expressed in terms of knowngenerators. The simplest case is N f = 0, where most of the right-hand sides vanish:([ P i (Φ) V α mon ] + F i (Φ)) i =1 , , , = (cid:18) , , − m , (cid:19) . (5.27)For N f = 1, the answer is:([ P i (Φ) V α mon ] + F i (Φ)) i =1 , , , = (cid:18) m , m , − f m + 13 m , f m − m (cid:19) . (5.28)Note that we have not checked whether this is a complete set of equations, i.e., whether theCoulomb branch is a complete intersection, though it should be possible to do so from amore careful analysis of the relations. However, these equations are locally independent, sothe Coulomb branch is at least a local complete intersection. Having explained how our formalism can be used to derive the Coulomb branch chiral ringsof the gauge theories under study, we now turn to a more refined observable: the OPE ofthe Coulomb branch 1D sector. As explained in [36, 37], the OPE O i ( ϕ ) O j (0) ϕ → − −−−→ (cid:88) k c ijk O k (0) r ∆ i +∆ j − ∆ k (5.29)can be interpreted as a noncommutative star product O i (cid:63) O j = (cid:88) k c ijk O k r ∆ i +∆ j − ∆ k (5.30)on the chiral ring that reduces to ordinary commutative multiplication of the correspondingholomorphic functions as we take r → ∞ : O i (cid:63) O j | O ( r ) = O i O j = O j O i . (5.31)Here, O i ( ϕ ) denotes a twisted CBO on S ϕ , and the S radius r keeps track of differencesin conformal dimension. The topological property of cohomology classes of Q C ensures that(5.30) is position-independent. This star product has the interpretation as a quantizationof the ring of holomorphic functions on the Coulomb branch, with 1 /r serving as the quan-63ization parameter. In particular, the terms of order 1 /r in the OPE are interpreted asthe Poisson bracket of the holomorphic functions from the r → ∞ limit, induced by theholomorphic symplectic form on M C :[ O i , O j ] ∗ | O ( r − ) = {O i , O j } . (5.32) SU (2) with N f Fundamentals and N a Adjoints
To illustrate that the OPE indeed gives more information than the chiral ring, let us presentan example where distinct 3D theories have the same Coulomb branch chiral ring but differentstar products. Such an example was in fact already encountered in Section 5.1.1: it is the SU (2) gauge theory with N f fundamental and N a adjoint hypermultiplets. In the previoussubsection, we showed that the Coulomb branch is a D N f +2 N a singularity, so it depends onlyon the combination n = N f + 2 N a . We now show that the OPE does not depend only onthis combination, so that for any fixed n , we obtain (cid:100) n/ (cid:101) distinct quantizations of the ringof holomorphic functions on the cone over the D n singularity.We restrict to the case N f + 2 N a >
2, where the theory is good, and to N f >
0, whereall bubbling coefficients can be set to zero. The operators of dimension ∆ U = N f + 2 N a − V = N f + 2 N a −
1, and ∆ W = 2 whose flat-space limits are given in (5.13) are U = 2 N f − ( M + M − ) , V = − i N f − Φ( M − M − ) , W = Φ . (5.33)Using the corresponding shift operators obtained from (4.5), we then find V + U (cid:63) W (cid:63) U = P ( W ) + 2 r U (cid:63) V , (5.34)where all products are understood to be star products and P ( W ) = W N f +2 N a − + O (1 /r ) isthe following polynomial in W : P ( W ) ≡ (cid:16) √W + ir (cid:17) (cid:16) √W + ir (cid:17) N f − (cid:104)(cid:16) √W + i r (cid:17) (cid:16) √W + i r (cid:17)(cid:105) N a √W + ( i ↔ − i ) (5.35)(despite appearances, this expression is indeed a polynomial). To leading order in 1 /r , wereproduce (5.10). We can also compute various OPEs such as the antisymmetrized OPEs of64he Coulomb branch chiral ring generators (5.33):[ U , W ] (cid:63) = 4 r V − r U , [ V , W ] (cid:63) = − r W (cid:63) U − r V , [ U , V ] (cid:63) = − r U + Q ( W ) , (5.36)where Q ( W ) is a polynomial in W given by Q ( W ) ≡ i (cid:16) √W + ir (cid:17) N f − (cid:104)(cid:16) √W + i r (cid:17) (cid:16) √W + i r (cid:17)(cid:105) N a √W + ( i ↔ − i ) (5.37)(this expression is again a polynomial in W , despite its appearance).We see that (5.34) and (5.36) do not depend only on the combination N f + 2 N a thatdetermines the Coulomb branch, thus providing an example of different quantizations of thesame chiral ring. Note, however, that the 1 /r terms in (5.36), like the chiral ring relation(5.10), do depend only on N f + 2 N a : thus the Poisson structure on D N f +2 N a is the same forall of the distinct quantizations.For other examples where our formalism can be used to determine the quantization ofthe Coulomb branch chiral ring, see Appendix E. G with N f Fundamentals
Let us make a few comments on the theory with gauge group G , which appeared as one ofour earlier examples. At the very least, the same two Casimirs and six primitive monopolesof minimal charge α mon are expected to generate the noncommutative algebra A C : f = Φ + Φ , f = Φ (Φ − ) ,m = M α mon , m = [Φ M α mon ] , m = [Φ M α mon ] ,m = [Φ M α mon ] , m = [Φ M α mon ] , m = [Φ M α mon ] . (5.38)They satisfy the same relations as in (5.25), (5.26), with the left-hand side written in termsof the star product and the right-hand side receiving 1 /r corrections. One may ask, however, whether a change of basis for the generators U , V , and W could render (5.34)and (5.36) dependent only on N f + 2 N a . For changes of basis where we only allow ourselves to redefineeach operator by adding operators of strictly lower dimension multiplied by appropriate factors of 1 /r , it isimpossible to make (5.34) and (5.36) depend only on N f + 2 N a . A C as a noncommutative algebra, or alternatively (but not equivalently ingeneral), as a commutative Poisson algebra. Namely, we find that m i (cid:63) f − f (cid:63) m i = − ir √ m i +1 − r m i , i = 0 , . . . , , (5.39)implying that it is enough to have f , f , and m to generate the rest of the algebra throughstar products. The above equation also implies the Poisson bracket { m i , f } = − i √ m i +1 . Inorder to compute star products, we must use the bubbling factor derived in Section 4.4.3. We now demonstrate the utility of the shift operator formalism for computing correlationfunctions of twisted CBOs, with applications to non-abelian 3D mirror symmetry [78–80].We first review the general setup for the computation of correlation functions before givingan example.
The three ingredients for computing correlation functions are the vacuum hemisphere wave-function, the gluing measure, and the shift operators. The vacuum hemisphere wavefunctionΨ ( σ, B ) (where σ is valued in the Cartan of g and B in the coweight lattice) can be readoff from (2.15) by setting b = 0:Ψ ( σ, B ) ≡ Z ( (cid:126) σ, B ) = δ B,(cid:126) (cid:81) w ∈R √ π Γ( − iw · σ ) (cid:81) α ∈ ∆ 1 √ π Γ(1 − iα · σ ) , (5.40)where R denotes the weights of the hypermultiplet representation R of G and ∆ denotesthe roots of G . The gluing measure µ ( σ, B ) is as in (2.7), namely µ ( σ, B ) = (cid:89) α ∈ ∆ + ( − α · B (cid:34)(cid:16) α · σr (cid:17) + (cid:18) α · B r (cid:19) (cid:35) (cid:89) w ∈R ( − | w · B |− w · B Γ (cid:16) + iw · σ + | w · B | (cid:17) Γ (cid:16) − iw · σ + | w · B | (cid:17) (5.41)(note that (cid:81) α ∈ ∆ + ( − α · B = e πiρ · B where ρ is the Weyl vector). The shift operators aregiven by (3.18) combined with (3.21) (see also Appendix A). Without loss of generality, we66ork in the North picture, where M bN = (cid:81) w ∈R (cid:104) ( − ( w · b )+ r | w · b | / (cid:0) + irw · Φ N (cid:1) ( w · b ) + (cid:105)(cid:81) α ∈ ∆ (cid:104) ( − ( α · b )+ r | α · b | / ( irα · Φ N ) ( α · b ) + (cid:105) e − b · ( i ∂ σ + ∂ B ) , Φ N = 1 r (cid:18) σ + i B (cid:19) , (5.42)and therefore drop the N subscripts.With these ingredients, the matrix model expression for the correlator of twisted CBOs O i ( ϕ i ), i = 1 , . . . , n , inserted at points ϕ i obeying 0 < ϕ < · · · < ϕ n < π , takes the form ofan inner product (see also [38]) (cid:104)O ( ϕ ) · · · O n ( ϕ n ) (cid:105) S = 1 |W| Z S (cid:88) (cid:126)B (cid:90) d(cid:126)σ µ ( (cid:126)σ, (cid:126)B )Ψ ( (cid:126)σ, (cid:126)B ) (cid:98) O · · · (cid:98) O n Ψ ( (cid:126)σ, (cid:126)B ) , (5.43)where (cid:98) O i are the shift operators corresponding to O i and Z S is the vacuum S partitionfunction by which we divide to obtain a normalized correlator: Z S = 1 |W| (cid:88) (cid:126)B (cid:90) d(cid:126)σ µ ( (cid:126)σ, (cid:126)B )Ψ ( (cid:126)σ, (cid:126)B ) . (5.44)This is a special case of the gluing formula of Section 2.2. From here on, we drop the hatson the shift operators and therefore do not make a notational distinction between a shiftoperator and the twisted CBO that it represents. N = 8 Example
As a concrete example, let us consider the U ( N c ) gauge theory with one adjoint hypermul-tiplet and one fundamental hypermultiplet. This theory has N = 8 SUSY enhancement(being IR dual to N = 8 U ( N c ) SYM) and is therefore self-mirror [81]. This theory is uglyin the sense of Gaiotto and Witten [30], so the monopoles of lowest dimension saturate theunitarity bound ∆ = 1 / S partition function is Z S = 1 N c ! (cid:90) N c (cid:89) I =1 dσ I (cid:81) I 2) and tr Φ, M ± (1 , , M (1 , − , M ( ± , (with ∆ = 1). Particular linear combinations of these operators comprise the chiralring generators, namely M ± (1 , , M ± (1 , , and −M (1 , − − i tr Φ. They satisfy the single The adjoint hyper contributes a sign ( − | B IJ | = ( − ( B IJ ) + +( − B IJ ) + to the (cid:81) I 0) in U ( N c )SQCD with N a = 1 and N f ≥ 1, but for N f > 1, the insertion on the Coulomb branch side does not simplifyso easily, and correlators on the Higgs branch side also become difficult to compute. N = 4 theory on asphere rather than by studying it in an Ω-background: the latter route to quantization hasa less straightforward connection to SCFT operators.Finally, on our way to achieving these goals, we gained an improved understanding ofmonopole bubbling phenomena, which are crucial nonperturbative effects in the descriptionof magnetic defects. Our approach to bubbling is purely algebraic in nature, based onsymmetries and algebraic consistency of the OPE. It avoids the technicalities of previousanalytic bubbling computations [53,54,56,58], which involve equivariant integration over themoduli space of bubbling solutions to the Bogomolny equation, therefore serving as a goodcheck and testing ground for them. Our approach further allows for the determination ofpreviously unknown bubbling coefficients, such as in theories without minuscule monopoles.While the focus of this paper is mostly on developing the general formalism, we alsoprovide some explicit applications and examples in theories of small rank. In Section 4, wederive the “abelianized bubbling coefficients” for a large family of rank-one and rank-twogauge theories, which can be used to extract data on the Coulomb branch operators of thesetheories (including the algebra A C and its correlators) in a completely straightforward andalgorithmic fashion. We then illustrate these results in Section 5. While the abelian exam-ples in [38] provide quantizations of A N singularities, we present in Section 5 the exampleof SU (2) gauge theory with fundamental and adjoint matter, resulting in many inequiva-lent quantizations of the D N singularity. For the purpose of illustration, we also apply ourformalism to the G gauge theory, as no other techniques are available in this case. Look-ing ahead, shift operators provide a method to potentially determine previously unknownCoulomb branch chiral rings and their quantizations, such as those of bad theories. Finally,we use our matrix model for correlation functions of twisted CBOs to derive, in some cases,how Higgs and Coulomb branch chiral ring operators map across non-abelian 3D mirrorsymmetry. Using our formalism, we are able to derive the precise normalization factors inthe mirror map and distinguish operators that could mix on the basis of symmetries. Furtherapplications are gathered in the appendices. At a computational level, there exist numerous directions in which the discussion of Section5.3 could be generalized. The most well-known families of mirror theories are those of ADE type [78], as well as the higher-rank counterparts of those of A - and D -type [79, 80]. In [38],the abelian A series (i.e., the mirror duality between SQED N f and the affine A N f − quiver)71as analyzed in our formalism, resulting in a derivation of the precise mirror map as arefinement of the known mapping of charge matrices [80]. The self-mirror duality consideredin Section 5.3.2 is only a special case of the non-abelian generalizations of the A -type mirrorsymmetries derived in [79]. Aside from the ADE examples that we have not investigated,more examples can be found in [82], and further examples can be generated via the procedureof [83], in the same spirit as the constructive approach of [84] to abelian mirror symmetry.It remains to be seen what further lessons for 3D mirror symmetry can be extracted fromour formalism.Besides further applications of our formalism to gather more data on various 3D N = 4gauge theories (in particular SCFTs), or to check or discover new dualities, there are anumber of conceptual questions that present interesting avenues for future work: • It would be interesting to extend our construction to more general gauge theories, namelygauge theories that also have charged matter in half-hypermultiplets, those that involveboth ordinary and twisted multiplets at once, and/or theories with Chern-Simons cou-plings. Understanding the moduli spaces of vacua, their quantization, and the corre-sponding correlation functions in such theories, if possible, are among the outstandingquestions to address. • It would be interesting to compare the bubbling terms obtained using our method tothose coming from the dimensional reduction of the 4D bubbling terms computed in [54].We performed a few preliminary comparisons (summarized in Appendix D) and foundthat the two agree up to operator mixing and various normalization factors, but a moresystematic study is needed. Furthermore, it has been observed (already in [54], and laterin [56]) that the results of [54] sometimes involve discrepancies with those obtained usingthe AGT correspondence, particularly in 4D N = 2 superconformal QCD ( SU ( N c ) with N f = 2 N c ). A fix was recently proposed in [58]. Based on our preliminary checks, itappears that all of the subtleties in 4D involving integration over monopole moduli spacedisappear upon reduction to 3D, and it would be nice to understand why. Likewise, therelation of our construction to the Moyal product of [54] and its implications for the lineoperator OPE in 4D remain to be understood. • It would be interesting to recast our construction of shift operators and bubbling coef-ficients (or equivalently, abelianization) in a way that uses the mathematical definition Namely, the U ( N c ) n necklace quiver with v i ≥ U ( N c ) v necklace quiver where v = (cid:80) i v i and for every i , there is a fundamental charged under the j th gauge group where j ( i > 1) = (cid:80) i − (cid:96) =1 v (cid:96) and j (1) = (cid:80) n(cid:96) =1 v (cid:96) . 72f the Coulomb branch [60–63]. It could also be of interest to understand whether theabelianized bubbling terms introduced in this work, which provide an algebraic decompo-sition of the Weyl-averaged bubbling terms considered heretofore in the literature, have acorresponding geometric interpretation in terms of a decomposition of monopole modulispace. • It would be interesting to understand more conceptually whether there exists a relationbetween quantization on S and quantization via the Ω-background [42–44] (see also [85]).Similar relations are abundant in various dimensions for problems involving a supercharge(equivariant differential) Q such that Q is a vector field with fixed points. See, forinstance, the recent work [86] for the case of isolated fixed points. • More broadly, our work fits into the larger program of constructing and classifying de-formation quantizations arising from 3D N = 4 quantum field theories. While ourconstruction is certainly derived starting from a Lagrangian description, one may won-der whether it can be generalized to non-Lagrangian theories (such as various classes ofSCFTs from [87]), and/or whether Lagrangian theories play a special role in the broaderclassification program of deformation quantizations. Acknowledgements We thank T. Daniel Brennan, Anindya Dey, Tudor Dimofte, Pavel Etingof, Amihay Hanany,Petr Kravchuk, Dominik Miketa, Hiraku Nakajima, and Takuya Okuda for various discus-sions and correspondence. YF thanks the members of QMAP, UC Davis for their hospitality.The work of MD was supported by the Walter Burke Institute for Theoretical Physics andthe U.S. Department of Energy, Office of Science, Office of High Energy Physics, underAward No. de-sc0011632, as well as the Sherman Fairchild Foundation. The work of YF wassupported in part by the NSF GRFP under Grant No. DGE-1656466 and by the GraduateSchool at Princeton University. The work of SSP was supported in part by the US NSFunder Grant No. PHY-1820651, by the Simons Foundation Grant No. 488651, and by anAlfred P. Sloan Research Fellowship. The work of RY was supported in part by a grantfrom the Israel Science Foundation Center for Excellence, by the Minerva Foundation withfunding from the Federal German Ministry for Education and Research, and by the ISFwithin the ISF-UGC joint research program framework (grant no. 1200/14). We thank T. Dimofte for this last remark. We thank P. Etingof for a discussion about this topic. Conventions Here, we summarize our conventions and notation. Unless otherwise stated, G is assumedto be a simple gauge group, g its Lie algebra, t its Cartan subalgebra, and t C = t ⊗ C its complexification. The root system is denoted by ∆, the weight lattice by Λ w , and thecoweight lattice (the weight lattice of L G ) by Λ ∨ w . The matter representation is R ⊕ R .The abelian North pole shift operator (2.25) is denoted by M bN , while its South poleanalog is denoted by M bS . These operators do not incorporate monopole bubbling effects.Sometimes, we simply write M b , in which case it is assumed to be the North pole operator.Here, b ∈ Λ ∨ w is a coweight of G . The abelianized shift operator (including bubbling) isdenoted by (cid:102) M bN , with the same remark concerning N/S: (cid:102) M b = M b + (cid:88) | v | < | b | Z ab b → v (Φ) M v , (A.1)where the sum is over coweights shorter than b and Z ab b → v (Φ) are abelianized bubbling coef-ficients. The commutative ( r → ∞ ) limit of the shift operator M bN is denoted by v b , andthe same for M bS , as the N/S distinction disappears in the commutative limit. Similarly,the abelianized bubbling factor in this limit is denoted by z b → v (Φ), and the abelianizedcommutative shift operator is: (cid:101) v b = v b + (cid:88) | u | < | b | z b → u (Φ) v u . (A.2)We deal with a number of objects that involve sums over Weyl orbits. If a quantity F ( b )depends on the coweight b , then we employ the following convention in summing over itsWeyl orbit: (cid:88) b (cid:48) ∈W b F ( b (cid:48) ) ≡ |W b | (cid:88) w ∈W F (w · b ) , (A.3)where W b ⊂ W is the stabilizer of b . In particular, we use it to define the Weyl-averagedshift operator, the bare monopole operator, and the dressed monopole operator: M b = (cid:88) b (cid:48) ∈W b M b (cid:48) , M b = (cid:88) b (cid:48) ∈W b (cid:102) M b (cid:48) = 1 |W b | (cid:88) w ∈W M w · b + (cid:88) | v | < | b | Z ab b → v (Φ w ) M w · v , P (Φ) M b ] = 1 |W b | (cid:88) w ∈W P (Φ w ) M w · b + P (Φ w ) (cid:88) | v | < | b | Z ab b → v (Φ w ) M w · v . (A.4)As explained in the main text, Φ w = w − · Φ, and Φ takes values in t C = t ⊗ C . If a monopoleof GNO charge b cannot bubble, then M b = M b . The Weyl-averaged shift operator M b defined here does not appear in the main text, but it plays a certain role in the appendices.The dressed commuting monopole operator is defined as:[ P (Φ) V b ] = 1 |W b | (cid:88) w ∈W P (Φ w ) (cid:101) v w · b . (A.5)Using the transformation property Z abw · b → w · v (Φ) = Z ab b → v (Φ w ) , (A.6)we might also introduce Z b → v mono (Φ) = (cid:88) b (cid:48) ∈W b Z ab b (cid:48) → v (Φ) , (A.7)so that the bare monopole becomes M b = M b + (cid:88) | v | < | b | Z b → v mono (Φ) M v . (A.8)In the appendices, we sometimes omit brackets [] around dressed monopoles when no risk ofconfusion is present. B Twisted-Translated Operators Here is a brief, qualitative review of twisted operators and their corresponding topologicalsectors. Let us first recall the setup in R .In 3D N = 4 SCFTs, half-BPS operators are labeled by their charges (∆ , j, j H , j C ) underthe bosonic subalgebra so (3 , ⊕ su (2) H ⊕ su (2) C of the 3D N = 4 superconformal algebra osp (4 | j = 0) and can be classified as either HBOs (∆ = j H , j C = 0) or CBOs (∆ = j C , j H = 0), which we write abstractly with su (2) H/C spinor indicesas O ( a ··· a jH ) and O ( ˙ a ··· ˙ a jC ) . Hence su (2) H and su (2) C are spontaneously broken on theHiggs and Coulomb branches, respectively. In a Lagrangian theory, the vector multipletcontains adjoint scalars Φ ˙ a ˙ b in the triplet of su (2) C and the hypermultiplet contains scalars75 a , ˜ q a in the doublet of su (2) H and in R , R of G . Then HBOs are precisely gauge-invariantpolynomials in q a , ˜ q a while CBOs consist of Φ ˙ a ˙ b and (dressed) monopole operators M b ˙ a ··· ˙ a jC .The key fact is that twisted HBOs/CBOs, defined as O ( x ) = u a ( x ) · · · u a jH ( x ) O a ··· a jH ( x ) , O ( x ) = v ˙ a ( x ) · · · v ˙ a jC ( x ) O ˙ a ··· ˙ a jC ( x ) (B.1)with appropriate position-dependent R-symmetry polarization vectors u and v , have topolog-ical correlation functions when the coordinate x is restricted to a line in R . This is becausethey represent equivariant cohomology classes of certain supercharges Q H/C ∈ osp (4 | Q H/C ) , and operators O ( x ) at different x are relatedby Q H/C -exact operations called twisted translations . Hence the Q H/C -cohomology class ofa twisted-translated operator O ( x ) is independent of its position x along the line. It followsthat each supercharge Q H/C has an associated 1D topological sector of cohomology classes:the OPE of these twisted HBOs/CBOs is an associative but noncommutative product, sincethere exists an ordering along the line.The setup on S , where we localize with respect to Q C , is essentially the same (up tosubtleties involving the “branch point” at infinity, discussed at length in [38]): the distin-guished line is stereographically mapped to a great circle S ϕ , so that twisted operators areparametrized by ϕ rather than x in (B.1), and the deformation parameter r (implicit in thedefinitions of Q H/C ) becomes the S radius. Taking g YM → ∞ at fixed r gives an SCFT on S whose correlators are equivalent to those of the IR SCFT in flat space by stereographic pro-jection. The non-conformal 3D N = 4 superalgebra s contains the su (2) (cid:96) ⊕ su (2) r isometriesof S as well as u (1) (cid:96) ⊕ u (1) r R-symmetries. The supercharges Q H/C each contain terms fromboth su (2 | 1) factors of s , as required by the fact that they square to isometries with nontrivialfixed points. The corresponding twisted translations take the form P ϕ + R H/C = {Q H/C , . . . } .Finally, the embedding of s into osp (4 | u (1) R-symmetries into su (2) H and su (2) C .These twisted operators are interesting for at least two reasons: • Their two- and three-point functions fix those of HBOs and CBOs in the full 3D theory,by conformal symmetry and R-symmetry (roughly, conformal symmetry suffices to putany two or three operators on a great circle). • At any fixed ϕ , twisted operators in the cohomology of Q H/C are in one-to-one corre-spondence with elements of the Higgs/Coulomb branch chiral ring. The R-symmetrypolarization vector u or v fixes a complex structure on the corresponding branch, so that76he operators are chiral with respect to an N = 2 superconformal subalgebra of osp (4 | Q C -cohomology classes, namely: thetwisted scalar Φ( ϕ ) = v ˙ a ( ϕ ) v ˙ b ( ϕ )Φ ˙ a ˙ b ( ϕ ), twisted bare monopoles M b ( ϕ ), and twisted dressedmonopoles [ P (Φ) M b ( ϕ )] (composite operators formed by monopoles and scalars). C Matrix Nondegeneracy and Abelianized Bubbling Here, we prove that the matrix determining the linear system (3.20) is nondegenerate, thusimplying that (3.20) has a unique solution.The Weyl group might not act freely on the orbit of a general (dominant) coweight b ,meaning that |W b | = dim C ( ρ b ) < |W| . Each w · b ∈ W b has a possibly nontrivial stabilizerStab w · b ≡ W w · b ⊂ W , and as a result, (cid:102) M w · b in Equation (3.20) is multiplied by (cid:80) w (cid:48) ∈ Stab w · b P i (Φ w (cid:48) w ). For brevity, let us denote P i (Φ w ) averaged over Stab w · b by P i (Φ w ). Let us alsopick representatives w , . . . , w dim( ρ b ) of classes in W / Stab b , so that the basis of ρ b is givenby M b = M w · b , M w · b , . . . , M w dim( ρb ) · b (we assume that w = id represents the trivial class).Then Equation (3.18) can be written in matrix form as M b (cid:2) P M b (cid:3) ... (cid:2) P dim( ρ b ) M b (cid:3) = |W b | − P (cid:102) M w · b (cid:102) M w · b ... (cid:102) M w dim( ρb ) · b , (C.1)where P = P (Φ w ) P (Φ w ) P (Φ w ) · · · P (Φ w dim( ρ b) ) P (Φ w ) P (Φ w ) P (Φ w ) · · · P (Φ w dim( ρ b) )... ... ... . . . ... P dim( ρ b ) (Φ w ) P dim( ρ b ) (Φ w ) P dim( ρ b ) (Φ w ) · · · P dim( ρ b ) (Φ w dim( ρ b) ) . (C.2)In fact, this matrix is nondegenerate, meaning that its determinant is given by a polynomialin Φ that is not identically zero, as we now show. By construction, |W b | − (cid:80) w ∈W P i (Φ w ) M w · b for i = 1 , . . . , dim( ρ b ) form a basis over C [ t ] W . This implies that the rows of the matrix P are linearly independent over C [ t ] W , i.e., over Weyl-invariant polynomials.Let us assume that the matrix is nonetheless degenerate: this means that one of the rows,77ay the j th row, is a linear combination of the other rows with coefficients being rational andgenerally non-Weyl-invariant functions Q i : P j (Φ w a ) = (cid:88) i =1 ,..., dim( ρ b ) i (cid:54) = j Q i (Φ) P i (Φ w a ) for all a = 1 , . . . , dim( ρ b ) . (C.3)This should hold as an identity for all a = 1 , . . . , dim( ρ b ), with Q i (Φ) independent of a .Acting with an element of the Weyl group w ∈ W on (C.3) should give another valid identityfor all a = 1 , . . . , dim( ρ b ). On the other hand, doing so simply permutes the columns of P .Thus it permutes the equations in (C.3), at the same time replacing Q i (Φ) by Q i (Φ w ). Doingthis for every element of W and averaging implies that we can replace Q i (Φ) in (C.3) byits Weyl-averaged version. So we may assume that Q i are Weyl-invariant rational functions.Every such Q i is a ratio Q i (Φ) = A i (Φ) B i (Φ) (C.4)of polynomials A i and B i . Let us consider D (Φ) = (cid:89) i =1 ,..., dim( ρ b ) i (cid:54) = j B i (Φ) , (C.5)which is the common denominator (not necessarily the minimal one) of all the Q i . Even ifthis D i is not Weyl-invariant, one can define another polynomial that is: D W (Φ) = (cid:89) w ∈W D (Φ w ) . (C.6)If we now multiply relation (C.3) by this D W (Φ), we obtain: D W (Φ) P j (Φ w a ) = (cid:88) i =1 ,..., dim( ρ b ) i (cid:54) = j D W (Φ) Q i (Φ) P i (Φ w a ) for all a = 1 , . . . , dim( ρ b ) . (C.7)This cancels all denominators of Q i . Furthermore, since both Q i (Φ) and D W (Φ) are Weyl-invariant, their product D W (Φ) Q i (Φ) is a Weyl-invariant polynomial. So (C.7) says that therows of P are linearly dependent over the ring of Weyl-invariant polynomials C [ t ] W . This isa contradiction, which proves that the matrix P is nondegenerate.78aving proven that P is nondegenerate, we can solve (3.18): (cid:102) M w · b (cid:102) M w · b ... (cid:102) M w dim( ρb ) · b = |W b | P − M b (cid:2) P M b (cid:3) ... (cid:2) P dim( ρ b ) M b (cid:3) . (C.8)Since the Weyl group simply permutes the columns of P , it is enough to have an expressionfor (cid:102) M b , with all other (cid:102) M w · b obtained as Weyl images thereof. Remembering that the leadingterm in (cid:2) P i (Φ) M b (cid:3) takes the form |W b | − (cid:80) w ∈W P i (Φ w ) M w · b , we write the solution as: (cid:102) M b = M b + dim( ρ b ) (cid:88) i =1 ( P − ) i (cid:88) | v | < | b | (cid:88) w ∈W V b → vi (Φ w ) M w · v . (C.9)Introducing the notation Z ab b → v (Φ) for the second term, the solution takes the form (3.21). D Bubbling Coefficients from 4D In U ( N ) gauge theories, there exist known methods for computing monopole bubbling co-efficients in the 4D N = 2 context [54], the results of which can be used to infer bubblingcoefficients in the corresponding 3D N = 4 theories in a specific basis. However, the na¨ıve di-mensional reduction prescription sometimes requires supplementing these known results withnontrivial normalization factors to ensure polynomiality, at least for monopoles of sufficientlyhigh charge. Here, we comment on these subtleties, leaving a more complete understandingfor future work. D.1 The IOT Algorithm A systematic procedure for computing monopole bubbling coefficients relevant to the lineoperator index of 4D N = 2 U ( N ) gauge theories with fundamental or adjoint hypermul-tiplets was developed in [54] and adapted to S × S in [55]. We refer to it as the “IOTalgorithm.” It produces a function that we call Z b → v mono, IOT as follows. Consider the quantity Z R mono ( b, v ) defined in [55]: with all flavor symmetry fugacities η i set to 1, it is a functionof the thermal fugacity x (related to the size of the thermal circle by x = p = e − β ) and the Alternatively denoted by Z N mono ( b, v ) or Z S mono ( b, v ), the square of which is Z S mono ( b, v ) in [55]. λ i (related to ours by λ i = βσ i ). We set Z b → v mono, IOT ( β, σ ) ≡ Z R mono ( b, v ; x, λ, η = 1) . (D.1)The bubbling coefficients Z b → v mono, IOT , and those considered in previous literature on 4D N = 2theories, share the property that they are Weyl-invariant in b but not in v : Z b → v mono, IOT ( β, σ ) = Z w · b → v mono, IOT ( β, σ ) = Z b → w · v mono, IOT ( β, w · σ ) . (D.2)Hence they should be identified with our abelianized bubbling coefficients, which obey (2.19)and are Weyl-invariant with respect to neither b nor v , only after Weyl-averaging over b asin (A.7). As described in Section 2.6, this identification involves a dimensional reductionin which we keep only the leading term in Z b → v mono, IOT as a power series expansion in β ,which occurs at order β ∆ b − ∆ v . We then make an appropriate substitution β → − i/r (withthe constant of proportionality determined empirically) to restore dimensions. However, our Z b → v mono is a function of Φ that is defined to multiply monopole shift operators on the left, while Z b → v mono, IOT as given above is a function of σ . To adjust for this discrepancy, we substituteΦ for σ after subtracting the B -dependent term evaluated in the appropriate monopolebackground, i.e., the flux created by the charge- v monopole that the bubbling coefficientmultiplies. In the end, we arrive at the prescription Z b → v mono (Φ) = Z b → v mono, IOT ( β, σ ) | O ( β ∆ b − ∆ v ) ,β →− i/r,σ → r Φ − iv/ (D.6) To illustrate the difference between Weyl-averaged and abelianized bubbling, consider a monopole whosecharge is a simple coroot and that can bubble only into the identity: M b = M b + Z b → idmono (Φ) . (D.3)The bar denotes Weyl averaging, and the bubbling term is a Weyl-symmetric rational function: Z b → idmono (Φ) = Z b → idmono (Φ). If Z b → idmono (Φ) were the whole story, then it would be natural to guess that[ P (Φ) M b ] ! = P (Φ) M b + P (Φ) Z b → idmono (Φ) (D.4)where P (Φ) is a (not necessarily Weyl-invariant) polynomial. However, such a guess turns out to be incon-sistent with polynomiality of the Coulomb branch algebra. Properly defining a dressed monopole insteadrequires a more elementary bubbling term Z ab b → id (Φ) satisfying Z ab b → id (Φ) = Z b → idmono (Φ):[ P (Φ) M b ] = P (Φ) M b + P (Φ) Z ab b → id (Φ) , (D.5)where in general, Z ab b → id (Φ) (cid:54) = Z b → idmono (Φ). Z b → v mono, IOT to our Weyl-averaged bubbling coeffi-cients Z b → v mono . It should be kept in mind that the Weyl-averaged bubbling coefficients (D.6)computed via the IOT algorithm single out a basis of Coulomb branch operators, whosestar algebra must then be consistent with polynomiality. By contrast, our polynomiality-based approach, which applies to a far more general class of theories, determines bubblingcoefficients up to a choice of basis. D.2 Bubbling Patterns It turns out that (D.6) is not always sufficient to guarantee that the resulting 3D monopolessatisfy polynomiality. We conjecture that in general, this prescription should be supple-mented by signs and combinatorial factors unrelated to Weyl symmetrization, the latter ofwhich involve dividing by an integer that depends on the number of simple roots subtractedto get from the bare monopole b to the bubbled monopole v . The pattern of combinatorialfactors depends on the “depth” to which the b monopole can bubble, i.e., the maximum ofthe number of simple roots that must be subtracted from b to obtain any monopole chargeto which b can bubble. On the other hand, the possible dressing signs depend on the rankand matter content of the theory, as well as on the monopole charges. Schematically, we seefrom a number of examples that one possibility for the general bubbling pattern is M b = M b + (cid:15) , c , (cid:88) v ∈W v Z b → v mono (Φ) M v , (D.7) M b = M b + (cid:32) (cid:15) , c , (cid:88) v ∈W v + (cid:15) , c , (cid:88) v ∈W v (cid:33) Z b → v mono (Φ) M v , (D.8) M b = M b + (cid:32) (cid:15) , c , (cid:88) v ∈W v + (cid:15) , c , (cid:88) v ∈W v + (cid:15) , c , (cid:88) v ∈W v (cid:33) Z b → v mono (Φ) M v (D.9)at depth 1 , , 3, respectively. Here, the subscript i on v i indicates the number of simple rootsby which it differs from b , the c i,j are positive integers with c , = c , = c , = 1 , | c , , | c , , | c , , (D.10)and (cid:15) i,j ∈ {± } . It would be interesting to determine the pattern for arbitrary depth, butwe are limited to relatively small charges b by our implementation of the IOT algorithm. The IOT algorithm for U ( N ) constructs Z b → v mono, IOT as a sum of functions labeled by N -tuples of Youngdiagrams, and the bottleneck lies in enumerating these diagrams. Namely, given b, v in the Cartan of U ( N ) D.3 Examples D.3.1 U (2) with N a = N f = 1We start with U (2) SQCD with one adjoint and one fundamental hypermultiplet, whichappears as a prototypical example throughout these appendices. Our conventions for U ( N )are as in Section 5.3.2. For convenience, we first record some useful star products of CBOs(to be derived below). The free sector is generated by the ∆ = 1 / M ( ± , , whosequadratic star products generate the ∆ = 1 operators M ( ± , and M (1 , − :( M ( ± , ) = M ( ± , , (D.12) M ( − , (cid:63) M (1 , = M (1 , (cid:63) M ( − , + 2 r = M (1 , − . (D.13)On the other hand, the ∆ = 1 operators tr Φ and M ± (1 , satisfy the quadratic relations M ± (1 , (cid:63) M ± (1 , = M ± (2 , , (D.14) M ± (1 , (cid:63) M ∓ (1 , = (cid:18) r ± i Φ (cid:19) (cid:18) r ± i Φ (cid:19) = 14 r ± i r tr Φ + 12 [tr Φ − (tr Φ) ] , (D.15)[tr Φ , M ± (1 , ] (cid:63) = ± ir M ± (1 , , (D.16)where we have defined the commutator [ · , · ] (cid:63) with respect to the star product. In the mixedsector, we have the relations M ± (1 , (cid:63) M ( ± , = M ( ± , (cid:63) M ± (1 , = M ± (2 , , (D.17) M ± (1 , (cid:63) M ( ∓ , ± r M ( ± , = − i [Φ M ( ± , ] , (D.18) M ( ∓ , (cid:63) M ± (1 , ∓ r M ( ± , = − i [Φ M ( ± , ] , (D.19) where b bubbles into v , we define the matrix K = diag( K , . . . , K k ) byTr e πibν = Tr e πivν + ( e πiν + e − πiν − 2) Tr e πiKν , (D.11)where ν is a dummy variable. This condition can be understood by fixing a Weyl ordering of v and orderingthe entries of K from greatest to least. Then the algorithm requires finding all N -tuples of Young diagramswith k boxes in all, colored with the numbers s = 1 , . . . , k , such that K s = v α ( s ) + j s − i s where α ( s ) = 1 , . . . , N labels the diagram to which s belongs and i s , j s are the row and column positions. 82s well as the miscellaneous relations M (1 , − (cid:63) M (1 , = M (1 , (cid:63) M (1 , − + 2 r M (1 , = M (2 , − , (D.20) M (2 , (cid:63) M (1 , = ( M (1 , ) = M (3 , , (D.21) M (1 , − (cid:63) M (1 , − = M (2 , − − r M (1 , − . (D.22)Implicit in the above relations is a choice of basis (monopole bubbling coefficients), whichwe now specify.In U (2) gauge theory, a monopole of charge ( b , b ) can bubble if and only if | b − b | ≥ M (1 , , M (1 , , M (2 , . (D.23)To examine some monopoles of small charge that can bubble, we compute that the IOTalgorithm yields the following bubbling coefficients in U (2) SQCD with N a = N f = 1: Z (2 , → (1 , = 2 − σ ) + O ( β ) , (D.24) Z (1 , − → (0 , = β (cid:20) − − iσ ) − iσ ) − i ( σ + σ ) (cid:21) ( σ + σ ) + O ( β ) , (D.25) Z (3 , → (2 , = 3 − 39 + 4 σ + O ( β ) , (D.26) Z (2 , − → (1 , = iβ (cid:20) − i ( σ + σ ) + iσ − − iσ − iσ ) + 1 + iσ iσ ) (cid:21) + O ( β ) . (D.27)In this theory, the leading power of β in Z ( b ,b ) → ( v ,v )mono, IOT is ∆ ( b ,b ) − ∆ ( v ,v ) = ( | b | + | b | −| v | − | v | ). Applying the prescription (D.6), we obtain Z (2 , → (1 , (Φ , Φ ) = 2 − r Φ ) , (D.28) Z (1 , − → (0 , (Φ , Φ ) = − i (cid:20) − − ir Φ ) − ir Φ ) − ir tr Φ (cid:21) tr Φ , (D.29) Z (3 , → (2 , (Φ , Φ ) = 3 − − ir Φ ) − ir Φ ) , (D.30) Z (2 , − → (1 , (Φ , Φ ) = 12 r (cid:20) − ir tr Φ + 2 ir Φ − − ir Φ − ir Φ ) + 3 + 2 ir Φ ir Φ ) (cid:21) , (D.31)83hich enter into the shift operators M (2 , = M (2 , + Z (2 , → (1 , (Φ , Φ ) M (1 , , (D.32) M (1 , − = M (1 , − + Z (1 , − → (0 , (Φ , Φ ) , (D.33) M (3 , = M (3 , + Z (3 , → (2 , (Φ , Φ ) M (2 , + Z (3 , → (2 , (Φ , Φ ) M (1 , , (D.34) M (2 , − = M (2 , − + Z (2 , − → (1 , (Φ , Φ ) M (1 , + Z (2 , − → (1 , (Φ , Φ ) M (0 , . (D.35)Using these shift operators, we can then reproduce the star products for this theory givenin (D.12), (D.13), (D.20), (D.21). Hence these results of the IOT algorithm are consistentwith polynomiality.All of the bubbling monopoles considered in the previous paragraph bubble to depth 1,and clearly satisfy (D.7) with (cid:15) , = 1. To see some more complicated examples, let us firstconsider monopoles whose charges take the form ( a, − a ). We have already seen how M (1 , − bubbles. For the star product relation (D.22) to hold, it turns out that we must define M (2 , − = M (2 , − + Z (2 , − → (1 , − (Φ , Φ ) M (1 , − + Z (2 , − → (1 , − (Φ , Φ ) M ( − , + 12 Z (2 , − → (0 , (Φ , Φ ) (D.36)where Z (2 , − → (1 , − (Φ , Φ ) and Z (2 , − → (0 , (Φ , Φ ) are obtained via (D.6) from Z (2 , − → (1 , − = β (cid:20) i + σ + σ − σ + σ σ ) (cid:21) + O ( β ) , (D.37) Z (2 , − → (0 , = β (cid:20) σ + σ + 4 σ σ + 4 i ( σ + σ ) − − σ σ )32(4 + σ ) (D.38)+ 15 − i ( σ + σ ) − σ σ σ ) (cid:21) + O ( β ) . (D.39)The factor of 1 / M (3 , − ,we must define M (3 , − = M (3 , − + Z (3 , − → (2 , − (Φ , Φ ) M (2 , − + Z (3 , − → (2 , − (Φ , Φ ) M ( − , + 18 [ Z (3 , − → (1 , − (Φ , Φ ) M (1 , − + Z (3 , − → (1 , − (Φ , Φ ) M ( − , ]+ 124 Z (3 , − → (0 , (Φ , Φ ) (D.40)84o ensure closure of the star algebra, namely for the relation( M (1 , − ) = M (3 , − − r M (2 , − + 4 r M (1 , − (D.41)to hold. The bubbling coefficients are obtained from Z (3 , − → (2 , − = 3 β (cid:20) σ + σ + 3 i − σ + σ σ ) (cid:21) + O ( β ) , (D.42) Z (3 , − → (1 , − = 3 β (cid:20) σ + 2 σ + 6 σ σ + 12 i ( σ + σ ) − 15 + 9(2 − σ σ )8(9 + σ ) − σ σ − 1) + 24 i ( σ + σ )8(4 + σ ) (cid:21) + O ( β ) , (D.43) Z (3 , − → (0 , = 3 β (cid:20) σ + σ + 9 σ σ ( σ + σ ) + 9 i ( σ + σ + 4 σ σ ) − 894 ( σ + σ ) − i + 225( σ + σ )(5 − σ σ )1024(9 + σ ) + 3(108 i + 35( σ + σ ))(3 − σ σ )128(4 + σ )+ 9(480 i + 683( σ + σ ) − iσ σ − σ σ ( σ + σ ))1024(1 + σ ) (cid:21) + O ( β ) . (D.44)These examples exhibit the combinatorial factors in (D.8) and (D.9), with all signs (cid:15) i,j = 1.To substantiate this pattern, we now consider monopoles with charges of the form ( a, M (1 , ) a = M ( a, fixes the combinatorial factors. For M (4 , , wemust define M (4 , = M (4 , + Z (4 , → (3 , (Φ , Φ ) M (3 , + Z (4 , → (3 , (Φ , Φ ) M (1 , + 12 Z (4 , → (2 , (Φ , Φ ) M (2 , (D.45)where the bubbling coefficients are obtained from Z (4 , → (3 , = 4 − 14 + σ + O ( β ) , (D.46) Z (4 , → (2 , = 12 − i i + σ ) − i i − σ ) − i i + σ ) − i i − σ ) + O ( β ) . (D.47)85or M (5 , , we must define M (5 , = M (5 , + Z (5 , → (4 , (Φ , Φ ) M (4 , + Z (5 , → (4 , (Φ , Φ ) M (1 , + 14 [ Z (5 , → (3 , (Φ , Φ ) M (3 , + Z (5 , → (3 , (Φ , Φ ) M (2 , ] (D.48)where the bubbling coefficients are obtained from Z (5 , → (4 , = 5 − 525 + 4 σ + O ( β ) , (D.49) Z (5 , → (3 , = 40 − i i + 2 σ ) − i i − σ ) − i i + 2 σ ) − i i − σ )+ O ( β ) . (D.50)For M (6 , , we must define M (6 , = M (6 , + Z (6 , → (5 , (Φ , Φ ) M (5 , + Z (6 , → (5 , (Φ , Φ ) M (1 , + 18 [ Z (6 , → (4 , (Φ , Φ ) M (4 , + Z (6 , → (4 , (Φ , Φ ) M (2 , ]+ 124 Z (6 , → (3 , (Φ , Φ ) M (3 , (D.51)where the bubbling coefficients are obtained from Z (6 , → (5 , = 6 − σ ) + O ( β ) , (D.52) Z (6 , → (4 , = 120 − i i + σ ) − i i − σ ) − i i + σ ) − i i − σ ) + O ( β ) , (D.53) Z (6 , → (3 , = 480 − i i + σ ) − i i − σ ) − i i + σ ) − i i − σ ) − i i + σ ) − i i − σ ) + O ( β ) . (D.54)All of these examples are consistent with (D.8) and (D.9) for bubbling to depths 2 and 3,again with all signs (cid:15) i,j = 1. To explore higher monopole charges, it would be useful to havea more efficient implementation of the IOT algorithm.Contrary to the general expectation from (D.2), nearly all of the expressions for Z b → v mono, IOT that we have encountered so far in this section are manifestly symmetric in σ ↔ σ : anexception is (D.27). 86 .3.2 U (2) with N a = 0 and N f ≥ This class of theories is ugly when N f = 3 and good when N f > 3. The dimension of the( a, b ) monopole is ∆ ( a,b ) = N f | a | + | b | ) − | a − b | . (D.55)Bubbling occurs to monopoles of smaller | a − b | , and we see that a monopole bubbles intomonopoles of equal or lower dimension only when a and b have opposite signs. In particular,the ( n, 0) monopoles have minimal dimension ∆ ( n, = | n | ( N f / − 1) for their topologicalclass, yet nonetheless bubble.For example, we find that the relation ( M (1 , ) n = M ( n, holds only after accounting forbubbling into higher-dimension operators. For n > 0, the monopoles M ( n, that bubble todepth at most 2 are given by( M (1 , ) = M (2 , − r ( ir Φ + 1)( ir Φ − M (1 , , (D.56)( M (1 , ) = M (3 , − r ( ir Φ + 2)( ir Φ − M (2 , − r ( ir Φ + 1)( ir Φ − M (1 , , (D.57)( M (1 , ) = M (4 , − r ( ir Φ + 3)( ir Φ − M (3 , − r ( ir Φ + 1)( ir Φ − M (1 , + 6 r ( ir Φ + 2)( ir Φ + 1)( ir Φ − ir Φ − M (2 , , (D.58)( M (1 , ) = M (5 , − r ( ir Φ + 4)( ir Φ − M (4 , − r ( ir Φ + 1)( ir Φ − M (1 , + 10 r ( ir Φ + 3)( ir Φ + 2)( ir Φ − ir Φ − M (3 , + 10 r ( ir Φ + 2)( ir Φ + 1)( ir Φ − ir Φ − M (2 , . (D.59)Using (D.6), all of these rational functions are accounted for by bubbling coefficients for M ( n, derived from the IOT algorithm if we set (cid:15) , = (cid:15) , = − (cid:15) , = +1, and c , = 2 in Heuristically, monopole bubbling is similar to operator mixing in that both effects are related to renor-malization (in the case of bubbling, a renormalization of the GNO charge), and both relate operators withthe same global symmetry charges (e.g., topological charge). However, bubbling differs from mixing in thatit does not necessarily occur to monopoles of equal or lower dimension, and the bubbling coefficients thataccount for differences in dimension are generally rational functions of Φ. M ( − , (cid:63) M (1 , = M (1 , − + ( − N f +1 r N f − (cid:20) ( ir Φ − / N f ir Φ ( ir Φ − 1) + ( ir Φ − / N f ir Φ ( ir Φ − (cid:21) , (D.60) M (1 , (cid:63) M ( − , = M (1 , − + ( − N f +1 r N f − (cid:20) ( ir Φ + 1 / N f ir Φ ( ir Φ + 1) + ( ir Φ + 1 / N f ir Φ ( ir Φ + 1) (cid:21) . (D.61)The rational function appearing in the first line (D.60) is precisely the bubbling coefficient Z (1 , − → (0 , computed via the IOT algorithm and (D.6), after adjusting for a minus sign: M ( − , (cid:63) M (1 , = M (1 , − = M (1 , − − Z (1 , − → (0 , (Φ , Φ ) . (D.62)The second line (D.61) simply differs from the first by a polynomial. Similarly, we find itnecessary to define M (2 , − = M (2 , − − Z (2 , − → (1 , − (Φ , Φ ) M (1 , − − Z (2 , − → (1 , − (Φ , Φ ) M ( − , + 12 Z (2 , − → (0 , (Φ , Φ ) (D.64)for star products involving M (2 , − to close, where the relevant bubbling coefficients followfrom applying (D.6) to Z (2 , − → (1 , − = β N f − N f − (cid:20) σ + i ) N f σ ( σ + 2 i ) + ( σ ↔ σ ) (cid:21) + O ( β N f − ) , (D.65) Z (2 , − → (0 , = β N f − N f − (cid:20) (2 σ + i ) N f σ ( σ + i ) (cid:18) (2 σ + 3 i ) N f σ + 2 i + 2(2 σ + i ) N f σ − i (cid:19) + ( σ ↔ σ ) (cid:21) + O ( β N f − ) . (D.66)Hence the shift operators for M (1 , − and M (2 , − fall in line with the general patterns (D.7)and (D.8).As an aside, examining some of the simplest relations involving M (2 , − and dressings of M (1 , − allows us to determine the abelianized bubbling coefficients for M (1 , − in the “IOT Setting Φ = x + y ir and Φ = y − x ir , we check that [ M (1 , , M ( − , ] (cid:63) is proportional to the polynomial( x − x + y + 1) N f − ( y − x − N f ] + ( x + 1)[( y − x + 1) N f − ( x + y − N f ] x ( x − x + 1) (D.63)in x and y (moreover, this polynomial is even in x , which guarantees that it is Weyl-invariant with respectto Φ ↔ Φ ). M (1 , − ) = M (2 , − + [ P (Φ , Φ ) M (1 , − ] , (D.67)where we have defined the dressed and bubbled monopole[ P (Φ , Φ ) M (1 , − ] ≡ P (Φ , Φ ) M (1 , − + P (Φ , Φ ) M ( − , − R (Φ , Φ ) (D.68)(note the minus sign in front of R , reflecting (D.62)) with P being the polynomial P (Φ , Φ ) = ( − N f r N f − ( ir Φ + 1) × (cid:20) ( ir Φ − / N f − ( ir Φ − / N f ir Φ − ( ir Φ + 1 / N f − ( ir Φ − / N f ir Φ + 2 (cid:21) (D.69)and R being a rational function whose explicit form we omit for brevity. We also have M ( − , (cid:63) M (2 , = M (2 , − . (D.70)On the other hand, reversing the order of the star product gives M (2 , (cid:63) M ( − , = M (2 , − + [ P (Φ , Φ ) M (1 , − ] + Q (Φ , Φ ) (D.71)where we have defined the dressed and bubbled monopole[ P (Φ , Φ ) M (1 , − ] ≡ P (Φ , Φ ) M (1 , − + P (Φ , Φ ) M ( − , − R (Φ , Φ ) , (D.72)again accounting for the minus sign for bubbling at depth 1 in this theory. Here, the dressingfunction P is a non-Weyl-invariant polynomial, Q (which appears in the star product) is aWeyl-invariant polynomial, and R is a rational function, none of which we write explicitly.We have seen that the correct dressing prescription is to associate a bubbling coefficient toeach abelian (non-Weyl-averaged) monopole shift operator. So we have (cid:102) M (1 , − = M (1 , − + R (Φ , Φ ) = ⇒ M (1 , − = M (1 , − + R (Φ , Φ ) + R (Φ , Φ ) (D.73)for some ( a priori , non-symmetric) rational function R (Φ , Φ ). Given the relations stated89bove, the bubbling function R (Φ , Φ ) must satisfy the overconstrained system of equations R (Φ , Φ ) + R (Φ , Φ ) = − Z (1 , − → (0 , (Φ , Φ ) , (D.74) P i (Φ , Φ ) R (Φ , Φ ) + P i (Φ , Φ ) R (Φ , Φ ) = − R i (Φ , Φ ) ( i = 1 , , (D.75)which require that R (Φ , Φ ) = P i (Φ , Φ ) Z (1 , − → (0 , (Φ , Φ ) − R i (Φ , Φ ) P i (Φ , Φ ) − P i (Φ , Φ ) ( i = 1 , 2) (D.76)where Z (1 , − → (0 , and R i are symmetric but P i is (in general) not. These equations indeedhave a solution: using the known expressions for P i , R i ( i = 1 , R (Φ , Φ ) = − (1 − ir Φ ) N f N f r N f − ( ir Φ )(1 + ir Φ ) for N f ≥ , − Z (1 , − → (0 , (Φ , Φ ) for N f = 0 , . . . , . (D.78)In particular, R (Φ , Φ ) is symmetric for N f = 0 , . . . , 4, in which case R i (Φ , Φ ) = (cid:20) P i (Φ , Φ ) + P i (Φ , Φ )2 (cid:21) Z (1 , − → (0 , (Φ , Φ ) (D.79)for i = 1 , D.3.3 U (3) with N a = 0 and N f ≥ 0A few new lessons can be learned by going to higher rank. For U (3) SQCD with fundamentalflavors, we will be brief and discuss only the monopoles M (1 , , − , M (2 , − , − , M (1 , , − , whichbubble to depth at most 2. The IOT results for the Weyl-averaged bubbling coefficients ofthese monopoles are as follows (these will be useful references for our analysis of the SU (3)theory with the same matter content). For M (1 , , − , we have Z (1 , , − → (0 , , = (cid:18) − iβ (cid:19) N f − (cid:88) i =1 ( iσ i − / N f (cid:81) j (cid:54) = i iσ ij ( iσ ij − 1) + O ( β N f − ) . (D.80) If P i is symmetric, then the above expression is indeterminate and we have simply P i (Φ , Φ ) Z (1 , − → (0 , (Φ , Φ ) = R i (Φ , Φ ) . (D.77) M (2 , − , − , we have Z (2 , − , − → (1 , , − = β N f − N f − i + 2 σ (cid:18) N f − ( i + σ ) N f i + 2 σ + ( i + 2 σ ) N f − i + 2 σ (cid:19) + O ( β N f − ) , (D.81) Z (2 , − , − → (0 , , = β N f − N f − (cid:88) { i,j,k } ( i + 2 σ i ) N f ( i + 2 σ j ) N f σ ij ( i + σ ij ) σ ik ( i + σ ik ) σ jk ( i + σ jk ) + O ( β N f − ) , (D.82)where { i, j, k } runs over permutations of { , , } and in the first line, the correspondingexpressions for other Weyl orderings of the charges (1 , , − 1) are obtained by taking permu-tations. For M (1 , , − , we have Z (1 , , − → (1 , , − = Z (2 , − , − → (1 , , − | σ → σ , (D.83) Z (1 , , − → (0 , , = Z (2 , − , − → (0 , , . (D.84)Polynomiality shows that in addition to using (D.6) to reproduce the expected bubblingcoefficients Z b → v mono , we must set c , = 2 in (D.8) for bubbling to depth 2. Moreover, onecan show that the shift operators for M (1 , , − and M (2 , , − must be defined by setting (cid:15) , = 1 and (cid:15) , = (cid:15) , = 1 in (D.7) and (D.8), respectively, whereas the shift operators for M (2 , − , − and M (1 , , − must be defined by setting (cid:15) , = (cid:15) , = − , , − 2) bubbles not only into (1 , , − 1) and (0 , , , − , − 1) and (1 , , − E More (Quantized) Chiral Rings In this section, we use our formalism to compute the quantized chiral rings of some simpletheories.Existing approaches to deriving the Coulomb branch chiral rings of 3D N = 4 quivergauge theories or their quantizations include the Hilbert series [82, 88], abelianization [59,70], combinations of the aforementioned techniques [89], and incorporating half-BPS localoperators into the type IIB brane/S-duality realization [90] of 3D mirror symmetry [91]. Inparticular, the Hilbert series can be used to infer the quantum numbers of the generatorsand their relations (such as for U , U Sp , and SO gauge theories, for which the Coulombbranch is a complete intersection [82]), but it does not specify numerical coefficients.When the moduli space is a hyperk¨ahler cone, these known techniques for extracting91enerators and ring relations work well. In other situations, such as in bad theories [46–49],the chiral ring has not been as thoroughly studied. In particular, the Coulomb branch of SU ( N c ) gauge theory with N f fundamental flavors has no global symmetry, and the case N c > N c = 3 below. Weexpect such theories to present good opportunities for applications of our formalism.All of the non-abelian examples in this section take the form of U ( N c ) or SU ( N c ) gaugetheories with fundamental and adjoint matter ( N f and N a ). In these theories, the dimensionof a monopole with GNO charge ( b , . . . , b N c ), computed in the UV, is∆ = 12 N f ( | b | + · · · + | b N c | ) + ( N a − (cid:88) i 0, or = 1 / 2, respectively (in bad theories, unitarity-violating monopole operators arerealized by free scalar fields in an IR dual description [46, 47]). For SU ( N c ), the monopoleof smallest dimension has charge (1 , − , (cid:126) 0) and ∆ = N f + 2( N c − N a − N f + 2( N c − N a ≥ N c − . (E.2)It is never ugly because ∆ is an integer. On the other hand, for U ( N c ), the monopoles ofsmallest dimension have charge ( ± , (cid:126) 0) and ∆ = N f / N a − N c − N a = 1 and N f > 0, both the SU and U theories are never bad. E.1 SQED N versus U (1) with One Hyper of Charge N Before presenting the non-abelian examples, we start by providing another example of twotheories that have the same Coulomb branch but different quantizations. These theories areSQED N and U (1) gauge theory with a single hyper of charge N (the Z N gauge theory of afree hypermultiplet), which we denote by U (1) + N . The CBOs M ± , Φ in either theory arerepresented by the shift operators M N = ( − N r N/ (cid:18) − B iσ (cid:19) N e − i ∂ σ − ∂ B in SQED N , ( − N r N/ (cid:18) − N B iN σ (cid:19) N e − i ∂ σ − ∂ B in U (1) + N , (E.3)92s well as by M − N = e i ∂ σ + ∂ B r N/ , Φ N = 1 r (cid:18) σ + i B (cid:19) × N ,N in U (1) + N . (E.4)In both theories, let X = 1(4 π ) N/ M − , Y = 1(4 π ) N/ M , Z = − i π Φ . (E.5)Then we compute that X (cid:63) Y = (cid:18) Z + 18 πr (cid:19) N in SQED N , N − (cid:89) k =0 (cid:18) Z + 2 k + 18 πr (cid:19) in U (1) + N , (E.6)where multiplication on the RHS is understood to be (cid:63) . The two theories have identicalCoulomb branches ( C / Z N ) and chiral ring relations, as can be seen in the r → ∞ limit of(E.6), but different star products. E.2 Theories on D2-Branes The first few non-abelian examples that we study can be realized as worldvolume theorieson N c D2-branes [82]. These theories, which all have at least one adjoint hypermultiplet, areparticularly straightforward to analyze because the matter contribution cancels the denom-inator in the abelianized chiral ring relations (2.30). We will see explicitly how to reproducethe formalism of [59] for such theories, working our way up in complexity. E.2.1 U (2) with N a = N f = 1The Coulomb branch of this theory is known to be Sym ( C ), which has complex dimensionfour. To exhibit the Coulomb branch chiral ring, let us denote the generators of each C by x i , y i for i = 1 , 2. The coordinate ring of the symmetric product is generated by the fivesymmetric polynomials x + x , y + y , x x , y y , ( x − x )( y − y ) (E.7)93the S = Z by which we quotient acts as 1 ↔ x + x ) − x x ][( y + y ) − y y ] = [( x − x )( y − y )] . (E.8)This relation is precisely that of C / Z , whence Sym ( C ) ∼ = C × ( C / Z ).We would like to identify the combinations (E.7) with Coulomb branch operators. Recallfrom Section 5.3.3 that the candidate such operators aretr Φ , M ( ± , , M ± (1 , , M (1 , − , M ( ± , . (E.9)While the monopoles M ( ± , and M ± (1 , do not bubble and are therefore represented bythe na¨ıve (unbubbled) shift operators (5.42), the monopoles M (1 , − and M ( ± , do bubbleand are most conveniently constructed as star products of M ( ± , : see (D.12) and (D.13).Thinking of x i , y i as having topological charge ∓ 1, respectively (these conventions are naturalfrom the point of view of correlation functions, as in [38] and Appendix F), we would expectto identify x + x and y + y with M ∓ (1 , , x x and y y with M ∓ (1 , , and ( x − x )( y − y )with a linear combination of M (1 , − and tr Φ. Thus from (E.8), we would expect a chiralring relation of the form( M ( − , − M ( − , − )( M (2 , − M (1 , ) = ( M (1 , − + c tr Φ) (E.10)for some constant c , where the products are commutative (not (cid:63) ). Since the star productgives a quantization of the chiral ring product, we would expect (E.10) to hold with respectto the star product after appropriate ordering, and up to subleading O (1 /r ) terms. Indeed,letting () (cid:63) denote symmetrization with respect to the star product (e.g., ( O O ) (cid:63) = ( O (cid:63) O + O (cid:63) O )), we find that with c = 2 i ,(LHS of (E.10) − RHS of (E.10)) (cid:63) = − r M (1 , − − ir tr Φ + 5 r r →∞ −−−→ , (E.11)which is equivalent to (E.10) in the chiral ring ( r → ∞ ). To summarize, we identify x + x ↔ M ( − , , y + y ↔ M (1 , , x x ↔ M ( − , − , y y ↔ M (1 , , ( x − x )( y − y ) ↔ −M (1 , − − i tr Φ (E.12) There are many different ways of writing the star products that reduce to the same chiral ring relations.We find the “democratic” symmetrization convenient. 94n the chiral ring, as claimed in (5.50). Given what we have said so far, the last identificationwould be consistent with either sign in ± ( M (1 , − + 2 i tr Φ): we explain the above choice ofsign in our discussion of the case N f > E.2.2 U (2) with N a = 1 and N f ≥ ( C / Z N f ). We denote the generatorsof each copy of C / Z N f by x i , y i , z i , which satisfy the relations x i y i = z N f i (E.13)for i = 1 , 2. As in our discussion of the case N f = 1, the coordinate ring is generated by thenine symmetric polynomials x + x , y + y , z + z , x x , y y , z z , ( x − x )( y − y ) , ( x − x )( z − z ) , ( y − y )( z − z ) , (E.14)which satisfy the five relations ( x x )( y y ) = ( z z ) N f , (E.15) [( x + x )( y + y ) + ( x − x )( y − y )] = z N f + z N f , (E.16)[( x + x ) − x x ][( y + y ) − y y ] = [( x − x )( y − y )] , (E.17)[( x + x ) − x x ][( z + z ) − z z ] = [( x − x )( z − z )] , (E.18)[( y + y ) − y y ][( z + z ) − z z ] = [( y − y )( z − z )] . (E.19)The RHS of (E.16) can be written in terms of the generators z + z and z z using Newton’sidentities.To physically interpret the Coulomb branch chiral ring, note that the lowest-dimension(∆ = N f / 2) monopoles are M ( ± , . The relation (D.12) holds equally well when N f ≥ N f = 1, while the relation (D.13) generalizes to M ( ± , (cid:63) M ( ∓ , − M (1 , − = ( − N f N f +2 r N f (cid:20) (2 ir Φ ± N f (2 ir Φ ± ir Φ ( ir Φ ± 1) + (2 ir Φ ± N f (2 ir Φ ∓ ir Φ ( ir Φ ∓ (cid:21) . (E.20)One can check that the expressions M ( − , (cid:63) M (1 , and M (1 , (cid:63) M ( − , differ by a Weyl-95nvariant polynomial in tr Φ = Φ + Φ that vanishes in the chiral ring limit ( r → ∞ ). Wework in a basis where M ( − , (cid:63) M (1 , = M (1 , − . The coordinates x i , y i correspond totopological charge ∓ N f / 2, while the z i correspond to zero topological charge and∆ = 1. Hence we still expect to have x + x ↔ M ( − , , y + y ↔ M (1 , (E.22)with ∆ = N f / 2, and from the relation M ± (1 , (cid:63) M ∓ (1 , = (cid:20)(cid:18) ∓ r − i Φ (cid:19) (cid:18) ∓ r − i Φ (cid:19)(cid:21) N f (E.23)(which is derived in our discussion of arbitrary N c below), we also expect to have x x ↔ M ( − , − , y y ↔ M (1 , , z z ↔ − Φ Φ = 12 [tr Φ − (tr Φ) ] (E.24)in the chiral ring, where the first two generators have ∆ = N f and the last has ∆ = 2. Theremaining generators can be deduced from charge and dimensional considerations, as well asthe relations (E.15)–(E.19). However, we will take a simpler approach. Rather than directlyidentifying which Coulomb branch operators are chiral ring generators, we postulate thefollowing more elementary correspondences between coordinate ring generators of ( C / Z N f ) (before the quotient by S = Z ) and non-Weyl-averaged Coulomb branch operators: x , x ↔ M ( − , , M (0 , − , y , y ↔ M (1 , , M (0 , , z , z ↔ − i Φ , − i Φ . (E.25)The basic identifications (E.25) are consistent with all of those stated earlier, and allow usto determine the missing ones. For instance, we have12 ( M ( ± , M (0 , ± + M (0 , ± M ( ± , ) = C (Φ , Φ ) M ± (1 , (E.26)where C (Φ , Φ ) ≡ r Φ r Φ r →∞ −−−→ , (E.27) This happens to coincide with the “IOT basis,” in which M ( − , (cid:63) M (1 , = M (1 , − = M (1 , − + Z (1 , − → (0 , (Φ , Φ ) , (E.21)generalizing (D.29) for N f = 1. However, we do not need any of the explicit results of the IOT algorithm inthis section (or, indeed, in this paper). x x ↔ M ( − , − and y y ↔ M (1 , in the chiral ring (since the M are not gauge-invariant chiral ring operators, the above product is not a star product anddoes not obey polynomiality). We now easily deduce ( x − x )( y − y ) by computing12 (( M ( − , − M (0 , − )( M (1 , − M (0 , ) + (( − , ↔ (1 , −M (1 , − + R (Φ , Φ ) (E.28)where R (Φ , Φ ) ≡ − i ( p − (Φ ) − p − (Φ )) + r Φ (3 + 4 r Φ )( p + (Φ ) + p + (Φ ))2 N f +3 r N f +1 (1 + r Φ )Φ , (E.29) p ± ( x ) ≡ ( − − irx ) N f ± − irx ) N f . (E.30)We then have that R (Φ , Φ ) r →∞ −−−→ − i Φ ) N f + ( − i Φ ) N f ]= ⇒ ( x − x )( y − y ) ↔ −M (1 , − + 2[( − i Φ ) N f + ( − i Φ ) N f ] , (E.31)which is consistent with (E.16) and reduces to precisely the expected result for N f = 1. Wefurther compute that − i M ( − , − M (0 , − )Φ + Φ ( M ( − , − M (0 , − ))= (cid:18) r − i Φ (cid:19) M ( − , + (cid:18) r + i Φ (cid:19) M (0 , − ; (E.32)in the limit r → ∞ , this becomes the dressed monopole − i Φ M ( − , , which we identifywith ( x − x )( z − z ) in the chiral ring. Similarly, we compute that − i M (1 , − M (0 , )Φ + Φ ( M (1 , − M (0 , ))= − (cid:18) r + i Φ (cid:19) M (1 , − (cid:18) r − i Φ (cid:19) M (0 , ; (E.33)in the limit r → ∞ , this becomes the dressed monopole − i Φ M (1 , , which we identify with( y − y )( z − z ) in the chiral ring. We have now used (E.25) to identify all of the gen-erators (E.14) with physical Coulomb branch operators; by construction, the appropriatelysymmetrized star products of the latter reproduce the chiral ring relations (E.15)–(E.19).97 .2.3 U ( N c ) with N a = 1 and N f ≥ U (2) with N a = 1 and N f ≥ N c ( C / Z N f ), we first identify certain elementary non-Weyl-invariant operators with thecoordinates x i , y i , z i of ( C / Z N f ) N c ( i = 1 , . . . , N c ), along the lines of (E.25). The Sym N c operation corresponds to Weyl averaging. One then easily checks that these operators sat-isfy the “abelianized” relations x i y i = z N f i , upon which it automatically follows that thefull (Weyl-invariant) chiral ring generators satisfy the relations of Sym N c ( C / Z N f ). This isessentially the approach advocated in [59]. Of particular interest are the non-bubbling monopoles M ± (cid:126)k with charges (cid:126)k ≡ (1 , . . . , (cid:124) (cid:123)(cid:122) (cid:125) k , , . . . , , k = 1 , . . . , N c , (E.34)which are natural candidates for chiral ring generators ( k = 1 corresponds to the free sectorfor N f = 1). The case k = N c is special because the sum over Weyl reflections is trivial, soin this case, the star products M ± (cid:126)k (cid:63) M ∓ (cid:126)k contain no monopoles and are expressible purelyin terms of Φ. Namely, in this theory, we have M ( − ,..., − N = e (cid:80) I ( i ∂ σI + ∂ BI ) r N c N f / , (E.35) M (1 ,..., N = ( − N c N f r N c N f / (cid:34)(cid:89) I (cid:18) − B I iσ I (cid:19)(cid:35) N f e (cid:80) I ( − i ∂ σI − ∂ BI ) , (E.36)from which we compute that M ± (1 ,..., (cid:63) M ∓ (1 ,..., = (cid:34)(cid:89) I (cid:18) ∓ r − i Φ I (cid:19)(cid:35) N f r →∞ −−−→ (cid:34)(cid:89) I ( − i Φ I ) (cid:35) N f , (E.37)generalizing (D.15). One can write the symmetric polynomial in Φ I on the RHS in terms oftraces of powers of Φ using Newton’s identities. The same reasoning implies that the Coulomb branch chiral ring of U ( N c ) with N a ≥ N f is simply Sym N c ( C / Z N f +2( N a − ). One can also use our polynomiality results for bubbling coefficientsin low-rank theories where the IOT prescription is unavailable to construct the quantized Coulomb branchchiral rings for other theories on N D2-branes [82]. These include U Sp (2 N ) with one antisymmetric and N f fundamentals (whose Coulomb branch is Sym N ( C /D N f )), and SO (2 N + 1) with one symmetric and N f fundamentals (whose Coulomb branch is the same as that of U Sp (2 N ) with one antisymmetric and N f + 3fundamentals). .3 Theories with No Adjoints We now turn to theories with no adjoints, for which the chiral ring relations do not simplifyso easily. We restrict our attention to theories with SU gauge group (which lack minusculemonopoles) and fundamental matter.As mentioned in Sections 4.3 and 5.1.1, we can obtain SU ( N c ) gauge theory by gaugingthe topological U (1) symmetry of a U ( N c ) theory with the same matter content. At the levelof local operators in the U ( N c ) theory, this is equivalent to restricting to the sector of zerotopological charge and setting tr Φ = 0, and indeed, the Coulomb branch of SU ( N c ) gaugetheory can be obtained as a hyperk¨ahler quotient of the U ( N c ) case by U (1) top [47, 49].Correspondingly, our strategy to obtain the dressed monopoles with the proper abelian-ized bubbling coefficients in SU ( N c ) gauge theory is to start with a U ( N c ) gauge theory,where the minuscule monopoles and their dressed versions do not bubble, and then take starproducts of these minuscule monopoles and descend to SU ( N c ). For example, in U ( N c ) with N f fundamentals, the basic monopole shift operators are given by M ( − ,(cid:126) = 1 r N f / − N c +1 N c (cid:88) i =1 (cid:81) j (cid:54) = i ir Φ ij e i ∂ σi + ∂ Bi , (E.38) M (1 ,(cid:126) = ( − N f − N c − r N f / − N c +1 N c (cid:88) i =1 (1 / ir Φ i ) N f (cid:81) j (cid:54) = i ir Φ ij e − i ∂ σi − ∂ Bi . (E.39)Their star products (generalizing (D.60) and (D.61)) are M ( ± ,(cid:126) (cid:63) M ( ∓ ,(cid:126) = M (1 , − ,(cid:126) + ( − N f + N c − r N f − N c +2 N c (cid:88) i =1 ( ir Φ i ± / N f (cid:81) j (cid:54) = i ir Φ ij ( ir Φ ij ± , (E.40)and we work in a basis where M (1 , − ,(cid:126) = M ( − ,(cid:126) (cid:63) M (1 ,(cid:126) . In the SU ( N c ) theory with N f fundamentals, the operator M (1 , − ,(cid:126) is then given by the RHS of (E.40) with the bottomsign, after imposing (cid:80) N c i =1 Φ i = 0. This prescription for gauging the U (1) top of U ( N ) results in SU ( N ) gauge theory as opposed to any ofthe other global forms of su ( N ), and hence can be performed regardless of matter content. This is simplybecause the gauging restricts the lattice of GNO charges of monopoles in the resulting theory, as a sublatticeof the coweight lattice of su ( N ), to be the coroot lattice. Hence the magnetic gauge group is P SU ( N ), whichhas trivial center. When possible, we construct the generators for the SU ( N c ) chiral ring as products of dressed U ( N c )minuscule monopoles in such a way as to agree with the bubbling coefficients produced by the IOT algorithmdescribed in Appendix D. This choice of operator basis simply facilitates comparison with the results of [54];it is no more privileged than those used in the main text.For SU theories of low rank, the bubbling coefficients constructed in this way are all consistent with those .3.1 SU (2) with N f ≥ Revisited We first revisit SU (2) with N f fundamental flavors to demonstrate how to reproduce theresults of Section 5 for the chiral ring and its quantization using the trick of gauging U (1) top (compare to the analysis in Appendix A of [49]). The chiral ring relation can be written as x + zy + z N f − = 0 ( N f > ,x + zy + y = 0 ( N f = 0) . (E.42)Accounting for monopole bubbling is crucial to obtaining the modified relation for N f = 0.In general, we must also account for operator mixing relative to the na¨ıve identifications x ∼ Φ M (1 , − , y ∼ M (1 , − , z ∼ tr Φ (E.43)of the generators with the dressed monopole, bare monopole, and Casimir invariant.We start with the operator tr Φ and the minuscule monopoles M (1 , and M (0 , − in U (2) with N f flavors. From them, we construct M (1 , − = M (0 , − (cid:63) M (1 , , Φ M (1 , − = Φ M (0 , − (cid:63) M (1 , , (E.44)where ∆ (1 , − = N f − 2. We reduce to SU (2) by setting Φ = − Φ = Φ. We then computeusing the corresponding shift operators that for all N f ,(Φ M (1 , − ) − M (1 , − (cid:63) tr Φ (cid:63) M (1 , − − ir M (1 , − (cid:63) Φ M (1 , − = P (Φ) M (1 , − (E.45)where P (Φ) M (1 , − = P (Φ ) M (0 , − (cid:63) M (1 , | Φ =Φ and P (Φ) is a polynomial of degree N f : P (Φ) = (cid:18) − (cid:19) N f +1 (1 − ( − N f )(2 ir Φ + 1) N f − + (2 ir Φ + 1) N f + (2 ir Φ − N f r N f . (E.46)To deduce the quantized chiral ring relation from (E.45), we must write P (Φ) M (1 , − interms of generators. Let us see how to do so explicitly for N f = 0 , , , derived in the main text using polynomiality. For example, setting N c = 2 and Φ = − Φ = Φ gives M (1 , − = M (1 , − + 12 N f r N f − (cid:20) (1 − ir Φ) N f − − (1 + 2 ir Φ) N f − ir Φ (cid:21) . (E.41)The bubbling coefficient in square brackets is a polynomial in Φ when N f ≥ 1, as it must be according toSection 4.3. • When N f = 0, we find that P (Φ) M (1 , − = −M (1 , − . (E.47)In the chiral ring, we have x + zy + y = 0 with x = Φ M (1 , − , y = M (1 , − , z = − 12 (tr Φ ) . (E.48) • When N f = 1, we find that P (Φ) M (1 , − = i Φ M (1 , − + 12 r M (1 , − . (E.49)In the chiral ring, we have x + zy + 1 = 0 with x = 2Φ M (1 , − − i, y = M (1 , − , z = − . (E.50) • When N f = 2, we find that P (Φ) M (1 , − = 12 tr Φ (cid:63) M (1 , − − r M (1 , − . (E.51)In the chiral ring, we have x + zy + z = 0 with x = Φ M (1 , − , y = i (2 M (1 , − + 1) , z = 18 tr Φ . (E.52) • When N f = 3, we find that P (Φ) M (1 , − + i (cid:63) Φ M (1 , − + 14 r tr Φ (cid:63) M (1 , − − i r Φ M (1 , − − r M (1 , − = 0 . (E.53)In the chiral ring, we have x + zy + z = 0 with x = Φ M (1 , − + i , y = √ i M (1 , − , z = 14 tr Φ . (E.54)101 .3.2 SU (3) with N f ≥ SU (3) gauge theory with N f ≥ N f ≥ 5. For comparison,the U (3) theory with the same matter content is ugly for N f = 5 and good for N f > SU (3) theory has beenidentified in [88]. Choosing convenient Weyl orderings of the GNO charges, it consists of the(dressed) monopoles∆ = N f − M (1 , , − = M (0 , , − (cid:63) M (1 , , , (E.55)∆ = 2 N f − M (2 , − , − = M (0 , − , − (cid:63) ( M (1 , , ) , (E.56)∆ = 2 N f − M (1 , , − = ( M (0 , , − ) (cid:63) M (1 , , , (E.57)∆ = N f − + Φ ) M (1 , , − = (Φ + Φ ) M (0 , , − (cid:63) M (1 , , , (E.58)∆ = 2 N f − + Φ ) M ( − , − , = (Φ + Φ ) M ( − , − , (cid:63) ( M (1 , , ) , (E.59)∆ = 2 N f − + Φ ) M ( − , − , = (Φ + Φ ) M ( − , − , (cid:63) ( M (1 , , ) , (E.60)∆ = 2 N f − + Φ ) M (1 , , − = (Φ + Φ ) M (0 , , − (cid:63) M (0 , , − (cid:63) M (1 , , , (E.61)∆ = 2 N f − + Φ ) M (1 , , − = (Φ + Φ ) M (0 , , − (cid:63) M (0 , , − (cid:63) M (1 , , , (E.62)which are straightforward to write in terms of non-bubbling monopoles of U (3) (as we havedone above), the dressed monopolesΦ M (1 , , − , Φ M (1 , , − , Φ M (1 , , − , Φ M (1 , , − , (E.63)which are less straightforward to construct from elementary dressed monopoles, and thescalars tr Φ and tr Φ , where we write Φ = diag(Φ , Φ , − (Φ + Φ )) in the reduction to SU (3). Constructing the explicit dressings in (E.63) is a delicate procedure, so we elect touse the generators∆ = N f − g (1 , , − ≡ M (0 , , − (cid:63) Φ M (1 , , = Φ M (1 , , − + P N f − (Φ , Φ ) , (E.64)∆ = N f − g (1 , , − ≡ M (0 , , − (cid:63) Φ M (1 , , = Φ M (1 , , − + P N f − (Φ , Φ ) , (E.65)∆ = N f − + Φ ) M (1 , , − = (Φ + Φ ) M (0 , , − (cid:63) M (1 , , , (E.66) To obtain this generating set, we consider the N R = 0 case of the results in [88]: (8.17) yields the threebare monopole operators, while (8.25), (8.26), (8.27), (8.28), (8.29) yield their dressed versions; tr Φ andtr Φ are the relevant Casimir invariants after passing from U (3) to SU (3) (8.20). See Section 8.4.2 for theHilbert series. 102 = N f − + Φ ) M (1 , , − = (Φ + Φ ) M (0 , , − (cid:63) M (1 , , (E.67)in their stead: along with the Casimir invariants, they clearly generate (E.63). The degrees d of the unspecified polynomials P d in (E.64)–(E.67) follow from the dimension formula (E.1).This basis differs slightly from that of [88] and has the benefit that the corresponding shiftoperators are easier to construct.To make sense of the formulas above, first recall that in defining dressed monopoles, theWeyl group actions on the GNO charges of the abelian summands and on the dressing factorsare opposite to each other. For example, writing (cid:102) M (1 , , − = M (1 , , − + Z ab(1 , , − → (0 , , (Φ , Φ , Φ ) , (E.68)we have P (Φ , Φ , Φ ) M (1 , , − = P (Φ , Φ , Φ ) (cid:102) M (1 , , − + P (Φ , Φ , Φ ) (cid:102) M (1 , − , + P (Φ , Φ , Φ ) (cid:102) M ( − , , + P (Φ , Φ , Φ ) (cid:102) M (0 , , − + P (Φ , Φ , Φ ) (cid:102) M (0 , − , + P (Φ , Φ , Φ ) (cid:102) M ( − , , , (E.69)where W = S in our case. A given dressed monopole can also be written in a number ofdifferent ways, depending on how the components of the GNO charges are Weyl-ordered: forexample, P (Φ ) M (1 , , , P (Φ ) M (0 , , , and P (Φ ) M (0 , , are all equivalent. Starting with(E.55), we can construct M (1 , , − dressed by any polynomial that is symmetric in the firsttwo arguments by writing P (Φ , Φ , Φ ) M (0 , , − (cid:63) M (1 , , = P (Φ , Φ , Φ ) + P (Φ , Φ , Φ )2 M (1 , , − . (E.70)Similarly, for P symmetric in the last two arguments, the dressing of M (1 , , − by P is givenby the leading term of M (0 , , − (cid:63) P (Φ , Φ , Φ ) M (1 , , (E.71)(as P is now acted on by the shift operators in M (0 , , − ). Starting with (E.56), we find that P (Φ , Φ , Φ ) M (0 , − , − (cid:63) ( M (1 , , ) = P (Φ , Φ , Φ ) + P (Φ , Φ , Φ )2 M (2 , − , − . (E.72)Unlike in the case of M (1 , , − , this procedure for constructing dressings of M (2 , − , − iscompletely general because M (2 , − , − , like M (0 , − , − , is sensitive only to the part of the103ressing polynomial that is Weyl-symmetric in the last two arguments. Similar statementshold for M (1 , , − . Note that in this theory, all non-bubbling monopoles have nontrivialstabilizers under the Weyl group. In particular, dressing the minuscule monopoles in (E.55)individually is not enough to extract the abelianized bubbling coefficient Z ab(1 , , − → (0 , , in(E.68). However, by dressing both M (0 , , − and M (1 , , at the same time, one can inprinciple extract Z ab(1 , , − → (0 , , itself (we will not need to do so).Moving on to the chiral ring, we have listed 14 generators ((E.55)–(E.62), (E.64)–(E.67),and the Casimir invariants), while the Coulomb branch has complex dimension 4. Hence wemust find at least 10 relations. If the moduli space is not a complete intersection (meaningthat the relations could be redundant at generic points but may all be needed to describethe whole variety), then we will find strictly more than 10 relations. In this case, we cannotread off the degrees of the relations and generators from the Hilbert series. By contrast,the moduli space of U with fundamental hypers is known to be a complete intersection [82].To proceed, one can write all possible relations according to dimension and solve for thecoefficients. Rather than presenting an exhaustive analysis, let us simply determine thelowest-dimension relation that relates operators of different GNO charges. Clearly, thisrelation has ∆ = 2 N f − 6. As a further simplification, we work with the commutative limitof the shift operators (in which the order of the multiplications in (E.55)–(E.62) and (E.64)–(E.67) is immaterial), as our interest is in the chiral ring and not its quantization. Then wefind that0 = M (2 , − , − + M (1 , , − + ( g (1 , , − ) − g (1 , , − [(Φ + Φ ) M (1 , , − ] + [(Φ + Φ ) M (1 , , − ] (E.73) − 12 tr Φ ( M (1 , , − ) + P (Φ , Φ ) M (1 , , − in the chiral ring, where P is a polynomial of degree N f − 2, expressible in terms of theCasimirs, which we do not write explicitly (note that P necessarily vanishes for N f = 0 , Indeed, from (8.38) of [88], although we see that the difference between the degrees of the denominatorand the numerator of the Hilbert series equals the dimension of the moduli space in this case, the degreesappearing in the denominator only include the dimensions of tr Φ , tr Φ , and the bare monopoles (1 , , − , , − , − , − Correlation Functions F.1 Mirror Symmetry Check for an N = 8 SCFT In this section, we give more details on the derivation of the mirror maps (5.51)–(5.56) inthe main text.When discussing an N = 8 SCFT in N = 4 language, it is useful to embed the N = 4superconformal algebra osp (4 | 4) into the N = 8 superconformal algebra osp (8 | so (8) R ⊃ su (2) L ⊕ su (2) R ⊕ su (2) ⊕ su (2) , (F.1)where su (2) L ⊕ su (2) R is the N = 4 R-symmetry algebra and su (2) ⊕ su (2) is a flavorsymmetry from the N = 4 point of view [35]. For 3D N = 8 theories, the 1D topologicaltheory has a global su (2) F symmetry [35], which can be identified with su (2) for the Higgsbranch TQFT and su (2) for the Coulomb branch TQFT. To write correlation functionsconcisely, it is convenient to organize operators in the 1D theory into representations of this su (2) F symmetry and to contract their su (2) F indices with commuting su (2) F polarizationvectors z i ( i = 1 , 2) transforming under su (2) F as a doublet. For an operator O i ··· i j ( ϕ ) inthe spin- j representation of su (2) F , we define O ( ϕ, z ) ≡ O i ··· i j ( ϕ ) z i · · · z i j . (F.2)We thus label operators in the 1D theory by O (∆ ,j ) ( ϕ, z ) and subscripts H, C dependingon whether they belong to the Higgs or Coulomb branch TQFT, respectively. The label ∆corresponds to the scaling dimension of the 3D operator from which O (∆ ,j ) originates, and j is the su (2) F spin. We further normalize the two-point functions as (cid:104)O (∆ ,j ) ( ϕ , z ) O (∆ ,j ) ( ϕ , z ) (cid:105) = (cid:104) z , z (cid:105) j (sign ϕ ) , (F.3)with all other two-point functions vanishing, where we have defined the su (2) F invariant (cid:104) z A , z B (cid:105) ≡ (cid:15) ij z iA z jB , (cid:15) = − (cid:15) = 1 , (F.4)denoting the su (2) F singlet that can be formed from two polarizations z A and z B . With thenormalization (F.3), three-point functions are fixed by the su (2) F symmetry as follows: for The subscripts on the right are also called H, C, F, F (cid:48) , respectively, in [92]. j , j , j satisfying the triangle inequality, (cid:104)O (∆ ,j ) ( ϕ , z ) O (∆ ,j ) ( ϕ , z ) O (∆ ,j ) ( ϕ , z ) (cid:105) = λ (∆ ,j ) , (∆ ,j ) , (∆ ,j ) × (cid:104) z , z (cid:105) j (cid:104) z , z (cid:105) j (cid:104) z , z (cid:105) j (sign ϕ ) ∆ (sign ϕ ) ∆ (sign ϕ ) ∆ , (F.5)where j abc ≡ j a + j b − j c (the correlator vanishes otherwise). The sign factors are fixed byconformal symmetry, while su (2) F symmetry requires that the polarizations appear as theydo by power counting.In the following, we specialize to the IR limit of U ( N c ) SQCD with N a = N f = 1. F.1.1 U (2) with N a = N f = 1The 1D Higgs branch theory in the case N c = 2 was partially analyzed in [37]; our discussionhere is self-contained. On the Higgs branch side, there is a single j = 1 / (cid:101) O (1 / , / H ( ϕ, z ) = z tr Q ( ϕ ) + z tr ˜ Q ( ϕ ) (F.6)and two distinct operators with j = 1: (cid:101) O (1 , H, ( ϕ, z ) = ( z ) (tr Q ) ( ϕ ) + ( z ) (tr ˜ Q ) ( ϕ ) + 2 z z tr Q tr ˜ Q ( ϕ ) , (cid:101) O (1 , H, ( ϕ, z ) = ( z ) tr Q ( ϕ ) + ( z ) tr ˜ Q ( ϕ ) + 2 z z tr Q ˜ Q ( ϕ ) . (F.7)The tildes indicate that these operators are unnormalized and do not necessarily obey (F.3).To find which linear combinations obey (F.3), we compute their two-point functions using(5.49). Performing Wick contractions in the theory at fixed σ using (5.48), we obtain (cid:104) tr Q ( ϕ ) tr ˜ Q ( ϕ ) (cid:105) σ = − sign ϕ πr , (cid:104) (tr Q ) ( ϕ )(tr ˜ Q ) ( ϕ ) (cid:105) σ = − (cid:104) tr Q tr ˜ Q ( ϕ ) tr Q tr ˜ Q ( ϕ ) (cid:105) σ = 18 π r , (cid:104) tr Q ( ϕ ) tr ˜ Q ( ϕ ) (cid:105) σ = − (cid:104) tr Q ˜ Q ( ϕ ) tr Q ˜ Q ( ϕ ) (cid:105) σ = 116 π r (cid:20) ( πσ ) (cid:21) , (F.8) (cid:104) (tr Q ) ( ϕ ) tr ˜ Q ( ϕ ) (cid:105) σ = (cid:104) tr Q ( ϕ )(tr ˜ Q ) ( ϕ ) (cid:105) σ = − (cid:104) tr Q tr ˜ Q ( ϕ ) tr Q ˜ Q ( ϕ ) (cid:105) σ = 116 π r . Since the natural index structure is Q ij and ˜ Q ij , when we write tr Q ˜ Q , we really mean tr Q ˜ Q T . we obtain that the orthonormalized operators O (1 / , / H ( ϕ, z ) = √ πr (cid:101) O (1 , H, ( ϕ, z ) , O (1 , H, free ( ϕ, z ) = √ π r (cid:101) O (1 , H, ( ϕ, z ) , (F.12) O (1 , H, int ( ϕ, z ) = √ π r (cid:20) (cid:101) O (1 , H, ( ϕ, z ) − (cid:101) O (1 , H, ( ϕ, z ) (cid:21) obey (F.3). The subscripts “free” and “int” indicate that this theory flows to the product ofa free sector and an interacting sector. Similarly, the nontrivial three-point functions with( j , j , j ) = (1 / , / , 1) follow from (cid:104) tr Q ( ϕ ) tr Q ( ϕ )(tr ˜ Q ) ( ϕ ) (cid:105) σ = (cid:104) tr ˜ Q ( ϕ ) tr ˜ Q ( ϕ )(tr Q ) ( ϕ ) (cid:105) σ = − (cid:104) tr Q ( ϕ ) tr ˜ Q ( ϕ ) tr Q tr ˜ Q ( ϕ ) (cid:105) σ = sign( ϕ ϕ )8 π r , (F.13) (cid:104) tr Q ( ϕ ) tr Q ( ϕ ) tr ˜ Q ( ϕ ) (cid:105) σ = (cid:104) tr ˜ Q ( ϕ ) tr ˜ Q ( ϕ ) tr Q ( ϕ ) (cid:105) σ = − (cid:104) tr Q ( ϕ ) tr ˜ Q ( ϕ ) tr Q ˜ Q ( ϕ ) (cid:105) σ = sign( ϕ ϕ )16 π r , (F.14)while those with ( j , j , j ) = (1 , , 1) each involve six distinct contractions: for example, (cid:104) tr Q ( ϕ ) tr ˜ Q ( ϕ ) tr Q ˜ Q ( ϕ ) (cid:105) σ = − sign( ϕ ϕ ϕ )64 π r (cid:20) ( πσ ) (cid:21) . (F.15) Some useful integrals are as follows. The partition function is Z S = 132 (cid:90) dσ dσ sinh ( πσ )cosh ( πσ ) cosh( πσ ) cosh( πσ ) = 116 π . (F.9)Denoting Z S with an extra insertion of f ( σ , σ ) in the integrand by Z S [ f ( σ , σ )], we have Z S [ σ σ ] = − π , Z S [ σ + σ ] = 124 π , Z S (cid:20) ( πσ ) (cid:21) = 196 π . (F.10)In particular, if (cid:104)(cid:105) σ is a constant, then (cid:104)(cid:105) = (cid:104)(cid:105) σ , while (cid:104) tr Q ( ϕ ) tr ˜ Q ( ϕ ) (cid:105) = − (cid:104) tr Q ˜ Q ( ϕ ) tr Q ˜ Q ( ϕ ) (cid:105) = 796 π r . (F.11) su (2) F -symmetric way as (cid:104)O (1 / , / H ( ϕ , z ) O (1 / , / H ( ϕ , z ) O (1 , H, free ( ϕ , z ) (cid:105) = λ free(1 / , / , (1 / , / , (1 , (cid:104) z , z (cid:105)(cid:104) z , z (cid:105) sign( ϕ ϕ ) , (cid:104)O (1 , H, free ( ϕ , z ) O (1 , H, free ( ϕ , z ) O (1 , H, free ( ϕ , z ) (cid:105) = λ free(1 , , (1 , , (1 , (cid:104) z , z (cid:105)(cid:104) z , z (cid:105)(cid:104) z , z (cid:105) sign( ϕ ϕ ϕ ) , (cid:104)O (1 , H, int ( ϕ , z ) O (1 , H, int ( ϕ , z ) O (1 , H, int ( ϕ , z ) (cid:105) = λ int(1 , , (1 , , (1 , (cid:104) z , z (cid:105)(cid:104) z , z (cid:105)(cid:104) z , z (cid:105) sign( ϕ ϕ ϕ ) , (F.16)where we have defined the structure constants λ free(1 / , / , (1 / , / , (1 , = √ , λ free(1 , , (1 , , (1 , = ( √ , λ int(1 , , (1 , , (1 , = √ , (F.17)while (cid:104)O (1 / , / H ( ϕ , z ) O (1 / , / H ( ϕ , z ) O (1 , H, int ( ϕ , z ) (cid:105) = (cid:104)O (1 , H, free ( ϕ , z ) O (1 , H, free ( ϕ , z ) O (1 , H, int ( ϕ , z ) (cid:105) = (cid:104)O (1 , H, free ( ϕ , z ) O (1 , H, int ( ϕ , z ) O (1 , H, int ( ϕ , z ) (cid:105) = 0 (F.18)because the free and interacting sectors are decoupled.On the Coulomb branch side, the monopoles M ( ± , and M ± (1 , do not bubble, so theirshift operators M take the form of M in (5.42) averaged over the Z Weyl group. The shiftoperators for bubbling monopoles can be constructed from these by taking star products(i.e., by composition), as described in Appendix E.2.1: see in particular (D.12) and (D.13).We then find using (5.43) that, for instance, π (cid:104)M ( − , ( ϕ ) M (1 , ( ϕ ) (cid:105) = − sign ϕ πr , (F.20) π (cid:104)M ( − , − ( ϕ ) M (1 , ( ϕ ) (cid:105) = − (cid:0) − i π (cid:1) (cid:104) tr Φ( ϕ ) tr Φ( ϕ ) (cid:105) = π r , (F.21) (cid:0) π (cid:1) (cid:104)M ( − , ( ϕ ) M (2 , ( ϕ ) (cid:105) = − (cid:0) π (cid:1) (cid:10)(cid:0) M (1 , − − r (cid:1) ( ϕ ) (cid:0) M (1 , − − r (cid:1) ( ϕ ) (cid:11) The symmetry of the integral (F.9) under σ , ↔ − σ , sometimes allows us to write these correlatorsin terms of Z S with simple insertions, such as (cid:104)M ( − , ( ϕ ) M (1 , ( ϕ ) (cid:105) = Z S [ − sign ϕ ] rZ S , (cid:104)M ( − , − ( ϕ ) M (1 , ( ϕ ) (cid:105) = Z S [1 − σ σ ]4 r Z S , (F.19)which can be evaluated using (F.10). π r , (F.22) (cid:0) π (cid:1) (cid:0) π (cid:1) (cid:104)M ( − , ( ϕ ) M (1 , ( ϕ ) (cid:105) = (cid:0) π (cid:1) (cid:0) π (cid:1) (cid:104)M ( − , − ( ϕ ) M (2 , ( ϕ ) (cid:105) = − (cid:0) π (cid:1) (cid:0) − i π (cid:1) (cid:10)(cid:0) M (1 , − − r (cid:1) ( ϕ ) tr Φ( ϕ ) (cid:11) = π r . (F.23)It is straightforward to check that these correlation functions, as well as the various three-point functions, agree precisely with those of the 1D Higgs branch operators given in (5.51)–(5.55), as extracted from the two-point functions of (F.12) and the three-point functions(F.16). This provides a derivation of (5.51)–(5.55).Here is an alternative method of deriving the mirror map. Given the relations (D.12)–(D.22), we may view the “basic” operators on the Coulomb branch side as M ( ± , and M ± (1 , , for which we have already justified the mirror map in (5.51) and (5.53). The mirrormap (5.52) for M ( ± , then follows from tr Q (cid:63) tr Q = (tr Q ) and (D.12). Next, using Wickcontractions to define composite operators, we compute on the Higgs branch side thattr Q (cid:63) tr ˜ Q + N c πr = tr ˜ Q (cid:63) tr Q − N c πr = tr Q tr ˜ Q (F.24)for arbitrary N c , which, in light of (D.13), is consistent with the identification (5.54) for M (1 , − when N c = 2. Finally, on the Higgs branch side, we find thattr Q (cid:63) tr ˜ Q + 12 πr tr Q ˜ Q = tr ˜ Q (cid:63) tr Q − πr tr Q ˜ Q = tr Q tr ˜ Q + 18 π r . (F.25)Using (D.15) and (5.53), we deduce the mirror map (5.55) for tr Φ as well as18 π (cid:20) tr Φ − (tr Φ) − r (cid:21) ↔ tr Q tr ˜ Q . (F.26)Using (F.26) and the mirror map (5.55), we can further identify what tr Φ corresponds to:on the Higgs branch side, we compute thattr Q ˜ Q (cid:63) tr Q ˜ Q = (tr Q ˜ Q ) − π r , (F.27)so in light of tr Φ (cid:63) tr Φ = (tr Φ) and (F.26), we get that18 π (cid:18) tr Φ − r (cid:19) ↔ tr Q tr ˜ Q − Q ˜ Q ) . (F.28)109ne can make further consistency checks of the identifications that we have derived bymatching one-point functions of these composite operators. As a consistency check of (5.54),we see from (F.24) that (cid:104) tr Q tr ˜ Q (cid:105) = 0 for any N c , so we expect that (cid:104)M (1 , − (cid:105) = r , whichis indeed the case. As a consistency check of (5.55), we have that (cid:104) tr Q ˜ Q (cid:105) = 0, which isconsistent with (cid:104) tr Φ (cid:105) = 0 , (cid:104) tr Φ (cid:105) = Z S [ σ + σ ] r Z S = 23 r , (cid:104) (tr Φ) (cid:105) = Z S [( σ + σ ) ] r Z S = 712 r (F.29)(as follows from (F.10)), where the notation Z S [ f ( σ , σ )] is the same as in Footnote 42. Asa consistency check of (F.28), we may use (F.25) and (F.27) to rewrite (F.28) in terms ofstar products of elementary operators:18 π (cid:18) tr Φ + 12 r (cid:19) = tr Q (cid:63) tr ˜ Q − Q ˜ Q (cid:63) tr Q ˜ Q + 12 πr tr Q ˜ Q. (F.30)Taking the expectation value of both sides of (F.30) and using (F.11) and (cid:104) tr Q ˜ Q (cid:105) = 0 resultsin (cid:104) tr Φ (cid:105) = 2 / r , precisely as expected from (F.29). F.1.2 U ( N c ) with N a = N f = 1While we do not study the case N c > U ( N c )with N a = N f = 1 can be analyzed in the same way for all N c (compare to the analysis of U (3) with N a = N f = 1 in [92]). First consider the Higgs branch side. Letting tildes denoteunnormalized operators, we set (cid:101) O (1 / , / H, free ( ϕ, z ) = z tr Q ( ϕ ) + z tr ˜ Q ( ϕ ) , (F.31)so that all operators in the free sector of the 1D theory are simply powers of this operator: (cid:101) O ( j,j ) H, free ( ϕ, z ) = [ (cid:101) O (1 / , / H, free ( ϕ, z )] j . (F.32)The basic result is (cid:104) (tr Q ) m ( ϕ )(tr ˜ Q ) m ( ϕ ) (cid:105) = (cid:104) (tr Q ) m ( ϕ )(tr ˜ Q ) m ( ϕ ) (cid:105) σ = m ! (cid:18) − N c sign ϕ πr (cid:19) , (F.33)110y counting m ! equivalent contractions. We compute the two-point functions (cid:104) (cid:101) O ( j,j ) H, free ( ϕ , z ) (cid:101) O ( j,j ) H, free ( ϕ , z ) (cid:105) = (2 j )! (cid:18) N c πr (cid:19) j (cid:104) z , z (cid:105) j (sign ϕ ) j . In terms of the normalized operators O ( j,j ) H, free ( ϕ, z ) = 1 (cid:112) (2 j )! (cid:18) πrN c (cid:19) j (cid:101) O ( j,j ) H, free ( ϕ, z ) , (F.34)we then compute the three-point functions (cid:104)O ( j ,j ) H, free ( ϕ , z ) O ( j ,j ) H, free ( ϕ , z ) O ( j ,j ) H, free ( ϕ , z ) (cid:105) (F.35)= λ free( j ,j ) , ( j ,j ) , ( j ,j ) (cid:104) z , z (cid:105) j (cid:104) z , z (cid:105) j (cid:104) z , z (cid:105) j (sign ϕ ) j (sign ϕ ) j (sign ϕ ) j for j , j , j satisfying the triangle inequality, where λ free( j ,j ) , ( j ,j ) , ( j ,j ) = j ! j ! j ! (cid:112) (2 j )!(2 j )!(2 j )! (cid:18) j j (cid:19)(cid:18) j j (cid:19)(cid:18) j j (cid:19) (F.36)(compare to (F.17) for N c = 2). We claim that the corresponding operators on the Coulombbranch side are given by O (1 / , / C, free ( ϕ, z ) = (cid:114) rN c ( z M ( − ,(cid:126) ( ϕ ) + z M (1 ,(cid:126) ( ϕ )) . (F.37)To see this, one can match two-point functions. The shift operators are M ( − ,(cid:126) N = 1 r / N c (cid:88) I =1 (cid:81) J (cid:54) = I ( B IJ − iσ IJ ) (cid:81) J (cid:54) = I ( − iσ IJ + B IJ ) e i ∂ σI + ∂ BI , (F.38) M (1 ,(cid:126) N = − r / N c (cid:88) I =1 ( − B I + iσ I ) (cid:81) J (cid:54) = I ( − B IJ + iσ IJ ) (cid:81) J (cid:54) = I ( iσ IJ − B IJ ) e − i ∂ σI − ∂ BI , (F.39)from which we obtain (cid:104)M ( ∓ ,(cid:126) ( ϕ ) M ( ± ,(cid:126) ( ϕ ) (cid:105)| ϕ <ϕ = Z S [ I ± ] Z S , I ± ≡ ± r N c (cid:88) I =1 ( + iσ I ) (cid:81) J (cid:54) = I ( + iσ IJ ) (cid:81) J (cid:54) = I ( iσ IJ )(1 + iσ IJ ) . Z S without insertions is invariant under σ I ↔ − σ I , inserting I ± isequivalent to inserting I ± ( σ , . . . , σ N c ) + I ± ( − σ , . . . , − σ N c )2 = ± N c r . (F.40)It follows that (cid:104)M ( − ,(cid:126) ( ϕ ) M (1 ,(cid:126) ( ϕ ) (cid:105) = − N c sign ϕ r , (F.41)thus substantiating the stated map. F.2 An Abelian/Non-Abelian Mirror Symmetry Example Let us end with a simpler example where we can derive the correspondence between chiralring generators in mirror dual pairs. It is known that SU (2) SQCD with three fundamentalhypers is dual to U (1) SQED with four charged hypers, because both theories are mirrordual to the U (1) necklace quiver gauge theory [78]. Their Coulomb branch is given by C / Z : it has three holomorphic generators X , Y , and Z subject to the chiral ring relation X Y = Z , whose quantization is X (cid:63) Y = ( Z ) (cid:63) + O (1 /r ). The generators have dimensions∆ Z = 1 and ∆ X = ∆ Y = 2. Let us identify X , Y , and Z in the SQCD theory.To compute correlation functions, we use that the vacuum wavefunction (5.40) isΨ ( σ, B ) = δ B, [ π Γ( − iσ )Γ( + iσ )] π Γ(1 − iσ )Γ(1 + 2 iσ ) = δ B, sinh( πσ )4 σ cosh ( πσ ) (F.42)and the gluing measure is µ ( σ, B ) = ( − | B | (4 σ + B ) . (F.43)Using |W| = 2, this gives the S partition function Z = 12 (cid:90) dσ µ ( σ, ( σ, = 112 π , (F.44)in agreement with the S partition function of the four-node quiver theory and SQED withfour flavors (see, e.g., [38]).The Coulomb branch chiral ring operators are gauge-invariant products of Φ and GNOmonopole operators with b ∈ Z . The smallest-dimension such operator is the GNO monopole M (1 , − . This operator has ∆ = 1, so it should correspond to Z in the four-node quiver112heory. Matching the normalization of the two-point function gives Z = 14 π M (1 , − . (F.45)There are three operators with ∆ = 2: M (1 , − (cid:63) M (1 , − , tr Φ , and the dressed monopoleΦ M (1 , − . Clearly, M (1 , − (cid:63) M (1 , − = (4 π ) Z (cid:63) Z , so we expect to obtain X and Y as linearcombinations of tr Φ and Φ M (1 , − . We find that X = 164 π (cid:18) tr Φ − M (1 , − (cid:63) M (1 , − − r + 4 i (cid:18) Φ M (1 , − − i r M (1 , − (cid:19)(cid:19) , (F.46) Y = 164 π (cid:18) tr Φ − M (1 , − (cid:63) M (1 , − − r − i (cid:18) Φ M (1 , − − i r M (1 , − (cid:19)(cid:19) (F.47)obey the following relations:[ X , Z ] (cid:63) = 14 πr X , [ Y , Z ] (cid:63) = − πr Y , X (cid:63) Y = (cid:18) Z + 18 πr (cid:19) (cid:63) . (F.48)These are precisely the relations obeyed in the four-node quiver theory. In addition, one cancheck that (cid:104)X (cid:105) = (cid:104)Y(cid:105) = (cid:104)Z(cid:105) = 0, just as in the four-node quiver theory. The last relationin (F.48) shows that the Coulomb branch is indeed C / Z . References [1] V. Borokhov, A. Kapustin, and X.-k. Wu, “Topological disorder operators inthree-dimensional conformal field theory,” JHEP (2002) 049, arXiv:hep-th/0206054 [hep-th] .[2] D. T. Son, “Is the Composite Fermion a Dirac Particle?,” Phys. Rev. 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