JHEP01(2021)086 = 4 N d wreathed Springer 3 June 30, 2020 : January 15, 2021 November 4, 2020 : November 20, 2020 : : , Received Revised Published Accepted Published for SISSA by which produces Coulomb branches of https://doi.org/10.1007/JHEP01(2021)086 [email protected] , quiver folding . 3 2005.05273 The Authors. Field Theories in Lower Dimensions, Global Symmetries, Supersymmetric c

We study two types of discrete operations on Coulomb branches of , a.bourget@imperial,ac.uk Theoretical Physics, Department of Physics,South Imperial Kensington College Campus, London, London, SW7E-mail: 2AZ, U.K. [email protected] Open Access Article funded by SCOAP ArXiv ePrint: generators and relations including mass deformations. Keywords: Gauge Theory, Solitons Monopoles and Instantons Abstract: quiver gauge theories using bothprevious abelianisation work and on the discretequiver monopole quotients formula. theories. of We generalise Coulomb Wenon-simply branches further laced and quivers. study introduce Our methods novel explicitly describe Coulomb branches in terms of Antoine Bourget, Amihay Hanany and Dominik Miketa Quiver origami: discrete gauging and folding JHEP01(2021)086 41 43 45 48 54 61 62 14 58 wreathing 23 37 4 52 S – 1 – 33 39 17 gauging 41 7 2 29 53 51 S 44 2 2 47 C 11 G 2 46 51 C min min 21 17 → → 47 51 13 16 )min 3 4 . 2 A D 38 G (n discrete gauging discrete gauging 7 2 3 3 4 → → S Z min min 4 B 3 45 A D → 5 with 4 flavors and abelianised rings 5.2.3 5.2.4 Mixed5.2.5 folding and D 5.1.2 Discrete5.1.3 gauging Folding min 5.2.1 Initial5.2.2 quiver Folding 4.2.2 min 5.1.1 Initial quiver 4.2.1 5.3 U(2) 5.2 4.3 Non-simply laced quivers 5.1 3.7 Mirror symmetry and discrete gauging 4.1 Action on4.2 the Coulomb branch Monopole formula: examples 3.2 Action on3.3 the Coulomb branch Wreathed quivers 3.4 Monopole3.5 formula for wreathed quivers HWG for3.6 wreathed quivers Higgs branch of wreathed quivers 2.2 The monopole2.3 formula Abelianisation 2.4 Construction of Coulomb branch multiplets 3.1 Wreath product 2.1 Folding of Dynkin diagrams B Folded Lie algebras areC the same Computation as of discretely Hilbert gauged series Lie with algebras A 5 Examples 4 Quiver folding 3 Discrete gauging Contents 1 Introduction 2 Preliminaries JHEP01(2021)086 ) 2 1 d . 3 C n, -type ] and 1 7 (2 − discrete theory n sl 2 A -type quiver Coulomb 2 and exhibiting -type quivers are folded 4 G Dynkin diagrams, can be 3 F n D ] argued that their Coulomb 3 -type quiver Coulomb branches, n ] generated similar results on the . Assuming simply laced unitary quivers node with adjoint matter. D 6 , ) ] that the Coulomb branch monopole 5 1 n theory -type quivers. 1 d U( 4 − n B , can be summarised as follows. -type quivers while 1 4 – 2 – D nodes by a . Previous work [ symmetry. Balanced U(1) ]. Both aspects expand on our previous work in [ ) relates Coulomb branches of pairs of simply laced and non- n C 6 , 5 n, (2 usp wreathed quivers ], and our paper may be viewed as a new entry. quiver folding ]. We also develop a second, related but distinct discrete operation which assuming the quiver is understood as a 12 4 – 8 ] can be extended to quivers in the form of non-simply laced framed Dynkin 2 , produces Coulomb branches. It has been known since [ In contrast, The main concepts, presented inQuivers figure with an automorphism possess a discrete symmetry relating gauge groups. In the rest of this text we will elideA the node distinction is between balanced folding when a the contributions quiver of theory gauge and and folding matter to its the RG flow of the gauge coupling Discretely gauging string backgrounds is of course an old idea which has generated a lot of discussion, = 4 -type quivers with -type quivers. The folded spaces are fixed points under the group action induced by 3 1 2 6 n Coulomb branch, butcircumstantial wish evidence for to the be former. clear thatexactly we cancel out present solidwithout evidence loops, this only amounts for(gauge to or the the matter) condition latter nodes’ that ranks. twice and the at node’s best rank equals the sum of all surrounding branches can be similarlyE obtained from the quiver automorphism and we show that they are symplecticfor leaves example of in spaces [ obtained quivers, ie. framed linear quiverssymmetry satisfying on the balance the condition CoulombC branch, can beie. “folded” Coulomb into branches of Coulomb balanced“folded” branches framed into quivers of Coulomb shaped balanced branches like of balanced simply laced quiverbranch gauge and theories. conjectureHowever, To that we be one have clear, can beencompute we unsuccessful view the show in it Higgs our the as branch attempts action of one to folded on effect write theories. the of down Coulomb Coulomb the an branches path action of integral balanced on or the and we demonstratequotient that of this the resultsthe original, in quiver where a automorphism the theory (orgauging action whose subgroup by Coulomb thereof). which branchCoulomb we This is branch quotient operation, by a replacing is which discrete directly we induced call by likewise present an effective approachcomplex for mass explicit deformations computation of of generators, Coulomb relations branches and with which we solve several novel cases. By analogy with continuous gauge groups, it, or any of its subgroups, can be gauged, time what to makebranches of result the from novel aquivers. multiple discrete link. We folding independently operationabelianisation Recently derive [ on [ and Coulomb illustrate branches thewas of previously same simply claim studied laced through in the [ method of The purpose of thisN work is to clarifyformula the relation [ between several conceptsdiagrams. relating However, to while alland the flavor ingredients nodes, of hypermultiplet a links — simply are laced readily Dynkin interpretable, diagram it — was gauge unknown at the 1 Introduction JHEP01(2021)086 m and , the S 3 o Γ S i i through ] of the 1 Q Q 13 Q . = 2 m — and since the Poisson ) ) Γ n n identical subquivers U( U( m and 0 1 0 1 0 k 0 k , here depicted for 0 k Q Q Q Q m Q . We claim that, since discrete gauging is 2 through – 3 – node. (Top right) Wreathed quiver. (Bottom right) 0 1 ) ], in which orbits in the small affine Grassmannian n Q 14 U( m 1 Q Q . We show that it has a Poisson structure and, since the fixed Γ generic subquivers k ) affine quiver) and their folded spaces coincide; as a result, there are “fewer” n 4 U( D . (Top left) Folding, on the other hand, reduces the Coulomb branch to the fixed subspace under Actions of both discrete gauging and folding on the Coulomb branch are readily in- are connected to a common central 0 1 k 0 of the Q Q m 3 the same group action subspace is (a singular)folded part space of is a the subdiagram corresponding oforbit orbifold, the folds orbifold’s the to Hasse Hasse another diagram. diagram nilpotentof In a orbit [ all general known (of phenomenon cases the identified a folded in nilpotent [ algebra). This situation is reminiscent structure respects this group action,the the original orbifold inherits space a is naturalnot a symplectic always, nilpotent structure. a orbit nilpotent If of orbitunder some of the the algebra folded relevant then folded algebra’s the algebra, action. orbifold but is it is sometimes, in but any case symmetric Z folded than wreathed quivers. terpreted through a geometricimplemented lens, by see restricting figure the chiralresulting ring space to is an invariants of orbifold of a the symmetry initial group Coulomb branch action under Non-simply laced quiver. The multiple link has valence by discretely gauging theirsubgroups respective of original the Coulomb quiver branches. automorphism can In give some identical cases sets distinct of fixed points (eg. Figure 1 Q JHEP01(2021)086 . ], 1 18 , 17 factorises as G which are the union quotient of the affine 2 Z node without affecting the . Flavorless simply laced quiver U(1) Reeder pieces symmetry. (middle) Coulomb branch 2 ], following the construction [ Z 16 crossing fixed points — respectively the middle and 2 Z – 4 – factor contributing a (geometrically uninteresting) ] for a friendly introduction addressed to physicists) 15 symmetry. (right) Coulomb branch of the folded quiver, the U(1) 2 Z ; see [ G , one which can be identified with a ] is realized on the quiver as leg permutations like in figure symmetry. G 14 (the subvariety of the affine Grassmannian corresponding to the 2 Z G . Coulomb branches of framed unitary ADE quivers were identified 2 is somewhat constrained, the allowed options are in practice equivalent to the Coulomb branch, which is discarded by convention. Crucially, while 1 , with the decoupled S U(1) global involution — here these orbits would be depicted as the left portion of involution of [ features a third discrete action called × 2 3 2 1 U(1) Z R . (left) Initial Coulomb branch with highlighted Z × . Some of these orbits can be mapped to so-called 2 ]. Table 19 U(1) Coulomb branch. The situation ison modified opposite for sides non-simply laced of quivers, theone where directed ungaugings is multiple the link discrete give rise quotientdo to of not pairs the study of other. it Coulomb further We branches in listin where this this [ paper. case for The the reader could sake instead of consult completeness, the but recent treatment theories possess a certainG/ freedom of reparametrisation:factor the of gauge group the choice of and one can in particular choose to ungauge any given As a consequence, several ofof the view examples discussed from below these followproviding previous from quivers the for works; geometric each the point ofseries present the and paper three HWGs spaces, sheds of and a their giving new closures. formulas to light compute on the this Hilbert topic by of two nilpotent orbitsGrassmannian of slice, and theright other parts as of the figure with slices in theand associated the affine Grassmannian [ for an algebraic group so-called small coweights of possess a figure of the discretely gaugedform quiver a depicted singular as subspace an undersubspace orbifold the fixed of under the the original space. Note that bold edges Figure 2 JHEP01(2021)086 node . 2 U(1) node on the (dimension (dimension preserving) preserving) preserving) Fixed points Operation on (not dimension Coulomb branch Discrete quotient Discrete quotient U(1) ) occur in Coulomb The same pattern is 2 4 . ? 2 action String Orbifold Orientifold background and ungauging another quiver quiver Resulting Wreathed quotient corresponds to gauging a n Non-simply Non-simply laced quiver laced quiver n – 5 – ] the authors identified that discrete quotients Z ] up to low but non-trivial rank. The lines painted 20 7 2 b 1 b S o In [ b . Discrete actions on the quiver. 1 a a . a → 3.5 → → Table 1 b b b b quotient. However the HWGs are under control, and are discussed b 1 2 2 ] showed that cases 5, 6 and 7 (yellow in table Z Quiver description 19 1 a a a ] provide more examples of discrete and non-discrete quotients in nilpotent orbits. 21 A recent work [ Case number 8 still presents a challenge, and we are not aware of any quiver realisation theory Folding gauging Ref. [ Discrete Crossing on gauge 4 Operation of the corresponding briefly at the end of section branches of non-simply laced quivers.on The the “long” end“short” of end. an The edge quiver of realisations of multiplicity all eight known cases are collected in table series and abelianisation methods as inin [ green (cases 2, 3,in 4 and red, 9) stands correspond apart tospace wreathed can because be simply of described laced algebraically quivers. the usingof abelianised Case non-simply variables, 1, the the laced painted explicit monopole initial implementation formula quiver;investigations. for although the non-simply moduli laced wreathed quivers is postponed for future Kostant-Brylinski reductions. of certain minimal nilpotent orbitsorbits were of equivalent to other (generically algebras;observed non-minimal) in their nilpotent discrete results gauging areof and summarised their we cases in claim 1,2,3,4 table that and our 9. construction is We empirically a confirmed physical this realisation conjecture using both Hilbert

JHEP01(2021)086

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1

1 2

3

1 [1] 2 S o

, [3 1]

2 4

3 2

D 9 5 ⊕ G ·O S

C C

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1

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2

4 4

2

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8 8 8 2 4 D

3 2 1 1 2 3 2 1 1 2

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4 4 2 4

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C

Z

16 4 B

1 1

2 2 2 1 2 2 2 2

1 , [3 ]

n n

2

n 2 3 −

B 6 D 2 − n 2 ·O

C

Z

··· ··· n 2 n D

1 1

1 3 2 2 1 1

[3]

2 2 3

3 Λ ⊕ A G 5 ·O

C C

Z

3 2 3 2 A

1

2 – 6 –

2

1 S o 1] [2 3 2

] 1 , 2 , [3

1 3 2

4 6 2 7 8

11 F E 4 ·O

C

Z

26 4 F

1 2

1 1 1

··· 1

2

S o 1] 1 1 [1

] 1 , [2

n 1 − n 2

2 4 − n 2 2

1 − n 2 / ) (Λ C A 3 ·O

C C

Z

n 2 2 n

C

···

···

1 1 1

2 2 2 1 1 S o [1] 2 2 1

] 1 , [3

n +1 n

2 2 − n 2

1 − n 2 B D 2 ·O

C

Z

··· +1 n 2 ··· n B

1 1 1

1 1 1 2 1 2

] 2 , [3

2 3

2

4 G B 1 O

C

7 2 G

1 1

M dim M V uvrfor Quiver for Quiver g g g g H ˜ ˜ JHEP01(2021)086 , , . 1 3 ≤ f − S n = (2.5) (2.1) (2.2) (2.3) (2.4) 2 i, a or of the π A i 2 ], which ≤ ◦ S 1 22 f and so on, [ -th node. The j counterpart. is then folded to 2) = 2 3 folding g B − n (2 f despite lacking an obvious as acting on indices 2 G f (2) = i f ± . is the Cartan matrix. Throughout , E ia κ , which is in practice either ia κ . Γ folds to i κ . 3 = graph automorphisms, while the special 3 ± H 1) = 1 ) 2 B a -th node is mapped to the j ( = 0 S i − and think of π ] = 0 ] = ] = j ) i i b n i are step operators and the indices n ± ( ]). The associated algebra ± − ) = π the (2 i i C E ,H – 7 – ( 22 κ f ± ,E ,E a ji π π i : a κ E H E π − [ H [ 1 [ ] Chevalley-Serre basis folds to its · taking as input nodes of the unfolded Dynkin diagram , (1) = 4 i f f ± D E [ ]. 7 has a (up to some choices of sign) canonical Chevalley-Serre . In a unique case, ; in other words, rotations in eight dimensions are restricted 3 g which folds to 3 S acts as . 1 B . We write 6 be a complex simple Lie algebra associated to a Dynkin diagram − f Γ E n g 2 ∈ A or let π 5 +1 n D , . 1 n is a Dynkin diagram automorphism translates into the following invariance of − span the Cartan subalgebra, n satisfy this constraint as they possess π 2 a is invariant under 6 range over nodes of the diagram as in table ) = algebra folds into A 4 H ) E n 4 ( g D ( We define the folding function First let us denote the set of automorphisms by Dynkin diagrams can be folded if there is a graph automorphism such that no node is For simplicity, f D or Results for semi-simple Lie algebras follow the same pattern. 5 by the following recipe. n ˜ and mapping them toAs appropriate an nodes example, in take theoriginal folded linear diagram. diagram. Consequently, but to express that under thefact automorphism that the Cartan matrix under the action of g and its elements by linked to its ownD image under the automorphism.case In particular, thegraph diagrams automorphism for (see figure 30.14 in [ where rank this work we will oftenwe use follow a the matrix construction realisation of of [ the Chevalley-Serre basis, in which case to seven. Moreover, we show the of type basis obeying 2.1 Folding of Dynkin diagrams Some pairs of simple Lieacts algebras on can an be algebra’s related Dynkinthe diagram by an and its operation internal called structure. In a prototypical example, 2 Preliminaries JHEP01(2021)086 1 1 1 n 1 − − 1 2 1 7→ n n 7→ 7→ 3 4 2 1 7→ 7→ 7→ 2 4 7→ 7→ , 5 4 3 − n 7→ 7→ 7→ 1 + 1 , , , 1 1 3 7→ n , 1 2 3 6 1 2 2 7→ 1 2 ± − 7→                   , projection    1 . . . n n 1 . . . n n, n : : : 2 2                               3 S S : : S 2 2 S S ˜ g 1 1 1 2 1 2 3 4 1 2 − − 1 2 n n 1 2 n n last row is a special case treated in several places 2 – 8 – S Dynkin diagram of 1 2 g − − + 1 n n 1 5 4 2 2 n 4 3 − + 1 6 3 1 2 n 2 n 1 2 n 1 3 1 2 n 1 2 1 − n Dynkin diagram of 2 2 n 4 n ˜ g F G G C B . Foldable simple Lie algebras. Note that numbers label nodes and do not indicate gauge 1 − +1 4 3 6 n n g 2 E B D A D in the main text. Table 3 groups as these are not quiver theories. The JHEP01(2021)086 . 2 C , the (2.8) (2.9) (2.6) (2.7) there 1 . (2.10) (2.11) (2.12) → n − n 3 n C C 2 . A A 23 E . The Cartan 3 − ) is in C 12 3 E 2.11 ˜ 2 E ˜ E 32 κ 23 ]) = κ 2 = folding to = . In the case of 3 ,E ) with a bit more effort. To . According to the definition ˜ g 3 ˜ and only comes into play for ] 2 5 E 3 ˜ E 2 2.4 E n A − . Consider the case of − B ,E                  ) 2 . . . g ] + [ = = E ( → 2 which satisfies ( [ 1 0 ) for 3 4 f . . . 1 j ) and ( − .      − E ,E E b +1 − ± 1 1 2.2 n → . . . n 0 0 0 0 0 0 0 0 1 0 1 0 0 0 0 0 = H − E − 2.3 2 − E i = 0 . Then it follows that D g a 3 2 A 2 ([ . . . 23 . 1 0 2 2 ...... )–( )= E )= E 4 E j E b − − − ( in ( F − 1 X − X JX f n f . . . 2.1 : = 1 2 : − b j 2 0 C – 9 – X + 2 → ] = − and . . . ] = ] = 2 ˜ ] = = 6 E J 3      4 in 2 i 1 0 0 T a E . . . E ,E 0 1 0 0 0 0 0 0 0 0 0 0 0 0 ± = ,E ˜ − 3 X H X ,E ˜ 4 + E and 1 κ E                  H 2 and 3 E [ + = 2 E + ,E 1 − J 2 3 G + E [ = H H 1 → − . E 12 4 n E assumes the following form: ] = [ ] = [ = C D ] = 3 2 2 1 , J ˜ ˜ E E 2 ˜ ˜ E E , , , G 2 3 1 ˜ ˜ ˜ H H E [ [ such that for every → [ 3 − J B is = , 3 n 12 C C ˜ E → 1 In our convention Folded Lie algebras sometimes preserve additional tensors. In the case of The interested reader can easily check ( Special care must be taken when folding non-simple roots. Sometimes a sign change is The folding procedure is now easily stated: − includes two elements n 2 3 We can also reverse this statement: every exists a tensor and A illustrate the typical calculation,matrix we of will confirm ( In this specific case itthe is second, clear that is the mappedthat sign such flips to a because the scenario the first, will third never node, which occur which comes comes in after before the case the second. Likewise it is clear folded algebra is indeed required to preserve the algebraA homomorphism of folding just given, This defines the Chevalley-Serre basis for the folded algebra JHEP01(2021)086 ij δ (2.15) (2.16) (2.17) (2.18) (2.19) (2.13) (2.14) ) = folding j 3 X B ( ∗ i X ab preserving the δ can be expressed = = 1 2 ab G δ SO(7) b 567 can only occur if both . φ # ) d b − as square matrices of the j = 0 joint choices of (identical) 1 # ∗ i b c = X = 0 ∗ B c i 3 X . # c = ˜ 4 X X H 347 0 D cd φ is defined to obey + δ abc − b } a B φ 1 ∗ i j ∗ = tr ( b )= ˜ )= = 0 ∗ }| ± H d c X i c basis elements - i ( ( H X X ∗ { E i f f : a 2 : = i 246 c d + G φ )= ,X ) = )= D b 4 b b j j 0 j 1 ( − ( X b ( # X H f f X : : f b X h j = + c defined as : = is a folding of 0 i a i = 1 i j 1 3 D D – 10 – 4 3 ab # # ∗ 235 d i |{ φ B : j φ H H , which is the subalgebra of φ = = 0 ,H = b 2 + + = i c ∗ ,X X a i ∗ ∗ ± G i ˜ D D H 3 1 H ˜ # + h E of node 145 X b a H H a h φ 0 )= a )= = = , and that this can happen for − d c ( ( b X X ) = f f G B 2 3 : = : j bc c d 0 = ˜ ˜ H H a X b a ( φ 136 1 ∗ i # φ 0 and represent the evaluation as the linear extension of a X multiplicity − X = i i = X ∗ b in the same way that ˜ H 3 127 , a B φ . Given our choice of Chevalley-Serre basis the tensor is defined as ˜ 2 H h G : 4 in the Chevalley-Serre basis. In practice we realise denotes the ∈ D i i . X X # d despite a lack of graph automorphisms. This is easily elucidated with a quick detour 2 We close this section with a brief discussion of the aforementioned case of For example: The dual Chevalley-Serre bases of “parent” and folded algebras are related: The dual Chevalley-Serre basis of linear forms The other case of this type is G and where we decorate eachtrated Cartan - generator and with the aas pattern subscript a holds denoting folding up its of for algebra. remaining As illus- c to through where the second-to-last equalityfold follows to from the same the node, fact ie. that where same dimension as with the remaining values either fixed by antisymmetry orfor equal all to 0. following rank 3 antisymmetric tensor for all JHEP01(2021)086 V . | v,j (2.22) (2.23) (2.24) (2.25) (2.20) (2.21) m stands for − a magnetic with global ) v,i ∅ m m v, f | 6= ]. The formula v F =1 2 = ( n and , X j 0 1 r , e v is associated a gauge S =1 n ) i X γ V V 2 ) t ∈ X v ∈ m . The total gauge group is − ) v v 2∆( | − n t . An edge . global symmetries arising from v,i F } U( det (1 m | × m ) ) is better suited for generalisation f a subgroup of m V n ( = U(1) . v ) ⊂ H 2.25 W X =1 n v m . 0 v ∈ i X · n n γ 0 v E r S E g n | Z U( ∈ V ) V ∈ . For H X ∈ V X ∈ r Y . To each vertex m X v v ∈ v,f ) Z Y ∈ | ( v – 11 – V = = g 1 ∈ ( + W = { | 2 r | 0 W m S G ,i is associated a hypermultiplet in the bifundamental of 0 v ) = ⊂ ) = E t m ). The form of this formula slightly differs from the one m ( ( is defined by E ∈ . . Finally, we have a set of flavor vertices − W ) ) H e 0 3 2.22 , and a set of edges m v,i F v, v m ∆( | ∈ 0 ) = HS v = ( t =1 f 0 n X i e v for HS( =1 n i X ) f E ∈ n where ) ]; in particular note the absence of the Casimir symmetry factors and the 0 ) 2 X 0 v,v v , and to each edge SU( ( ) n v is the Weyl group ( n U( ) = W U( × m ) A magnetic charge is an element Consider a simply laced quiver. The underlying graph is made of a set of vertices v hypermultiplets in the fundamental representation of n 2∆( f where presented in [ summation on the wholedifferences the magnetic formulas lattice. are equivalent.to One The wreathed form can quivers in ( show section that despite these superficial Then the (unrefined) Hilbertmonopole series formula, for which the can Coulomb be branch written of as the quiver is given by the The conformal dimension charge, we define the stabiliser The Weyl group is n and it has rank and a set ofgroup (unoriented) edges U( symmetries The monopole formula is anchiral important computational rings tool in which the wascalculates study of the introduced Coulomb chiral branch and ring subsequentlyconformal Hilbert generalised dimension series by and, in counting [ optionally,unitary monopole gauge action operators nodes. under labelled by their 2.2 The monopole formula JHEP01(2021)086 v (2.28) (2.26) (2.27) : v z | . ) v,i m m | ) f γ 2∆( 2 t n t symmetry. Non-simply  v ) =1 n − n extra fugacities i X m 0 ( | A v E q v V -valent directed edge from ∈ symmetry. | z ) n 2 X v det (1 v,f G ( Q s  t + or ) ) -symmetry spin 1. The quaternionic undirected edges | i 0 m 4 z R ( n ,i ( 0 F s v W X , p ∈ n m 0 We briefly describe the method by which γ C ≥ ,v r 0 Z Z , v X ∈ n ∈ κ s X m B counting charge under a Cartan subalgebra of | − – 12 – | i . The Hilbert series expands as v,j ) = z 1 g i W v,i are connected by an | m 0 m 0 t, z v − . We only need introduce ) = are connected by v,v v v v,i 0 κ q HS( | and v . m 0 | t, z g v v ( =1 0 n v X i =1 n if and X j ,W v =1 1 n v v i ref X ) to =1 n − i X if E ∈ V = ) n 0 ∈ 2.24 X v X v − 0 v,v v ) = HS ( − v = ] further improved the formula with the addition of non-simply laced quiv- v 1 0 n, κ t, z ) = v ( are characters of − κ m ) ref i = = 0 = 2 z 0 0 is defined as follows: ( v HS 2∆( s vv vv vv κ p to κ κ κ We first state the general strategy for a nilpotent orbit, whose coordinate ring is The only difference introduced by non-simply laced quivers to the monopole formula Reference [ This formula can be further refined by labelling each monopole insertion with its charge • • • where generated by a single (co)adjoint representationdimension with of each Coulomb branch is easily calculated by summing up gauge ranks, which we extract chiral ring relationsHilbert from series the is Coulomb refined branchthe Hilbert with Coulomb series. branch fugacities symmetry Assume algebra that the The similarity toabelianised Cartan formalism. matrices is, of course,Reading relations not off coincidental the and Hilbert reappears series. in the where quiver resembles. For example, balancedlaced linear balanced quivers quivers exhibit were found to have is a modification of ( ers to the world ofas quiver Lagrangian gauge theories), theories. it was Whilesuch relatively they that, straightforward were when to not modify computed explicitlylowed the the constructed for pattern monopole (say, non-simply of formula their laced simply-lacedquivers’ cousins. quivers, Coulomb In the branch particular, results symmetry it is enhances made well according known sense that to balanced and the fol- Dynkin diagram which the under the topological symmetry JHEP01(2021)086 ) ]) g 24 (2.30) (2.29) adj( n , where i,a ϕ Sym s t and ) i . If the space is a z g ( − i,a : for the present paper s u 2 r generators subject to G , g =1 ∞ X s + i,a u − dim s t ) i z | | ( s i,a i,a g -th symmetric product w are characters of α | | n i ) i =1 ∞ i X s z are expressed as weights in the weight ( is the number of fundamental flavors on s = w, ~ϕ α, ~ϕ w r h h ] the two labels are swapped. i ] of its coordinate ring. They are always f Φ ]. One first defines the abelianised ring and  . 23 N ∈ ∈R i k i 26 , α and w , 28 G , z ,  ) Q Q k i is finite and they correspond respectively to the 25 n f – 13 – t z =1 s i n − 27 (  , r s n,N abelianised relations 4 Q g = HS and weights whereas in [ =  ,...M and − i,a 1 , α u G s 1 [01] g . For example, the (closure of the) minimal nilpotent orbit log + i,a t ,M ) u n k k G ( µ 6 rank =1 ∞ ]. X k (whose coordinate ring is generated by one coadjoint generator [ n, 23 g . Then, whenever feasible, we verify that the full set of relations are 8 ) = t i . We will sometimes blur the distinction between the three types of i to the character representation of the , . . . , ϕ or . The variables satisfy t, z G contributes several basic variables to the ring: 1 n , 2 6 i 1 i,a t t x ϕ G is the Möbius function and  rank ) index the vertices and hence gauge group factors of a quiver gauge theory. Each k = i ( ≤ PL(HS)( ~ϕ µ a -th node, and both the roots Let This procedure is only slightly modified in the few isolated cases in this paper where We could then expand the highest weight generating function, comparing (polynomial) i This paper differs from the present paper in the simple root convention for 6 ≤ where the basis of the theory’s gauge group the (co)adjoint representation goes by leaving only then reduces it to a particular Weyl-invariant sub-ring. gauge node 1 variables by dropping all identifying information except for the node and gauge indices, visible in the PL. 2.3 Abelianisation Coulomb branch chiral rings ofstructed simply-laced following unitary the prescriptions quiver of theories [ can be explicitly con- higher order, identified: we calculate thethe Hilbert relations series in of question a and ring compare defined it by to tabulatedthe expressions. Coulomb branch isgenerators, not which a are nilpotent in orbit. these particular The cases chiral also ring coadjoint. is then Their generated contribution by will more be generators and relations ofpresent the in Coulomb the branch. first few The ordersof minimal of any set simple algebra of relationsis is described typically by anecessarily set quadratic of in Joseph the relations coadjoint [ generator. In more general cases we go to slightly where complete intersection, the list of look up the space in [ coefficients of and find missing representations suggestingthe same the computation existence in of a relations. more Or elegant we fashion can using perform the plethystic logarithm: is unaffected by discrete gauging. Knowing the dimension and global symmetry, we can JHEP01(2021)086 ] i - ] C R [ C [ C C (2.32) (2.33) (2.31) (2.34) (2.35) — the 0 ≥ and g Z defines a is simple. } ∈ i j g 1) P − SU(4) , . In this paper 0 g ] = , ), generate every C 0 [ , C 2.30 1; 0 form a closed Poisson , ) ) 2 algebra ] while the Poisson bracket (0 C 2 j, b j, b 7 [ ] ). The Coulomb branch chi- C C [ ,... 6= ( -symmetry weight C of the Coulomb branch and all ) ) = ( 0) 2.30 R , | | g 0 i, a i, a i,a , i,a ( ( } . We leverage information from the 0 w α | | , G i if if 1) i 1 -symmetry, , W ] 1 − R − w, ~ϕ and whose symmetry algebra α, ~ϕ j,b -symmetry weights as 1; ( h h ± j,b abel , ϕ , . The true Coulomb branch chiral ring 2 R u Φ ] ] C − [ ∈ (0 and the last four belonging to ∈R 1) C . ± i,a [ , α C i,a w u − factor acting non-trivially on the Coulomb branch. An abel ϕ C A 1) , Q – 14 – C Q [ . ij − (1 with well-defined U(2) κ C , ± i,a i,a { 0 ∂ ± 0 u can be read off from ( SU(2) , O . (The remaining Poisson brackets vanish.) U(2) ∂ϕ 0    ± ± i,a , Φ = and therefore operators in u = = = 2.2 2 } } } 0; 0 , − ± j,b i,a ± i,a − j (1 + i , u , u , u ] have weight 2 under the C ± i,a i,a i,a + [ ,... u u ϕ with 4 fundamental flavors comes with the following matter and C i,a { { 0) { ϕ , . Weights of operators assemble into a single (coadjoint) representation of 0 2 → , 2 necessarily assemble into tensors of the . ] − i 0 i U(2) ] i , C ] [ C 1 [ C [ C − C is the vector space of all Coulomb branch chiral ring operators with C : 0; i , ] is defined as in section C ·} -symmetry is assumed to be the (1 [ , ij { R C κ = {O Any Coulomb branch operator These Poisson brackets, along with the abelianised relations ( The Coulomb branch is a symplectic space and its chiral ring carries a Poisson bracket, For example The R 7 chiral rings are generatedConsequently, by operators in moment map of the symmetry — which has a matrix realisationoperator’s for weight all is cases twice in its this conformal article. dimension. map algebra. This algebra isoperators precisely in the symmetry algebra we focus almost exclusively on good (in fact, balanced) theories whose Coulomb branch Abelianised variables scales with weight ral ring of any good orwhere ugly theory is graded by symmetry weight is a sub-ring ofHilbert the series Weyl-symmetrised ring to identify thewith correct the subring, correct picking representation-theoretic Weyl-invariantring, abelianised properties. and operators a For specific more example, details see on appendix the2.4 abelianised Construction of Coulomb branch multiplets where element of the “abelianised” chiral ring where the charges are associated to which descends from a bracket defined on the abelianised ring: gauge representations: with the first two charges belonging to JHEP01(2021)086 , . . A N m I N N X . exact ] (2.38) (2.36) (2.37) C klm [ ]), all at are dual c ] and an C 6 ∗ 1 k m , , and let 5 X P M I . And since . If ] = N l m O ,X k klm X c [ ]. We construct moment m 7 . P M i in [ ] l /I = ] , are replaced by symbolic com- } l ,X N ϕ [ k satisfying O C , X g [ k and = . ∗ k N, {O A ± h to form components of X u /I k is explicitly constructed as G = )] 8 O , for highest weight fugacities. W ] µ i} i,a ). Then we claim that N l k synthetic method X acts as a Lie algebra homomorphism: – 15 – , ϕ abel C = 2.30 ·i such that [ ± i,a N,X u k C h form a basis of N ( N, , O h i g N k [ ∈ (similar to but distinct from that studied in [ C k , independently check its relations and also calculate a positive simple root operator to fix the remaining choices, at least in N,X X N ] = {h C operators transforming in the coadjoint representation along + i,a ). In this last step abelianised relations and components of operators, also in the coadjoint representation; the following [ u ) 2 satisfies certain matrix relations, which can be inferred from the C j a formed by ] 4 ] , the moment map ϕ C 2 C [ P kl N ] is heavily constrained — enough, in fact, to allow us to select an ]. Let [ δ − C 4 ) with matrix relations also expressed in terms of components of C C [ k i ij = C ϕ O N ( i l / discrete gauging set of operators from 1 ,X 9 (ie. ∗ k X N h ] two of this paper’s authors developed a prescription for coadjoint chiral ring 7 , ie. k This combined approach is called the It is in effect a change of variables to a set which is well-defined even on the non- The moment map The choice of This definition guarantees that In [ We reserve the usual symbolIt for is moment enough maps, to declare X 8 9 operation called the level of individual and explicitly constructed operators. the quivers we consider. which is forever off-limits torepresentation any theory method front relying and solely centre. on Hilbert series, allmaps while for keeping all examples intotal this agreement paper between predictions and of explicitly Hilbertquivers, check series the and we expected our explicit relations. will construction By of argue finding folded for specific interpretations of the multiple link from [ abelian locus of therepresentation brings moduli with space. itit a On to major the construct advantage: other the it generator coefficients hand, is the in very explicit less-than-perfect these and abelianised relations rigid. (including We can dependence use on mass parameters) — something Note that the third objectbling contains factors no of abelian expressionsboth (and expressed hence in no terms potentially ofponents trou- abelianised of variables generates the Coulomb branch chiral ring, we have foundHilbert a series set of of the generators Coulombstand for branch. for Let the us ideal denote of the abelian ideal relations they ( form as almost unique There is a basisto of ring is generated by with another setdiscussion of straightforwardly generalises for such cases. generators following [ We will also consider two cases (one in the main text, one in an appendix) in which the JHEP01(2021)086 nodes identical wreathed U(1) m ) connected factors. 0 j Q 1 , or a subgroup node with adjoint m U(1) 2 Ours differs in sev- 2 ) S S S o n 10 o and ]. U( [1] 1] 6 , gauge nodes will combine 5 1 2 U(2) 1 U(1) × 1 [1 U(2) One discretely gauges the 12 . (and potentially other legs nodes (which form a i for two examples. We support this claim by Q pivot 3 – 16 – 11 U(1) n 1 1 simultaneously exchanging . For example, three legs composed solely of 2 S copies of arbitrary gauge groups or “legs” of the quiver. 1 n 1 1 identical legs node, while two legs with 3 m , with 2 1 1 2 . Quivers on the left wreathe into quivers on the right. S o S o 1 1 U(1) U(1)) Figure 3 ]) to acting on × 1 1 6 , 5 , ie. one with 1 to get over this difficulty; see figure (U(2) Our strategy in this section consists of the following steps. We first demonstrate the It is possible to discretely gauge any quiver of the type depicted in the top left corner In contrast to our treatment, these works didA not leg can claim have toIt arbitrary discretely shape may gauge and be the in possible theory, particular to but need restricted discretely not gauge be quivers linear. without a pivot node but we do not have a successful 10 11 12 quivers their claims only to effects on the Coulomb branch. case to present. existence of a well-definedconsistent with orbifolding existing operation literature. onas Then a we the suggest whole Coulomb that in branch,results the a of operation giving way this acts results that operation on cancannot should the be be quiver be viewed deduced written as from quivers down the in using Coulomb their branch existing own action right, notation; even and if we that they the introduce often the concept of thereof, which permutes thegauged gauge the factors quiver’s associated automorphism withwill each arrange leg. into a We sayto that give we have matter in [ of figure to a single common nodelegs which by we extending call the the overall gauge group with the symmetric group Our first example ofbranch a by discrete a quiver subgroup operation, ofon discrete the the Coulomb gauging, quiver’s branch as orbifolds automorphisms. an theeral orbifold, Another respects: was Coulomb operation, previously it studied which preserves in also the [ allows dimension acts of for the consistent Higgs and branch successive aseralises well discrete beyond as gauging acting the on of Coulomb a branch, nodes collection into of “larger” nodes and gen- 3 Discrete gauging JHEP01(2021)086 Γ is G o ) ] r k (3.5) (3.3) (3.4) (3.1) (3.2) r U( , which Γ . In this , or more o ) ) ··· k r r G of a group U( 1 U( Γ r [ o . is an ordered list ) G i Γ ( 1 o − 0 σ × · · · × G g ) i ∈ 1 g r ) = U( i , (parameters) rather than ring g, σ )) ( 0 Γ . is a unitary group g ]. As a set, we define . The group multiplication law is ( × ,b wreath product Γ ) the direct product of groups. There 29 G j ! , which can in addition be permuted gσ ( ∈ i , we have ( , and more generally + π . G ,a G Γ u σ m ) not numbers i o i,a S ,a ( ) n ) =1 π i Y i r M ( is understood in the notation ∈ with x + π

, π n u U( = ) : 0 ) = copies of , by ) – 17 – Γ = for i,a ) = Γ . An element of o n i,a , σσ o Γ M act on them, ie. G + j,b ) x o 0 ( G ( u G g π r π ( ≡ + i,a ∈ not u ) gσ Γ ( 0 (the integer n π o , σ n 0 does S G g ) = ( 0 ( π . ⊆ Γ , σ of the group 0 o Γ g and n if there is a risk of confusion) [ ( )) · , and we write G together with a permutation k Γ Γ r ) r o n G o G U( g, σ is the direct product of G of ( ∈ ,..., i 1 Γ ) g o G denotes the Cartesian product of sets, action permuting them. For any G g, σ × · · · × × ( m ) S 1 r copies elements . Since each node contributes several variables to the abelianised chiral ring, there is an In this paper we consider wreath products where n (U( Γ n Note that mass parameterselements should (VEVs); therefore be treated as Action on more complicated (polynomialby or action rational) on functions of indices these of variables the is full defined expression. For example We will first study thisgoal procedure is through to the show lens that the ofof Coulomb Coulomb the branch branch can quiver. abelianisation. be reduced The to an orbifold by an automorphism induced case, in particular in thereplaced quivers, by we the extend rank thefor usual shorthand notation in which 3.2 Action on the Coulomb branch Intuitively, by generally a direct product of finitely many unitary groups where the are of given, for by a permutation group we could denote well as calculating aknown few Higgs wreathed branch quiver operation Higgs is branches. the We 3d also mirror3.1 conjecture to that this operation a Wreath on well product theWe Coulomb pause branch. for a moment to introduce the notion of the generalising the monopole formula to this family of quivers and computing an example, as JHEP01(2021)086 : , for (3.6) (3.7) (3.8) (3.9) 3 m (3.12) (3.13) (3.10) (3.11) S  } of − i,a Γ ,u ) + i,a 1  u } M { ) is compatible with  ± j,b − π are Coulomb branches 5 ,u ˜ = ϕ C 2.35 ± i,a | | )( u 4 )–( i,a i,a { ϕ α and w   | | 2 i } π C − i ) 2.33 . ) 5 ± i,a ~ϕ i = ~ϕ ( . ϕ ( Q ,u ,b ( ,a ) to arbitrary subgroups ) . We investigate one such example ,b j . i − ) i,a ( ) ( orbifold of the original space. This w,π j m · . m π α,π ( π ϕ h ( ) h then the mass deformation breaks the Z S ± π { ϕ } π m Φ . The third line is similarly preserved M u  5 i ∈R )) = S ∈ − m ,a π 1 α m w α ) M = S ,a x, y i ) ) ∈ ( X { = Q Q i M /S 5 π ± π ( ( 6= i,a ↔ π C ,a ! u ,a π − 1 ) M ) ) i 1 ϕ i M i ( 1 m ( = to preserve connectedness while the second line – 18 – ( = M ∂ ± π − ij π ϕ π ˜ C u κ } 5 π α )( ) ) = ∂ϕ ϕ ± ) = 2 ± y · ( i,a ϕ ( )( = = = or cyclic group 4 P and M } , π } } − is physical. ϕ ( ) m ,b ,a i 1 ,a ) ) ) π x ) i − i j A ( ϕ w ( ( ( O ( 5 π ± π − π ± π ( { ϕ − ↔ theory which gauges to the bottom right quiver in figure P ( ( ,u ,u ,u ,a 5 ) π − ,a ,a i ) ) ) i ( i i A , the two mass parameters must be equal to preserve symmetry ( ( ( is a consequence of the automorphism. In fact, it is noteworthy π = π + π ± π − 5 w ) = ϕ u u ij u − 5 − 1 { { { κ u + 5 u u + 5 = = + 1 = u ) = } } u } ) ) ) ( . j π ( ) = ± j,b ± i,a − i,a -invariant operators are physical. This is easily done through the use of a π − 1 u u u ) m i symmetry. ( ( ( u ( S π 2 + 1 5.2.5 ,π ,π ,π κ S u ) ) ) . This is a sensible constraint: if ( π ± i,a i,a i,a + π u u ϕ Note that nothing prevents generalisation from The effect on the Coulomb branch is then transparent. If To implement the quotient on the Coulomb branch chiral ring, it is enough to declare The first line is clearly compatible with the action. The second line also succeeds with We should check that the form of the Poisson brackets ( ( ( ( π π π { { { construction leads to newthat they Coulomb are branches orbifolds which of were known Coulomb previously branches. unknown,instance provided the alternating group in section of, respectively, the original quiver andby discretely gauged quivers, the two spaces are related ie. the discretely gauged Coulomb branch is a that only projector: Every operator of the form a simple relabelling: because that the third lineenforces forces identical the gauge action and of matter content on each leg this action in the sense that Since under quiver’s To see this consider the In fact, this constraint forces JHEP01(2021)086 ) 4 D 2.4 (3.17) (3.14) (3.15) (3.16) )–( i 1 , 2.1 0 , 0 , 0 ∗ h i 1 . The projector E i ± i 0 , , 1 quivers as studied 1 , 0 , 0 ± , 0 , i , 0 1 2 , 1 0 ± ± AD h , i , ∗ . h− we show what happens e 0 1 1 ,
1 ± E i , ∗ h± i + P α 0 ± 1 0 , , i i i , which replicates ( E ∗ − 1 1 ± quiver then 0 1 1 0 , ) , , , , 2 i ± ± 0 E 1 0 1 0 ] , , i ∗ 1 h , , , 1 , α 1 1 0 0 1 1 1 C , i − , , , ± E [ , − 0 1 ± 0 0 0 n , , ) ∗ h± 1 ∗ ∗ ∗ h h h
h− e 2 i C 0 1 ± , ∗ h e 1 ± α E E E E , ± 0 A , , ) + i i i ∗ 2 ± − E i , 0 0 ∗ 1 1 0 4 , ) ∗ , , , h 0 1 ± ∗ . α E ∗ , , i 0 n 4 1 0 1 ∗ n ) , , , h i H ± 1 0 1 E E , , C 0 1 1 0 ) ) α H B , , , , 1 ± 1 E α 4 , i h± 0 0 0 0 ) N ) . Select the Chevalley-Serre basis of − , ± 0 ± 1 e h h h N h e , i , , 4 e 1 e ··· e e ( ( 0 1 1 ± 0 ( 7 h , , h ± + , 1 ± ± 1 P P e , , + + + + P ) = 0 2 Φ ( ∗ ± 0 0 + ) = h 1 , , i i i i h h± ∈ X + + 0 0 1 1 0 0 3 − + P e e E , , , , , h α +1 i n i ( ( ∗ 0 h 0 1 0 1 3 e n – 19 – ∗ 2 , , , , α 1 h 0 ( + , ( 1 1 2 1 + P P e A D , , , , H ± 0 E i , , ( 0 0 1 1 ) P ∗ ) i 1 1 ∗ ∗ ∗ ∗ 1 + h h h h N N 3 + + α P ∗ ( ( 4 ± ± ± ∗ 3 H h E E E E i i , , + , e i i i i i ( 0 1 0 1 0 P ( P H , , i + 1 0 0 1 h H h , , , , 0 ± 1 ± 0 4 P e , , , , i P 1 0 1 0 4 4 , , , , h 1 0 ± 1 0 0 , , , , h 2 1 1 1 ≤ + , , , , + ± 1 1 0 1 i + , , ∗ h± 1 0 0 1 + i 2 ∗ 1 ± 1 ± X h h h h 2 ≤ + Φ , , ∗ h± 0 ∗ E 1 e e e e 3 , 0 0 ∈ 3 X , ) ∗ ∗ acts component-wise. Similarly, h± h H 0 E , i α 0 ) H h ) ∗ ∗ + + + + h±± h = 0 1 E E 2 , i 3 P ) ) i i i i 1 1 ± + 4 E E h h i i , 1 0 0 0 + is the moment map for a type ) ) , , , , ( 0 1 ± 1 ∗ D i , , i i ∗ 1 0 0 1 h± ∗ 2 , , , , + 1 0 ± 0 ± 0 0 P , , , , , , 1 0 1 1 N H E , , , , − 1 0 1 1 0 1 ∗ H 2 ) i , , 1 1 1 1 n i 2 + ∗ ∗ ∗ ∗ 0 h h h h 1 ± ± ± 0 ± 2 , , , , , , H h h ∗ 0 1 ± 1 1 1 0 1 A E E E E 2 , , , ( i i i i 0 0 0 h , N + 1 0 0 0 H h± h± h h± h± h P , , , , 1 ) e e e e e e 1 0 1 0 ∗ 4 1 , , , , 1 + ( ( ( ( ( ( 1 0 1 1 h± ≤ h , , , , ∗ H i 1 P P P P P P e ( 1 1 1 1 1 X h h h h ≤ H h 1 P e e e e + + + + + + + 1 h + + + + = = moment map and ) = = 4 n 4 D ]. In fact, if To see this action on an example, and to illustrate why its action on moment maps is The projector acts remarkably simply on moment maps of type C D 7 N ( N P acts on operators: where the dots stand for the negatively charged terms, eg This sum above contains 28to terms, most and of to them: exhibit the action of letters, reserving lower case lettersIn for this notation, operators the with moment appropriate map commutation is relations. so simple, consider theand top its left operator counterpart, quiver ie. inwith the figure Poisson basis of brackets. operators We in will denote the algebra elements and their duals with capital in [ is a JHEP01(2021)086 i 0 = , 1 i ± , (3.18) (3.19) (3.20) (3.21) 1 ± a,b,c , h i 1 ˜ e 0 , ∗ h± 0 , i 1 E or 1 i i ± ± , 1 1 , i 1 1 ± ± , , ± 0 ∗ 0 h± , , , 1 0 1 i E , a,b,c,c 0 ± i 0 ± h , , ∗ , h 0 1 0 , e i i 1 ∗ h 0 i E 1 1 ± , ) , 1 ) i 1 E h± 4 ± ± 1 2 , , 1 i = ± , e ± , i 1 1 ϕ 1 ± , 2 ± , 0 1 , 1 , ± ± ± 2 i 0 + ϕ , , 0 , 1 ∗ ± h c − , i , 2 2 1 i 2 h± 0 1 ± 0 , E , 0 , + ± ± ± e ,a + 4 , , , 0 , ± 1 0 i 2 , h 1 ∗ 1 1 h 1 1 u 1 a,b, , 0 e + h + 4 , ϕ ± ˜ ∗ h± 2 ± , ± + 3 E i ∗ ∗ , ˜ h± h± 1 e , u i i 1 2 0 2 ϕ )( u E 0 + 1 1 ± h , E E , ± 3 ± i , 1 i i ,a e i , + ± ± 0 2( 2 , , 0 1 + ∗ 1 1 + 2 ϕ 2 2 h , ± 0 0 , + 3 i , , 1 u ± ± + ± ± E 0 − , , i , , i h± 0 0 0 − h u , , ± i 1 1 h 1 1 1 0 e , ) , 1 0 e 1 2 e 0 + ∗ ± ± ± ± 0 4 h ,a , , , , , , ∗ ± = h± ± 2 i , 0 , 2 2 2 + 1 0 1 ϕ + E ˜ h ∗ + ˜ 0 1 h i 0 E ∗ , ϕ i , ± ± 4 ± ± i e 1 i , , ˜ , i , 1 ± ( otherwise. Notice a feature common to com- , 1 1 2 E + 0 , + 2 1 1 0 0 0 , 0 i H , ∗ ∗ i ± 1 h h , ± ± 2 1 i , 3 , , , 2 ± + 1 0 0 , ˜ h± h± , 0 1 , 0 1 E E 2 ± , ∗ ϕ i , ± 2 h± + ± 0 , 0 e e i , i , 1 ± 1 0 =1 h X – 20 – 1 , , 1 1 ± 0 ∗ 0 ∗ h± E 3 a , , ∗ e ± 0 h 1 ± ± , acts non-trivially: the prefactor from the operator + + ∗ , ± ± 1 0 h ) a,b, , , , , E ˜ 0 h 0 ± H i E 0 i i , , 1 0 1 ∗ = i h + = 2( E 1 h i e ) h h± 1 1 0 0 P 1 ∗ , 0 ± ± h ) e i i 4 e e ± , , , E 4 ± ± 0 i , 0 1 ± , i , 1 0 0 ∗ , , , i = h h E 1 0 h h h 1 1 1 , 1 1 0 1 + ˜ + ˜ + ± ) , , , i e e ± 1 , E ± ± ± , i 1 0 1 ± + 0 i , , i , i + 0 , , , i 0 ± 1 h 2 1 1 1 1 0 , , 0 0 ± 1 0 , 3 + ˜ + 3 , , ∗ h h e 3 1 1 ± ± ± ± ± ± 0 , a,b, 1 ± 1 h , , , i , , i , e h e , ˜ h ± 0 1 1 1 0 1 1 0 1 , , H 0 ± h± , , + ˜ ˜ e , , h± 0 0 3 ± 1 ± ± 0 ± e , 0 0 = = = h , , i , , , ∗ ∗ h± h± e , + ( h 0 ˜ h 1 0 0 ± 1 0 1 e 0 h i i , , , ∗ 3 h e ∗ + E E h 2 2 1 2 1 0 1 e + , , i i , ˜ h ∗ ∗ ∗ + h± h± h± e and i + E 1 0 1 1 2 1 i + H , , 0 i ˜ ˜ ∗ ∗ h± h± ∗ i , 2 + E E E 0 0 0 2 1 ± ± 2 , i + 0 1 i , i , i ˜ ∗ h h h± ˜ , i E E 1 0 h 2 ± 1 1 1 0 H i , ˜ ˜ e e 1 ± 0 , i i , ˜ 6= 0 , , E ± 0 1 2 ± 1 ± 0 ± ± 0 0 , , ± 1 1 , , i , , , , , + , c ˜ h 1 1 ± 1 ± 1 0 1 1 1 0 1 , 1 ± ± , , , 1 , , ± ± 1 ± 1 ± 0 ± ± ± 0 ± if , , ± 1 0 , , , , , , , , + ∗ ± , , 1 0 ± 1 ± 1 1 1 1 1 1 1 i , , 0 0 , ∗ 1 0 0 h h± h 0 ,c H h± h h 3 h± h h± h± h± h± h± h± h± e e e ˜ 0 1 H e e e e e e e e e e e e B 1 ± a,b, ˜ N h h + ˜ + ( + ( + + + ˜ + ˜ + ˜ + + + + + ( + + h e = = = + i 0 a,b,c, h and the remaininge operators follow theponents of same the moment pattern map on which where we defined JHEP01(2021)086 (3.24) (3.22) (3.23) and the with the t ) modulo no ] N . The methods 3.24 -terms. 4 30 F lm D N ∗ obey several relations 4 3 H ijklmno  B 4 h N 3 = 0 = 0 = 0 ∗ 4 + ) 2 3 # lmno 4 N 3 H t with the HWG [ N N 3 h 2 1 P ∧ + 3 µ # ∗ + 3 B N − independent relations, or generators ∗ 3 H , so does ) 4 H ∗ 3 4 1 )(1 4 D ˜ 3 h 2 H . h t satisfies the identities in ( 3 3 ∗ other 3 # 2 + + 3 ˜ µ # # H 3 . B 3 3 3 h − h ˜ ˜ h h N (7) (1 = = ( = = – 21 – satisfying the relations (computed using standard so ∗ 4 M H ], albeit for the next-to-minimal orbit of ) = ) form the complete set of relations for the next-to- i ) 4 31 h ( t, µ 3.24 P -symmetry weight appearing in the exponent of + R ∗ 3 HWG( H ) [000] :[002] : [010] : tr 3 4 4 6 t t t h . Note that this is an instance of Case 2 of the Kostant-Brylinski ( 13 nilpotent orbit. The space should be an orbifold of the minimal orbit P (7) 3 B so satisfies certain matrix relations which identify the space it parametrises should be understood as the contraction 4 N D N ∧ N . , so it should in particular have the same quaternionic dimension, namely 5. That is 2 4 One can check that the moment map Just as A similar set of relations appears in [ D 13 automorphism. We will now arguebranch that of the the orbifold original can also quiver be after recovered gauging as its theemployed Coulomb automorphism. therein can It bethe extended is quiver to for natural the which to it present ask is case: the if given Higgs a the branch general and nilpotent look for orbit, matrix one relations can implied construct by the computation shows that indeedminimal ( orbit of table 3.3 Wreathed quivers The previous section establishes that some Coulomb branches can be orbifolded by a quiver notation rank 7 antisymmetric invariant tensor of abelianised relations. To show that therefor exist no that matter, one can calculate the Hilbert series of the ring as described below. This We describe a relationglobal by symmetry its representation in which itthe transforms. tensorial This often, form but of not always, the specifies relation, which we provide on the other side of the colon. The This space is parametrised by aplethystic matrix techniques) of precisely the dimension of the next-to-minimal orbit of as the (closure ofappropriate the) for minimal a nilpotent orbit of becomes the inverse multiplicity required in the definition of the new dual basis, e.g. JHEP01(2021)086 ] a 6 . See , node, (3.29) (3.30) (3.25) (3.26) (3.27) (3.28) n 5 , much 2 S } S 2 o , 1 ∈ { U(1) ) a . For illustrative 2 , , 3 ,a ϕ , 3 2 ϕ ) 3.12 with adjoint matter. − 2 , ) wreathing group 3 ,a n 2 ϕ and ϕ . U( ) ) − ,a )( ) 2 ± 3 , ,a 1 3 1 , , ) 2 2 u 3 3 ϕ 2 , ϕ ϕ and ϕ ϕ 2 − 2 ( ϕ − − n ) ) − 2 ,a S 2 , , ,a ,a − 2 2 o ,a 2 3 1 2 3 ϕ ϕ ϕ is slightly more subtle. The first and ϕ ϕ ϕ ϕ )( )( )( 2 3 − with adjoint matter: at the level of the − )( )( ,a 1 ) 1 , U(1) 1 1 1 1 2 , ) , 2 , ,a 2 2 ϕ ϕ M with an associated 3 3 n ϕ ϕ ϕ ϕ ϕ . n ( − − − − U( )( S The Coulomb branch of a wreathed quiver can − − ,a ,a ,a 1 o – 22 – , ,a 2 2 3 1 1 2 3 , ) · ϕ ϕ ϕ ϕ 3 ϕ ϕ ( ( ( ( ( ϕ )( ( − − − 2 − − . However, the pattern of abelianised relations, which } ], we present two particularly simple examples depicted ,a M = = = 2 = 3 6 , , node with adjoint matter. This explains why in [ − 3 − ,a ,a ϕ 1 ,a 5 − 1 − 2 ( u − 3 ,a u u + 3 u 2 − ,a ,a ∈ { u wreathed quivers ,a ϕ + 1 + 2 + 3 ( U(2) = a u u u , − ,a − 3 ,a 3 u = . The third node brings six variables and all physical operators are invariant under it. nodes combined into ϕ ,a ,a + 3 3 a − 2 u u and ,a U(1) + 2 ,a u ± 3 n u node would. The wreathing group acts similarly to a Weyl group in that it . 3 for two prototypical examples. The top right quiver has a single wreathed node , is as follows: U(2) 3 3 The case of the bottom right quiver in figure There are very few new elements in the wreathed quiver theory depicted in theAbelianised top relations on the middle node read Traditionally a quiver theory is described by a quiver diagram in which nodes represent figure Coulomb branch, there is no difference between second gauge nodes, whichsix are variables inside thecan scope be of determined a by two-fold wreathing, consistency each with come the with discrete gauging of the bottom left quiver in Interestingly, the latter can beor read as in two ways: either as the relation of a which is appropriate for“bouquet” a of and the relations on the third node are essentially unchanged: right quiver of figure like a permutes the index abelianised calculations performed on discretelythis gauged non-wreathed amounts quivers. to In practice, non-wreathed keeping quiver the while indices, imposingpurposes, Poisson invariance and under and to the draw abelianised a projector linkin chiral to figure [ structure from the figure while the bottom right quiver is anAbelianisation example of of wreathed a quivers. quiverbe with studied a through longer abelianisation wreathedwrite with leg. them relatively down minor in changes, but full it generality. is We cumbersome find to much greater clarity in (entirely equivalent) original, ungauged theory.notion This of is a indeed quiver possible, theory to albeit at thegauge cost or flavor of groups generalising andtheories the links add represent wreathed appropriately legs charged denoted matter. by Wreathed quiver resulting theory is also a quiver theory which could be studied without reference to the JHEP01(2021)086 (3.44) (3.38) (3.39) (3.40) (3.41) (3.42) (3.43) (3.31) (3.32) (3.33) (3.34) (3.35) (3.36) (3.37) invariant. ) m , is the wreath Γ ) ∆( 3 W ϕ − 2 , 2 . ϕ again acts on this index, ) 3 ) . )( 2 γ 3 ϕ and leaves 3 r 2 ) S 2 ϕ ) t 3 ϕ S 1 G m , ϕ − − − ± 3 − ⊆ ) 2 u , − 1 2 2∆( : , Γ 2 , 2 ) t , 2 2 2 Γ , 2 ± 2 ϕ , W 2 ϕ 1 ϕ u 2 , )( ) det (1 ϕ ± 3 1 )( , ϕ 1 3 3 2 ⊆ + , ) , u 2 ± 2 )( ) , ± 3 2 ϕ ϕ 2 2 1 u 3 2 3 m , , + , ) u ( 3 ϕ ± 2 2 ± 2 2 3 , ϕ ϕ ϕ 2 1 − − Γ , 3 , u ϕ u ± 1 ± 3 ϕ ϕ ± 2 2 − 1 X 1 1 2 2 W ϕ u , , , , , − − − u u ϕ − ± 3 ± 2 2 2 1 ± 1 ∈ 2 2 , , − 1 1 1 2 1 ( , γ , , , , , u u u ± 1 1 2 ϕ ϕ ± 2 , ± 3 ± 2 2 2 1 2 ± 3 2 r 2 , u , , 2 ϕ u − – 23 – u )( )( u Z 2 u ϕ ϕ ϕ ± 2 ± 1 ) + 2( ) + 2 1 1 2 ϕ 1 ) 2 ∈ , , , 1 2 , ϕ u u , , , X )( )( )( “stretches” across several nodes (but not the mass 2 + ± 2 2 2 ± 1 of the gauge group ± 2 , 2 m 3 2 1 1 1 , )( ,W 1 , , , + u u ϕ ϕ 1 u a ± 1 | ϕ ϕ 3 2 2 2 , 2 2 1 1 ϕ , , , Γ 2 3 u Γ , 1 W ϕ ϕ ϕ ϕ ± 1 ± 2 , ± 1 − − ± 2 1 o + + ϕ ϕ ± 2 ± 3 u u + u W u 2 2 | , , 1 1 − − − − u u , , 1 1 1 − − 1 , W 1 + + 1 2 , 1 1 1 1 , 1 , , , , ϕ ϕ ± 1 1 1 ± 2 1 1 ϕ ϕ 1 1 2 1 , , by the wreathing group ( ( , , ϕ = u ( ( 1 2 u ) = ± 1 ± 2 ± 3 ϕ ϕ ϕ ϕ t ( ( ( ( ϕ ϕ Γ − − u u u + + ( W Γ W = = 2( ======) with an appropriate discrete group, necessarily a subgroup of i i i i i i i i i 1 2 3 HS 1 1 1 1 0 0 1 0 1 , , , h h h 1 ± ± ± ± 0 1 ± ± , , , , , , , 2.25 2 1 2 2 1 ± 0 ± 1 , , , 0 1 ± ± ± ± 0 ± , , , , , h± h h 2 1 0 1 0 e e e h h± h e e e h± h± h± e e e symmetry and the moment map parametrises the next-to-minimal nilpotent ) generalises readily for such a group, setting 3 C 2.25 Note in particular that the index which contains the Weyl group r This is the monopole formula for the wreathed quiver. Formula ( Consider now a wreathedthe quiver. Weyl To group compute in theS ( monopole formula, weGenerically, need several to such discrete replace groups exist.product Our of choice, the which Weyl we dub group 3.4 Monopole formula for wreathed quivers variable, which is sharedand by invariance all under legs). it isbranch The has a wreathing necessary group prerequisite fororbit of operator this physicality. algebra. The Coulomb Its components include: JHEP01(2021)086 , ) 4 2 D W 3.44 (3.50) (3.47) (3.48) (3.49) (3.45) (3.46) × 1 . | W 2 a, = ) m Γ γ 2 − W t . For such a task, 1 t − a, ), via 1 m . | ), then this group can 2 . We now illustrate the 3.45 } ) Γ . det (1 − o 2.22 ) , ) ) . The magnetic charges are . The total gauge group is | ) γ a, e G m 2 ( 2 ( m t a,i S { n S = 1 X m − = = ∈ 1 | 2∆( } as in ( e t γ 0 e ) n ! + ]. This is done in the usual way of { W E 1 | W 2 n m d = ; = t det (1 = which contains exactly one element of d ( m ) F ). One of the four rank-one nodes is a Γ n ) = r Γ The monopole formula in the form ( m } − n ( W Z ) W Γ = 11 P a,i X W c Γ) a, d } ∈ m o n and the conformal dimension is given by ( | γ G , | 5 ) – 24 – , . . . , m = + Γ Z 1 | 1 b X c W a, c m ∈ n Weyl( | ( ; m a, b, c, d ∈ ) Consider the quiver corresponding to the affine , , t { d ) ( m − n ) = . The Weyl group is . = S = 2 , m 3 a, b a,i P m c ( ) = a S V ; does not decompose in general as a direct product of sym- t { = 5 m 2 ( n | can be written as a product of two groups t , m Γ r Γ ( = b ) := + Γ Γ W | N E HS W b , m 2 P m a, − , m is a principal Weyl chamber for the group , . . . , m 1 a,i with rank ) can then be rewritten 1 a, 3 . In the context of Weyl groups, or more generally Coxeter groups, this is m Γ) | m Γ m ( o ; 3.44 2 W 2 t =1 G U(1) ( i = ( X ) n × m denotes the central node, and U( ) = P Weyl( a m For wreathed quivers, This procedure involves finding a subset of = U(2) 2∆( The corresponding ranks are G elements where Example 1: subgroupsDynkin diagram of (see theflavor first node, so column we of can table define the graph by the vertices where procedure on three examples and most explicitly on the third. metry groups. One can introduce symmetry factors exactly as in ( The formula ( for the Casimir factorspresenting as the defined monopole in formula. the appendix of [ each orbit of called a fundamental chamber.be For instance, used if to orderthe the identity magnetic charges in increasing order for each node. Then one uses is very time-consuming to evaluateit numerically is in preferable a to preprocess series itIn expansion somewhat, in particular, using if the high the levelthen group of one symmetry can that it split presents. the summation into two sums. A comment on computational complexity. JHEP01(2021)086 ): 3.44 (3.51) (3.52) (3.53) (3.54) as a sub- , next to 2 subgroups S 2 S SO(8) to be invariant, ). Clearly, in this ∆ 1 ) ) and 8 8 1 2 2 4 4 4 8 3 6 t t 12 24 ) 2 (this is also known as the 8 ) , and out of these 9, 6 con- + + t ], and the subregular orbit Z 8 have to contain 3 t 6 6 , 12 S + t t 14 Γ 3 Cardinality [ + D 6 Z × t 2 6 , t 2 G 3 + 17 + 10 S . S + 3 4 4 4 ( + 3 t t 4 S (14) 10 10 t 3 4 ) ) 10 , t , 10 2 2 S ) (13) ) t t , (13) (124) 2 (13)(24) (34) 2 + 20 + 48 t , , t , , ) the groups + 20 + 6 − − — 2 2 (12) (13) t t − 2 2 − (123) (1234) t t , (1 (1 (12)(34) (1 (12) 3.43 (12) (1 Generators (1234) (123) – 25 – (12) (12)(34) . Subgroups of )(1 + 10 )(1 + 3 )(1 + 17 )(1 + 3 , and we can readily evaluate the expression ( 2 2 2 2 Γ t t t t (1 + (1 + (1 + (1 + which are subgroups of Table 4 Γ = = = = 4 3 3 2 2 4 4 3 2 3 4 S Z Z Z S S S Z Z A × × × Dih 2 Name 2 2 Trivial HS Z Z Z Normal Klein HS HS HS Non-normal Klein ), and moreover to satisfy ( Double transposition , double cover of the subregular orbit of 12 includes 156 subgroups which can be gathered into 19 conjugacy classes. D 5 S SO(7) as a subgroup. However there are two equivalent but non-conjugate 2 S , and we have to pick one of them. We are then left with 4 classes of subgroups, which . 2 12 G D which identify the spaces asminimal the of (closure of the)of minimal nilpotent orbit of We end up with 4 inequivalent groups group. Out of the 19tain classes a of subgroups, 9 areof subgroups of can be identified with thesimple example 4 this classes analysis of is subgroups slightly of superfluous and the result could have been guessed. The group These 19 classes arewe partially have ordered to select and those formdihedral groups group a Hasse diagram. For JHEP01(2021)086 , 4 S 4 ...... + ...... (3.55) + + 8 ... + ... t + + + + 8 8 ... 8 t t + 8 8 8 8 t + t t t t + 8 8 t 8 t t 56 784 8400 396 21 165 586 − +522 − − +0 − − +7 − +273 − 6 6 6 6 6 6 t t t t 6 6 6 6 6 t t t t t t t 6 7 49 55 25 70 10 215 PLog +7 − − − − 4 − − − − +140 +833 4 4 4 t 4 t t t 4 4 4 4 4 4 t t t t t t t − . We also give the first 2 36 +12 t +12 +13 +5 106 C +35 2 +24 +38 +24 − 2 2 2 t ). The results are gathered 2 t 14 t t − 2 2 2 2 8 t 8 t t t t 2 14 9 15 9 t 10 16 21 3.44 28 symmetry of the quiver explicit, 4 ...... S ...... + + ...... + + + + + 8 8 + + 8 8 8 8 t t + + quiver by acting on the legs by all subgroups 8 t t t t 8 8 t 8 8 t t 4 t t D U(2) which is studied in the previous example. +5096 +8918 × +2184 +3094 +2898 +4522 3 +771 +896 +2324 6 6 4 +1485 +2270 6 6 6 6 t t S U(1) 6 6 6 t t t t – 26 – 6 6 t t t t t U(1) +539 +693 +685 +985 +210 +223 +515 +1155 +1925 We now consider the same quiver as in the previous +356 +497 4 4 4 4 term gives the dimension of the isometry group of the 4 4 4 4 4 t t t t 4 4 t t t t t t t 2 . t quiver, but we use the fact that the gauge group of the 4 +48 +48 +90 S ], where the cycle index technique was used. The group 4 +104 +118 +69 +125 +83 +160 +195 +300 2 2 2 2 2 2 2 2 2 t t t 2 2 D t t t t t t 32 t t Perturbative Hilbert series 1+8 1+8 1+9 1+9 . Part of the results presented here already appear in unpublished 1+10 1+14 1+14 1+15 1+16 4 1+21 1+28 S acts diagonally. This form makes the contains as a subgroup the 4 4 4 3 4 4 3 2 U(1) , where they are arranged in the shape of the Hasse diagram of the subgroups S S S S Z Z . The coefficient of the A Dih 4 Trivial 5 . Wreathed quivers obtained from the affine Subgroup . We give some details about the computation in appendix . Normal Klein For each subgroup, one can compute the Hilbert series ( 4 4 S Non-normal Klein S Double transposition in table Coulomb branch. where we give a name to each class ofin subgroups. figure of orders of the series expansions of these Hilbert series, along with their plethystic logarithms, example, namely the affine theory is really and this Following the approach of this section,class one of can subgroups define of a wreathedsummer quiver work for by each conjugacy Siyul Leeadmits [ 30 subgroups that can be organized into 11 conjugacy classes, as listed in table Table 5 of Example 2: subgroups of where the

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(8) Trivial 1 so JHEP01(2021)086 . , 2 Γ S . W × 16 (3.59) (3.57) (3.58) (3.56) t . With 2 ) where 4 S + × 14 Dih 3.47 t 3 S , which acts by 4 × + 14 S 2 12 S is then the product associated with the is t         Γ × 2 2 becomes ( explicitly as generated 10 = e d 1 0 S W 4 (10) S ≤ ≤ Γ X + 106 1 1 so ≤ 2 e d = Dih 10 − We now consider the quiver ), and one finds the Hilbert t ) + × . 2 W t 2 3 < 2 e d         3.44 S 1 1 ≤ ≤ + 454 X × 1 1 (1 + 8 e d 2 . The group t ) 20 S         ) 2 e 2 2 2 3 d 0 0 1 0 0 0 0 1 1 0 0 0 0 1 0 0 × c t , e 1 ≤ 1         + 788 2 − . We can describe S c , e 4 X 6 . So there are only two allowed groups of index 2. Let’s focus on this second group. 2 ≤ t (1 S c 3 e 1 = c , d 2 ⊂ W , 1 Γ – 28 – b 0 d ≤ ( W and Γ of         X + 454 1 b b 2 4 d Γ ⊂ t a and the Weyl group is 2 W are unaffected by the permutation, so we omit them in X ) for S 1 a W × 0 1 0 0 1 0 0 0 0 0 1 0 0 0 0 1 + 106 2 −→ 3.44 = 10 2         in S t 10 r 3 Z S ∈ X m × | where 2 1 + 14 , we need a symmetry of the quiver, which is given by permutation Γ 0 S 1 ∆ Γ = W | × . 4 × 1 and an extension 3 S . The letters in red give our assignment of magnetic charge for the various S 14 Dih ] 2 W × × 3 1 2 , S = 4 S × Γ 2 [2 × S W 1 × 1 S S However, the sum involved in the computation is difficult to evaluate in practice, and The factors = HS Γ column of table it is useful to usethe the present symmetries case, to the avoidthe sum unnecessary sum in repetitions, over as ( the explained Weyl chamber above. is In given by: series for the Coulomb branch of the wreathed quiver, The corresponding HWG and other data concerning this space are gathered in the middle This group is isomorphicthis to the explicit dihedral description, group it of is order now 8, that possible we to denote evaluate by ( permuting the four magneticW fugacities by the following two permutation matrices: In order to preserve of the two legsnamely containing the nodes the matrix discussion that follows. The commutant of this part in whose Coulomb branch ispartition the closure of thegauge nilpotent groups. orbit of The rank is Example 3: wreath product of non Abelian groups. JHEP01(2021)086 . ) ) ) 4 ∈ 1 α i R ) δ 2 ( > 3.61 3.59 ) (3.62) (3.63) (3.60) (3.61) , e ). This 1 α of order ) , e α α 2 . Inside the , d ) 1 2 dim ker( , d , e 3 1 dim ker( , c , e 2 2 . , c , d 1 ) ,..., 1 ). The condition ( 0 d 1 x , c ( . 2 = 1 > Γ ≤ , b i / 1 x C . For every element a, b 6 and and we need to say what we mean 0 = ( ) y ). Let’s now explain the summation 2 2 α e d m = , y 3.46 ( 0 , we get the conditions listed in the last 0 dim ker( 6 ≥ or ≥ ) comes from the cases where R 0 m m · · – 29 – 3.62 of a quiver, which can be wreathed by a finite α α , with some of their properties. Without entering δ y < y C ). ). The first three sums in the right hand side of ( 6 . The principal Weyl chamber is then defined by 1 1 ⇔ 6 e can be read from table d ) 0 3.59 3.56 Γ , of order 8, or the group of symmetries of the square 4 , y eigenspace can have dimension 0 W x 1 ( is an element of Dih − ≤ m · ) α (the ). The subtlety in ( δ . However, in the present case the order 2 elements don’t necessarily x, y 4 α . The goal is to compute the HWG for ( are listed in table R Γ 3.61 0 Γ ). 3.59 , seen as a group of endomorphisms of its representation space, we pick a basis is the dihedral group Γ 0 . A simple choice, which we adopt here, is to pick the lexicographic order W The elements of order 2 in Γ denoted with the letters as in ( ≥ which satisfy the inequalities 10 3.5 HWG forWe wreathed now quivers explain how tois perform the the HWG orbifold at ofpermutation the group the level Coulomb of the branch HWG. The starting point by Doing this for everycolumn order of that 2 figure. elementrange Combining in in all table ( these conditions together, we obtain the summation is done in the third column of table which generalizes ( When this is the case, We leave the study offuture the work. general case, and the connection with Coxeter group2 theory, in for of the kernel of this endomorphism in a consistent way (with m for every simple root fix a hyperplane in then has to be replaced by a more general condition, which we now explain on our example. The elements of into the details of theare theory delimited of by Coxeter subspaces groups, fixedgroup let by of us the a order note simple 2 that Lie elements the algebra, in Weyl chambers the the in group. principal Weyl Formally, the chamber Weyl is defined as the set of charges Z exploit in the standardsame way the principle symmetric is groups, used whichsum, to allow one get to of the order course summation therange introduces charges. range for symmetry over the The factors indices last ( four indices in ( The notation here should be clear, with the charges JHEP01(2021)086 (3.64) (3.65) 2 1 1 e e d ≤ ≤ ≤ 1 2 1 or or or 2 d d 2 d d e 2 2 2 ≤ ≤ and and 1 and < e < e < e 1 1 e 2 2 − d 2 2 1 1 e e e e Inequality d d = = = ). On our three examples, 2 1 1 e d d 1 0 3.70 − 3-cycles 2-cycles         1 2 3 1 1 1 1 1 2 Id 1 1 1 1 1 1 − 1 0 1 1 1 0 − − 3 − − − S 1 1 1 1 1 1 1 1 0 0 ε eigenspace 1 2 1 1 0 0 1 0 1 0 1 0 0 1 − 0 0 0 0 0 0 , and whose rows are labelled by irreducible 1 − − − 2   Γ Γ = − Z       Cardinality – 30 –  1 to keep the discussion concrete. We first recall that representation. The last column displays the condition Γ = 3 ) 1 2 2 2 2 2 4 4 2 Cardinality S . In the first column, they are presented as permutations, 2 2 , e 4 ω Order 1 2 and , e Dih ω ω ω ω 2 1 1 ω 3 d d , d Z 1 1 1 1 1 1 1 1 1 1 , d 2 2 1 → → 2 ( e e e 2 1 Z 3 e e Z ↔ ↔ ↔ 2 2 . The second column is the order of the element, the third gives a basis f f 1 1 2 2 d e → → ) Γ = e d d 2 . For our three examples, these character tables are 2 2 , , , Γ = Γ d d ↔ ↔ , e 2 1 2 1 d e e 1 1 Cardinality ). e Identity d → → , e 2 1 2 ↔ ↔ ↔ Permutation e e , d has a well defined character table, which is a square matrix whose columns are 1 1 1 3.62 1 d d d d Γ → → eigenspace in the ( 1 1 . Elements of the group 1 d d − In the following, we give the general prescription, and at the same time we illustrate with three examples the group labelled by conjugacy classesrepresentations of of elements of of the imposed by ( Table 6 acting on These character tables containclass in in each the entry correspondinginstead the representation. of trace the trace, One of the waythese the list to matrices. (unordered, matrices refine and We of with will this repetitions the need information allowed) these conjugacy of is eigenvalues the in to eigenvalues equation of give, ( JHEP01(2021)086 for , in ) 0 (3.66) (3.70) (3.68) (3.69) (3.67) k R,j 0 K . For a λ i ρ is finite is ) in C in irreducible j the number of ,...,M 0 1 } C R n } 1 the list of eigen- can be written M } 1 ( their cardinalities, − ) gives the charac- 1 ) , | C HWG( are denoted j 1 R,j C Λ , with 3.67 | . and R,j }{ ) ) 0 ) 2 = Λ 0 ), ( k }{ } {− k K j HWG( 1 1 c M M , . . . , n ω, ω ,j . 3.66 0 0 3-cycles 2-cycles k R,j , not necessarily irreducible, of ) k R 0 . ) }{ λ = 1 Γ λ 0 k k }{ }{ 1 ,...,M ) is the trivial representation), and j ) 0 1 , 1 1 1 2 3 − − 0 Id M } of k k M 1 1 { { } 1 1 1 1 , 0 M ρ { 0 (1 1 (1 M M 0 ( − k R, , so that the HWG is R k R 0 t =1 λ K =1 − − Q λ 0 Q k K HWG. We say that 3 } {− }{ k the list of eigenvalues for − − (1 (1 S 1 1 1 1 1 , and and { { 0 × ε . The elements of 1 2 (1 =1 =1 (1 i,j K Q 0 Q j K . k R R 0 k c Λ =1 k Γ = K =1 finite Q 0 Q 2 k K – 31 – k n =1 Cardinality Z , . . . , n X j ) =  1 | C 1 Γ ) = | = 1 for all Γ = C i } Cardinality and the variable } } the conjugacy classes ( 2 2 1 ) ω = 1 j ω HWG( , such that the numerator and the denominator of the above l Γ) = 0 1 µ / C ( HWG( K C k R, }{ }{ }{ 2 λ 1 can be written in that way; this is a non-trivial assumption, as ω ω ω is the class of the identity element), ω ) and 1 C C }{ }{ }{ in the representation HWG( K 1 1 1 1 1 1 1 is a symmetry group translates into the fact that to the above ex- j { { { Γ C which is not irreducible, we similarly denote by HWG( . This list is easily obtained from the decomposition of ). Let us call 3 R R Z f f 1 3.65 dim Γ = ), ( Cardinality ,..., 3.64 The fact that We now show how to compute the HWG for an orbifold Coulomb branch based on their dimensions. Finally we denote by the irreducible representations ( = 1 i i From this expression itorbifold is then straightforward to write the conjectured HWG for the pression are associated tworespective representations dimensions expression transform according to these representations. Then We assume that it is known that many Coulomb branches don’t satisfy it. there exist two liststhe of highest monomials, weight fugacities that we denote representation values of the class k representations. Note that an initial Coulomb branch that admits a Of course in each caseters ( the sum ofconjugacy the classes, eigenvalues listed and in ( ρ d we get JHEP01(2021)086 . . 4 2 C (3.75) (3.76) (3.77) (3.71) (3.72) (3.73) (3.74) . . ) fugacities, is affine quiver. ) 10 8 ) i i 4 t t 4 4 4 4 ) 2 2 2 t t t ) D 2 D 2 2 2 4 µ µ t t 3 1 µ µ 4 has HWG equal to 2   εµ − µ 1 µ 4 ) ) µ − 1 . One can check that 4 4 ] +  − + t t D i − . )(1 2 2 2 4 . An alternative way of 2 12 6 )(1 t t − t t µ µ 2 t 2 2 2 2 2 4 ) )(1 2 t )(1 (8) 6 8 µ µ µ 2 2 t t µ εµ )(1 2 3 1 1 t 3 1 2 2 1 µ so 2 2 µ t µ µ µ + + 2 1 )(1 + )(1 + + µ 3 − ) ) 2 2 2 µ 2 4 4 2 µ − − − t t t t is a weighted hypersurface in t t t − 2 2 2 2 1 2 2 − 1  20 )(1 2 2 12 µ µ µ t 6 µ µ 2 µ t )(1 )(1 )(1 Z correspond to the external nodes of 2 2 t t εµ 1 f − )(1 2 2 4 6 4 orbit of 1 − − 4 1 / 2 t µ t t − − 4 µ 3 2 t + + 2 1 6 1 3 1 2 1 µ µ + 2 1 µ i 1] 1 3 1 ) C µ µ µ µ )(1 2 2 , written in terms of 3 1 2 , µ )(1 )(1 − µ t t µ 2 4 )(1 )(1 t 2 2 2 12 2 t 1 1 quotient, we charge them under the three 1 2 2 1 ω − − − − t t t 2 F − 2 t t µ µ ω µ 1 1 6 1 µ , 2 2 and 1 2 2 µ ). This way one gets the HWG 1 − 1 f 2 . Consider for instance the HWGs written in µ µ µ Z µ µ [3 + 3 − h h h )(1 )(1 )(1 1 1 − 4 4 2 4 µ − t t t 11 t − − )(1 , 3.70 ) 2 2 1 2 – 32 – 2 )(1 1 2 4 (1 t µ µ µ )(1 2 (1 (1 )(1 + )(1 + µ 1 t µ 2 2 2 2 2 t 1 ) ) µ − − − t t 1 i + ) = PE ) = PE ) = PE 2 2 1 1 t t µ 2 3 C C C ω ωµ µ µ 1 1 2 )(1 )(1 )(1 µ  µ µ 2 2 2 − 1 − − − ]. t t t +  2 1 2 − − 2 1 (1 19 (1 (1 (1 µ µ µ − µ (1 (1 + + + 2 We can apply similar methods to the eighth line of table − − − =0 (1 i X   ) to obtain + ( 1 (1 (1 1 3 (1 1 6 1 2 2 . . The weights t ± ] . The identification of the irreducible representations is as follows: 2 2  = 2 3.70 = = = t X 2 µ 4 )  t ) = ) = ) = 4 2 1 3 3 2 µ µ ± Z Z PE[ = /S / / + X + 1 C C C  3 ::: HWG( HWG( HWG( 2 t µ 1 4 2 3 3 2 S Z Z + µ fugacities. The closure of the minimal nilpotent orbit of subgroups and apply formula ( 2 2 + HWG( 2 HWG( HWG( µ G 2 Z t + 1 1 µ µ 2 We illustrate how this formula works in practice on the example of the  which evaluates to this is indeed the HWGseeing for the the same closure computation of relies the on the fact that The HWG forPE[( the minimal nilpotentthe orbit Dynkin of diagram. In orderdistinct to perform the This reproduces the results in [ Eighth case of table We then use equation ( All HWGs and quivers are gatheredterms in of table PE JHEP01(2021)086 ) Γ 1 − ∈ . We D γ 4 (3.78) (3.79) (3.80) . The 1 for the S Γ − o y γty nodes. This theories, but fugacities are − U(1) y 4 1 ) = 4 U(1) 2 , and the resulting − and 4 N y i S 2 z d ) det ( t 3 1 y , − − be a subgroup of ) )d 2 Γ , of ADE type. Specifically, y )(1 πiy 2 2 γtyD factors and fugacity y − ) makes it manifest that quiver, and compute the Higgs 2 , y, t, γ i t − . Let (1 4 z SU(2) 4 ( 4 − 4 [ D 3.78 1 4 U(1) z F πiz ) d )(1 2 2 t , y 3 i ) det ( 3 , that can be identified using the degrees z − z ( 7 D groups, therefore the Haar measure is not πiz d 1 µ ). It follows that the Higgs branch Hilbert (1 2 . The integral over the ) for the − d ,E )
4 2 6 4 2 ,y , 2 i . Note that ( z – 33 – 3 U(1) γty z , z U(1) ,E γt , πiz d 3 Z 6 2 2 − Γ − , , z = 1 . 1 4 ∈ 2 4 ,D | 1 X a finite subgroup of γ 1 5 4 1 z y | , z | = 1 πiz d J D 1 1 Γ 2 | i z = ( = det | i ) det ( i z z J ) = ) = | t , with Diag( ( , y i H Γ z /J γtyD ], where groups were extended by outer automorphisms, following ( ]. Here the context is different but the techniques spelled out 2 µ − C HS 33 34 d 4 1 is non-trivial) a disconnected gauge group, as follows directly from the Γ det ( ). Disconnected gauge groups have been considered in the context of the ) are readily evaluated for each of the 11 subgroups of ) = 3.1 3.78 the diagonal matrix ] apply. In fact, the case considered here is particularly easy to handle because the factor (after ungauging a diagonal , y, t, γ D i We make a few comments on the results. First, the Hilbert series coincide with those We focus on a simple but rich example, the affine z 33 ( [ F of Du Val singularities four instances occur, namely of invariants of the correspondingdimension of groups. the In Higgs particular, branches this of shows all that these quivers the is quaternionic 1. with performed over the contours can be considered asintegrals a ( discrete fugacityHilbert for series the are disconnected presented gauge in group figure and where the measure is in [ groups which are beingmodified. wreathed are We pick all fugacities U(2) series is obtained via a Molien-Weyl integral which is written explicitly as consider the wreathed quiverproduces defined (when by this groupdefinition acting ( on theplethystic four program in [ a formula of Wendt [ such quivers, using anaction averaging of procedure. wreathing Finally, on itCoulomb the branch. allows Higgs the branch study and of contrasts the it geometric withbranch the of parallel all action the on wreathed quivers the that appear in figure In this subsection, we turn tothe the rest Higgs branch of of the wreathed paper, quivers.it which serves This is focuses several in on purposes. contrast the with a First, Coulomb well-defined it branch hyper-Kähler demonstrates of quotient, that the whichgauge wreathed can quivers group be do is indeed associated disconnected. provide with a Secondly, gauge we theory explain whose how to compute the Hilbert series of 3.6 Higgs branch of wreathed quivers

JHEP01(2021)086

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4 Trivial 1 D JHEP01(2021)086 , . 4 Γ 31 D then Y , we , and (3.84) (3.85) (3.81) (3.82) (3.83) k 5 = So gen- = 23 = 0 Y 14 i 3 then one finds = X gauge groups. k 12 . If + . ) for the affine Y 6= 4 2 2 ] ’s. The generators 12 A i i U(1) X Y k being a row vector. 12 α t α ’s or more. − i + 4 i . The generators with subject to the relations − B B α 1 3 . If = 4 6 )( S X 4 t 4 A and that . Finally, there is a relation r A A 6= 4 i i + 3 3 ji , and study the action of the α α S 4 α 4 Y 4 t 1 . B − α ··· i, j, k 2 . . The F-term relations imply that 1 i = U(1) α 4 = ( = 0 PE[2 3 α i ij 4 and α = 0 4 Y ) we obtain 1 A B i A . i k α k α . 4 ) subjected to α 6= 4 . 3.81 3 B 4 = 0 =1 α , . An irreducible gauge invariant is one that i X j i 7 i i 2 = 0 = α have weight 6). To deal with the wreathed , , B 4 B 4 j ,B 4 i groups, say i representation of B 3 ij A A A – 35 – A and , ε 6= 4 Y 3 = 1 2 = . Note that B i α = , 4 permuting the three remaining =1 i 4 4 i X being a column vector and i ( U(1) and we’re back in the previous case. 4 = 0 3 α A α i 4 A are j 4 2 i S = 1 k group are with A i α α α i A α i i i 4 ) of i, j, , As a check of the computations presented in figure 4 4 α α α α 4 A 4 α 4 6= Γ 1 , k For U(1) l 3 B U(2) α B α + , 4 j 2 k = α B = , i α with α ij i transform in the ) and combining with ( = 4 4 + Y X B 3 = 1 ij A i l Y α X i α 3.82 ( 2 j i the scalars in the chiral multiplets transforming as bifundamentals of are α α X i i 4 i 1 α B B 4 have weight 4 while the α , for where the indices can not take the value X i − B i 4 − A X = . The results are gathered in table and r = i Γ U(1) i 4 4 α A A A k i ’s are built from α α ··· i and j j 1 α The F-term equations on α α i i i Consider for instance α α α 14 4 4 4 The spectrum of operatorsfor on trivial the Higgs branch is a subset of the oneB determined above B Putting all this together,quiver we (the obtain the Hilbertquivers, series we have to impose the additional gauge invariance under the discrete factor containing one transform in the irreducibletwo two-dimensional representations of which shows that the between the two families, for instance in the form In particular an irreducibleerators gauge of invariant can the Higgs not branch contain coordinate three ring contain either one or two Gauge invariants are paths in theabove, quiver using of the the form shorthand notation can not be writtenB as a product of other non-trivial gauge invariants, so it can be written Taking the trace of ( The F-term equations on the briefly show how the sameWe results call can be obtainedU(2) from a counting ofFor invariant simplicity, operators. we ungauge onewreath of product the by a subgroup Gauge invariant operators. JHEP01(2021)086 ] it , as 4 6 6 7 6 , quiver 5 /E /E /D /D 4 2 2 2 2 ) = 1 D C C C C 2  2 Geometry . In [ S o 6 1 + 2 = 0 2 · [1] 2 2 = 0 z z z 2 (3 1 2 z + action into a Coulomb (c) ) = − 3 ) = + 2 y y y 3 4 S 1) y · + − 4 1 2 x 2 +27 Relation − (  x · y ( 4 C 3 x xy x xy = 4 27 = (4  − 4 ) D 2 2 ) H 2 X , of which one dimension is a free factor X +2 dim 1 1 2 +2 (b) 1 ) = 3 X 2 2 2 2 2 X ) )( 2 X X 2 2 · – 36 – )( + + 1 2 2 X X Consider the case depicted in figure 2 2 2  X X quiver (left) is orbifolded by an − + 2 1 X 2 X X C 12 12 1 1 4 12 + + 1 1 1 Y Y X X (1 + 2 Y 1 1 D X X X X X ( = = X = = X = − + + = 12 y )(2 x ) y x z 2 = 2 1 1 2 Y 2 y )(2 x 2 : : : X X X = 6 4 4 X Generators z t t t − = = quiver therefore has . 2 + 2 1 − y x 3 · : 4 : : 1 X S  ( 4 8 D 10 X t t t 12 1 1 Y 1 + 2 = ( · z = 2 2 1 z (a) · : : : : : 6 8 : 12 t t t 8 12 18 t = (2 1 t t  . The Coulomb branch of the Γ . Generators and relations for operators on the Higgs branch of the affine C U(2) loop 3 1 2 3 H S S S The (quaternionic) dimension of the Higgs branch, when there is complete Higgsing, Z from the trivial factor in the adjoint loop. We can be more precise and compute the Group Table 7 wreathed by subgroups of which is the casegauge here, multiplets. is The equal todo the all number the wreathed of quivers. matterdim The multiplets quiver minus (b) the hasH number a of Higgs branch of quaternionic dimension Comparison with adjoint matter. was pointed out that thebranch of Coulomb the branch quiver of (a). the(c) We quiver is have (b) argued also is that that the an very Coulomb orbifold same branch of orbifold. of the the Let’s Coulomb wreathed look quiver at the Higgs branch of quiver (b). Figure 6 branch shared by two distinct quivers. JHEP01(2021)086 is n 4 [210] C D (3.87) (3.86) 2 mirror S . In fact d min 1 5 3 O C = 3 B 1 ] (labelled C 35 7 min . O n Discretely Gauged = and the nilpotent cone in H ] 2 1 , 2 4 − D . n share the same Coulomb branch, ] 1 [2 under which the generators of the C 6 12 O t min 8 which can be discarded. The second − O class S theory on a Riemann surface of = 5 1 Sp(1) t H ] and equation (7.12) for the hypersurface H 2 1 0 Higgs Quiver A 37 + 2 2 1 4 t +1 A D – 37 – 2 n S 2 D S min o ]PE[3 t = [1] 3 orbit of dimension 7, see table 12 in [ B PE[2 ] 2 2 1 , 1 min [4 . O n algebra, namely the transverse slice between the maximal orbit 1 Discretely Gauged = 3 C C quiver theories admit a dual description as a theory whose Higgs branch 4 1 1 D = 4 ] where the identification as a transverse slice is made, as well as an explicit 2 1 N min 38 and one puncture). See section 7.2 of [ . Illustration of the relation between i) discrete gauging’s effects on the Coulomb branch d = ]. This family appears in the context of trivertex theories where the rank of 3 n 1 C 36 In summary, the two quivers on the right of figure [ Coulomb Quiver n dimension, as one would expect from discrete3.7 gauging. Mirror symmetryMany and discrete gauging is the original’s Coulomb branch and vice versa; this property is known as but only the wreathed quiver’s Higgs branch shares the original quiver’s Higgs branch equation. The same familywork also of appears [ as aform Coulomb of branch the of hypersurface the equation. mirror The quiver Hasse in diagram the is a rare occurrence ofit a has symplectic been singularityintersections which suggested of is that Slodowy also all slices aC hypersurface of hypersurface the symplectic in nilpotent singularities orbit interpreted of as the dimension genus 2 ofgenus a are Riemann surface ( term can be identifiednilpotent as cone the in Hilbert the series(of for dimension an 9) intersection and oftherein). the a The Slodowy global slicechiral and symmetry ring the on transform this in space the is [2] and the [1] representations, respectively. This space makes Hilbert series using the hyper-Kähler quotient, finding The first term comes from a free contribution Table 8 and ii) discrete gauging’s effects on the Higgs branch of a corresponding electric quiver. JHEP01(2021)086 . We claim this 1 1 represents the discrete 1 2 1 . Its dual is depicted in enlarged 2 4 S . The Coulomb branch of D 8 ∼ = 1 1 1 2 Z ∼ = nilpotent orbits would resemble the (1) O but they act differently. On the left quiver BCFG 15 – 38 – 1 1 1 nilpotent orbits could be recovered as Coulomb branches 1 1 2 1 1 ] through the use of the monopole formula. It was already well ADE . Quivers on the left fold into quivers on the right. ]. The new gauge node 1 21 1 1 ” in [ Figure 7 ] and is a consequence of S-duality for theories with brane realisations. 1 1 (1) 41 – ] that many O 2 39 [ If the enhancement of the mirror is discrete, so must be the original’s. Moreover, since Let us consider the paradigmatic case of quivers in table We are not aware of a physics reference for discrete gauging on the Higgs branch but believe it to be 15 fairly well known among physicists interested in Higgs branches of quiver gauge theories. of unitary quiver theoriesand and the that resulting there nilpotent is orbit.like a the In desired robust particular, symmetry connection the algebra’s quiver betweenquivers Dynkin should diagram. choice whose be of Consequently Coulomb balanced one quiver branches and might assume shaped reproduce that 4 Quiver folding The next discrete operation allowswhich for were a natural identified interpretation in of [ established non-simply [ laced quivers, an automorphism is gauged; on the right we gaugediscrete a gauging background vector. ofmirror background dual. vectors is a genuine action on quiver theories, so is its the quiver into the oneas depicted in “the the case fourth column;symmetry such of an operation the was gauged firstis vector. reported the In mirror this dualboth case of procedures the the are gauge process called group covered discrete in is gauging the previous section. Somewhat confusingly, turns out, it is already known. the quiver in the firstthe column third is column the of minimalsymmetry the nilpotent on same orbit the figure; of flavor the node.be symmetry Each viewed of matter as its hypermultiplet Higgs a is branch background coupled is vector to the a multiplet. same mass This as which vector can the can in turn be gauged, turning symmetry One should therefore expect to be able to find the mirror dual of discrete gauging. As it JHEP01(2021)086 i + 4 1 , u 0 , 0 (4.3) (4.1) (4.2) ] and h = ˜ e 45 + 3 u They have ]. Some of 3 quiver in the 16 4 D in the former’s HWG ]) that the multiple link ] folds into the quiver in folding map 3 ) quivers fold into spaces 50 +1 i < N n for examples. We first utilise D 7 ( for ] or gauge-vortex duality [ n ) 1 ) i and in general use hats to denote y 44 y − µ ( ( ˆ n C 2 F ] conjectured a minimal modification F 7→ 1 A ]. A mathematical treatment of folding m i ] and our results. ) 3 ) + 46 − x ; see figure ( i x N ( I,a 2 spaces. Although the conjecture was highly # ˆ x cF F µ 2 ) the discretely gauged quiver’s operator G – 39 – ) = ) = ) = . A space is folded by restricting it to the subspace n y + 4 3.20 i,a y and u x + m ( 4 + x F ( F quiver folding cx + 3 ( F u F there is no more need to track each individual wreathed = ] on ˆ i C 1 , 42 0 ] can be reinterpreted as folding the five-dimensional theories’ Dynkin diagrams. Ref. [ , ], precise details of multiple links remained elusive. 0 , 0 43 h 48 , e . Recall that in ( ]. + 7 and so on; we denote this space 47 i 0 49 4 BCFG , 1 ϕ ], Jan Troost suggested that quivers of this type might be understood as folded simply , ]. 0 1 , 3 0 = , h 1 e ) symmetry and derive the effects of folding on the monopole formula. We then 3 ] as can be guessed by mapping n ϕ B 19 ( n C As long as we stay on Note that the examples below focus on nilpotent orbit quivers only because they are In this section we show (using an alternative approach to [ According to [ -algebras associated to them were studied in [ 16 laced quivers, an idea that ultimately finds validation in [ across folded legs; sometimesas this a removes result, all even independent though mass the parameters original but space one isvariable. and, mass-deformable, To the reduce folded to space a is minimal not. necessary set we introduce the bottom left of figure is defined as fixed under the action of the symmetry,as which in well this as case generates thevariables constraints on the folded space. Note that mass parameters must be set to identical values 4.1 Action onAlthough the one Coulomb can branch fold a quivergauged directly, Coulomb the operation branch. can The alsoa be prerequisites quiver performed for with on an folding a automorphism. and discretely We discrete start gauging yet are again with identical: the example of a most easily studied usinggeneral tools operation. we have For developed. example,(7.1) the We of quiver expect [ of folding section toand 4.1.2 be comparing in a to [ the completely HWG of the latter quiver. abelianisation to show thatwith Coulomb branches of reinterpret folding as anlaced action quivers; on in the particular, quiver ourthe itself, analysis form of showing in the that [ monopole it formula produces on non-simply examples reproduces W and Coulomb branches ofthe non-simply laced phenomena quivers in was [ magnetic recently quivers provided [ in [ can be interpreted as the result of to the monopoleearlier formula tentative which results reflected ofsuccessful [ the in enigmatic its multipleallowing goal, link, further study giving checking [ support against made to an appearance the in the existence study of of little string non-simply theory [ laced quivers and non-simply laced JHEP01(2021)086 . . }  ) } i , we ˆ g ( (4.8) (4.5) (4.6) (4.7) (4.4) 2 ] π ˆ ˆ f, C [ {  C ) = π j ∈ ( i π = ˆ ) : O } } . ˆ i g j 1 j,b { , ˆ j 0 f, and , , x { 0 h 2 ] e i,a ˜ C x [ = ˆ { ): , ie. = min C ( ] ) ˆ + 3 C I π 4 ∈ [ u i 3.20 ϕ C ˜ O − ) = ˆ )) = i Q. ) ,a } ∈ 1 , 2 + 4 ˆ k,c g 0 . , u ϕ x k k 0 ˆ ( ( f, h ˜ + 3 Aut ˆ O 2 O { e F u F k (˜ 2 k ) ( ⊂ ) 2) ,a ij 4 3 ij F + 2 3 / π c / c ϕ Γ 2 ˆ 3 u 2 ϕ ( ˆ + 3 k ϕ k + 3 ∈ − ˆ X u − F X ˆ ) = u − π ,a ) ,a ,c ,a ,a i 2 = + 2 ) ∀ = ) + 2 2 1 ,a k ˆ , i , u ϕ – 40 – } 2 ˆ ( u of the Poisson algebra satisfy this property: } ϕ we require 2 1 ( , , j ,a π j ϕ ) 1 1 ] ˜ , , i (2 F ( ˆ x O ˆ ( O C 0 0 i,a ( , [ h h 2 2 2 , π ˆ , , , i x is a complex number and i e e F x C ˜ (˜ ( . As a result, ˆ O c O =1 =1 =1 X X X { + 3 a a a F F ∈ { 2 = ˆ u ) = ˆ g = = = = = i,a k,c ˆ i f, x 2 x ) = , ( 1 + 4 , 0 F u h denotes the number of nodes that fold onto the same node as node ( ˆ e i = F } # j,b ) = + 3 , x u ( i,a x F { are arbitrary operators, multiplicity y is the subgroup by whose action we fold. Γ and Now we re-establish contact with discrete gauging. For So we have in our hands two pieces: a “folded” subspace (with its coordinate ring) and If the folded space is to retain the original’s hyper-Kähler property, the symplectic While abelianised variables fold in a completely trivial manner, composite operators The folding map has a simple interpretation. Abelianised variables of the initial, x , and therefore the relationsthe in folded particular subspace: hold on the automorphism’s fixed point, which is a Poisson bracket onrestrict this to space. the subspace, If weits we have symmetry? assume a new that hyper-Kähler the space complex to structures study. also Whathave properly is it? What is where we restrict to the folding locus where property in particular must be preservedclose. and the In Poisson other brackets on wordsIt the for folded is any space enough must to show that generators variables are normalised by nodeis multiplicity. a In restricted other subspace words, of the the folded discretely Coulomb gauged branch quiver’sare Coulomb more branch. interesting. For example, let’s fold the operator in ( i In particular, unfolded quiver, partition intoevery orbits single of the variable automorphism. in that The orbit folding map to merely the sets same value; for convenience, basic abelianised where the JHEP01(2021)086 B (4.9) (4.13) (4.10) (4.11) (4.12) )-symmetric 2 q n folds into the z B 3 ( 4 q the partially and n D  2 C y x nodes. G i  ϕ 1 q ) U(1) non-simply laced quiver, 2 and xy
3 ( G 1 4 3 3 | t B 3 ϕ B i q | despite the lack of an obvious + ϕ + = | 2 2 3 t 6 . . 3 q ) G 3 2 2 B 3 . 2 2 − r t 2 ϕ 4 z q | = t 1 , with 4 − 1 + 8 r + 4 3 ) = | + x 2 2 D i µ |
by folding two (1 3 folding locus the following holds: t q 2 2 2 ϕ B 1 t r − t 2 9 2 1 − ϕ q G − | and and that all non-simply laced quivers can be 1 (1 + = r 1 + | | 1 1 + 6 → 4 – 41 – + q | | 4 D 4 = 3 t 1 , can fold into r ϕ | D 1 Z t 7 ∈ µ 1) = 1) = 3 Z = , , ∈ ,q 4 1 1 2 2 X , in some special cases a , identically vanish, the two algebras have identical struc- D 3 ,r ,q X t, 1 as if ) quiver’s Coulomb branch folds into a 1 1 ( ϕ k r q t, ˆ 2 C quiver theory O +1 2 3 ( 2.1 . Of course this space is just the Coulomb branch of a non- = ) ) n G 3 A 2 2 4 HS 2 t t D B D 1 ( 1 1 C HS ϕ − − 1 − be the Hilbert series of the initial and folded quivers, respectively: (1 (1 n 2 shape by simply reversing the folding procedure. Let us denote the C min A 4 ) = ) = HS D → is not 0 everywhere on the folded space so we conclude that folded spaces 3 and k A t, x, z ˆ ( O A t, x, y, z C quiver can fold to ( 3 A HS min HS B HS Let As was hinted in section In particular, a quiver into a 3 The unrefined Hilbert series are: 4.2.1 The first check will be done on quivers in table So the 4.2 Monopole formula:To examples show that folded quiversand become conjecture non-simply that the laced, pattern we generalises. compute two explicit examples variables of that quiver’sfully Coulomb folded branch counterparts. e.g. Then at the eg. the bottomsymmetry. right quiver There in is figure involved one manner. major As difference a prerequisite, however:same the rank multiplicities “short and root” are number (i.e. assignedB of third) in flavors gauge as node a the must more “vector have the root” (i.e. first node). We can unfold the minimal nilpotent orbit of simply laced quiver, andexplored we folding claim as this anone action is facet on no of the coincidence: anunderstood Coulomb action although as branch, on folded we we the simply have conjecture laced so it quivers. far is in only fact merely ture constants and areshows in that fact isomorphic ashave Lie the same algebras. continuous A symmetries as simple their proof discretely in gauged appendix space counterparts. of strictly lower dimension and the minimal nilpotent orbit of Therefore, unless some folded JHEP01(2021)086 , 3 2 3 2 q q to ↔ ) A 2 z + i 2 1 + 1 3 i q 3 t . We q u , (4.15) (4.14) ,q ( 2  i 2 − average ,q ,q y x 3 1 i 1 q q  2 h + , as can be 1 1 while linear e y and q q ) 3 2 with opposite 0 q O q ) y 1 z xy 1 ( = ± | − 3 1 3 q q | q : theory. “Flipping” a − + | ) 1 t, x, y, z 3 2 q 2 q ( t y C , respectively. − A 3 2 q − precisely one of them q | + HS 1 + 1 + (1 | q 2 3 x q q |  3 − q 1 1 y | = q . Since all operators in the chain symmetry exchanges | + 3 1 | i . + + 2 q 3 q i | 3 2 q 1 S y . ,q q C − + (of which precisely one exists), which i | 2 2 or  2 t t ,q 1 q 1 1 | i 1 3 − 1 2 Z q q − 1 q + ∈ , picking up the desired contributions by h 3 | and in two steps. 3 2 q e 2 q ,q . The = + 3 action on all the monopoles in the product 1 + 2 1 C i } X − = 0 q 1 A Y 1 ,q 1  , q q 1 2 , | y q HS unchanged or changes both by of operators in the product produces a monopole 0 = ,q + – 42 – | y  3 3 1 +1 πiy q → d q ,q | 3 2 which is crucial in the following calculations. By t 2 2 q come from the charge sublattice A + 2 | ∈ { + ,q Z i 3 1 1 I A y ∈ q 1 y and q q Sequential 3 ) HS . 1 2 O ,q dressing factor O . More generally, a bare monopole monomial in the monopole formula better count 1 + HS − t q 2 X 2 − ) i 2 1 ,q y or − 1 y 4 q + 1 in any number C | q  u 3 (ˆ U(1) q . To set up later generalisation we further slightly modify the y . We need only multiply the entire expression with 2 + y y 1 and 1 2 ˆ πiy πiy ϕ d d q , + 1 2 2 ) = (1 y 2 3 = ,q q I I is preserved. and integrate around 3 , depending on the parity of 3 q 2 2 1 3 2 = + ϕ t, x, z − − − q 1  ( ) ) 3 1 q . It follows that in this chain of flips there is an operator all fold to . Note that 2 q 2 2 1 y q 1 C = i j + t t + O 2 q ,q ): terms constant in 1 ) 1 1 2 q + i − − q , HS + 3 ,q ϕ 2 y 3 P u q ,q  (1 (1 4.10 ( theory which folds to the exact same monopole in the 1 2 1 2 1 2 O = +1 and acting with it on 3 3 i j q 2 = = i 1 A q + and 1 + 1 ,q operators at order q i 2  And indeed: We conjecture that these two modifications areTo sufficient implement to them, represent we the multiply action the of (unsummed) monopole formula by the kernel The second step corrects for scalar dressing: one extraneous scalar field must be re- To accomplish this we must extract only the terms constant and linear in We will now derive theAt action the level of bare monopole operators, many become duplicate. For example, + 3 ,q O u i 3 q 1 + 1 πiy h 2 R.h.s. the residue theorem. Finally we multiply by the scalar factor moved since remove the newly duplicate folding on the Hilbert series. seen from ( terms all satisfy prescription to an equivalent form:over we will extract every operator at order or fold to the same operator,will the pick either translates to selecting only monopoles with e in the single operator in this way leaves signs so that produces theory can be expressed (not necessarily uniquely) as a product of generators where comparison with known Hilbert(closures of series, the) we minimal find nilpotent orbits that of the two Coulomb branches are the u Note the unusual fugacity JHEP01(2021)086 + y ( 2 node, r 1 2 (4.16) 2 z r 1 2 r z r 1 x z r | symmetry. 1 U(2) 2 x 2 r / | q 2 legs folds by 2 x ) z ) = 1 + | / A 2 y +1) , z 1) 2 / ( +1 1 q 2 r 3 − f 1) ( z q 1 | 3 − , y r U(1) − q ( + 1 1 | | 1 r . − q 2 ( n + | 1 / | y q 2  + 3 2 | / y q minimal nilpotent orbit 1 2 r 2 3 +1) t, x, y ) 1) + / y y z q 1 4 ( 1 1 1 r − 1) + q 4 ( r − 1 1 D + x − r D n x q − | ( 1 | 2 2 1 3 x r 2 r − | q H ( | y y / | 2 3 ) 1 , which have an − r + q + 2 the fugacity for the | r i | | | 2 | + 2 r y 3 , the novel feature in non-simply + + | r | | + | , y q z , . . . , y | 2 2 3 2 1 + − 1 1 2 − | r y q r 2 y / 2 y 2 / − ( − 2 q 1 r ( | 2 symmetry. r 2 nodes. This parametrisation is chosen + 1 f 1) − q / + | − − | − 2 1 2 − 2 2 2 2 / n + 1 y r 1 2 | q r A | r +1) A f 2 r y ( z − 1 | q | + + χ 1 πiy +1) | 1 r d U(1) ( r − + q 1 2 2 | | | 1 n 1 1 r r x 1 2 ( q + 1 y | + | | − / | | 2 2 1 gauge nodes of the 2 2 1 r + + r fundamental representation. – 43 – 1) | y | / q / + 2 z | 2 1 1 πiy d − t 1 1 q r / − 1 1 | | 2 r 1 − Z r t y +1) r + ( x n 1 | ∈ | | | U(1) Z  +1) r ) = 1 + 3 t 2 I 2 + ( 1 A ∈ 1 / | | r ,q 1 3 r 2 3 t 1 1 2 ( 2 − ) X r r | ,q | | − 2 n +1) t ,q 2 t t 1 X t 1 Z r ) ) ,q q +1) | 1 (2 Z Z Z − q +1) + ∈ | (2 (2 (2 Z 1 1 y X , . . . y ∈ ∈ r ∈ ,r | (2 πiy 1 ) = 1 + 2 d 1 1 1 y X t Z 2 y 2 ∈ ,r ,r πiy X X ,r d ( 1 G ∈ Z X Z Z Z 2 2 , y ,r f ) = (1 ∈ ∈ ∈ r I ∈ ] for a systematic look at the link between folding and the modified ]. 1 2 Z , which are related by an 2 2 2 3 1 2 2 y r r I 1 ,r ∈ r X ( 1 2 − − 2 2 2 − min 2 r r ) ) f y − − − − the fugacities for the three 2 2 2 2 t, x, z ) ) ) ) t t ( − − 2 2 2 2 → 2 2 1 ) ) t t t t y − − 2 2 G 4 and t t x − − − − is the character of the (1 (1 , D y HS − − 2 1 1 2 1 2 1 (1 y y − 1 2 + +(1 +(1 + +(1 n x A f min , = (1 = (1 = 1 χ xy . We conjecture that the monopole formula of a quiver with The steps outlined above can be generalised to longer legs, larger gauge groups and, Note that this kernel is a natural generalization of the previous case The folding equation becomes In particular note the appearance of 2 in ) 1 − monopole formula of [ where presumably, to completelyourselves, arbitrary we legs. refer to [ However, rather than undertaking this task y integration over the kernel appeared as with We now look atquiver. the folding We of again three and assign fugacities toso the that nodes: folding call correspondsNote that to this an prescription integration generalises the over previous the example, where the “folding fugacity” laced quivers’ monopole formulas. 4.2.2 JHEP01(2021)086 is 2 2 ˆ / Q / 2 (4.20) (4.17) (4.18) (4.19) ˆ + 2 ϕ u − 2 ˆ 2 / / 3 + 1 1 x ˆ u ˆ ϕ = 2 1 ˆ 12 -th node of x ϕ but the relations

). Then i κ , ) 8 ± 2 } , i,a ˆ u + 3 j ˆ = 2 ϕ , u 2 is an auxiliary function 3 ϕ + 2 ) ϕ . Keeping with the term’s u and 1 | { w − ˆ + 1 Q ( ϕ 1 i i,a u , + | ± i,a to node g w ϕ ˆ | ± 1 u } i ) i,a ˆ u 12 + 1 α w | ( κ i i and , u g ~ 1 i # i ϕ = 2 ~ # # ˆ which fold onto the ~ { ϕ/ / } ˆ ~ ϕ/ 2 + Q α, i,a = ( h ˆ ϕ , u w, Φ 3 h x 1 + ∈ α u = 1 ∈R { Q x w

– 44 – } 1 connects the node Q = 2 + 3 ϕ 2 i 3 4 u w x x # = ˆ into a simply laced quiver = = if otherwise + − 3 4 1 2 . x x ˆ x x i + 1 Q

= = = | } 1 2 , u x x + 4 ji 3 4 2

− i,a u κ 3 ϕ ˆ u | 1 defined precisely like the Cartan matrix of a Dynkin diagram. denotes a vector of ϕ , ϕ , ϕ +    + + i,a ± 4 ± 2 κ of node + ˆ ~ u # 1 + 2 u u i 1 ϕ ) = ˆ , u ~ ϕ/ { ϕ # + 3 w ( = = u i 3 1 on the subspace preserved by discrete action. For simplicity we present g } i + + 1 , ϕ , ϕ # + 1 ˆ u ± 3 ± 1 / , u u u 1 ]. The input data are a list of gauge nodes with optional fundamental matter { i,a ˆ 1 ϕ x { = multiplicity is defined as if the quiver were simply laced (ie. the multiple link were replaced } . Example of folding two “parallel” links which do not originate from the same node. = ˆ + 2 R ˆ is the Cartan matrix of the non-simply laced quiver. u i,a , x κ The derivation of Poisson brackets is slightly more subtle. As a concrete example, Each node still contributes three abelianised variables + 1 ˆ u { Similarly, and keeping to the same quiver for this example, Note that the factor of 2 comes from the two links which fold onto each other. consider a quiver with nodes4 are 1 connected, to and 4 3 (plus and possibly 4 others) fold such onto that 1 1 and and 2, 2 respectively , (see resp. figure 3 and defined as and where with one simple link), use in previous sections,called the the number of nodes of are slightly modified. Theythat can be derived bythem demanding in consistency the case with of folding; quivers recall with one multiple edge: It is possiblenon-simply to laced framed generalise quivers; abelianisation,achieved the in generalisation [ including of the theand monopole Poisson a formula connectivity structure, was matrix already directlyOne can to always unfold the quiver Figure 8 Note that folding does not introduce a multiple link4.3 in this case. Non-simply laced quivers JHEP01(2021)086 of (4.21) (4.22) (4.23) but also ) g parameters (essentially ) assemble into i,a ji M , κ coadj( also ij link multiplicity κ | i,a is the | w | j ij ) i,a = max( α w # | # ( i i / S ij g . Since there are precisely two ~ i # κ c j,b ˆ ~ , which can in turn be set to 0 by ± j,b ϕ # ˆ ~ + ϕ/ 3 ˆ u − ˆ ~ abelian moduli and . If the quiver is balanced, for the ϕ/ M α, i,a ± i,a i g h ˆ u , also exhibit an interesting pattern of w, # i,a = Φ M h / ∈ ϕ 1 9 α → i,a ∈R M ˆ Q ϕ w i,a j Q ij # M i – 45 – i,a # , ∂ # ˆ c ϕ . The resulting spaces are not as comprehensively ∂ S ij ± i,a + ˆ 2 i u κ ± ± i,a ϕ 5.2.5 defined as the number of its pre-images in the unfolded = # = = of the Coulomb branch symmetry. The chiral ring data is ) j } } } → g 2 ± j,b i,a ± i,a − C ˆ ˆ ˆ u u u i,a , , , and adj( ϕ i ± i,a i,a i,a + ' ˆ ˆ ˆ u u ϕ ) { { { )min g . (n → coadj( 3 A is a “simply laced” Cartan matrix defined as , ie. a complex mass deformation. However both discrete gauging and folding 3 S κ M min The quivers we choose, as exhibited in table Most spaces encountered in this section are nilpotent orbits; their coordinate rings Now that the procedure is clear it readily generalises: − 1 -type quivers tend to have very simple moment maps which can be presented in reasonably only appear in the abelian algebraunder as differences reparametrisations and as a resultmass the moduli parameters, space is the invariant M moduli space relationsremove can one be half of modified mass by parameters by terms forcing proportional to 5.1 A compact form, allowing us to present the action ofcomplex discrete mass gauging deformation. and As folding. a general rule, all examples studied in thiscoadjoint paper, (or sometimes we trivial) find representations that andOne the the such bulk remaining of case generators our appears techniquestabulated still in as applies. section nilpotent orbits andtheir we Hilbert generally series have or to highest turn weight to generating more functions. varied sources to find Coulomb branches in turn. are generated by anot single nilpotent coadjoint orbits: representation. theirby Coulomb But chiral branches there are ring generated are elements not a in only few other by cases representations which of are Coulomb branches are nilpotent orbits.which Their we chiral explicitly rings construct; arerepresentation generated recall by that moment maps, suchcompleted moment by maps providing transform a in setbranch the of symmetry. relations coadjoint We which discretely also form gauge representations and of fold the such Coulomb quivers and examine the resulting for quivers with one multiple edge. 5 Examples In this section we study several cases of nilpotent orbit quivers, ie. quiver theories whose where throwing away information about multiplicitythe of edge edges) between and nodes quiver. Remember that just as in the case of abelianised relations this form is appropriate JHEP01(2021)086 : 3 , (5.7) (5.1) (5.2) (5.3) (5.4) (5.5) (5.6) 2 ,         3 ) 1 = 1 3 2 M i ϕ ) ) 4 3 3 − +3 − 3 − 3 2 ϕ 1 u for u M ϕ − − 3 − 2 M − 2 i 2 )( u u u − 2 ϕ 1 − 1 ( + ϕ 1 , ϕ u 2 0 − 3 2 ± i − 1 M t ϕ u ( ϕ 2 1 ( − 2 µ 1 3 c 16 Folded 3 1 abcd + M ε 4 − 2 1 ) N 3 ) ) ) M 2 ) 3 2 3 − 2 3 ϕ M u ϕ 2 + − M ϕ M − 2 + 3 ] t − 1 1 − 3 3 u u u − ϕ − − − 30 1 µ ϕ 1 1 1 2 3 − 1 ϕ M 1 + µ ϕ ϕ ( M )( 4 2 ( )( )( 2 t 1 − 1 ϕ 1 2 2 16 1 2 2 S 1 3 2 0 M ϕ ϕ µ − o − − 3 − − − + 1] = = 1 2 ) = 1 2 3 2 M c c i 0 – 46 – ] t 2 4 b − ϕ ϕ ϕ b 2 1 1 ( ( ( N µ ) N t, µ M 0 − − − 3 [1 + 3 a a ϕ u = = = + − Discretely Gauged − 1 + 2 N + 2 2 2 u u u − 3 − 1 − 2 ϕ cd [ ( ϕ 0 u u u HWG( b 0 + 3 + 1 + 2 − + a u u u ) ) 1 ε 1 3 2 0 ϕ minimal nilpotent orbit and its discrete reductions. ,b C A 1 0 ( ( − X 3 a ) A 3 3 0 3 2 . ϕ M t 1 ) 2 − + 2 4 + 3 + 2 2 2 ϕ 3 1 t u u ϕ a µ Initial 3 − + 1 + 2 M + 1 1 )( µ 3 u u u 2 + ϕ 1 + 1 [020] : ϕ 1 Table 9 µ 2 1 u 4 − − − t µ 1 1 1 ( ϕ ϕ (         = ([101] + [000]) : 3 4 The Coulomb branch is generated by t A N or simply from the Joseph relations, which are obeyed by any minimal nilpotent orbit: and one can read its relations either from the HWG [ triplet of mass parameters. 5.1.1 Initial quiver To remind the reader we reproduce abelianised relations restricting the reparametrisation above. As a result only the original space can be deformed by one JHEP01(2021)086 ]         ) 30 3 (5.8) (5.9) ) − 3 ˜ ϕ (5.15) (5.16) (5.12) (5.13) (5.14) (5.10) (5.11) ) 3 ˜ u ) 3 ˜ − ϕ symme- − 2 2 − 3 ˜ ϕ ˜ u − ˜ ϕ 2 − 3 u 2 ˜ u − ϕ C − 2 − + ˜ )( ˜ 1 ˜ u 2 2 ˜ − 1 ϕ ˜ − 1 − 2 ˜ ϕ ϕ ˜ u u ˜ u − − − (˜ 1 ( . − 1 1 1 2 ˜ u ˜ 1 2 ϕ ϕ } ( ˜          ( 2 ) 1 2 ) − 2 2 2 = 0 ˆ u 1 2 1 ) ϕ 2 ϕ − 1 ) ˆ 2 ˆ − 1 3 ) ϕ 2 ˆ ˆ u 3 − 3 ˜ = 0 ≤ = 0 ≤ ˆ ϕ − u 1 2 − 1 − ˜ − 2 u 1 ϕ 1 2 u 1 JN ) ) − 2 2 ˆ u ) (ˆ − 2 ϕ − 2 ϕ ( ˆ ˜ . u ˜ ( ˆ + 2 + ˜ ϕ N N N N − ) ( ( u − 1 4 J + t ϕ = 0 1 − 2 + ˜ T 2 2 ( ˜ 2 ˆ 0 2 ˜ u ϕ 1 2 − 2 µ ˜ ϕ d + 1 rank rank d ϕ − 2 1 2 ˜ u 2 ˆ ˆ − u 2 u + 1 − − 1 N + 1 t (˜ − 2 − − 1 0 ˆ ˜ + u u ˜ 2 1 ϕ 1 2 1 c ˆ 2 ˆ ,N u u c 1 2 ˆ 2 ϕ µ ˜ 2 1 1 ϕ )(1 − N ˆ ϕ 0 ( 2 − − − b t ≤ b 1 2 − 2 1 1 ) N symmetry, which agrees with the minimal ) 2 µ 2 matrices as ordinary matrices, ie. without is a suitable candidate. Its HWG reads [ 3 N ) ϕ 2 ϕ + 3 ˜ ( 2 ϕ 2 − n ˜ u abcd ) C ) = ϕ + 1 − + ˆ C i − 1 C ε + 2 2 ˆ ˆ 2 u − 3 + 2 0 (1 ˆ ˜ ) + ˜ u u – 47 – 1 ˜ ϕ − d ˆ + 2 u u 3 0 1 2 ˆ 1 rank ˆ ϕ u c t, µ 0 ϕ ϕ 1 2 + 2 + ˜ − b ( ˆ 0 ) = u i 2 a + ˜ , − − 1 + 2 ˜ ε ϕ 1 u ˜ u 0 2 − (˜ ) ϕ + 1 t, µ 2 1 + 2 ,d 1 2 2 = 0 HWG( ( ˜ ˜ 0 u ˜ ˆ ϕ u + 2 ϕ ϕ 1 2 2 ,c 2 ˆ 2 ˆ 1 u ( 2 ˆ + 1 0 ) [02] : [20] : ˆ − 1 2 + 1 ϕ ˆ − u − ] 4 6 + 1 N X 1 ˆ u 1 | t t 1 2 1 2 u ˆ ϕ ) ˜ ) (ˆ ϕ 30 HWG( ( ˆ 3 b,c,d,b C − + 3 ˜ ϕ , ) ˜ u − − 3 ) + 2 ) 2 ˜ (4          ϕ ˜ u 3 ˜ + 2 ([00] + [01]) : ([00] + [01]) : ϕ − gl + 3 u ϕ 4 4 2 = ˜ t t u + ϕ ∈ 2 + 2 + ˜ + ˜ )( ˜ 2 ˜ u C 2 1 + 2 ˜ ϕ + 1 + 1 ˜ N tensor.) The second of these relations can be written equivalently as ˜ ϕ u N ϕ ˜ u − u { + 1 ( ˜ −  1 (˜ ˜ 1 u ˜ 1 2 ϕ 1 2 ϕ ( ˜ − ( 1 2 moment map discretely gauges to the following expression:         3 = A 2 C N This space satisfies slightly more stringent (Joseph) relations: The Coulomb branch hasnilpotent dimension orbit 2 with and HWG [ The folded moment map is similar: In other words, angauged explicit quiver algebraic is description of the Coulomb branch of the discretely 5.1.3 Folding (Note that in ourinsertion convention of we an multiply suggesting several relations: and the resulting space is expectedtry. to have The quaternionic next-to-minimal dimension nilpotent 3 orbit and of exhibit The 5.1.2 Discrete gauging JHEP01(2021)086 (5.18) (5.19) (5.20) (5.17)  2 spaces studied G 2 2 . . In other words, } G ,H 2 = 0 G 1 = 0 0 ] b b ,H 2 N 2 2 0 JN G a ± a 3 23 [ . − B ± which preserves a particular N ,E E J 2 1 3 2.1 T G ± B + 3 23 ,E B ± 3 3 123 3 2 12 B 3 B ,N ± 3 ± E G ± ; such spaces are necessarily nilpotent 1 B ± E H − E − ,E ≤ + [01] 2 3 12 ) 2 + 3 = 1 B 2 ± 1 3 1 B N ± G ± E ( B ± 4 H 234 E = ,E D ± – 48 – 2 = 3 E = rank 4 24 2 1 4 4 4 D 2 ± 4 G ± + D ± 3 D ± , 2 2 E H ,E E 3 12 4 124 2 − 3 123 : B 2 + ± D ± = 0 + 3 B 3 ± 4 3 4 2 23 2 E 1 3 3 E 2 2 4 3 B E B D ± ± D G ± ± B ± D ; for more details see section ± N = − E | H E φ = H E E )  34 and − = C 4 + = 123 2 + C , 4 D 4 4 ± 1 2 4 4 4 4 4 12 1234 1 2 12 coadjoint representation D D (4 ± ± D D D D D D E ± ± ± ± ± 2 gl E E E H H E E − 2 G is characterised as the subalgebra of ∈ ======span = = = [01] = [30] + [21] + [01][01] + [10] = + [40] [03] + [31] + [22] + [11] + 2[20] + [00] + 2[02] + [04] [01] = [20] + [00] + [02] G 2 3 4 2 2 2 2 2 2 2 2 2 1 2 1 2 N 2 2 3 2 12 . G G G G G ± ± ± ± 2 2 { 3 1 1 G G ± → 2 1 G G ± ± E E H H Sym Sym Sym G E ± 4 11 E E E D and min 10 Note that because the quiver has onlyWe flavor provide node the of rank first 1, few the symmetric products of the (co)adjoint representation for Recall that The goal of this subsection is to identify quivers whose Coulomb branches are generated is small yet non-trivial enough to serve as an excellent illustration of the techniques 2 reference: generators but is nottables a nilpotent orbit. The following sectionsbelow should cannot be be deformed read by alongside a complex mass. rank 3 antisymmetric tensor by operators in one orbits. We also study one related space whose coordinate ring is generated by coadjoint studied in this paper. Sinceprescription it from is only both fourteen-dimensional, we provide the complete folding an explicit algebraic description of the Coulomb branch of the folded quiver is 5.2 G The second of these relations can be written equivalently as

JHEP01(2021)086

. 5.2.4 section in describe as gauging

2 as iga o iptn risof orbits nilpotent for diagram Hasse a optdi [ in computed was G .Te[0 uvri osrce iha nsa obnto ffligaddiscrete and folding of combination unusual an with constructed is quiver [10] The ]. 51

cd ab ef

h ibr eisi h atclm r optduigMcua2 hyarewt h optto f[ of computation the with agree They Macaulay2. using computed are column last the in series Hilbert The . 0 = ε N N N .The ]. 23

abcdefgh

2

steadjoint. the is h relation The . ) , (7 ∈ N , the of representation 7-dimensional fundamental the in matrix a is N tnsfor stands 0 = N ∧ N ∧ N so g C

1 2 2 2 1 al 10 Table iptn ris euetecneto htteroot the that convention the use We orbits. nilpotent G . µ and fundamental the is µ that so long, is α root the and short is α

) t − (1

2 12 [22] tr( = ) N tr( 6 o ogt display to long too 0 = ) N

2 6

(1+ ) t (1+ ) t + t

2 2 8 4

3 [1] 2 S o

0 = N ∧ N ∧ N

) t − (1

1 2 1 1

2 2

10 2 µ 5 [02] t µ µ − t µ + t µ + t µ + t

10 3 8 2 6 3 4 2 2

) t + t +3 t +6 t )(1+3 t (1+

0 = ) N tr( 8 6 4 2 2

2

1

1 2

1 ≤ ) N ( rank

2 – 49 –

) t − (1

1 1 1

2

8 2

0 = N ∧ N ∧ N 4 [10] t µ − t µ + t µ + t µ

1

12 6 6 3 4 2 2

t 7 − t 7 − t +43 t +20 t 1+6

10 8 6 4 2

0 = ) N tr(

2

1

1 2

) t − (1

2

6 2 0 = N 3 [01] t µ

2 2

) t + t )(1+7 t (1+

4 2 2

1

0 = N 0 [00] 1 0

Relations Dimension Label ibr Series Hilbert PL(HWG) Quiver

JHEP01(2021)086

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6=1 n

2 3 4

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P n

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4 3 3 2

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2 1 1 2 1 1

2 2 2

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5

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2 1 2 1 2 1 2

2

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G G

2 4 2 2 12 2 2 8 2 6 1 2

1 2

2 3 3

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0 1 2 1

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1 1

iceeyGauged Discretely Initial Folded JHEP01(2021)086 ... + (5.27) (5.25) (5.26) (5.21) (5.22) (5.23) (5.24) , form quiver 8 } t i 4 moment  X 4 D 2 2 { µ D whose quiver + 2 4 2 G µ + 2 1 µ ] 2 2 = 0 = 0 quotient of the minimal µ defined either by folding 30 2 which is parametrised by as the common starting 3 [ ... N = 0 = 0 + S 4 N N 2 2 3 1 + ∧ 11 D N G µ 4 N 2 t N ∧ 2 2 µ Chevalley Serre basis . µ . N ) as 2 + i + 4 1 ∧ 4 G X 2 = 0 µ D 4 t N  2 2 D N ) µ 8 N + N t 6 2 2 : t µ tr ( 4  . One can either symmetrise the t 3 2 ∗ i − is known to be the = 1 + . µ 2 10 X 2 . To verify this claim we can look at the highest 2 – 51 – G ] for more details. t + )(1 G i 2 7 6 ) or, given the N 2 1 subject to the Joseph relations X t : : µ counterpart [01] 1 µ 3 1 4 4 2 4 = µ − t t M µ 10 3.12 2 of table D 1 t − folds into the minimal nilpotent orbit of 2 G + ([20] + [00]) 4 µ 3 1 N 3 1 )(1 [02] D ) = µ µ 4 t  t :: tr 1 2 minimal nilpotent orbit quiver or directly using the non-simply 4 6 + µ from its t t 1 + 4 4 2 t − D ] so it should be the Coulomb branch of the appropriate G HWG(  [00] [10] 5 2 2 under the label [ N )(1 µ 2 4 ([2000] + [0000]) ([0020] + [0002]) t 10 + 2 D 2 1 µ µ −  (1 + discrete gauging 2 3 t 2 S ) = moment map µ t 2 Two relations are needed this time: The highest weight generating function is Direct computation shows that the relation is satisfied by G = 1 + HWG( and both are satisfied by the coadjoint map using the projector definedthe in ( laced prescription. 5.2.3 The five-dimensional subregular orbit of nilpotent orbit of after discrete gauging, see row quadratic relation the moment map of the is depicted in table weight generating function of the minimal nilpotent orbit of or recall that the Joseph relations tell us that the coadjoint generator is constrained by the We refer the reader to its treatment in5.2.2 [ Folding The minimal nilpotent orbit of The next few examplespoint. share Its the Coulomb quiver brancha is on coadjoint the the (antisymmetric) minimal matrix left nilpotent of orbit table of 5.2.1 Initial quiver JHEP01(2021)086 2 G [02] ... (5.30) (5.29) ... (5.32) (5.33) (5.34) (5.28) (5.31) + 8 ]. How- + t
8 4 2 t in the 18 ) we define , µ  8 3 2 4 t + 3 2 µ 3.12 µ 1 + µ 2 2 + µ ) can be written as quiver. The quiver 2 2 1 2 µ µ 3 2 1 µ B 5.34 1 + + 2 orbit, written in terms of 2 ... = 0 ≤ µ µ 3 . ) = 0 ) 1 3 3 1 + 2 2 B N µ µ 8 10 t 4 N N + ∧ 3 2 t + ( 4 1 µ
1 4 1 µ 2 2 ) N i , which is itself four-dimensional. µ µ µ 3 ∧  + X + + rank B 3 6 2 6 t + t on row N B µ
2 2 1 6 3 2 N µ t µ µ 1 10 . We have checked analytically that the  + µ of the underlying + 2 3 tr ( 2 1 2 2 + µ 3 . The last relation ( µ µ ∗ i 2 4 1 B
t + 6 G 17 µ 2 X + ] t N ) = 0 2 – 52 – µ 2 3 1 + N t 1 i µ nb 2 30 ) µ µ X 1 2 is of rank at most 1 is of this type. 2 µ N 2 1 µ µ + − and project using the trick from the previous quiver 2 = µ cn 2 + 1 2 + t 2 N + 1 1 N 3 1 3 1 G G µ
µ gauging µ µ 4 12 N md t be recovered by using a specific and non-generic discrete =  t 2 of 2 1 6 1 + N 2 S µ } + t µ i = 1+( :: : tr( 2 2 4 am t µ can − ) t X − 4 6 8 1 2 t t t N { t µ 1  − 1 2 2 2 1 µ − ) µ [00] [10] [02] 2 ] − t nd + 2 30 1 2 1 µ N )(1 2 µ t cn −  1 µ N (1 + − mb 2 t (1 N 2 ] to be non-normal and hence not expected to be the Coulomb branch of any ) = µ t ) = am 23 t N ( In total the moment map is expected to satisfy three relations: Compared to the subregular nilpotent orbit we find an extra relation at The HWG of this orbit is given by [ We first construct the moment map We can compare this expression with the HWG for the minimal = 1 + HWG( HWG( m,n 17 The difference between the two expressions is fugacities, which reads [ and indeedP all are met by our coadjoint three relations above form a complete set of relations. representation. The condition that We depict the conjectured quiver theory in table This would make our construction the first non-normal Coulombhas branch no in obvious the automorphism literature. the so Chevalley-Serre rather basis than usingcalculation: the projector form in ( Midway between the twoknown previous [ examples liesquiver a since both nilpotent simply orbit andever non-simply of laced we dimension quivers conjecture are 4. that necessarilyoperation normal it It [ on is the minimal nilpotent orbit quiver of 5.2.4 Mixed folding and JHEP01(2021)086 . . , 3 ) 8 3 t Z S 10 t ( ⊂ (5.37) (5.38) (5.39) (5.40) (5.41) (5.42) (5.43) (5.35) (5.36) O 3 Z + symmetry 8 t 3 ] identifies a S 19 ⊂ [02]) but we find it is d 2 c − R S . For example: R is also the moment . b ) a R 1 and N R quiver automorphisms, ϕ 1 54 N m − S 3 = 0 = 0 = 0 = 0 = ϕ ([01] + [10] ] ] = 0 ( 2 2 } ] d N c − + 4 R N ) }} 6 u ∧ i 2 t N,R N,R 10 N 4 h N { ( symmetric gauging. Since ) + ˜ e b 4 [ , ∧ a 3 [ i ϕ ) S 2 01 N h − ˜ e N adjoint representation’s worth of operators 1 { 4 , ϕ t i ( ) and a different numerical factor in the last + 3 10 R u – 53 – . The plethystic logarithm of its highest weight h− ∧ ˜ e (3) 4 ) + D N is perfectly well defined. Here we consider the 3 −{ C ϕ 18 = m − operator which acts as the first simple root under action S i ([01] + [10] + [20] + [00]) } ∝ 4 4 01 4 h t ϕ − : : :( :: :[ tr ˜ e ( 4 6 8 8 4 6 6 t + 1 N,R ]; the latter paper reports matrix relations. In general folded t t t t t t u { is a quiver studied in this section. Its Coulomb branch was previously 31 i = 4 [02] [00] [10] [01] to generate an entire [00]) N, respectively. The lower coadjoint matrix 10 i ) 4 h i D 2 t − ˜ e 10 10 4 h 4 h N ˜ e ˜ e ] under the name (tr 19 ([20] + [00]) ([20] + [00]) + ([01] ∝ 2 3 t discrete gauging ] also follows the opposite root convention to the present paper. N 19 3 Z (eg ) = [01] 2 The plethystic logarithm suggests several relations between t Paper [ ( G 18 We are able tominimal set verify of all relations of as our them current symbolically, techniques but run against cannot a guarantee computational that limit. they form a relations follow the form oforiginal the relations original restricted quiver’s; indeed toin they the must folded as subspace. theyrelation, are we merely are Accounting the left for with several the coincidences following relations: not too helpful in thisAccordingly, we case. opt For for example, a itsnon-simply different syzygies laced approach obscure quiver to several with relations identify thethe at the same order relations. quiver Coulomb in Ref. branch, figure [ which is 8 itself of a [ folded version of of the moment map’sother components. root by And repeated justPoisson action bracket as on of one the can Liewhich “rotate” bracket, can it a be is simple bundled possible root together to to into form repeatedly any the act second with coadjoint the matrix there are operators inis this imposed. theory One which of are the removed simplest if operators the is remaining As its label suggests, PL This space is not aand nilpotent quartic orbit. order It in map is generated and by looks two precisely coadjoint like matrices the at quadratic one obtained by Although elsewhere in thediscrete paper gauging we by discretely adiscrete gauge gauging subset of or the of fold investigated in [ generating function was reported as 5.2.5 JHEP01(2021)086 13 , U(2) . ) gauge 12 C 2 , Z (9 = so 1 O ]. : only the initial 53 , antisymmetric matrix 5.1 30 nilpotent orbit quivers. 9 fugacities are also used. 5 × 5 D 9 D , demonstrating that discrete U(1) denotes antisymmetrisation over all indices, ]. The third line shows an electric quiver, by 53 N – 54 – ∧· · ·∧ N matrix effectively becomes a 10 × 10 , we have colored the terms of the HWG which are charged under the fugacities except in the first column where 14 follow the same pattern. The first line shows the unitary magnetic quiv- 4 -type moment map in the Chevalley-Serre basis is too long to print but – node comes with one flavor so the triplet of spaces exhibits interesting 4 B 14 D mirrors. Note in those electric quivers the appearance of an – symmetry required for both discrete gauging and folding. B 12 d 2 . Upon either discrete operation, all components along the last row and column 3 12 ) S U(2) → C group in these quivers as studied in [ 5 , present results of discrete gauging and folding on three D (1) (10 Note that notation of the form In tables Secondly, in the case of the next-to-next-to-minimal nilpotent orbit we wreathe a We draw the reader’s attentionFirstly, to a several interesting properties. Tables 14 action in violet. O so 2 or equivalently contraction with the appropriate Levi-Civita tensor. Z gauging generalises towreathed gauge ranks highercomplex than mass 1. deformationspace behaviour Finally, can in analogous be deformed the to byspoils complex same that mass, the example, of and turning section each on two inequivalent mass parameters unfolded moment map transforms inof the coadjoint (antisymmetric) matrixvanish and representation the originally padded by zeroes — and hence transforms in thenode coadjoint rather representation than of the simple and well understood case of group in the middle column.The The HWG use last lines show the Hasse diagrams, HWG and relations. both discrete gauging and folding have clear and discernible effects on it. The original, same Coulomb branch); our discrete gaugingan appears to be the unitarywhich analogue we of mean gauging a classicalstudy. quiver theory Several whose quivers Higgs branch maynot is be share the the this Coulomb branch property; under in particular the ones chosen here need We close off by studyingand discrete gauging and folding onThe a Hilbert family series, of HWGs quivers. and Tables quivers were originally reporteders. in The [ second line shows the equivalent orthosymplectic magnetic quivers (ie. with the 5.3

JHEP01(2021)086

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5 iia iptn ri uvradisdsrt eutos h oddsaei o-pca [ non-special is space folded The reductions. discrete its and quiver orbit nilpotent minimal D . 12 Table n hrfr ihu northosym- an without therefore and ] 53

0 = N : [0100] t

3 6

0 = N ∧ N : [00011] t 0 = N ∧ N : [0002] t

4 4

0 = N ∧ N : [0002] t

4

0 = N : [00000]) + ([20000] t 0 = N : [00000]) + ([2000] t

4 2 4 2

0 = N tr : [0000] t

2 4

4 1 2

B t ) µ + µ (

2

1 4 2 4 2

B t µ B t µ + t µ

4 2 2 2

5 2 D t µ

2

0 0

4

0 b

] 1 , [2

2 5 ] 1 , [3 ] 1 , [2 O

6 6 2

¯ O O

¯ ¯

(9) so

4 5 6 b d (9) (10) so so

6 1 a

7 7

1 1 1 1 Sp Sp O Sp

– 55 –

9 9 10 SO SO SO

3 1 2 2 1 3 1 3 1 2 2 1 3 1 SO Sp SO SO Sp SO Sp SO Sp SO O Sp SO Sp

Non-special

1 1 1 1 SO SO SO SO

1

2

1 2 2 1 2 1 S o [1] 2 2 2 1

1

1 1 1

iceeyGauged Discretely Initial Folded

JHEP01(2021)086

urn cp.IsHsedarmsol lob eadda conjecture. a as regarded be also should diagram Hasse Its scope. current

ntepeetpprne ob eeaie onncajitrpeettost eiyrltosbetween relations verify to representations non-coadjoint to generalised be to need paper present the in n uhwr sbyn the beyond is work such and R and N

ftetpqie.Tedsrtl agdsaei eeae yacajitmatrix coadjoint a by generated is space gauged discretely The quiver. top the of rnfrigas transforming R generator another and N h methods The . [2000]

5 2 , 1 1 , 1 al 13 Table r asprmtr soitdt h ao node flavor the to associated parameters mass are M and M reductions. discrete its and quiver orbit nilpotent next-to-minimal D .

4 4

2 , 1 1 , 1 2 , 1 1 , 1

N : [01000] t N ) M − M ( = N : [0100] t N ) M − M ( =

6 2 3 6 2 3 1 1

? ?

N : [00011] t 0 = N ∧ N : [0002] t 0 = N ∧

4 4

0 = N ∧ N : [0002] t

4

2 2

2 , 1 1 , 1 2 , 1 1 , 1

000 tr : [00000] t ) M − M ( = N tr : [0000] t ) M − M ( = N

2 2 4 2 2 4 1 1

1 1 4 1 2 1

B t µ − t ) µ + µ + (1 + t ) µ + µ (

8 2 4 2 2

1 1 1 1 1 4 2 4 2

+(1+ t µ B t µ − t µ + t ) µ + µ B t µ + t µ

2 4 2 2 12 4 6 2 4 2 2

1

2 5

D t µ + t µ

2 4 2

0

4 0

b

0

5 6

d

4

b

1 a

] 1 , [3

7 O

] 1 , [3

¯

6

7 O

1

a

(10) so ¯

6 (9) so

7 7 1 1 a a

7

8 1 a

– 56 – 1 a

8

1 1 1 1 1 1 1 O Sp O O Sp O Sp

9 10 9 SO SO SO

1 3 1 3 1 3 1 2 2 1 3 1 3 1 2 SO Sp SO Sp SO Sp SO Sp O Sp SO Sp SO Sp SO

ntanloetorbit) nilpotent a (not ?

3 1 2 2 SO SO SO SO

1

2 1 2 2 2 2 2 2 2 2 S o [1] 2 2 2 2

1

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JHEP01(2021)086

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5 5 4 al 14 Table r asprmtr soitdt h flavor the to associated parameters mass are M and M reductions. discrete its and quiver orbit nilpotent next-to-next-to-minimal D .

1 ≤ ) N ( rank : [0200] t

8 2

6! · 2

0 = N ∧ N ∧ N : [0010] t ) N ∧ N ( ? = N ∧ N ∧ N : [00011] t N ∧ N ∧ N : [0010] t 0 =

6 6 6 5 4 ) M − M 3i(

16

5 4

200 : [20000] t 0 = N : [0100] t 0 = N : [0100] t 1 ) M − M ( = N

3 6 2 2 4 2 4 1

0 = ) N tr( : [0000] t

2 4

4 4 3 1 2

µ ( B t ) µ + µ ( + t ) µ +

4 2 2

3 4 1 4 3 1 2 3 1 2

t µ + t µ µ + t ) µ + µ ( + t µ t µ + t µ t µ µ −

4 2 2 12 2 2 8 2 6 4 2 2 2

5 5 4 2

D t µ µ + t µ

4 2

0 0

0

4 b

4 b

6

5

d

] 1 , 2 , [3

] 1 , [2

2 2

2 4 O

O

¯

¯

6

1] , [2 (9) so (10) so

4 O

1 a ¯

2 (9) so b

7 2 7 b

8

3 d

3 d

8

10 10 2 c

– 57 –

2 2 1 2 Sp Sp O Sp

10 9 9 SO SO SO

2 4 2 1 4 2 4 1 2 1 2 1 2 4 ] SO SO [ SO Sp SO Sp SO Sp SO SO Sp SO Sp Sp

Non-special

2 2 SO SO

1 2

2 S o 1] 2 1 [2 3 1 3 2 1 2 3 2 1

1 2

iceeyGauged Discretely Initial Folded JHEP01(2021)086 ] ) abel C with extra (A.1) 2.35 (A.2) M [ , so we C ] on their U(2) C ] [ subject to C abel i,a implies that the C . ϕ [ ] G and C W ] abel
C and ) [ is included in ], where it is introduced C abel 4 2 ± i,a C ) [ ≤ 2 u ), b) this ring contains C ϕ b G may appear to be fairer, since ϕ is constructed as the ring W − either.) theory with 4 flavors. Ele- ] ] G 6= 1 G ), and even if one restricts to the W ϕ a ] b abel ( W ] abel ϕ ϕ C / [ C 1 ) U(2) [ abel C i = abel C [ C the Coulomb branch chiral ring, since a) C a [ M C 2 ϕ are inequivalent, take the theory abelianised relations C ) 19 ( ] − 2 not / ] 1 ϕ and abel ] ϕ C − ( -bosons) never appear in [ roots 4 1 ) modulo relations between abelianised monopoles. C ∈ abel C ≤ ) j W i ϕ b } ( M ≤ 1 ϕ [ 1 and − C ) − ] – 58 – Q a W j ϕ − abel C ( M / ( = 1 M { [ , C } − 1 a u ϕ ). + 1 { despite any confusion it may cause when compared to the similarly , u } ± A 2.35 u (inverse masses of { [ ) )–( C 2 ϕ 2.33 − ] = . (A comparison between ] 1 ϕ abel C abelianised ring abel ( C as an active computational precursor to the Coulomb branch chiral ring and therefore / [ M ]. [ 1 ] ) and ( C 4 C abel with 4 flavors and abelianised rings C 2.30 [ C U(2) In the interest of concreteness consider the case of a First, we set the stage. Starting with abelianised variables, the Poisson bracket ( We stress that this notion of the abelianised ring is a departure from that of [ We use To see that the two “abelianised” rings 19 4 flavors (as inbut the absent remainder from ofboth this appendix). objects are Then Weyl-symmetric,element the but in element it question makes does no not difference belong since to thereserve Weyl-invariant ring it the name denoted ring in [ scalar operators and allThe inverse masses authors (i.e. follow with clarificationthe that inverse this masses is are emphatically “discriminant not locus” defined of the everywhere Coulomb onelements. branch it (that is, (i.e. the when points where in (4.9) (with minor notational differences) as In essence, this “abelianised” ring is the Weyl-invariant part of a ring generated by all abelianised monopoles, ments such as own. On the other hand, the abelianised relation concerns in this appendix. generates new elements ofmultiplying the basic abelianised variables. ring The whichunderlying cannot full the be abelianised Poisson expressed ring algebra byrelations generated adding ( by and abelianised variables generate a set offrom abelianised the Hilbert operators series, and select certain then,of Weyl-invariant combinations to guided the form ring by tensor generators and representation-theoretic verify data a that deep they dive satisfy into theleave abelianisation correct such relations. discussion and out The the of method precise the does relation main not body between require of this paper and address some of the potential Theory for their hospitality, and the MIT-Imperial College London SeedA Fund for support. The main text describes a two-stage method to explicitly construct the chiral ring: first The authors are gratefulimportant to point Julius about Grimmingeranonymous discrete and reviewer Zhenghao gauging for Zhong insightful andstudentship for feedback. useful grant clearing D.M. comments ST/N504336/1 up isST/P000762/1. on an while supported the A.B. by We draft STFC and are DTP and grateful A.H. research to to are an organisers supported of by the STFC 11th grant Joburg Workshop on String Acknowledgments JHEP01(2021)086 , 2 1 + + ϕ ϕ has and ) as − 2 − 1 + must u u (A.4) (A.3) (A.5) 1 → ·} ] ·} + 1 + 1 , ϕ , C 2 2.30 u must be u [ 2 , {· ] ϕ C ϕ ] is free of ± 2 C [ 4 u + to C + ] 1 ) ϕ ± 1 { − 2 u abel u C [ + C . operators is finite so is not in the Coulomb ) − 1 i 2 2 ) u ) 3 ϕ M t 2 )( ( ϕ + 2 = − O u 1 2 − ∆ = 2 ϕ ϕ + 1 ( i ϕ + 1 ] suggest in their section 6.3 that ( u Q 4 . ] which only uses 1 ( would fail it. We thank an anonymous 4 ϕ − G − + 2 and single-node elementary symmetric ) 2 2 W i u ] 2 ϕ ) ± i,a M + 2 abel u − 2 ϕ etc. We have observed its validity in every C − [ a 2 1 ϕ C 1 − ϕ ([4] + [2] + [0]) + i,b P + 1 ϕ 4 1 u − ϕ ( t Poisson algebra – 59 – which includes an extra contribution of i ϕ ). There is nothing else left to do without using ] = ( i,a = = ) increases dimension by 1 while Q C [ ϕ − 2 } } 20 A.3 C − u ·} [2] + + 2 − 2 , 2 u u t a

x , . . . , ~x j , . . . , ~x 6= 0 1 i = )) j ~x ~x x ~x n ( (

= x..~x,. . , ~x ) f f j ( ~x n f

− − i 1 j ) ~x ~x ~x ) ) , . . . ~x n . . ε 1 n n k k is a constant, which contradicts the ε = = = ˜ ˆ ~x , . . . ~x ) O O ( 0 i 1 0 1 k k ,i n f ~x ~x ~x , . . . ~x ij ij 1 ( 6=1 1 c c ~x 0 j f , . . . , ~x , . . . , ~x ~x i i 1 ∇ ~x ( k k ~x

, . . . ~x X X f εe j 1 1 ∇  = ( ~x ~x = = + ) ( i – 61 – ~x i 0 n ∇ } } f X , . . . , ~x = j j n i j ˜ ˆ ~x O O

εe , , ) = i i j , . . . ~x ) = ˜ ˆ + 0 1 , . . . , ~x O O . In particular these operators vary across the moduli )) 1 i 2 { { ~x n ] ( ~x ~x ˜ ( ( C f vanishes identically. We will now prove that this does not [ 0 1 f f , so the operator becomes C ~x k x..~x,. . , ~x 0 0 ~x ( x..~x,. . , ~x , . . . ~x ˆ vanishes, ie. ∇ O ( 1 → → f ε ε  ~x f ~x ~x = ( ~x = i j ∇ = lim = lim = f ∇ ~x i ~x j

~x ) )) is non-constant. It follows that both the discretely gauged and folded n ∇ n ( k ˜ labels the leg. So we can rewrite the operator as O i , . . . ~x , . . . ~x 1 1 ~x ~x ( ( f i f i where ~x is a non-constant symmetric function in variables attached to wreathed legs; call ~x i which form a basis of ∇ k ∇ ~x i ( ˜ However all the summands are identical under the restriction: O ˜ O . At the fixed point unless assumption that spaces have isomorphic Lie algebras and hence share the same continuous symmetry. so Then n everywhere. Then This does not necessarilyr.h.s. mean terms that could the vanishhappen. if two Lie algebras are isomorphic as some ofthem the for space. Restricting to the folded subspace, we find As mentioned in the main text, the Lie algebra of the discretely gauged space is given by B Folded Lie algebras are the same as discretely gauged Lie algebras JHEP01(2021)086 . | a = 0 m (C.4) (C.5) (C.6) (C.7) (C.1) (C.2) (C.3) 2 | 2 a, , m |− under the Γ e ) which are µ m | C.3 + | d as well. m | have to be collected is straightforward to 15 + Γ i | ), one can set orbits under c the orbit of Σ ) can be done in principle ) m | µ µ 4 ( 3.55 ( ) , + . Using this formula, all the n 4 | i m ) S b γ O 2 m 2∆( . | t , t ) i ) e i + j − ) 1 | µ )Σ j e wreathing i , m ; µ d 2 µ m ( we call t 4 Range ; Γ ( ∈ 2 − 4 S ) det (1 Γ , m t O e a c Z ( ) P ) Γ µ ) µ ,m m ∈ ˜ ( =1 | X µ d P , m a Γ( j ( =1 n b X X j µ ∈ n + ,m 6 γ | =1 c – 62 – i X d | ), that we recall here, , m ) = ,m a 1 Γ ) = b m | µ ( µ . For m m ) = − ( 4 ; 3.46 4 t 2 S a ( S t . The exact value of the sums ) = Γ = ( =0 ( O m a µ ∞ | Γ of X ; ), but using the gauge group ( ˜ 15 m m HS P 2 + Γ t | ( = c Γ i , and the Casimir factors for the group 3.48 m 4 P Σ S − a are evaluated in a fraction of a second on a standard computer. m 4 | + are representatives of these orbits, (not uniquely!) determined by the | b 4 Z m and the conformal dimension can be expressed in terms of is any element satisfying the condition Range is defined in table a − ∈ 4 i a j m . This orbit can be written as a disjoint union of Z ). However it is often useful to massage this formula until a more manageable 4 µ m | ≡ ∈ S i 1 3.44 µ Range a, ) = m Let’s now pick a subgroup Using the notations of ( m quivers are simply where Hilbert series of figure The rationale behind thisadapted definition to is the full that group accordingly. we have This evaluated being the done, sums the ( Hilbert series for the Coulomb branch of the wreathed one can define the modified Casimir factor where the above equation. Using the notation ( where compute (note the absence of Casimir factors!) andaction is of given in table 2∆( One then computes the auxiliary sums and as The computation of the exactusing Hilbert ( series presented inform figure can be used inthe practice. case In of this the appendix, quiver we at give hand. the result Derivations use of such simple manipulations algebra in and are not detailed here. C Computation of Hilbert series with JHEP01(2021)086 , , = 4 ]. 274 N , (2019) 261 ]. SPIRE 02 IN ][ theories SPIRE IN JHEP , ) ][ = 4 Nucl. Phys. B ) Coulomb branches , ]. 10 ]. t ) 10 2 Nucl. Phys. B 12 t 4 ]. ) ) + t t , d N 8 8 8 + = 4 + t t t 8 3 ]. t 1+ + + 10 3 3 2 2 3 2 t 6 6 SPIRE +4 N ) ) ) ) ) ) t t )( SPIRE 6 2 2 2 2 2 2 +3 2 t t t t t t t IN 6 SPIRE t +2 IN t +3 +2 8 + IN ][ t 4 4 i t SPIRE ][ t t (1+ (1+ (1+ (1+ (1+ (1+ +13 +9 arXiv:1309.2657 6 5 5 4 4 3 ][ 4 4 Σ IN ), and exact values of these sums. +6 [ ) ) ) ) ) ) t t 1+ +7 +4 6 2 2 2 2 2 2 t 2 2 t t t t t t Coulomb branch and the moduli ][ t t )( − − − − − − 2 C.3 +18 +10 t +5 ICTP summer school in high-energy arXiv:1807.03221 2 2 (1 (1 (1 (1 (1 (1 4 t t + t 4+7 2+3 [ t ( ( -dimensional 4 2 − 3 +5 , in t t 1 2+9 2 Strings on orbifolds Strings on orbifolds. ( 4+16 ( t (2014) 005 4 ( t 6 t 01 Monopole operators and Hilbert series of Coulomb The Coulomb branch of arXiv:1408.6835 (2018) 158 arXiv:1807.02784 [ e e e e e e e e [ – 63 – arXiv:1503.04817 hep-th/9804208 08 [ m m m m ]. JHEP , < m < m = < m < m = = = ]. d d d d d d d d i m m m m JHEP SPIRE (2014) 103 , (2018) 157 ), which permits any use, distribution and reproduction in Coulomb branches of quiver gauge theories with symmetrizers IN < m < m < m = = = < m = Discrete quotients of ]. Tree level constraints on gauge groups for type I superstrings [ (2017) 671 SPIRE c c c c c c c c Nilpotent orbit Coulomb branches of types AD 12 08 Range IN Discrete gauging in Coulomb branches of three dimensional m m m m ][ 354 < m < m = < m = < m = = SPIRE gauge theories , (1997), pg. 128 [ b b b b b b b b IN JHEP ]. ]. [ JHEP m m m m m m m m , (1982) 97 , = 4 CC-BY 4.0 Lectures on orientifolds and duality This article is distributed under the terms of the Creative Commons 119 SPIRE SPIRE i 1 2 3 4 5 6 d N 3 IN IN [ [ arXiv:1807.11491 . Definitions of the ranges involved in the sums ( [ (1985) 678 (1986) 285 physics and cosmology 113 Phys. Lett. B Commun. Math. Phys. supersymmetric gauge theories via the cycle index space of instantons branches of arXiv:1907.06552 A. Hanany and D. Miketa, N. Marcus and A. Sagnotti, L.J. Dixon, J.A. Harvey, C. Vafa and E. Witten, M. Bullimore, T. Dimofte and D. Gaiotto, A. Hanany and A. Zajac, A. Hanany and M. Sperling, S. Cremonesi, A. Hanany and A. Zaffaroni, H. Nakajima and A. Weekes, S. Cremonesi, G. Ferlito, A. Hanany and N. Mekareeya, L.J. Dixon, J.A. Harvey, C. Vafa and E. Witten, A. Dabholkar, [7] [8] [9] [4] [5] [6] [2] [3] [1] [10] [11] any medium, provided the original author(s) and source areReferences credited. Table 15 When there are two possible ranges, this means that theOpen two choices Access. lead to theAttribution License same ( sums. JHEP01(2021)086 d , 3 J. ]. , 22 Ann. , , , ] ]. ]. SPIRE IN quiver gauge Int. J. Math. ][ , SPIRE SPIRE (2019) 75 = 4 IN -dimensional IN 3 23 ][ ][ d N , unpublished 3 The affine ]. ]. arXiv:1503.03676 [ ]. Adv. Theor. Math. Phys. 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