Factorised 3d \mathcal{N}=4 orthosymplectic quivers
Mohammad Akhond, Federico Carta, Siddharth Dwivedi, Hirotaka Hayashi, Sung-Soo Kim, Futoshi Yagi
FFactorised 3d
N = orthosymplectic quivers Mohammad Akhond, a Federico Carta, b Siddharth Dwivedi, c Hirotaka Hayashi, d Sung-Soo Kim, e and Futoshi Yagi f a Department of Physics, Swansea University,Singleton Park, Swansea, SA2 8PP, U.K. b Department of Mathematical Sciences, Durham University,Durham, DH
LE, United Kingdom c Center for Theoretical Physics, College of Physical Science and Technology, Sichuan University,Chengdu, 610064, China d Department of Physics, School of Science, Tokai University,4-1-1 Kitakaname, Hiratsuka-shi, Kanagawa 259-1292, Japan e School of Physics, University of Electronic Science and Technology of China,No.2006, Xiyuan Ave, West Hi-Tech Zone, Chengdu, Sichuan 611731, China f School of Mathematics, Southwest Jiaotong University,West zone, High-tech district, Chengdu, Sichuan 611756, China
E-mail: [email protected] , [email protected] , [email protected] , [email protected] , [email protected] , futoshi [email protected] Abstract:
We study the moduli space of 3d
N =
N = a r X i v : . [ h e p - t h ] F e b ontents E × E sequence 103.2 The E × E sequence 133.3 The E × E sequence 163.4 The E ′ × E ′ sequence 183.5 The E ′ × E sequence 213.6 The E × E sequence 223.7 The E × E sequence 243.8 The E ′ × E ′ sequence 263.9 The E ′ × E sequence 273.10 The E × E sequence 293.11 The E × E sequence 313.12 An outlier: the E × E theory 34 E × E sequence 36B List of HWG of unitary quivers 37C Unrefined Coulomb branch Hilbert series for low rank OSp quivers 38D Central charges and 3d mirrors for D-type class-S theories 41 D.1 Central charges 41D.2 3d Mirrors 41
Gauge theories in three spacetime dimensions are strongly coupled in the IR, determiningtheir low energy dynamics is therefore generically difficult. One arena in which one canovercome this difficulty is the realm of 3 d N =
N =
N = d N = d N = which are exchanged under mirror symmetry. In certain cases,it can be shown that the magnetic quiver of a given theory, is the 3d mirror of the toruscompactification of that theory to 3d [7, 17–19]. Another interesting aspect of 3d N = theories is the enhancement of their global symmetry in the infrared (IR) limit. Gaugetheories in 3d possess a topological (or magnetic) symmetry, which is valued in the centreof the Langland dual of the gauge group G . Classically, this is an abelian symmetry, butin the IR this is typically enhanced to a non-abelian global symmetry.It will be convenient for us to make a distinction between quivers which are made en-tirely of unitary gauge nodes, and those which can also have orthogonal and/or symplecticgauge nodes in addition. We will refer to the former as unitary and the latter as orthosym-plectic (OSp) quivers respectively. The primary focus of this paper is orthosymplecticmagnetic quivers.A convenient tool to study the moduli space of 3d N = One of the main results of this paper is the refined Hilbert series, and thereforethe enhanced magnetic symmetry of the OSp quivers under our study. Together with theother tools of the Plethystic programme [23, 24], the Hilbert series can be used to give analgebraic description of the moduli space as a variety. We will in particular make use ofthe notion of highest weight generators (HWGs) developed in [25] in order to write downclosed form expressions for the Coulomb branch Hilbert series of the OSp magnetic quiversunder our consideration.In this paper we uncover an interesting phenomenon which is common to all modelsunder our consideration; the moduli space of the OSp quivers that we study genericallyfactorizes into two decoupled sectors, each of which has an alternative description, in termsof the moduli space of a single connected unitary quiver. An upshot of this result is that One can also consider mixed branches, but we will not explore that in this work. This difficulty is related to the notion of hidden FI parameters in OSp quivers, see e.g. [22]. – 2 –e can write exact highest weight generating functions (HWGs) encoding the Coulombbranch Hilbert series of several families of orthosymplectic quivers using known resultsfor the individual factors. This result ultimately follows from fact that the OSp quiversthat we study serve as magnetic quivers to 5d
N = × SU(2), and a spinor and conjugatespinor transform under different SU(2) factors, each such theory can be reformulated asa product of two decoupled theories, each of which has a gauge group that is a productof SU(2)s. The theories containing SO(4) factors can be engineered using a single typeIIB brane web with the inclusion of O5-planes, which can then be used to obtain an OSpmagnetic quiver [2, 5]. On the other hand the formulation in terms of the product oftheories with SU(2) factors is engineered by two independent brane webs, giving rise totwo magnetic quivers, which will be unitary by construction [11].We note that an analogous factorization phenomenon happens for 4d N=2 theories ofclass-S of D-type. In this context, it is well known that there are cases in which a singlethree-punctured sphere describes the direct sum of 2 SCFTS, each of which also admitsa realization in A-type class-S [26–29]. Indeed, we identify some 4d N=2 theories of D-type class-S which exhibit this factorization, and for which the orthosymplectic 3d mirrortheories correspond to the magnetic quiver derived from the 5-brane webs with O5-plane.Likewise, the 3d mirror of the two A-type factors also corresponds to the unitary magneticquiver derived from the 5-brane web without O5-plane.For ease of presentation, we tabulate a list of all the orthosymplectic quivers appearingin this work, along with their Coulomb branch symmetry and the refined highest weightgenerating functions of the Coulomb branch in table 1.The organization of this paper is as follows. In section 2 we review the relevant toolsthat we will need from the plethystic programme. We discuss the monopole formula for theCoulomb branch and the Molien-Weyl formula for the Higgs branch and review the notionof highest weight generating functions. Section 3, contains our main results. Here we willpresent the product exceptional sequences of OSp quivers, their 5d origin from brane webswith O5 planes and their unitary counterparts as well as their 5d origin from ordinary branewebs. In this section we also state the Hilbert series results for all the quivers. Finally, insection 4 we discuss potential applications of our results and state open problems whichwe find deserve further investigation. Appendix A contains additional details for some ofthe Higgs branch Hilbert series computations. We collect the results of HWGs for theunitary quivers in the table 9 of Appendix B. In Appendix C, we give more details onthe computations of the Coulomb branch Hilbert series for the OSp quivers. Appendix Dcontains a review of the class S technology that is used within the main sections.– 3 – able 1 : Summary of the orthosymplectic quivers, their Coulomb branch symmetryand the associated HWG. The corresponding fugacities are denoted by subscripts in thesymmetry groups. Note that for N =
1, there is an enhancement in the symmetry asdetailed in the later sections.Quiver Symmetry PL[HWG] ⋯ N − N SU(2 N ) µ × SU(2 N ) ν N ∑ k = ( µ k µ N − k + ν k ν N − k ) t k ⋯ N − N SU(2 N ) µ × SU(2 N ) ν × U(1) q N ∑ k = ( µ k µ N − k + ν k ν N − k ) t k + t +( q + q − ) ν N t N + − ν N t N + ⋯ N − N − SU(2 N ) µ × SU(2 N ) ν × U(1) q N ∑ k = µ k µ N − k t k + N − ∑ k = ν k ν N − k t k + t +( ν N + q + ν N − q − ) t N + − ν N + ν N − t N + ⋯ N − N SU(2 N ) µ × SU(2 N ) ν × U(1) q × U(1) q N ∑ k = ( µ k µ N − k + ν k ν N − k ) t k + t +( q + q − ) µ N t N + − µ N t N + +( q + q − ) ν N t N + − ν N t N + ⋯ N − N − N − SU(2 N ) µ × SU(2 N ) ν × U(1) q × U(1) q N − ∑ k = ( µ k µ N − k + ν k ν N − k ) t k + t +( µ N + q + µ N − q − ) t N + − µ N + µ N − t N + +( ν N + q + ν N − q − ) t N + − ν N + ν N − t N + ⋯ N − N − SU(2 N ) µ × SU(2 N ) ν × U(1) q × U(1) q N ∑ k = µ k µ N − k t k + N − ∑ k = ν k ν N − k t k + t + ( q + q − ) µ N t N + − µ N t N + +( ν N + q + ν N − q − ) t N + − ν N + ν N − t N + ⋯ N N SU(2 N + µ × SU(2 N + ν N ∑ k = ( µ k µ N + − k + ν k ν N + − k ) t k – 4 – ⋮ N − N − N − N −
21 2
SU(2 N + µ × SU(2 N + ν N − ∑ k = ( µ k µ N + − k + ν k ν N + − k ) t k ⋮ N − N − N − N −
21 1
SU(2 N + µ × SU(2 N + ν N ∑ k = µ k µ N + − k t k + N − ∑ j = ν j ν N + − j t j ⋮ N N SU(2 N + µ × SU(2 N + ρ × SU(2) ν × U(1) q N ∑ i = ( µ i µ N + − i + ρ i ρ N + − i ) t i +( ν + ) t + ν ( µ N q + µ N + q − ) t N + − ν µ N µ N + t N + ⋮ N − N − N − SU(2 N + µ × SU(2 N + λ × SU(2) ν × SU(2) η × U(1) q N ∑ i = ( µ i µ N + − i + λ i λ N + − i ) t i + ( ν + ) t + ν ( µ N q + µ N + q − ) t N + − ν µ N µ N + t N + + η t
22 2 21 2 N N ⋯ SU(2 N + µ × SU(2 N + η × SU(2) ν × SU(2) λ × U(1) q × U(1) r N ∑ i = ( µ i µ N + − i + η i η N + − i ) t i + ( ν + λ + ) t + ν ( µ N q + µ N + q − ) t N + + λ ( η N r + η N + r − ) t N + − ν µ N µ N + t N + − λ η N η N + t N + – 5 – ⋮ N + N +
24 22 22
SU(2 N + µ × SU(2 N + λ × SU(2) ν × SU(2) ν × SU(2) ρ × SU(2) ρ N + ∑ i = ( µ i µ N + − i + λ i λ N + − i ) t i + ( ν + ν + ρ + ρ ) t + t + ν ν µ N + ( t N + + t N + ) + ρ ρ λ N + ( t N + + t N + ) − ν ν µ N + t N + − ρ ρ λ N + t N + N N + N N + N N ⋮ N SO(4 N + µ × SO(4 N + ν N ∑ k = ( µ k + ν k ) t k ⋮ N + N + N + N N SO(4 N + µ × SO(4 N + ν × U(1) q N ∑ i = ( µ i + ν i ) t i + t +( µ N + q + µ N + q − ) t N + – 6 – N + N + N + N + ⋮ N + SO(4 N + µ × SO(4 N + ν × U(1) q × U(1) q N ∑ i = ( µ i + ν i ) t i + t +( µ N + q + µ N + q − ) t N + +( ν N + q + ν N + q − ) t N + N + N + N + N + ⋮ N + SO(4 N + µ × SO(4 N + λ × SU(2) ν × SU(2) ρ N + ∑ i = ( µ i + λ i ) t i + t + ( ν + ρ ) t + νµ N + ( t N + + t N + ) + ρλ N + ( t N + + t N + ) + ( µ N + + λ N + ) t N + − ν µ N + t N + − ρ λ N + t N + In this section we review the material we need for the computation of the Hilbert series. Thediscussion will be minimal and will cover only those aspects necessary for the subsequentsection. For more details the reader can consult the original papers. The literature on thismaterial is vast, but we will mostly follow [6, 20, 21, 23, 25]. In the following subsectionswe consider a 3d
N = G of rank r and n H hypermultipletstransforming under the representation R i of G ( i = , ⋯ , n H ). The bosonic fields in a 3d
N =
N =
N = V m of magnetic charge m and the adjoint scalar φ . The Hilbert series– 7 –HS) for the Coulomb branch of a good or ugly (in the sense of [16]) 3d N = ( t, z ) = ∑ m ∈ Λ G ∨ m /W G ∨ z J ( m ) t ( m ) P G ( t, m ) , (2.1)where the sum is over the magnetic lattice of the gauge group G , we refer to [6] for a detailedaccount. ∆ ( m ) is the conformal dimension of the monopole operator with magnetic charge m , and is given by ∆ ( m ) = − ∑ α ∈ ∆ + ∣ α ( m )∣ + n H ∑ i = ∑ ρ i ∈ R i ∣ ρ i ( m )∣ , (2.2)where ∆ + is the set of positive roots and ρ i are the weights of the representation R i . Theclassical dressing factor P G ( t, m ) counts invariants built out of the adjoint scalars φP G ( t, m ) = r ∏ i = − t d ( i ) , (2.3)where d ( i ) are the degrees of the Casimir invariants of the gauge symmetry H ⊂ G which isthe unbroken part of the original gauge symmetry in the presence of a monopole operator V m of charge m . Finally z denote the fugacities of the topological symmetry whose exponent J ( m ) denotes the topological current. The Higgs branch of a 3d
N = G . The Higgs branch operators are therefore those constructed by consideringall symmetrised tensor powers of these irreps. The symmetrisation is done in order to beconsistent with Pauli statistics. To avoid overcounting, the relations among these scalaroperators due to the superpotential need to be imposed. One will then need to project ontothe gauge singlet states in order for the resulting operators to be well defined gauge invariantoperators. The Higgs branch Hilbert series is therefore computed using the Molien-Weylformula [21] HS H ( t ) = ∫ G dµ G PE [∑ n H i = χ R i ( x ) t ] PE [ χ Adj ( x ) t ] , (2.4)where χ R i ( x ) is the character of the representation R i of G under which the scalars in the i -th hypermultiplet transform, χ Adj is the character of the adjoint representation of G , therepresentation carried by the relations. The function PE [⋅] is the plethystic exponential,defined via PE [ f ( z , ⋯ , z r )] = exp ( ∞ ∑ k = k f ( z k , ⋯ , z kr )) , (2.5)it is a symmetrising function that generates the characters of the symmetrised tensor powersof χ R i . Finally the projection onto gauge invariant operators is achieved by integratingover the group manifold using the Haar measure. In a suitable basis, the Haar measurecan be taken to be ∫ G dµ G = ( π i ) r r ∏ i = ∮ ∣ x i ∣= dx i x i ∏ α ∈ ∆ + ( − r ∏ k = x α k k ) . (2.6)– 8 – .3 Highest weight generating functions The refined Hilbert series for a given theory can be generically expanded as Taylor series in t such that the coefficients are sum of the characters of representations of global symmetryof the theory. In general, given a global symmetry of rank r , the refined Hilbert series canbe expanded as: ∞ ∑ n = ∞ ∑ n = ⋯ ∞ ∑ n r = χ [ f ,f , ⋯ ,f r ] t η , (2.7)where each of f , . . . , f r and η can be some polynomial function in variables n , . . . , n r .The notation χ [ f , ⋯ ,f r ] is the character of the irrep of the global symmetry whose highestweight is f Λ + . . . + f r Λ r with f i as the Dynkin labels. A convenient way to repackage thesame information is in terms of highest weight generating functions or HWGs [25]. Oneintroduces a set of fugacities { µ , . . . , µ r } , called highest weight fugacities, and one writesthe characters in terms of µ i according to the map χ [ f ,...,f r ] ↔ µ f . . . µ f r r . (2.8)With this map, the Hilbert series becomes a formal power series which can be resummed,the corresponding generating function is termed as its HWG:HWG = ∞ ∑ n = ⋯ ∞ ∑ n r = µ f . . . µ f r r t η . (2.9) In this section we present sequences of orthosymplectic magnetic quivers whose modulispace is the product of two decoupled sectors, each of which enjoys a description as themoduli space of a unitary quiver. Each sequence is parameterised by an integer N whichis the sequence number, and labelled E m × E n , in accordance with the Coulomb branchisometry of the N = N =
1, some of the theories become particularly simplesuch that we can prove the equivalence of the orthosymplectic quiver with the two unitaryquivers. The E m × E n orthosymplectic quiver with sequence number 1 corresponds to themagnetic quiver for infinite coupling limit of 5d N = m − s and n − c . Correspondingly the dual unitaryquivers with sequence number 1 correspond to magnetic quivers for infinite coupling limitof 5d N = × SU(2) gauge theory with m − F , ) and n − , F ) representation of SU(2) × SU(2), where we denote by F thefundamental representation of associated gauge group. One can engineer these theoriesusing 5-brane webs with O5-planes [30], as well as using ordinary brane webs [31]. Thispattern generalizes for higher sequence numbers, namely one can provide an intuition forthe reason that the orthosymplectic quivers factorise into two decoupled sectors by viewingthem as magnetic quivers of a 5d theory. We will therefore employ this perspective inthe following. We will use EQ m,n to denote the 5d OSp electric quivers for the E m × E n sequence, while we use the notation EQ m to denote the 5d unitary electric quivers to which– 9 –he former factorise. Similarly, we will denote the OSp magnetic quivers of the E m × E n sequence by MQ m,n , while we denote the unitary components to which they factorise byMQ m . Occasionally there will be more than one generalisation of a given sequence forhigher sequence numbers, in which case we will distinguish the different sequences by aprime.Our conventions for the magnetic quivers are identical to those appearing in [2] whichwe briefly review in the following. We denote by a white, red and blue node, the groupsU ( n ) , SO ( n ) and USp(2 n ) respectively. A circular node is to be understood as a gaugegroup, while a square node denotes a flavour group. We use solid lines to denote bifunda-mental hypermultiplets, in the case where the solid line connects an orthogonal to a sym-plectic node there is a reality condition which renders the link to be a half hypermultiplet.We use a dashed link between two unitary nodes to denote a fundamental-fundamental hy-permultiplet and a jagged line between a unitary flavour and a gauge U ( ) node to denotea charge 2 hypermultiplet. E × E sequence Consider the 5-brane web constructed by collapsing 2 N NS5 branes on top of an O5-planethat is asymptotically O5 + as in figure 1. By resolving this web to go on the Coulombbranch one can identify the following low energy quiver description,EQ , = SO(4) − USp(0) − SO(4) − ⋯ −
USp(0) − SO(4)2 N − . (3.1)One can recast this theory, by using the accidental Lie algebra isomorphism so ( ) ≅ su ( ) × su ( ) as a product of two decoupled 5d quiver gauge theories:EQ , = ( EQ ) = ⎛⎜⎜⎝ SU(2) − SU(2) − SU(2) − ⋯ −
SU(2) N ⎞⎟⎟⎠ . (3.2)Since this description involves only special unitary gauge groups, we should be able toengineer it using (two copies of) ordinary brane webs, i.e. without using O5-planes. Itturns out there are a few possible candidates as there are distinct webs for SU(2) gaugetheory with discrete theta angles θ = θ = π . In order to identify the correct webdiagram, we scanned through all the possibilities and eliminated inconsistent choices bytrial and error, by counting the Higgs branch dimension of the different candidate webdiagrams and requiring the answer to match with the result of the same computationperformed on the brane web with O5-plane. We found that, modulo SL ( , Z ) and Hanany-Witten moves, the unitary web diagram in figure 1 is the unique choice satisfying theaforementioned criterion.At this point we need to clarify which aspect of the two theories (3.1) and (3.2) areexpected to be the same. This is because we used an isomorphism at the level of Lie algebra,ignoring any issues related to the global structure of the gauge group (with the exception of– 10 – + O5 + N − N ⋮ ( , )( , ) ( , − ) ( , − )⋰⋰ N − N N − N Figure 1 : Brane webs engineering EQ , (left) and EQ (right). The numbers near each5-brane denote the number of coincident 5-branes in the stack in that segment. Black dotsdenote 7-branes of charge ( p, q ) .the choice of discrete theta angle mentioned above). In particular, any information relatedto local operators in the two theories is likely to agree, while questions about e.g. line andsurface operators in general will be sensitive to the global structure of the gauge group.Our primary interest in these theories is in their Higgs branch, which is parameterised bylocal operators, and thus should agree regardless of any subtle differences such as thosealluded to above.Having constructed web diagrams for EQ , and EQ in figure 1, we can now proceedto derive their magnetic quivers, following the rules introduced in [2, 5, 11]. From theorientifold web in figure 1 we obtain the OSp magnetic quiverMQ , = ⋯ N − N , (3.3)while taking two copies of the unitary web in figure 1 leads us to conjectureMQ , = ( MQ ) = ⎛⎜⎜⎜⎜⎜⎝ ⋯ N ⋯ ⎞⎟⎟⎟⎟⎟⎠ . (3.4)Note that the set of balanced nodes in the unitary quiver implies an SU ( N ) × SU ( N ) symmetry, with each factor coming from the balanced nodes in one of the unitary quivers(3.4). This is consistent with the claim of Gaiotto and Witten [16], that whenever a chainof balanced unitary nodes terminate on a balanced symplectic node, the isometry of the– 11 –oulomb branch is doubled. A second consistency check, is that for N =
1, the two theoriesare obviously identical, the OSp quiver in this case is the so called T ( SO ( )) , while theunitary side is two copies of T ( SU ( )) . In other words for N = T ( SO ( )) ↔ T ( SU ( ) × SU ( )) ↔ T ( SU ( )) × T ( SU ( )) . (3.5)One upshot is that the HWG for the unitary quiver is straightforward to extract, given thatits refined Hilbert series can be computed. Indeed the unitary quiver has already appearedin previous studies, for instance in [3]. Therefore our conjecture implies that the HWG ofthe OSp quiver (3.3) is simply given by doubling the known result for (3.4), namely:HWG , = PE [ N ∑ k = ( µ k µ N − k + ν k ν N − k ) t k ] , (3.6)where µ and ν are highest weight fugacities for SU ( N ) × SU ( N ) . We can confirm thisproposal by computing the unrefined Hilbert Series for the OSp quiver (3.3) for small valuesof N . Some of the results are given in table 10.Even more remarkable, is that the agreement between the quivers (3.3) and (3.4) isalso valid on the Higgs branch. From the 5d perspective there is no a priori reason why thisshould be so, but it can be confirmed by an explicit calculation of the unrefined Hilbertseries (see appendix A for a derivation of this formula) N ∏ q = ( − t q ) ∫ dµ C N PE [ χ [ , , , ⋯ , ] CN t + χ [ , , ⋯ , ] CN t ] . (3.7)Let us evaluate this integral for N =
2. The measure over the USp ( ) group can be takento be ∫ dµ C = ∮ ∣ x ∣= dx π i x ∮ ∣ x ∣= dx π i x ( − x )( − x )( − x x )( − x x ) , (3.8)while the characters for the fundamental and second rank antisymmetric representation ofUSp ( ) are given respectively by χ [ , ] C = x + x x + x x + x ,χ [ , ] C = x + x x + + x x + x , (3.9)Thus the expression we need to evaluate is ∏ q = ( − t q ) ∮ ∣ x ∣= dx π i x ∮ ∣ x ∣= dx π i x ( − x )( − x )( − x x )( − x x )×× ( − x t )( − x x t )( − t )( − x x t )( − t x )( − x t ) ( − x x t ) ( − x x t ) ( − tx ) . (3.10) These characters are computed as follows. For a weight w = [ w , w ] ≡ w Λ + w Λ appearing inthe weight system of a representation R of C , the corresponding monomial in the character will be x w x w . For example, the weights appearing in the weight system of fundamental representation of C are {[ , ] , [− , ] , [ , − ] , [− , ]} where each weight is written in the fundamental weight basis. Thus thecharacter for fundamental representation will be simply x x + x − x + x x − + x − x . – 12 –his integral can now be evaluated using residues to arrive at the following Hilbert seriesHS H , ∣ N = = ( − t ) ( − t ) ( − t ) ( − t ) . (3.11)We recognise this as the Coulomb branch Hilbert series of two copies of U ( ) with 4 fun-damental hypermultiplets [20], which is the mirror of the N = E × E sequence In the previous subsection, we saw that the Higgs branch of the fixed point limit of 5 d SO(4) gauge theory, factorises to two copies of the Higgs branch of the 5 d pure SU(2) gaugetheory. It is natural to ask whether this pattern holds if we include matter transformingunder SO(4). The two matter representations which are straightforward to obtain from thebrane web are the vector of SO(4) and the two spinor representations of opposite chirality.Since the vector of SO(4) corresponds to bifundamental of SU(2) × SU(2), this will not leadto the desired factorised theory. However, each of the two spinor representations, denotedby s and c respectively, will only transform under one of the two SU(2) factors in SO(4).In this and subsequent subsections, we will exploit this well-known fact.Consider the orientifold web diagram presented in figure 2. The corresponding IRquiver gauge theory description is given by the electric quiverEQ , = [ s ] − SO(4) − USp(0) − SO(4) − ⋯ −
USp(0) − SO(4) − [ s ] N − . (3.12)By using the isomorphism so ( ) ≅ su ( ) × su ( ) , we can rewrite this theory as a productof the following electric quivers.EQ , = EQ × EQ = SU(2) − SU(2) − ⋯ −
SU(2) N × SU ( )∣ [ F ] − SU(2) − ⋯ − SU ( )∣ [ F ] N (3.13)The unitary brane web for EQ is given in figure 1, while the unitary brane web for EQ is presented in figure 2. The orientifold web in figure 2 admits two maximal subdivisions.Accordingly the Higgs branch of this theory is the union of two cones, given by the twoOSp magnetic quivers in table 2. On the other hand, we expect these magnetic quiversto be equivalent to the product MQ × ( MQ ( I ) ∪ MQ ( II ) ) , with the latter factor obtainedfrom the unitary web for EQ in figure 2. As a further non-trivial check, we can computethe Coulomb branch Hilbert series of the unitary and OSp quivers. The unitary quiversMQ and MQ ( II ) have known HWGs, but to our knowledge, the unitary quiver MQ ( I ) hasnot appeared previously in the literature. We will therefore spend some time to study itsCoulomb and Higgs branch Hilbert series.– 13 – + O5 + N − N (2,1)(2,-1) ⋮ ( , )( , ) ⋰⋰ N − N N − N Figure 2 : Brane webs engineering EQ , (left) and EQ (right). For the unitary braneweb engineering EQ , see figure 1.The Coulomb branch HS for MQ ( I ) in the case when N = ∞ ∑ n =−∞ ∞ ∑ n =−∞ ∞ ∑ n =−∞ ∞ ∑ m ≥ m =−∞ z n z m + m z n z n t ( n ,n ,n ,m ,m ) ( − t ) P U ( ) ( m , m ; t ) , (3.14)where the conformal dimension is given by∆ ( n , n , n , m , m ) = −∣ m − m ∣ + ∑ i = ∑ j = (∣ m i − n j ∣ + ∣ m i ∣) . (3.15)We want to expand (3.14) perturbatively as a power series in t . Upon transforming thetopological fugacities by the map z → x x , z → x x x , z → x x , z → qx , (3.16)the coefficients of t at each order will form the characters of some irrep of SU ( )× U ( ) , aspolynomial function in ( x , x , x ) and with U(1) charge given by the exponent of q . Inparticular the first few orders are1 + ( + [ , , ]) t + ( q + q − ) [ , , ] t + ( + [ , , ] + [ , , ] + [ , , ]) t + ( q + q − ) ([ , , ] + [ , , ]) t + ( + [ , , ] + [ , , ] + [ , , ] + [ , , ] + [ , , ] + ( q + q − ) [ , , ]) t + ( q + q − ) ([ , , ] + [ , , ] + [ , , ] + [ , , ]) t +([ , , ] + [ , , ] + [ , , ] + [ , , ] + [ , , ] + [ , , ] + [ , , ] + [ , , ] ++ ( q + q − ) ([ , , ] + [ , , ])) t + O( t ) . (3.17)– 14 –ased on this perturbative computation, we propose the following HWG for the Coulombbranch of MQ ( I ) HWG ( I ) = PE [ t + ( q + q − ) ν N t N + + N ∑ k = ν k ν N − k t k − ν N t N + ] , (3.18)where ν i are fugacities for SU(2 N ) and q is the U(1) charge.We are now in a position to write down the HWGs for the OSp magnetic quivers intable 2. The final result for the first cone readsMS OSp UnitaryMQ ( I ) , ⋯ N − N
11 1 ⋰ N ⋱
11 1 ⋰ N ⋱
11 1 MQ ( II ) , ⋯ N − N − ⋰ N ⋱
11 1 ⋮ N − N − N − ⋮
11 1
Table 2 : OSp and unitary representation of the two cones on the Higgs branch of EQ , .HWG (I)1 , = PE [ N ∑ k = µ k µ N − k t k ] PE [ t + ( q + q − ) ν N t N + + N ∑ k = ν k ν N − k t k − ν N t N + ] , (3.19)where µ and ν are the highest weight fugacities for SU ( N )× SU ( N ) while q keeps trackof the U(1) charge. The HWG for the second cone readsHWG (II)1 , = PE [ N ∑ k = µ k µ N − k t k ]× PE [ t + ( ν N + q + ν N − q − ) t N + + N − ∑ k = ν k ν N − k t k − ν N + ν N − t N + ] . (3.20)This can be verified upon comparison with the result of an unrefined Hilbert series com-putation on the OSp side. The results for low values of N are given in table 10.We can also compute the Higgs branch HS for MQ ( I ) , for N = , discussed around (A.2). We need– 15 –o evaluate the following integral ∏ q = ( − t q ) ∫ dµ C ∫ dµ U ( ) PE [ χ C [ , ] t + χ C [ , ] ( q + q − ) t + ( q + q − ) t ] (3.21)Evaluating this integral by finding the residues one arrives at the followingHS H , , ( I ) = ( − t + t )( + t )( + t + t + t + t + t + t )( − t ) ( + t ) ( + t ) ( + t + t + t + t ) . (3.22)This is to be compared with the product of the Higgs branch HS of the two unitary quiversappearing in the first row of table 2. We already know the result for one of these, whichis identical to MQ of (3.4). Its Higgs branch HS was discussed in the previous sectionand is given by the square root of the expression in (3.11). The Higgs branch HS of theother quiver in the first row of table, 2, which we dub MQ ( I ) is straightforward to compute.Specialising to the case N =
2, we need to evaluate the following integral ∫ dµ U ( ) ( x, q x ) ∫ dµ U ( ) ( u ) H T [ SU ( )] ( x ) H U ( ) glue ( x, q x )×× H [ ]−[ ] ( x, q x ) H [ ]−[ ] ( x, q x , u ) H [ ]−[ ] ( u ) H U ( ) glue ( u )= ∮ ∣ x ∣= dx π i x ( − x ) ∮ ∣ q x ∣= dq x π i q x ∮ ∣ u ∣= du π i u ( − t ) ( − t ) ×× PE [( x + + x − ) t + ( x + x − )( q x + q − x )( u + u − ) t + ( u + u − ) t + ( x + x − )( q x + q − x ) t ] . (3.23)Evaluating this integral by computing its residues results inHS H , ( I ) = ( + t + t + t + t + t + t )( − t ) ( + t ) ( + t )( + t + t )( + t + t + t + t ) . (3.24)Together with the result for the HB of MQ , this precisely reproduces the computation onthe OSp side (3.22). E × E sequence The E × E sequence corresponds to the fixed point limit of the electric quiverEQ , = ( EQ ) = SO ( )∣ [ s + c ] − USp(0) − SO(4) − ⋯ −
USp(0) − SO ( )∣ [ s + c ] N − . (3.25)The orientifold web which engineers this theory is given in figure 3. This brane web admitsthree maximal subdivisions leading to the three OSp magnetic quivers in table 3. It can beunderstood as a limiting case of the Y , N theory in [2]. One can provide a purely unitarydescription of this theory in terms of the following electric quiver:EQ , = EQ = ⎛⎜⎜⎜⎜⎜⎜⎜⎝ SU ( )∣ [ F ] − SU(2) − ⋯ − SU ( )∣ [ F ] N ⎞⎟⎟⎟⎟⎟⎟⎟⎠ , (3.26)– 16 – − O5 − N − N (1,1)(1,-1) ⋮ Figure 3 : Orientifold web for EQ , .which can be engineered by taking two copies of the unitary web shown in figure 2. Thisbrane web admits two maximal subdivisions whose magnetic quivers were discussed in theprevious subsection. Since we are taking two copies, a third cone arises when we take adifferent maximal subdivision for each web diagram. This leads us to the unitary magneticquivers in table 3.MS OSp UnitaryMQ ( I ) , ⋯ N − N ⎛⎜⎜⎜⎝ ⋯ N ⋯
11 1 ⎞⎟⎟⎟⎠ MQ ( II ) , ⋯ N − N − N − ⎛⎜⎜⎜⎜⎜⎝ ⋯ N − N − N − ⋯
11 1 ⎞⎟⎟⎟⎟⎟⎠ MQ ( III ) , ⋯ N − N − ⋰ N ⋱
11 1 1 ⋮ N − N − N − ⋮
11 1
Table 3 : Unitary and OSp magnetic quivers for the E × E sequence.– 17 –e can now infer the HWG for the OSp quivers in table 3 by taking those of thecorresponding unitary magnetic quivers as building blocks. This reasoning leads us toconjecture the following HWG for the three cones in table 3HWG (I)3 , ( t ) = PE [ t + ( µ N q + µ N q − ) t N + + N ∑ k = ( µ k µ N − k t k ) − µ N t N + ]× PE [ t + ( ν N q + ν N q − ) t N + + N ∑ k = ( ν k ν N − k t k ) − ν N t N + ] HWG (II)3 , ( t ) = PE [ t + ( µ N + p + µ N − p − ) t N + + N − ∑ k = µ k µ N − k t k − µ N + µ N − t N + ]× PE [ t + ( ν N + q + ν N − q − ) t N + + N − ∑ k = ν k ν N − k t k − ν N + ν N − t N + ] HWG (III)3 , ( t ) = PE [ t + ( µ N + p + µ N − p − ) t N + + N − ∑ k = µ k µ N − k t k − µ N + µ N − t N + ]× PE [ t + ( q + q − ) ν N t N + + N ∑ k = ν k ν N − k t k − ν N t N + ] . (3.27)Here µ and ν are the fuagicites for the two SU(2 N ) groups and p and q are the U(1)charges. We have verified this result by an explicit unrefined Hilbert series computation ofthe Coulomb branch of the OSp quivers which are presented in table 10 for low values of N . E ′ × E ′ sequence O5 + O5 − N N + ⋮ ( , )( , ) ⋰⋰ N − N NN + Figure 4 : Brane webs for EQ ′ , ′ (left) and EQ ′ (right).There is another sequence whose first member has an E × E symmetry. We will referto this as the E ′ × E ′ sequence. In figure 4 we present the orientifold web that engineers– 18 –he 5d electric quivers in this sequence, the IR quiver description is given byEQ ′ , ′ = SO(4) − USp(0) − SO(4) − ⋯ −
USp(0) − SO ( )∣ [ s + c ] N − ′ , ′ = EQ ′ = ⎛⎜⎜⎜⎜⎜⎜⎜⎜⎝ SU(2) − SU(2) − ⋯ − SU ( )∣ [ F ] N ⎞⎟⎟⎟⎟⎟⎟⎟⎟⎠ . (3.29)We present the unitary web engineering each EQ ′ factor in figure 4. Now we can use theunitary and orientifold webs in figure 4 to obtain unitary and OSp magnetic quivers for the E ′ × E ′ sequence, which appear in table 4. These MQs can be obtained by considering themaximal subdivisions appearing in figure 5. The first subdivision was already discussedin [2], while the second and third were overlooked. It was noticed in [9] that there shouldbe 2 additional OSp cones to match the analysis on the unitary side. We claim that themissing cones in this case correspond to the two new maximal subdivisions appearing infigure 5. O5 + O5 − N N ⋮ O5 + O5 − N − N − N − N − ⋮ O5 + O5 − N − N − N − N − ⋮ Figure 5 : Maximal subdivisions of the Higgs branch of EQ ′ , ′ at the superconformallimit. The subweb coloured in red is frozen and contributes only as flavour nodes to themagnetic quivers in table 4.Notice that for N =
1, the relation between the unitary and OSp quivers in the first rowof table 4 was already suggested in [32], our result generalises this to higher N . The HWGfor the unitary quiver is known and appears in [3]. Given the correspondence between theunitary and OSp magnetic quivers in table 4, we can use the results for the HWGs of theunitary quivers to obtain the HWGs for the OSp quivers. In order to do this, let us point– 19 –S OSp UnitaryMQ ( I ) ′ , ′ ⋯ N N ⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝ N N ⋱⋰ ⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠ MQ ( II ) ′ , ′ ⋯ N − N − N − N −
21 2 224 ⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝ N − N − N − N − ⋰ ⋱ ⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠ MQ ( III ) ′ , ′ ⋯ N − N − N − N −
21 1 22 1
N N ⋮⋮ N − N − N − N − ⋮ ⋮ Table 4 : Magnetic quivers for the E ′ × E ′ sequenceout the following useful fact; one of the unitary quivers appearing in the second and thirdrow of table 5, is itself a product of two unitary quivers: N − N − N − N − ⋰ ⋱ = N − N − N − N − ⋰ ⋱ × , (3.30)where the right hand side of the above is obtained after ungauging the overall decoupledU ( ) in the original quiver. The first quiver in the right hand side of the above is a height2 nilpotent orbit, whose HWG is presented in [3], while the second quiver is just N = N = ( I ) ′ , ′ = PE [ N ∑ k = ( µ k µ N + − k + ν k ν N + − k ) t k ] , HWG ( II ) ′ , ′ = PE [ N − ∑ k = ( µ k µ N + − k + ν k ν N + − k ) t k + ( ρ + λ ) t ] , HWG ( III ) ′ , ′ = PE ⎡⎢⎢⎢⎢⎣ N ∑ k = µ k µ N + − k t k + N − ∑ j = ν j ν N + − j t j + ρ t ⎤⎥⎥⎥⎥⎦ . (3.31)This proposal can be checked by a direct computation of the unrefined Hilbert series of theOSp quiver. For low values of N , results are given in table 10. E ′ × E sequence The E ′ × E sequence is the magnetic quiver for the fixed point limit of the 5d IR electricquiver EQ ′ , = SO ( )∣ [ s ] − USp(0) − SO(4) − ⋯ −
USp(0) − SO ( )∣ [ s + c ] N − . (3.32)It can be engineered by the orientifold web diagram presented in figure 6. Alternatively O5 − O5 − N N + ⋮ Figure 6 : Orientifold web for EQ ′ , .– 21 –e may reformulate the electric theory EQ , as a product of two unitary electric quiversEQ ′ , = EQ ′ × EQ = SU ( ) − SU(2) − ⋯ − SU ( )∣ [ F ] N × SU ( )∣ [ F ] − SU(2) − ⋯ − SU ( )∣ [ F ] N , (3.33)where EQ ′ is engineered by the unitary web in figure 4, while the unitary web engineeringEQ is the one in figure 7. Given these webs, the magnetic quivers can be extracted usingthe rules in [2]. The Higgs branch of EQ ′ , at the fixed point is the union of two cones,whose magnetic quivers are given in table 5. The HWG that we propose for the OSpquivers areHWG ( I ) ′ , = PE [ N ∑ i = µ i µ N + − i t i + ( ν + ) t + ν ( µ N q + µ N + q − ) t N + − ν µ N µ N + t N + ]× PE [ N ∑ k = ρ k ρ N + − k t k ] HWG ( II ) ′ , = PE [ N ∑ i = µ i µ N + − i t i + ( ν + ) t + ν ( µ N q + µ N + q − ) t N + − ν µ N µ N + t N + ]× PE [ N ∑ k = λ k λ N + − k t k ] PE [ η t ] (3.34)As a check, we have also computed the unrefined Hilbert series for low values of N whichare given in table 10. E × E sequence The E × E sequence are the magnetic quivers for the fixed point limit of the 5d electricquiverEQ , = [ + ] − SO(4) − USp(0) − SO(4) − ⋯ −
USp(0) − SO(4) − [ + ] N − . (3.35)We present the orientifold web that engineers this theory in figure 7. Alternatively we canwrite EQ , as the product of two unitary electric quiversEQ , = EQ = ⎛⎜⎜⎜⎜⎜⎜⎜⎜⎝ SU ( )∣ [ F ] − SU(2) − ⋯ − SU ( )∣ [ F ] N ⎞⎟⎟⎟⎟⎟⎟⎟⎟⎠ , (3.36)– 22 –S OSp UnitaryMQ ( I ) ′ , ⋮ N N
12 11 1
N N ⋮⋮ N N ⋱⋰ MQ ( II ) ′ , ⋮ N − N − N − N N ⋮⋮ N − N − N − N − ⋰ ⋱ Table 5 : Magnetic quivers for the two cones of the Higgs branch of EQ ′ , . O5 − O5 − N N + ⋮ ( , )( , ) ⋰⋰ NN +
11 2 N − N Figure 7 : Brane webs for EQ , (left) and EQ (right).where each copy of EQ can be engineered by the unitary web depicted in figure 7. Giventhe brane webs in figure 7, one can derive an OSp and a pair of unitary magnetic quiverswhose Coulomb branches are expected to describe the same unique 5d Higgs branch, leading– 23 –s to conjecture thatMQ , = N N ⋯ = ⎛⎜⎜⎜⎜⎜⎜⎜⎜⎝
11 1
N N ⋯⋯ ⎞⎟⎟⎟⎟⎟⎟⎟⎟⎠ (3.37)The HWG for the unitary quiver appearing here was evaluated in [33]. We can use thisresult to obtain an exact HWG for the OSp quiver by simply taking its square. Our claimis HWG , = PE [ N ∑ i = µ i µ N + − i t i + ( ν + ) t + ν ( µ N q + µ N + q − ) t N + − ν µ N µ N + t N + ]× PE [ N ∑ i = η i η N + − i t i + ( λ + ) t + λ ( η N r + η N + r − ) t N + − λ η N η N + t N + ] . (3.38)The explicit unrefined Hilbert series computation for N = N = E × E sequence The E × E sequence is obtained by taking the fixed point limit of the electric quiver givenby EQ , = [ s + c ] − SO(4) − USp(0) − SO(4) − ⋯ −
USp(0) − SO(4) − [ s + c ] N − , (3.39)which can be engineered using the following orientifold web: ⋮ O5 − O5 − N + N + . (3.40)– 24 –lternatively, we can rewrite EQ , as the product of two unitary electric quiversEQ , = EQ = ⎛⎜⎜⎜⎜⎜⎜⎜⎜⎝ SU ( )∣ [ F ] − SU(2) − ⋯ − SU ( )∣ [ F ] N ⎞⎟⎟⎟⎟⎟⎟⎟⎟⎠ , (3.41)each of which is engineered by taking one copy of the following unitary brane web ( , )( , ) ⋰⋰ NN +
11 2 NN +
11 2 2 1 . (3.42)From the brane webs in (3.40) and (3.42), we obtain the two corresponding magneticquivers which then implyMQ , = MQ = ⋮ N + N +
24 22 22 = ⎛⎜⎜⎜⎜⎜⎝ ⋯ N + ⋯
12 11 ⎞⎟⎟⎟⎟⎟⎠ . (3.43)The unitary quiver appearing here has been studied previously, and its HWG was given in[33]. We can now obtain the HWG for the OSp quiver by simply squaring that expressionto obtainHWG , = PE [ N + ∑ i = µ i µ N + − i t i + ( ν + ν ) t + t + ν ν µ N + ( t N + + t N + ) − ν ν µ N + t N + ]× PE [ N + ∑ i = λ i λ N + − i t i + ( ρ + ρ ) t + t + ρ ρ λ N + ( t N + + t N + ) − ρ ρ λ N + t N + ] . (3.44)– 25 – .8 The E ′ × E ′ sequence The E ′ × E ′ sequence is obtained by considering the fixed point limit of the following 5delectric quiver:EQ ′ , ′ = SO(4) − USp(0) − SO(4) − ⋯ −
USp(0) − SO(4) − [ s + c ] N − , (3.45)which can be engineered using the orientifold web: ⋯ N N + N + N + N N + − O5 + . (3.46)Alternatively we may rewrite EQ ′ , ′ as two copies of a single electric quiver with SU(2)gauge nodes, namely EQ ′ = ⎛⎜⎜⎜⎜⎜⎜⎜⎜⎝ SU ( ) − SU(2) − ⋯ − SU ( )∣ [ F ] N ⎞⎟⎟⎟⎟⎟⎟⎟⎟⎠ , (3.47)each copy of which can now be engineered using an ordinary brane web: ⋰ N N + N + N + N N . (3.48)– 26 –his leads us to conjecture the equivalence of the following 3d magnetic quiversMQ ′ , ′ = N N + N N + N N ⋯ N = MQ ′ = ⎛⎜⎜⎜⎜⎜⎝ ⋯ N NN N ⎞⎟⎟⎟⎟⎟⎠ . (3.49)We now want to write down the HWG for the Coulomb branch Hilbert series of MQ ′ , ′ , viathe conjectured relation to the unitary quiver. To the best of our knowledge, the unitaryquiver itself has not been studied previously. Thus we proceed to compute its HWG. Therefined Hilbert seires for the case where N = ′ , ′ ∣ N = = + [ , , , ⋯ , ] t + ([ , , , ⋯ , ] + [ , , , , , , ]) t + O( t ) , (3.50)from which we take as an ansatz for the HWG the following expressionPE [ µ t + µ t ] . (3.51)Now we expand this HWG in a power series in t and unrefine by replacing the characterof each irrep by its dimension1 + t + t + t + t + t + O( t ) . (3.52)This answer precisely coincides with the result of an unrefined HS computation. Extrapo-lating from this example we propose the following HWG for general N :HWG ′ = PE [ N ∑ k = µ k t k ] . (3.53)Consequently, the HWG for MQ ′ , ′ is obtained by squaring this result, namelyHWG ′ , ′ = PE [ N ∑ k = ( µ k + ν k ) t k ] . (3.54) E ′ × E sequence The E ′ × E sequence is obtained by taking the fixed point limit of the 5d electric quivergiven byEQ ′ , = [ s ] − SO(4) − USp(0) − SO(4) − ⋯ −
USp(0) − SO(4) − [ s + c ] N − , (3.55)– 27 –hich can be engineered by the following orientifold web diagram O5 − O5 − N N + ⋮ N + N + N + N + . (3.56)Alternatively we may reformulate EQ ′ , as the product of two unitary electric quiversEQ ′ , = EQ ′ × EQ = SU ( ) − SU(2) − ⋯ − SU ( )∣ [ F ] N × SU ( )∣ [ F ] − SU(2) − ⋯ − SU ( )∣ [ F ] N , (3.57)where EQ ′ is engineered by the unitary web in (3.48), and EQ is the IR quiver descriptionof the web diagram given by ⋰ N N + N + N + N + N + . (3.58)– 28 –eading off the OSp magnetic quiver from (3.56) and the unitary quivers from (3.48),(3.58), we arrive at the conjectureMQ ′ , = ⋯ N + N + N + N N == MQ ′ × MQ = ⋯ N NN N × ⋯ N + N + N +
11 1 (3.59)Obtaining the HWG for the Coulomb branch of MQ ′ , is now straightforward, given theabove relation. The HWG for MQ ′ was worked out in the previous subsection in (3.53)and the HWG for MQ apears in [33]. For the OSp quiver MQ ′ , in (3.59) we simply takethe product of these two resultsHWG ′ , = PE [ N ∑ i = ( µ i + ν i ) t i + t + ( µ N + q + µ N + q − ) t N + ] . (3.60)This can be verified by computing the unrefined Coulomb branch Hilbert series, the resultsof which are given in table 10 for low values of N . E × E sequence The electric quiver for the IR limit of the E × E sequence readsEQ , = [ s + c ] − SO(4) − USp(0) − SO(4) − ⋯ −
USp(0) − SO(4) − [ s + c ] N − . (3.61)It can be engineered using the following orientifold web ⋯ N + N + N + N + N + N +
21 O5 − O5 − . (3.62)Alternatively we may present the electric quiver as a product of two unitary quivers:EQ , = EQ = ⎛⎜⎜⎜⎜⎜⎜⎜⎜⎝ SU ( )∣ [ F ] − SU(2) − ⋯ − SU ( )∣ [ F ] N ⎞⎟⎟⎟⎟⎟⎟⎟⎟⎠ , (3.63)– 29 –here each copy is engineered by the web diagram in (3.58). We can then obtain themagnetic quivers from the brane webs in (3.62) and (3.58), that lead us to the conjecturethatMQ , = MQ = N + N + N + N + ⋯ N + = ⎛⎜⎜⎜⎜⎜⎜⎜⎜⎝ ⋯ N + N + N +
11 1 ⎞⎟⎟⎟⎟⎟⎟⎟⎟⎠ . (3.64)This has an immediate corollary which allows us to extract the HWG for the OSp quiverappearing above by squaring the known result [33] for the unitary quiver:HWG , = PE [ N ∑ i = µ i t i + t + ( µ N + q + µ N + q − ) t N + ]× PE [ N ∑ i = ν i t i + t + ( ν N + q + ν N + q − ) t N + ] . (3.65)Indeed the unrefined Hilbert series for the OSp quiver for N = N we were not able to perform anexplicit computation due to the high rank of the OSp quiver. This is one instance in whichour conjecture proves powerful, as it gives an exact expression for the Hilbert series of aquiver which would otherwise be very challenging to compute.We further point out an interesting fact about this theory, namely the existence ofa 4d N = D type in which a single three-punctured sphere realizes a product SCFT, whereboth factors are the E Minahan-Nemeschansky (MN) theory [34]. We recall that the E Minahan-Nemeschansky theory is a 4d
N = E , and central charges a E = , c E = , (3.66)We report the partitions labeling the punctures in table 6, together with their contri-bution to the effective number of hypermutiplets and vector multiplets.Nahm partition ( δn h , δn v )[ ] ( , )[ , ] ( , )[ , ] ( , ) Table 6 : Table containing the data defining the punctures for the 4d E × E theory.From this data it is easy to compute the central charges a and c of this theory , finding a E × E = , c E × E =
133 (3.67) For a small review see Appendix D.1 – 30 –s it should be for two copies of the E MN theory. By applying the procedure to writethe 3d mirror for this theory we find that the full puncture [ ] is associated to the quivertail (3.68) while the puncture [ , ] is associated to the quiver tail (3.69), (3.68) . (3.69)Gluing the three tails together results in the magnetic quiver for the 5d E × E depictedin (3.64). Therefore the magnetic quiver of the 5d E × E theory is the 3d mirror theoryof the 4d E × E theory above described. It is then tempting to conjecture that the5d E × E theory reduces to 4d to this D type class-S theory, giving two copies of E Minahan-Nemeschansky.Having derived the magnetic quiver for the E × E sequence from the brane web, forany N ∈ N , we can use the same argument as the paragraphs above to conjecture that allclass-S theories of D N + type given by a three punctured sphere with regular puncturesgiven by [ N + ] , [ N + , N + , , ] , [ N + , N + , , ] will be a factorized SCFT. Weconjecture that it will decompose into two copies of three punctured A N spheres, withregular punctures given by [ N + ] , [ N , ] , [ N , ] . It will be interesting to further checkthis proposal. E × E sequence The E × E sequence corresponds to the fixed point limit of the following IR quiverEQ , = [ s + c ] − SO(4) − USp(0) − SO(4) − ⋯ −
USp(0) − SO(4) − [ s + c ] N − . (3.70)It can be engineered using the orientifold web given by ⋯ N + N + N + N + N + N + − O5 − (3.71)This web can be converted to a magnetic quiver following [2], which results inMQ , = N + N + N + N + ⋯ N + . (3.72) For a small review see Appendix D.2 – 31 –ow we use the alternative description of the EQ , as a product of a pair of linear quiverswith SU(2) nodes, namelyEQ , = EQ = ⎛⎜⎜⎜⎜⎜⎜⎜⎜⎝ SU ( )∣ [ F ] − SU(2) − ⋯ − SU ( )∣ [ F ] N ⎞⎟⎟⎟⎟⎟⎟⎟⎟⎠ . (3.73)Each individual factor can be engineered using the following unitary 5brane web ⋰ N + N + N + N + N + N + . (3.74)The magnetic quiver one obtains from this unitary web leads us to the following conjectureMQ , = MQ = ⎛⎜⎜⎜⎜⎜⎝ N + N + N + N + ⋯ ⎞⎟⎟⎟⎟⎟⎠ . (3.75)The HWG for the unitary quiver appearing above was conjectured in [33]. We will use thisresult and square it to obtain the HWG for the OSp quiver MQ , (3.72):HWG , = PE [ N + ∑ i = µ i t i + t + ν t + νµ N + ( t N + + t N + ) + µ N + t N + − ν µ N + t N + ]× PE [ N + ∑ i = λ i t i + t + ρ t + ρλ N + ( t N + + t N + ) + λ N + t N + − ρ λ N + t N + ] . (3.76)We further point out an interesting fact about this theory, namely the existence of a 4d N = D type in which a single three-punctured sphere realizes a product SCFT, where bothfactors are the E Minahan-Nemeschansky theory [35]. We recall that the E Minahan-Nemeschansky theory is a 4d
N = E , and– 32 –entral charges a E = , c E = . (3.77)We report the partitions labeling the punctures in table 7, together with their contri-bution to the effective number of hypermutiplets and vector multiplets.Nahm partition ( δn h , δn v )[ ] ( , )[ ] ( , )[ , ] ( , ) Table 7 : Table containing the data defining the punctures for the 4d E × E theory.From this data it is easy to compute the central charges a and c of this theory, finding a E × E = , c E × E =
193 (3.78)as it should be for two copies of the E MN theory.By applying the procedure to write the 3d mirror for this theory we find that the fullpuncture [ ] is associated to the quiver tail (3.79), the puncture [ , ] is associated tothe quiver tail (3.80), and the puncture [ ] is associated to the quiver tail (3.81), (3.79) (3.80) . (3.81)Gluing the three tails together results in the magnetic quiver for the 5d E × E depictedin (3.72) for N =
1. Therefore the magnetic quiver of the 5d E × E theory is the 3d mirrortheory of the 4d E × E theory above described. It is then tempting to conjecture thatthe 5d E × E theory reduces to 4d to this D type class-S theory, giving two copies of E Minahan-Nemeschansky.Having derived the magnetic quiver for the E × E sequence from the brane web,for any N ∈ N , we can use the same argument as the paragraphs above to conjecturethat all class-S theories of D N + type given by a three punctured sphere with regularpunctures given by [ N + ] , [ N + , N + , ] , [ N + , N + ] will be a factorized SCFT.We conjecture that it will decompose into two copies of three punctured A N + spheres,with regular punctures given by [ N + ] , [ N + , N + ] , [ N, N, , ] . It will be interestingto further check this proposal. – 33 – .12 An outlier: the E × E theory While not explicitly written in [26], it is easy to use the methods of such paper to finda choice of punctures in the D theory, such that we realize the product of two copies ofthe E Minahan-Nemeschansky theory [35]. We recall that the E Minahan-Nemeschanskytheory is a 4d
N = E , and central charges a E = , c E = . (3.82)We report the partitions labeling the punctures which we believe engineer this productSCFT in table 8, together with their contribution to the effective number of hypermutipletsand vector multiplets. Nahm partition ( δn h , δn v )[ ] ( , )[ , ] ( , )[ , ] ( , ) Table 8 : Table containing the data defining the punctures for the 4d E × E theory.As a check that such 4d theory is really the product of two copies of the E Minahan-Nemeschansky theory, we compute the central charges from the data defining the punctures.We get a E × E = , c E × E =
313 (3.83)as it should be for two copies of the E MN theory. We also check that there exist no otherchoice of three punctures, in the D theory, that realizes these correct central charges.By applying the procedure to write the 3d mirror for this theory we find that the fullpuncture [ ] is associated to the quiver tail (3.84), the puncture [ , ] is associated tothe quiver tail (3.85), and the puncture [ , ] is associated to the quiver tail (3.86), (3.84) (3.85) . (3.86)Gluing the three tails together results in the quiver depicted in (3.87). Given thesimilarity of this case to the previous cases of E × E and E × E theory, discussedrespectively in sections 3.10 and 3.11, it is natural to pose the question whether there exista 5d E × E theory, whose magnetic quiver coincides with the one of (3.87), which we But surely noticed by the authors of such paper. See for example [27] and [28] for discussions aboutproduct SCFTs in class-S. – 34 –erived here from 3d mirror symmetry applied to the class-S construction of the 4d E × E theory, ⋯
10 10 12 8 8 ⋯ . (3.87) In this paper we studied 3d
N =
N = × SU(2), by rewriting the gauge theory in terms of the group SU(2) andthen taking the infinite coupling limit on both sides. The resulting theory in terms ofSU(2) gauge groups is generically comprised of two decoupled sectors, each of which wealso engineered using ordinary brane webs, without O5-planes. The ordinary brane webswere subsequently used to derive unitary magnetic quivers, which we then used to proposeas the components to which the OSp quivers factorise. We further used this correspondenceto extract highest weight generating functions for the Coulomb branch Hilbert series of theOSp quivers, relying on existing results for the unitary quivers. In some cases where theunitary quivers had not previously appeared in the literature, we also computed the highestweight generators. In order to test our proposal for the factorisation, and consequently theconjectured highest weight generators for the OSp quivers, we also computed the unrefinedCoulomb branch Hilbert series of the OSp quivers directly in a perturbative manner andfound an agreement with the proposed HWGs. We further illustrated the matching of theHiggs branch Hilbert series in two cases where we were able to perform the computationexactly on the OSp side. These too were in agreement with the results of the Higgs branchHilbert series of the unitary side.Although the higher dimensional intuition has led us to derive these results. It begsthe question, whether a truly three-dimensional logic can be used to argue for or providean explanation for the factorisation property of these OSp quivers. Moreover, all thequantitative checks performed in this paper probe the moduli space of the quivers studied.It would be interesting to ask whether the relationship between the OSp and unitary quiversin this paper are full-fledged dualities, or just a formal relation between their moduli spacesof vacua. One potential check that can be performed to illuminate this question would be tocompute other observables, such as the superconformal index, or the three-sphere partitionfunction for the theories in question. – 35 – cknowledgments
We thank Antoine Bourget, Julius Eckhard, Sakura Schafer-Nameki and Zhenghao Zhongfor stimulating questions, discussion and correspondence. SSK thanks APCTP, KIASand POSTECH for his visit where part of this work is done. The work of HH is sup-ported in part by JSPS KAKENHI Grant Number JP18K13543. FY is supported by theNSFC grant No. 11950410490, by Fundamental Research Funds for the Central Univer-sities A0920502051904-48, by Start-up research grant A1920502051907-2-046, in part byNSFC grant No. 11501470 and No. 11671328, and by Recruiting Foreign Experts Pro-gram No. T2018050 granted by SAFEA. F.C. is supported by STFC consolidated grantST/T000708/1. MA is supported by STFC grant ST/S505778/1.
A Higgs branch of E × E sequence In this appendix we give a derivation of the formula (3.7), for the Higgs branch Hilbertseries of MQ , (3.3). The Higgs branch of the quiver (3.3) can be computed using thegluing technique, following the discussion in appendix A of [36]. The first step is to breakdown the OSp quiver in (3.3) into pieces as follows ⋯ N − N − N − N N . (A.1)The Higgs Branch Hilbert series of the original quiver (3.3) is then obtained by takingthe product of the Hilbert series of the individual factors above, together with the gluingfactors associated with the U ( N − ) and USp ( N ) nodes which are to be gauged, allintegrated with the appropriate Haar measure for the aforementioned gauge groups ∫ dµ U ( N − ) ∫ dµ C N H T [ SU ( N − )] H ( N − ) glue H [ N − ]−[ C N ] H ( C N ) glue H [ C N ]−[ D ] . (A.2)The individual pieces in the above integrand are as follows H T [ SU ( N − )] = N − ∏ q = ( − t q ) PE [ χ SU ( N − )[ , , ⋯ , , ] t ] , H [ C N ]−[ D ] = PE [ χ C N [ , , ⋯ , ] t ] ,H [ N − ]−[ C N ] = PE [( χ SU ( N − )[ , , ⋯ , ] q + χ SU ( N − )[ , ⋯ , , ] q − ) χ C N [ , , ⋯ , ] t ] ,H ( N − ) glue = ( − t ) PE [ χ SU ( N − )[ , , ⋯ , , ] t ] , H ( C N ) glue = [ χ C N [ , , ⋯ , ] t ] . (A.3)Plugging these into the integral (A.2) we arrive at N − ∏ q = ( − t q ) ∫ dµ U ( N − ) ∫ dµ C N H [ N − ]−[ C N ] H ( C N ) glue H [ C N ]]−[ D ] . (A.4)Next we perform the integral over the U ( N − ) group, by counting the gauge invariantsof the free theory [ N − ] − [ C N ] , which is the only part of the integrand that sees this– 36 –ntegral (see appendix A of [36] for more details) ∫ dµ U ( N − ) H [ N − ]−[ C N ] = ( − t N ) PE [( + χ C N [ , , ⋯ , ] + χ C N [ , , ⋯ , ] ) t ] . (A.5)Substituting this back into (A.4) one obtains the desired formula (3.7) for the Higgs branchHilbert series of MQ , (3.3). B List of HWG of unitary quivers
In this appendix we tabulate the highest weight generating functions of all the unitaryquivers appearing in the previous sections.
Table 9 : List of the HWG for the unitary quivers appearing in the earlier sections. Thecorresponding fugacities are denoted by subscripts in the symmetry groups. Note that for N =
1, there is a possible enhancement in the symmetry which can be read from the setof balanced nodes in the quivers for N = ⋯ N ⋯ SU(2 N ) µ N ∑ k = µ k µ N − k t k ⋰ N ⋱
11 1
SU(2 N ) µ × U(1) q t + ( q + q − ) µ N t N + + N ∑ k = µ k µ N − k t k − µ N t N + ⋮ N − N − N − ⋮
11 1
SU(2 N ) µ × U(1) q t + ( µ N + q + µ N − q − ) t N + + N − ∑ k = µ k µ N − k t k − µ N + µ N − t N + N N ⋱⋰ SU(2 N + µ N ∑ k = µ k µ N + − k t k – 37 – N N ⋮⋮ SU(2 N + µ × SU(2) ν × U(1) q N ∑ k = µ k µ N + − k t k + ( ν + ) t + ν ( µ N q + µ N + q − ) t N + − ν µ N µ N + t N + N − N − N − N − ⋰ ⋱ SU(2 N + µ × SU(2) ν N − ∑ k = µ k µ N + − k t k + ν t ⋯ N + ⋯
12 11
SU(2 N + µ × SU(2) ν × SU(2) ν N + ∑ k = µ k µ N + − k t k + ( ν + ν ) t + t + ν ν µ N + ( t N + + t N + )− ν ν µ N + t N + ⋯ N NN N SO(4 N + µ N ∑ k = µ k t k ⋯ N + N + N +
11 1
SO(4 N + µ × U(1) q N ∑ k = µ k t k +( µ N + q + µ N + q − ) t N + + t N + N + N + ⋯ SO(4 N + µ × SU(2) ν N + ∑ k = µ k t k + t + νµ N + ( t N + + t N + ) + µ N + t N + + ν t − ν µ N + t N + C Unrefined Coulomb branch Hilbert series for low rank OSp quivers
In this appendix, we quote the results for the unrefined Hilbert series computed for theorthosymplectic quivers. These results match with the perturbative results obtained fromthe HWG listed in the table 1. – 38 – able 10 : The computation of Coulomb branch Hilbert series of orthosymplectic quiversgiven in main sections for small values of N . The full Hilbert series is given by summingover the integer and half-integer sublattices of the magnetic weights.Coulomb branch Hilbert seriesQuiver ⃗ m ∈ Z ⃗ m ∈ Z + / , ∣ N = + t + t + t + t + t +O( t ) not required 1 + t + t + t + t + t + O( t ) MQ , ∣ N = + t + t + t + t + t + O( t ) not required 1 + t + t + t + t + t + O( t ) MQ ( I ) , ∣ N = + t + t + t + t + t +O( t ) t + t + t + t + O( t ) + t + t + t + t + t + t + t + t + t + O( t ) MQ ( I ) , ∣ N = + t + t + t + t + t + O( t ) t + t + t + t + O( t ) + t + t + t + t + t + O( t ) MQ ( II ) , ∣ N = + t + t + t + t + t +O( t ) t + t + t + t + O( t ) + t + t + t + t + t + t + t + t + t + O( t ) MQ ( II ) , ∣ N = + t + t + t + t + t + O( t ) t + t + t + t + O( t ) + t + t + t + t + t + O( t ) MQ ( I ) , ∣ N = + t + t + t + t + t + O( t ) t + t + t + t + t +O( t ) + t + t + t + t + t + O( t ) MQ ( I ) , ∣ N = + t + t + t + t + t +O( t ) t + t + t + t + O( t ) + t + t + t + t + t + t + t + t + t + O( t ) MQ ( II ) , ∣ N = + t + t + t + t + t + O( t ) t + t + t + t + t + O( t ) + t + t + t + t + t + O( t ) MQ ( II ) , ∣ N = + t + t + t + t + t +O( t ) t + t + t + t + O( t ) + t + t + t + t + t + t + t + t + t + O( t ) MQ ( III ) , ∣ N = + t + t + t + t + t + O( t ) t + t + t + t + t + O( t ) + t + t + t + t + t + O( t ) MQ ( III ) , ∣ N = + t + t + t + t + t +O( t ) t + t + t + t + O( t ) + t + t + t + t + t + t + t + t + t + O( t ) MQ ( I ) ′ , ′ ∣ N = + t + t + t + t + t + O( t ) not required 1 + t + t + t + t + t + O( t ) MQ ( I ) ′ , ′ ∣ N = + t + t + t + t + t + O( t ) not required 1 + t + t + t + t + t + O( t ) MQ ( II ) ′ , ′ ∣ N = + t + t + t + t + t + O( t ) not required 1 + t + t + t + t + t + O( t ) – 39 –Q ( II ) ′ , ′ ∣ N = + t + t + t + t + t + O( t ) not required 1 + t + t + t + t + t + O( t ) MQ ( III ) ′ , ′ ∣ N = + t + t + t + t + t + O( t ) not required 1 + t + t + t + t + t + O( t ) MQ ( III ) ′ , ′ ∣ N = + t + t + t + t + t + O( t ) not required 1 + t + t + t + t + t + O( t ) MQ ( I ) ′ , ∣ N = + t + t + t + t + t + O( t ) t + t + t + t + t +O( t ) + t + t + t + t + t + O( t ) MQ ( I ) ′ , ∣ N = + t + t + t + t + t + O( t ) t + t + t + t + O( t ) + t + t + t + t + t + t + t + t + t + O( t ) MQ ( II ) ′ , ∣ N = + t + t + t + t + t + O( t ) t + t + t + t + t +O( t ) + t + t + t + t + t + O( t ) MQ ( II ) ′ , ∣ N = + t + t + t + t + t + O( t ) t + t + t + t + O( t ) + t + t + t + t + t + t + t + t + t + O( t ) MQ , ∣ N = + t + t + t + t + t +O( t ) t + t + t + t + t +O( t ) + t + t + t + t + t + O( t ) MQ , ∣ N = + t + t + t + t + t + O( t ) t + t + t + t + O( t ) + t + t + t + t + t + t + t + t + t + O( t ) MQ , ∣ N = + t + t + t + t + t + O( t ) t + t + t + t + t + O( t ) + t + t + t + t + t + O( t ) MQ , ∣ N = + t + t + O( t ) t + O( t ) + t + t + t + O( t ) MQ ′ , ′ ∣ N = + t + t + t + t + t + O( t ) not required 1 + t + t + t + t + t + O( t ) MQ ′ , ′ ∣ N = + t + t + t + O( t ) not required 1 + t + t + t +O( t ) MQ ′ , ∣ N = + t + t + t + t + t + O( t ) t + t + t + t + t + O( t ) + t + t + t + t + t + O( t ) MQ ′ , ∣ N = + t + t + t + O( t ) t + t + O( t ) + t + t + t + t + t + O( t ) MQ , ∣ N = + t + t + t + t + t + O( t ) t + t + t + t + t + O( t ) + t + t + t + t + t +O( t ) MQ , ∣ N = + t + t +O( t ) t + t + O( t ) + t + t + O( t ) – 40 – Central charges and 3d mirrors for D-type class-S theories
In this appendix we briefly review two properties of class-S theories of D N type. We restrictourself to the case in which the 6d ( , ) D N theory is compactified on the sphere, withregular (untwisted) punctures only. We discuss three instances of theories of this type inthe main text, in sections 3.10, 3.11 and 3.12. There, we consider examples in which asingle three-punctured sphere describes the 4-dimensional version of the E n × E n theory( n = , , ) , i.e. two copies of E n Minahan-Nemeschansky.The first property that we review in this appendix is the rule for the computationof superconformal central charges a and c , giving the data labeling the punctures [26].The second property is the prescription for finding the corresponding 3d N =
D.1 Central charges
The central charges a and c of a 4d N = T µµ = c π ( Weyl ) − a π ( Euler ) . (D.1)For Lagrangian theories, the central charges are related to the number of vector multiplesand hypermultiplets by a = n v + n h , c = n v + n h . (D.2)For non-Lagrangian theories, formula (D.2) still holds, but now n h and n v are interpretedas an effective number of hypermultiplets and vectormultiples. For the subset of theoriesof our current interest, it holds that n v = − N ( N − N + ) + ∑ α δn ( α ) v , n h = − N ( N − )( N − ) + ∑ α δn ( α ) h , (D.3)where g is the genus of the Riemann surface and α runs over the set of punctures. δn ( α ) v and δn ( α ) h are local contributions coming from the α -th puncture. Both formulae (D.3) andan algorithmic rule for the computation of δn h and δn v are derived in [26]. In the samepaper, the explicit values of δn h and δn v is listed for all the punctures up to N =
6. Wedefer the reader to such paper for further details.
D.2 3d Mirrors
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