Marcus Sperling等2023年7月在SciPost Phys.期刊发表论文.pdf 
Marcus Sperling等2023年7月在SciPost Phys.期刊发表论文.pdf
SciPost Phys. 15, 033 (2023) 3d N = 4 mirror symmetry with 1-form symmetry Satoshi Nawata1⋆ , Marcus Sperling2† , Hao Ellery Wang3‡ and Zhenghao Zhong4,5◦ 1 Department of Physics and Center for Field Theory and Particle Physics, Fudan University, 220, Handan Road, 200433 Shanghai, China 2 Shing-Tung Yau Center, Southeast University, Xuanwu District, Nanjing, Jiangsu, 210096, China 3 Yau Mathematical Sciences Center, Tsinghua University, Haidian District, Beijing, 100084, China 4 Theoretical Physics Group, The Blackett Laboratory, Imperial College London, Prince Consort Road, London, SW7 2AZ, UK 5 Mathematical Institute, University of Oxford, Andrew Wiles Building, Woodstock Road, Oxford, OX2 6GG, UK ⋆ snawata@gmail.com , † msperling@seu.edu.cn , ‡ yukawahaow@gmail.com , ◦ zhenghao.zhong@maths.ox.ac.uk Abstract The study of 3d mirror symmetry has greatly enhanced our understanding of various aspects of 3d N = 4 theories. In this paper, starting with known mirror pairs of 3d N = 4 quiver gauge theories and gauging discrete subgroups of the flavour or topological symmetry, we construct new mirror pairs with non-trivial 1-form symmetry. By providing explicit quiver descriptions of these theories, we thoroughly specify their symmetries (0form, 1-form, and 2-group) and the mirror maps between them. Copyright S. Nawata et al. This work is licensed under the Creative Commons Attribution 4.0 International License. Published by the SciPost Foundation. Received 19-01-2023 Accepted 30-05-2023 Check for updates Published 27-07-2023 doi:10.21468/SciPostPhys.15.1.033 Contents 1 Introduction 3 2 Gauging discrete 0-form symmetries 2.1 Abelian theories 2.1.1 SQED with higher charge 2.1.2 SQED with discrete gauge factor 2.2 An illustrative example 2.2.1 Gauge a subgroup with q|k 2.2.2 Gauge a discrete group Zq with k|q 2.3 T [SU(N )] theories 2.4 Tρσ [SU(N )] theories 2.5 T [SO(2N )] theories 2.6 Sp(k) SQCD and its orthosymplectic mirror 2.7 Sp(k) SQCD and its unitary D-type mirror quiver 4 4 4 6 9 10 10 11 16 21 23 24 1 SciPost Phys. 15, 033 (2023) 2.8 Examples of non-simply laced unitary quivers and their mirrors 2.8.1 C-type quivers 2.8.2 B-type quivers 2.8.3 A comment on F4 Coulomb branch quivers 2.9 Magnetic quivers and gauging discrete topological symmetries 2.10 Examples from 5d magnetic quivers 26 28 29 29 31 31 3 Discussion and conclusions 33 A Notations and conventions A.1 Hilbert series A.1.1 Monopole formula A.1.2 Higgs branch Hilbert series A.1.3 Gauging a discrete 0-form symmetry. A.2 Superconformal index A.3 Centre symmetries of classical Lie algebras 35 36 36 36 37 37 38 B Discrete gauging of T [SU(N )] B.1 Gauging discrete subgroup of topological symmetry B.2 Gauging discrete subgroup of topological symmetry revisited 38 39 42 C Mirror maps C.1 SQED and its abelian mirror quiver C.1.1 Standard mirror map C.1.2 Mirror map after gauging C.2 T [SU(N )] theories C.2.1 Standard mirror map C.2.2 Mirror map after gauging C.3 Examples for Tρσ [SU(N )] C.3.1 Example 1 C.3.2 Example 2 C.4 Sp(k) SQCD and its D-type unitary mirror quiver C.4.1 Standard mirror map C.4.2 Mirror map after gauging C.5 O(2k) SQCD and its C-type unitary mirror quiver 45 45 45 46 47 47 48 48 48 49 49 50 50 51 D Explicit Hilbert series results D.1 Linear Abelian quiver D.2 T [SU(N )] theories D.2.1 T [SU(3)] theories D.2.2 T [SU(4)] theories D.2.3 T [SU(5)] theories D.2.4 T [SU(6)] theories D.3 Some T [SU(N )] examples with higher charges D.4 Some Tρσ [SU(N )] examples D.5 T [SO(2N )] theories D.5.1 T [SO(6)] theories D.5.2 T [SO(8)] theories D.6 Sp(k) SQCD and orthosymplectic mirrors D.6.1 Sp(2) SQCD, 5 flavours D.6.2 Sp(2) SQCD, 6 flavours 51 51 53 53 53 54 56 59 60 62 62 62 63 63 64 2 SciPost Phys. 15, 033 (2023) D.7 Sp(k) SQCD and D-type mirrors D.7.1 D7 Dynkin quiver D.7.2 D8 Dynkin quiver D.8 O(2k) SQCD and C-type mirrors 65 65 68 71 References 1 73 Introduction Supersymmetric theories with 8 supercharges in space-time dimension 3 exhibit a rich set of intriguing features; One of the most prominent is 3d mirror symmetry [1]. Given a 3d theory that has a mirror dual theory, 3d mirror symmetry exchanges Coulomb branch and Higgs branch. In particular, this also implies the exchange of flavour symmetries G f (Higgs branch isometries) and the topological symmetries G t (Coulomb branch isometries). The notion of symmetries has been generalised to include novel types beyond the standard symmetries of local operators [2]. Among others, these include higher-form symmetries. Specifically for 3d theories, discrete 1-form symmetries can be generated by gauging discrete 0-form symmetries. The structure of generalised symmetries in 3d supersymmetric theories has been the focus of recent research, including [3–14] and references therein.1 Given the vast catalogue of 3d mirror pairs with trivial 1-form symmetry, one might wonder what mirror symmetry implies for 3d theories with 1-form symmetry. In this paper, we start with a known mirror pair (T , T ∨ ) of 3d N = 4 theories that admit UV quiver descriptions, and gauge a discrete Γ [0] subgroup of the 0-form symmetry to generate new theories with Γ [1] 1-form symmetry. Depending on whether Γ ≡ Γ [0] is a subgroup of the flavour or topological symmetry, the resulting mirror pair (T /Γ , (T /Γ )∨ ) changes. For Γ ≡ Γ f ⊂ G f , the field theory description of T /Γ f is straightforward, but for Γ ≡ Γ t ⊂ G t∨ , the description of T ∨ /Γ t is less transparent. In this paper, we consider Γ = Zq suitably embedded into a Cartan U(1) which enables us to derive explicit quiver descriptions for these cases and allows to specify the global form of the 0-form symmetries of (T /Γ , (T /Γ )∨ ). It is known that the resulting 0-form and the newly introduced discrete 1-form symmetry may not just be a direct product, but can form an extension, called 2-group symmetry [9, 17–21]. We comment on such extensions throughout this work. The remainder of this paper is organised as follows: in Section 2, we consider known mirror pairs and gauge discrete 0-form symmetries to generate mirror pairs with non-trivial 1-form symmetry. We first study abelian theories, followedQby non-abelian T [SU(N )] and Tρσ [SU(N )] theories with non-abelian product gauge groups i U(Ni ). This class of examples has the benefit that all 0-form symmetries are manifest in the UV description. Thereafter, SO(k) and Sp(k) gauge groups are considered by studying T [SO(2N )] theories, Sp(k) SQCD, and linear orthosymplectic quivers. While the flavour 0-form symmetry is manifest in this set of examples, the topological symmetry is at most accessible by discrete Z2 subgroups, which turns out to be sufficient for the intents and purposes here. Lastly, we consider mixed types: i.e. D and C-type Dynkin quivers composed of unitary gauge groups and their mirror Sp(k) and O(2k) SQCD theories, respectively. The advantage of this class of mirror pairs is that the flavour symmetry of the SQCD theories and the topological symmetry of the unitary Dynkin quivers are fully manifest. Before closing, some magnetic quiver examples are considered. 1 See also [15, 16] for recent review articles. 3 SciPost Phys. 15, 033 (2023) Conclusions are provided in Section 3. Several appendices complement the main text and provide computational details. Note added. During the course of this project, we were informed of a related work done by Bhardwaj, Bullimore, Ferrari, and Schäfer-Nameki [22]. We are grateful to them for coordinating the submission of our papers. 2 Gauging discrete 0-form symmetries In this section, mirror theories with non-trivial 1-form symmetry are constructed. Gauging discrete subgroups of the 0-form symmetry, which results in 1-from symmetries and a potential 2-group structure, has, for example, been considered in [9, 18–21]. The principle is simple: start from a known mirror pair (T , T ∨ ) and gauge discrete 0-form symmetries Γ (finite, cyclic) such that Γ ≡ Γ t ⊂ G t (T ) and Γ ≡ Γ f ⊂ G f (T ∨ ). This ensures that the resulting theories (T /Γ t , T ∨ /Γ f ) are mirror pairs with 1-form symmetry Γ . The aims here are (i) to provide explicit quiver descriptions for (T /Γ t , T ∨ /Γ f ) and (ii) to detail the resulting symmetries (0-form, 1-form, and 2-group). 2.1 Abelian theories As a first example, consider 3d N = 4 SQED with N hypermultiplets of charge 1 and its abelian mirror quiver theory [1], see Figure 1. The global 0-form symmetries are well-known: for SQED one finds U(1) t ×PSU(N ) f , while the abelian mirror quiver enjoys a U(1) f ×PSU(N ) t symmetry. 2.1.1 SQED with higher charge Suppose that one gauges a discrete Zq subgroup of the abelian U(1) factor of the global 0form symmetry. The resulting theories are straightforwardly derived. Gauging a Zq ⊂ U(1) t for SQED with N charge 1 hypermultiplets leads to SQED with N charge q hypermultiplets, see also [12]. Similarly, gauging a Zq ⊂ U(1) f of the abelian mirror quiver leads to an abelian quiver with a Zq 1-form symmetry. The two theories obtained are then mirror to each other, see Figure 1. The quiver notation is summarised in Table 3 of Appendix A. Consistency checks. The proposed mirror symmetry can be verified by Hilbert series techniques [23–26]. The Higgs branch Hilbert series is insensitive to the gauging of the Zq inside the topological symmetry of the SQED theory; similarly, the Coulomb branch of the mirror does not perceive changes upon gauging a discrete subgroup of the flavour symmetry. See Appendix A.1 for conventions. Performing the discrete gauging for the SQED theory reduces to a Zq Molien-Weyl sum over the Coulomb branch Hilbert series N 1 X m 1 N |m| HSC q=1 (w|t) = w t2 = PE[t + (w + w−1 )t 2 − t N ] , (2.1) SQEDN 1 − t m∈Z 1 HSCSQEDq (z|t) = q N q−1 X p=0 HSC q′ =1 SQEDN (w|t) 2πi 1 w=z q (ζq ) p ζq = e q ∈ Zq , 1 X m 1 N |q·m| z t2 = 1 − t m∈Z 1 = PE[t + (z + z −1 )t 2 qN − t qN ] . 4 (2.2) SciPost Phys. 15, 033 (2023) 1 N 1 mirror ... 1 1 0-form: U(1)t × PSU(N )f 1 1 1 0-form: PSU(N )t × U(1)f gauge Zq ⊂ U(1)t gauge Zq ⊂ U(1)f N Zq mirror q ... 1 1 0-form: U(1)t × PSU(N )f 1-form: Zq 1 1 1 0-form: PSU(N )t × U(1)f 1-form: Zq Figure 1: Gauging of discrete 0-form symmetries in SQED and its mirror. For SQED with charge 1 hypermultiplets, a Zqt gauging results in SQED with charge q hypermultiplets. These are indicated by an arrow with the label q. For the abelian mirror f quiver, the Zq gauging is realised by acting on the flavours. The fundamental flavours that are charged under the discrete Zq are connected to a grey node. See Appendix A for conventions. Likewise, one performs the Zq Molien-Weyl sum on the Higgs branch Hilbert series of the mirror theory HSH mirror ( y|t) = N −1 I Y N −2 X x 1 x b+1 dx a b PE + t 2 − (N −1)t 2πi x x x a b+1 b a=1 b=1 1 1 1 x N −1 x1 y2 y− 2 · PE + 1 + + t2 1 x1 x N −1 y2 y− 2 N = PE[t + ( y + y −1 )t 2 − t N ] , 1 HSH (z|t) = mirror/Zq q q−1 X p=0 HSH mirror ( y|t) (2.3) 1 1 = PE[t + (z + z −1 )t 2 qN − t qN ] . (2.4) y=z q (ζq ) p In summary, both results confirm the expectation and provide the explicit parameter map. As a remark, the superconformal index is equally well suited to probe such dualities; see for instance [12] for SQED with charge q = 2 hypermultiplets. Since either Higgs or Coulomb t/ f branch operators are unaffected by gauging a Zq , the Hilbert series is a more convenient tool. Symmetries. Using the techniques of [6], one can inspect the interplay between the discrete 1-form symmetry Zq ⊂ U(1) t and the global 0-form symmetry PSU(N ) f for the SQED theory. The centre symmetry Z F = ZN of su(N ) f is generated by α F = ζN , while the U(1) gauge group supports a ZG = ZN ·q centre generated by αG = ζN ·q . The diagonal α D = (αG , α F ) generates a E = ZN ·q ⊂ ZG × Z F . The 1-form symmetry Γ [1] = Zq is generated by αND = (αNG , αNF ) = (αNG , 1), 5 SciPost Phys. 15, 033 (2023) which acts trivially on the matter content. The short exact sequence 0 → Γ [1] = Zq → E = Zq·N → Z F = ZN → 0 (2.5) splits whenever gcd(q, N ) = 1, i.e. q and N are co-prime. In other words, gcd(q, N ) > 1 is a necessary condition for the existence of an extension to a 2-group structure. A sufficient condition is to have the non-trivial Postnikov class in H 3 (BPSU(N ); Zq ) [21], which is the image of the obstruction class for lifting a PSU(N )-bundle to an SU(N )-bundle, under the Bockstein map H 2 (BPSU(N ); ZN ) → H 3 (BPSU(N ); Zq ). In fact, the Postnikov class is nontrivial if and only if gcd(q, N ) > 1.2 Therefore, the short exact sequence (2.5) represents a non-trivial 2-group extension if and only if gcd(q, N ) > 1. See also [9, 12] for a recent discussion of SQED with 2 flavours of charge 2. Comments on lines. As explained in [9, 27, 28], 1-form symmetries and 2-group structures can be understood via equivalence classes of line defects.3 Here, we illustrate how the higherform symmetry is also realised on the line defects. Consider SQED with N hypermultiplets of charge q. A Wilson line of charge h with h ∈ {1, 2, . . . , q − 1} cannot end on a local operator because local operators are either constructed as polynomials in the fundamental hypermultiplets of charge q or are monopole operators, which are gauge singlets for 3d N = 4 theories. Thus, the 1-form symmetry Γ [1] (or its Pontryagin dual) is generated by the (q − 1) Wilson lines that cannot end. Refining with respect to the flavour symmetry shows that a Wilson line of charge q is equivalent to a flavour Wilson line transforming as [1, 0, . . . , 0]AN −1 . This however is not an allowed representation of G f = PSU(N ), and signals the existence of a 2-group structure. In fact, the N -th power of such a Wilson line is well-defined under G f , because the N -th tensor product of [1, 0, . . . , 0]AN −1 contains a singlet. Such lines generate the group Eb = Zq·N . Turning to the abelian mirror quiver, one can straightforwardly see that the fundamental Wilson lines, i.e. those of unit charge under a single U(1) gauge group factor, can end on a local operator constructed out of the hypermultiplets. Therefore, one needs to turn to the vortex lines to understand the 1-form symmetry. It is known [29] that the junctions between vortex lines are significantly more challenging than those between Wilson lines. It would be interesting to systematically address this in explicit examples. 2.1.2 SQED with discrete gauge factor Next, revert the logic: gauge a Zq subgroup of the PSU(N ) f symmetry of SQED. Conversely, on the mirror side, one gauges a Zq subgroup of the PSU(N ) t topological symmetry of the abelian quiver theory. For the abelian quiver theory, discrete gauging along a Cartan U(1) t of the topological symmetry alters the linear quiver theory by modifying the charges of the bifundamental hypermultiplets attached to a single gauge node. This follows from analogous arguments as for SQED with charge q hypermultiplets or the arguments used in Appendices B.1 – B.2. For the 2 We would like to thank A. Milivojević for providing the proof at mathoverflow. In brief, lines L1,2 are equivalent if there exists a local operator O at the junction between them. The set of equivalence classes {L}/ ∼ forms the Pontryagin dual b Γ [1] of the 1-form symmetry Γ [1] . Refining the equivalence relation by keeping track of 0-form symmetry representations R leads to the following equivalence relation: (L1 , R1 ) ∼ (L2 , R2 ) iff there exists a local operator transforming as R1 ⊗ R∗2 (or R∗1 ⊗ R2 ) at the junction of the lines. The equivalence classes give rise to Eb (Pontryagin dual of E ), which encodes the interplay between the centres of b → Eb → b gauge symmetry and 0-form symmetry. These groups fit into the short exact sequence 0 → Z Γ [1] → 0, which is the Pontryagin dual of the sequences discussed in the text, e.g. (2.5), (2.13), (2.15). Whenever these short exact sequences split, the 2-group is necessarily trivial. For non-split sequences, the Postnikov class controls whether the 2-group is trivial or not. 3 6 SciPost Phys. 15, 033 (2023) 1 N 1 mirror ... 1 1 1 0-form: U(1)t × PSU(N )f ... 1 1 gauge Zq ⊂ PSU(N )t 1 1 mirror k Zq 1 0-form: PSU(N )t × U(1)f gauge Zq ⊂ PSU(N )f N −k (1, −1) (1, −1) ... 1 1 −k) 0-form: U(1)t × ( SU(k)×U(1)×SU(N ) Zk ×ZN −k f (1, −q) (q, −1) 1 1 1 ... 1 −k) 0-form: ( SU(k)×U(1)×SU(N ) × U(1)f Zk ×ZN −k t 1-form: Zq 1-form: Zq Figure 2: Gauging of discrete 0-form symmetries in SQED and its mirror. The centre symmetries act with charges (−q mod k, q mod (N −k)) on the U(1) factor. For SQED, the Zq acts on k of the N hypermultiplets, which is indicated by k edges connected to a grey node. The remaining hypermultiplets are uncharged under the discrete group. One can use a global U(1) rotation to move the Zq -action onto the other set of hypermultiplets as well, which renders the entire setup symmetric, cf. Appendix C.1.2. For the abelian linear quiver, gauging along the Cartan U(1) t at the k-th gauge nodes leads to hypermultiplets with charge q under the k-th U(1), while still of unit charge under the adjacent gauge factors. This is indicated by an arrow with label, cf. Appendix A. SQED theory, gauging of a discrete flavour 0-form symmetry affects some of the fundamental flavours. To see this, one uses the original mirror map (C.7) between the parameters to identify which flavour fugacities are affected by gauging along a Cartan U(1) t factor in the abelian mirror. As a result, the flavours of the SQED split into two sets: one charged under Zq and the other is trivial. This is shown in Figure 2. Global symmetry: abelian mirror point of view. The global symmetry is affected as follows: suppose that one gauges a Zq ⊂ U(1)k ⊂ PSU(N ) t subgroup of the topological Cartan U(1) at the k-th node of the abelian quiver w1 1 ... 1 wk−1 1 wk (1, −q) 1 wk+1 (q, −1) ... wN −1 1 1 (2.6) 1 The 0-form symmetry algebra after gauging is su(k) ⊕ u(1) ⊕ su(N −k). As exemplified in Appendix D.1, the 0-form symmetry group is G t (2.6) = SU(k) × U(1)Q × SU(N −k) Zk × ZN −k , (2.7) where the centre symmetry Zℓ ⊂ SU(ℓ) acts on the fundamental representation [1, 0, . . . , 0]Aℓ−1 with charge +1 under Zℓ , for ℓ ∈ {k, N −k}, see also Table 4. Moreover, the Zk × ZN −k act with charge (−q mod k, q mod (N −k)) on the U(1)Q variable. Roughly Q ∼ w k , see (C.8) for details. 7 SciPost Phys. 15, 033 (2023) The global structure (2.7) can also be inferred directly from the set of balanced nodes in (2.6). The unbalanced gauge node U(1)k is connected to two balanced sets of gauge nodes, forming the Ak−1 and AN −k−1 Dynkin diagram. Generalising the arguments of [30], there are monopole operators transforming as [0, 0, . . . , q]Ak−1 × (1)Q (and its conjugate) and [q, 0, . . . , 0] × (1)Q (and conjugate). This follows because the U(1)k node is attached to the k − 1-th node of the Ak−1 Dynkin diagram and the 1-st node of AN −k−1 Dynkin diagram. Compared to the standard case of unit charge bifundamental hypermultiplets, the increased charge q modifies the appearing Aℓ representations accordingly. The existence of these monopole operators in the Coulomb branch chiral ring leads to the isometry (2.7). Global symmetry: SQED point of view. consider the SQED side: To illuminate this result, it is instructive to also N −k X̃d k Zq X a (2.8) 1 where the two distinct sets of fundamentals are denoted as X and X̃ (by convention, both have charge −1 under the U(1) gauge group). Here, as a different notation from elsewhere, we use the arrowed lines to symbolize 3d N = 2 chiral multiplets (inflow into the gauge node) and f anti-chiral multiplets (outflow from the gauge node). Computationally, gauging a discrete Zq is realised via the following flavour fugacities, see Appendix C.1 a = 1, x1 , xa Xa : a = 1, . . . , k : ya = ζq · Q 1 · x a−1 , 1 < a < k , (2.9a) 1 , a = k, x k−1 d = k + 1, u1 , ud−k X̃ d : d = k + 1, . . . , N : yd = Q 2 · ud−k−1 , k + 1 < d < N −k , (2.9b) 1 , d = N −k , uN −k−1 f with ζq ∈ Zq . The x a and ud are weight space fugacities for su(k) and su(N −k), respectively. The first observation is that if k|q then the Zk centre symmetry of SU(k) is gauged, such that a global PSU(k) f factor arises. Similarly, if (N −k)|q the ZN −k centre of SU(N −k) is gauged, leading to a PSU(N −k) f factor. For the general case, one fixes the two so far arbitrary U(1)Q 1,2 symmetries:4 ( ( N −k ! Q k1 · Q N2 −k = 1 , Q 1 = Q q·N , q ⇒ (2.9c) −k ! Q1 q·n , = Q , Q = Q 2 Q2 which agrees with (C.11) of Appendix C.1. Next, consider a gauge invariant operator O built from the fields {X a }ka=1 transforming as (ζq · Q 1 · [1, 0, . . . , 0]Ak−1 , −1) under flavour-gauge † transformations and fields {X̃ d† }Nd=k+1 transforming as (Q−1 2 ·[0, . . . , 0, 1]AN −k−1 , +1). Thus, X a X̃ d is U(1) gauge invariant. For Zq invariance, one also requires q-copies of X a in the form of Symq (X a ), which leads to the q-th symmetric representation Symq [1, 0, . . . , 0] of SU(k) f . As a consequence, one also requires q copies of {X̃ d† } in the form Symq (X̃ d ), which leads to the q-th 4 The definition of Q is a choice. Here, it is chosen such that the operator O in (2.10), as Higgs branch operator with lowest R-charge that is charged under the U(1), has the unit charge. 8 SciPost Phys. 15, 033 (2023) conjugate symmetric representation Symq [0, . . . , 0, 1] of SU(N −k) f . Such a gauge invariant operator has charges q O = Symqa1 ,...,aq (X a1 ) · Symd ,...,d (X̃ di ) ↔ Symq [1, 0, . . . , 0]SU(k) ⊗ Symq [0, . . . , 0, 1]SU(N −k) ⊗ 1 q Q1 q . Q2 | {z } =Q (2.10) The operator O has Zk ×ZN −k centre charges (q mod k, −q mod (N −k)). Hence, the Zk ×ZN −k transformations can be compensated by a global U(1)Q rotation if Q has charges (−q mod k, q mod (N −k)) under the centre symmetries. This confirms (2.7) as flavour symmetry G f . The operator O can be detected in the Hilbert series at R-charge q · 2 · 21 = q. Comments on lines. Returning to the quiver (2.6), consider a Wilson lines Wa of charge 1 under the a-th U(1) gauge group factor. For each a ̸= k, Wa can end on a local operator composed of concatenated bifundamental hypermultiplets. For a = k, Wk cannot end since the bifundamentals connected to the k-th gauge node are of charge q. Further, monopole operators cannot screen gauge Wilson lines, because monopole operators are gauge singlets for 3d N = 4 theories. Thus, the lines (Wk )h with h ∈ {0, 1, . . . , q −1} cannot end and generate the abelian group b Γ [1] = Zq . 2.2 An illustrative example One of the main messages of this paper is that gauging discrete Zq subgroups of the topological symmetry for quiver gauge theories T with unitary gauge nodes can result in theories T /Zqt which admit a simple quiver description. To illustrate this fact, consider U(k) SQCD with N ≥ 2k fundamental flavours N (2.11) k with the well-known 0-form symmetries: G f = PSU(N ), G t = U(1) t for N > 2k and G t = SO(3) for N = 2k. U(1)×SU(k) Next, express the gauge group as U(k) ∼ where Zk acts as centre on SU(k) = Zk and via Zk charge (k−1) on the U(1) factor. Rewriting U(k) magnetic fluxes m ∈ Zk into Sk−1 U(1) × SU(k) fluxes (h, l) requires the co-character lattice to be Γ = i=0 (Z + ki )k . Effectively, the SQCD theory can be understood as SU(k) × U(1) gauge theory with N copies of bifundamentals and an “unusual” magnetic lattice Γ . One can introduce a (topological) fugacity z that keeps track of the components of Γ . If w denotes the topological fugacity of T , one employs 1 w → zw k . Next, gauge a discrete Zq subgroup of the topological symmetry by performing a discrete Molien-Weyl sum over z. It is convenient to choose either q|k or k|q. One can show rigorously (e.g. using the superconformal index or the Coulomb branch Hilbert series, see Appendix B) the following: 9 SciPost Phys. 15, 033 (2023) 2.2.1 Gauge a subgroup with q|k If q|k then only the subgroup Zq ⊂ Zk is gauged. The theory becomes 1 k q −1 N /Z k , q with magnetic lattice [ q k Z+i· , k i=0 (2.12) SU(k) where the quotient Z k signals that this discrete group is not gauged, in the sense of [31]. q The resulting theory has a U(1) t × PSU(N ) f 0-form symmetry and a Zq 1-form symmetry. The potential interplay can be analysed via the action of the centre symmetries: defining k αG = ((ζk ) q , ζq·N ) ∈ Zk × Zq·N (because only a Zq ⊂ Zk is gauged) and α F = ζN ∈ ZN , the diagonal combination α D = (αG , α F ) generates a E = Zq·N group. The element k N N · α D = ((ζk ) q ·N , ζq·N , 1) generates a Γ (1) = Zq subgroup that acts trivial on the matter fields. By definition, this establishes the 1-form symmetry. The short exact sequence 0 → Γ (1) = Zq → E = ZN ·q → ZN → 0 (2.13) splits if gcd(q, N ) = 1. Whenever gcd(q, N ) > 1, this short exact sequence exhibits a non-trivial 2-group extension of Γ (1) and PSU(N ) f as discussed below (2.5). Symmetries via lines. One can again illustrate this higher-form symmetry by using line defects and their equivalence classes. A gauge Wilson line W in the representation [0, . . . , 0]Ak−1 × (−1) cannot end on any local operator; Neither polynomials of the hypermultiplets nor monopole operators, because of a mismatch in gauge charges. However, W q can end on the determinant operator O ∼ det(X ), obtained by contracting hypermultiplets X with the invariant ε tensor of SU(k). This operator has charges [0, . . . , 0]Ak−1 × k. Since q|k, O has the same centre charges as W q , such that W q can end on it. Therefore, the lines W a with a ∈ {1, 2, . . . , q−1} cannot end on any local operator and generate the abelian group b Γ [1] = Zq . Taking flavour charges into account, W q is equivalent to a flavour Wilson line transforming as ∧k [0, . . . , 0, 1]AN −1 , which follows from the flavour charges of O. This is not a representation of PSU(N ) f , but taking N -th tensor (W q )⊗N is equivalent to a singlet of the flavour symmetry. Thus, these lines generate the group Eb = ZN ·q and the 1-form symmetry potentially forms a 2-group with the flavour symmetry (depending on the gcd(N , q)). 2.2.2 Gauge a discrete group Zq with k|q If k|q then the SU(k) centre Zk is a subgroup of Zq and fully gauged. The theory becomes 1 q k N with magnetic lattice Zk , (2.14) SU(k) The difference is now that the N hypermultiplets transform as SU(k) fundamental with charge q k ∈ N under the U(1). This is indicated by the arrow, cf. Table 3. In terms of symmetries, the theory T /Zqt has a U(1) t topological symmetry, PSU(N ) f flavour symmetry, a Zq 1-form symmetry. Moreover, inspecting the gauge-flavour centre symmetries shows: αG = (ζk , ζN ·q ) ∈ Zk × ZN ·q and α F = ζN ∈ ZN . The diagonal generator 10 SciPost Phys. 15, 033 (2023) αd = (αG , α F ) spans a E = ZN ·q , and the element N · αd = (ζNk , ζNN ·q , 1) generates a Γ (1) = Zk·q subgroup, using that k|q. This subgroup acts trivial on the matter fields; thus, defining the 1-form symmetry Γ (1) . The short exact sequence 0 → Γ (1) = Zq → E = ZN ·q → ZN → 0 (2.15) splits if gcd(q, N ) = 1. In all other cases, there exists a non-trivial extension giving rise to a 2-group structure between Γ (1) and G f = PSU(N ). Symmetries via lines. Again, let us illustrate these structures with line defects. The gauge Wilson line W transforming as [0, . . . , 0]Ak−1 × (−1) under SU(k) × U(1) cannot end on a local operator, which either has to be a polynomial in the hypermultiplet X transforming as q [1, 0, . . . , 0]Ak−1 × (− k ) or has to be a monopole operator, which is gauge singlets. In contrast, the Wilson line W q can end on the local operator constructed as the determinant: i.e. the SU(k) gauge group is equipped with the invariant εi1 ,...,ik tensor. Contracting k hypermultiplets yields an operator O ∼ det(X ) which transforms as [0, . . . , 0]Ak−1 × (−q). Hence, the set of Wilson lines W a with a ∈ {1, 2, . . . , q − 1} cannot end and generate the abelian group b Γ [1] = Zq . If one also keeps track of the flavour symmetry representations, one finds that O transforms as ∧k [0, . . . , 0, 1]AN −1 which is not a representation of PSU(N ) f . Hence, this gauge Wilson line is equivalent to a flavour Wilson line and the centres of gauge and flavour symmetry intertwine to give rise to a 2-group structure. The following sections apply the analogous argument to other quiver gauge theories. The relevant questions are: (i) What is the resulting theory? (ii) What are its symmetries? (iii) What is the mirror dual theory? 2.3 T [SU(N )] theories Moving on to quiver theories with non-abelian gauge factors, consider the self-mirror T [SU(N )] theories [30], see Figure 3. The global 0-form symmetry group is given by PSU(N ) t ×PSU(N ) f . In the same spirit as above, one can gauge a discrete Zq 0-form symmetry inside, say, the topological symmetry. The mirror of the resulting theory is then obtained by gauging a Zq 0-form symmetry inside the flavour symmetry. The question is how the Zq is embedded inside the flavour symmetry, given that the Zq is embedded into a Cartan U(1) of the topological symmetry of the mirror. To answer this, one utilises the mirror map (C.15). In more detail, let us consider gauging a Zq ⊂ PSU(N ) t of a T [SU(N )] theory; one inquires about the nature of the resulting theory T [SU(N )]/Zqt . Analogous to Section 2.2, see also Appendix B.1, for a specific Zq embedded in the k-th topological Cartan U(1) factor, the resulting theories T [SU(N )]/Zqt are in fact related to versions of T [SU(N )] encountered in [31]. These quiver theories differ from T [SU(N )] as follows: the k-th node is replaced by U(k) → SU(k), and the flavour node becomes a U(1) gauge nodes with an N copies of bifundamental hypermultiplets between U(N −1) and the “new” U(1) gauge node. Restricting to the case that either 11 SciPost Phys. 15, 033 (2023) N N mirror 1 2 ⋯ ⋯ k N −1 0-form: PSU(N )f × PSU(N )t ⋯ k 2 1 0-form: PSU(N )t × PSU(N )f gauge Zq ⊂ PSU(N )f gauge Zq ⊂ PSU(N )t Zq mirror ??? ⋯ N −1 k N −k ⋯ N −1 ⋯ k 2 1 −k) 0-form: ( SU(k)×U(1)×SU(N ) × PSU(N )t Zk ×ZN −k f −k) 0-form: PSU(N )f × ( SU(k)×U(1)×SU(N ) Zk ×ZN −k t 1-form: Zq 1-form: Zq Figure 3: Gauging of discrete 0-form symmetries in T [SU(N )] theories. The centre symmetries act with charges (−q mod k, q mod (N −k)) on the U(1) factor. The quiver description for T [SU(N )]/Zqt , here denoted by ???, is provided in (2.16). The quiver f for T [SU(N )]∨ /Zq shows again a split of the N fundamental flavours into two sets: f k of them are charged under Zq , which is indicated by an edge of multiplicity k to the grey node; the remaining N −k flavours are uncharged. q|k or k|q, the resulting theory is given by q|k with d = k : q w1 w2 1 2 wk−1 ... k−1 magnetic lattice: d−1 [ i=0 k|q with q = a · k : w1 w2 1 2 ... wk+1 SU(k) i Γ+ d wk−1 k−1 magnetic lattice: Γ , k+1 wN −1 N N −1 v /Zd 1 , (2.16a) wk+1 SU(k) ... k+1 ... wN −1 N N −1 a v (2.16b) 1 wherein Γ denotes the standard integer lattice one assigns to the quiver based on [26]. The shifts by di are to be understood as in [31]. As a comment, restricting to k|q or q|k ensures that the theory after discrete gauging has a simple quiver description. If this constraint is relaxed, there may not be a simple quiver, but the gauging is perfectly well-defined on the level of Hilbert series and index. f The mirror theory T [SU(N )]∨ /Zq is obtained from T [SU(N )]∨ = T [SU(N )] by gauging a Zq ⊂ PSU(N ) f . The mirror map (C.15) dictates that this is realised by splitting the N fundamental flavours into two sets of k and N −k flavours, and gauging the Zq symmetry in the overall U(1) flavour symmetry of one of the two sets. For concreteness, consider gauging the 12 SciPost Phys. 15, 033 (2023) Zq on the set of k fundamental flavours: {yi }ki=1 Zq {yj }N j=k+1 N −k k N −1 ... (2.17) ... 2 k 1 and Appendix D.2 provides exemplary Hilbert series computations that confirm the mirror symmetry between the theories with non-trivial 1-form symmetry. f The mirror map between the parameters of T [SU(N )]/Zqt in (2.16) and T [SU(N )]∨ /Zq in (2.17) can be derived exactly. For concreteness, consider the case q = k, then the map between the parameters in (2.16) and (2.17) is established via ¨ yi w i = yi+1 , i ̸= k , (2.18) N v = yN . Further details on this map are provided in Appendix C.2. Global symmetry. Building on the understanding of the 0-form symmetry group (2.7) for (balanced) abelian quivers, one can utilise a similar logic for the balanced T [SU(N )] theories. Consider the quiver (2.16) the topological symmetry algebra is su(k) ⊕ u(1) ⊕ su(N −k). The global form is then given by Gt = SU(k) × U(1)Q × SU(N −k) Zk × ZN −k , (2.19) with Q has Zk × ZN −k charges (−q mod k, q mod (N −k)) , where the centre symmetries Zℓ act in the standard way on SU(ℓ). Note that for k|q there is a PSU(k) factor in the global symmetry. The examples in the next paragraph, as well as the explicit character decomposition in Appendix D.2, confirm this structure. This structure (2.19) is also apparent from the Higgs branch isometry of the mirror (2.17), i.e. denote the two distinct sets of fundamentals by Zq Xa N −k (2.20) X̃d N −1 ... 1 Here, as a distinctive notation from elsewhere, we use arrowed lines to represent the arrowed lines to symbolize 3d N = 2 chiral multiplets (inflow into the gauge node) and anti-chiral f multiplets (outflow from the gauge node). Analogously to (2.9), one can perform the Zq gauging by assigning (c.f. Appendix C.2) a = 1, x1 , xa Xa : a = 1, . . . , k : ya = ζq · Q 1 · x a+1 , 1 < a < k , (2.21a) 1 , a = k, x k−1 d = k + 1, u1 , ud−k X̃ d : d = k + 1, . . . , N : yd = Q 2 · ud−k−1 , k + 1 < d < N −k , (2.21b) 1 , d = N −k , uN −k−1 13 SciPost Phys. 15, 033 (2023) f with ζq ∈ Zq . The x i and u j are weight space fugacities of su(k) and su(N −k), respectively. The two appearing U(1) fugacities Q 1,2 effectively reduce to a single U(1)Q ; for instance by Q imposing i yi = 1, i.e. ( Q k1 · Q N2 −k q Q1 Q2 ! = 1, ! =Q, ⇒ ( N −k Q 1 = Q N ·q , (2.21c) k Q 2 = Q− N ·q , k f which agrees5 with (C.20) of Appendix C.2. Note also that for q|k the (ζq ) q ∈ Zq acts as the Zk ⊂ SU(k) x i centre symmetry; thus the global factor is PSU(k) x i in this case. The U(1)Q may transform non-trivially under the Zℓ ⊂ SU(ℓ) centre symmetries, depending on the charge of Q. To determine the charge, one again considers a specific gauge-invariant operator O build out of the two sets of fundamentals: X transforms as (ζq ·Q 1 ·[1, 0, . . . , 0]Ak−1 , N−1) and X̃ † transforms † as (Q−1 2 · [0, . . . , 0, 1]AN −k−1 , N−1). U(N −1) gauge invariance imposes Tr(X X̃ ), wherein the trace is taken over the gauge indices. Zq gauge invariance requires O = Symq Tr(X X̃ † ), where the symmetrisation acts on the flavour indices. The resulting operator transforms as q Q1 q q , (2.22) O : Sym [1, 0, . . . , 0]k ⊗ Sym [0, . . . , 0, 1]N −k ⊗ Q2 | {z } =Q such that the Zk × ZN −k centre charges are (q mod k, −q mod (N −k)). These can be compensated by a global U(1)Q rotation provided the centre charges of Q are (−q mod k, q mod (N −k)). This confirms (2.19) as flavour symmetry for the quiver (2.17). As a remark, the operator O can be detected in the Hilbert series as the first non-trivial term in Q. The R-charge of O is simply q × 2 · 12 = q. The appendix D.2 provides examples that illustrate this point. By analogous arguments as in Section 2.2, one can verify that theories (2.16) indeed have the expected Γ (1) = Zq 1-form symmetry. One finds that the centre generators of the combined gauge-flavour symmetry span a E = Zq·N group, such that there exists a non-trivial 2-group extension between Γ (1) and G f = PSU(N ) whenever gcd(q, N ) > 1. Similarly, the same conclusion is reached by inspecting the screening of Wilson lines. Example 1. For an illustrative purpose, let us consider N = 4. Gauging a specific Z2 0-form symmetry leads to a mirror pair: Z2 4 1 SU(2) 3 mirror 1 ←−−−−−→ (2.23) 2 3 2 1 The Hilbert series in (D.20) confirms that the Coulomb branch symmetry algebra for the left quiver (and the Higgs branch isometry algebra of the right quiver) is g = su(2) ⊕ su(2) ⊕ u(1). Moreover, the appearing SU(2) representations are all of integer spin; thus, suggesting the global form G = SO(3) × SO(3) × U(1). Choosing to gauge a specific discrete Z3 subgroup of the 0-form symmetry results in the 5 Again, the definition of Q is a choice. It is motivated by assigning the unit charge to the Higgs branch operator O in (2.22), which is the operator with the lowest R-charge that is charged under the U(1)Q . 14 SciPost Phys. 15, 033 (2023) pair: Z3 4 1 2 mirror ←−−−−−→ 1 SU(3) (2.24) 1 3 2 1 The explicit Hilbert series in (D.17) shows that the Coulomb branch symmetry algebra of the left quiver (and the Higgs branch isometry algebra of the right theory) is g = su(3) ⊕ u(1). Moreover, all appearing characters are neutral under the Z3 centre symmetry of SU(3); hence, the global form is G = PSU(3) × U(1). Example 2. Considering discrete symmetries of the type Zq with q = a·k allows us to uncover equivalent descriptions. Consider T [SU(5)] and gauge a Z6 0-form symmetry. Among the choices considered here, gauging a Z6 ⊂ PSU(5) t is realised by turning the U(3) gauge node into SU(3) together with charge 2 for the “new” U(1) node 2 w1 w2 1 2 w4 SU(3) 4 5 v 2 mirror ←−−−−−→ (2.25) 1 3 4 Z6 3 2 1 or by turing the U(2) node into SU(2) together with charge 3 for the “new” U(1) gauge factor w1 1 SU(2) w3 w4 5 v 3 4 3 Z6 2 mirror ←−−−−−→ (2.26) 1 3 4 3 2 1 Without the additional charges the theories are clearly distinct, for instance by the 0-form and 1-form symmetries, see (2.19) and Figure 3. However, with the modification, both become equivalent as, for example, the monopole formula in (D.44) confirms. Equivalently, on the mirror, one gauges a Z6 subgroup of the flavour 0-form symmetry, but one time acting on three fundamental hypermultiplets and one time on two. For (2.25) and (2.26), this is realised by { y1 , y2 , ζ6 y3 , ζ6 y4 , ζ6 y5 } , for (2.25), 5 { yi }i=1 → (2.27) or {ζ y , ζ y , y , y , y } , for (2.26), 6 1 6 2 3 4 5 f with ζ6 ∈ Z6 . But in both cases, the Z2 ×Z3 centre symmetries are gauged by the discrete gaugf ing of the Z6 0-form symmetry. Hence, the global symmetry is simply PSU(2)×U(1)Q ×PSU(3) for both. The observed equivalence can now be understood as follows: on the level of the Coulomb branch quivers, the global symmetry algebra arises from the split su(5) → su(3) ⊕ su(2) ⊕ u(1). Without the higher charge, the U(1) factor transform non-trivially under the centre symmetry, see (2.19). In fact, it transforms differently in both cases. However, the higher charge is just tuned such that the U(1) becomes independent of the discrete centre symmetries. This then also implies that the operators charged under the U(1) factor coincide in both theories. On the Higgs branch side, the equivalence is a simple consequence of a global U(1) rotation that takes the Z6 action from 2 fundamental flavours to the other 3 fundamental flavours. See also (C.19)–(C.20). 15 SciPost Phys. 15, 033 (2023) Comment. The considerations so far implicitly assume that the gauge node U(k) at which the discrete subgroup of the Cartan U(1) t of the topological symmetry is gauged has k > 1, see for instance Appendix B.1 and B.2. Gauging the Cartan U(1) t of the U(1) gauge node of T [SU(N )] is in spirit similar to Section 2.1.1. Concretely, after gauging Zqt at the U(1) node, the bifundamental between U(1) and U(2) is modified to have charge q under the U(1). Thus, the mirror pair becomes Zq w1 1 w2 q 2 ... y1 wN −1 mirror ←−−→ N −1 N {yj }N j=2 and the global symmetry becomes N −1 G t (left quiver (2.28)) = G f (right quiver (2.28)) = (2.28) ... N −1 2 1 U(1)Q × SU(N −1) , (2.29) ZN −1 where Q has ZN −1 charge q mod (N −1). 2.4 Tρσ [SU(N )] theories The class of linear quiver gauge theories with unitary gauge groups and fundamental or bifundamental hypermultiplets is given by the Tρσ [SU(N )] theories [30], where ρ, σ are two partitions of N . For σ = ρ = (1, . . . , 1) ≡ (1N ), the corresponding theory is simply (1N ) T(1N ) [SU(N )] = T [SU(N )] and the partition data can be dropped. Mirror symmetry exchanges ρ the partitions σ and ρ, i.e. Tρσ [SU(N )]∨ = Tσ [SU(N )]. Analogous to the cases considered so far, the gauging of a discrete 0-form symmetry (either inside the topological symmetry group G t or the flavour symmetry group G f ) leads to a theory with non-trivial 1-form symmetry. Again, consider the two options in turn. While the process of gauging a discrete subgroup of G t is by now understood (see Section 2.3 and Appendix B), determining the action of the discrete group on the flavour symmetry of the mirror theory becomes more challenging when G f is a generic product group. Thus, special attention is paid to determining the mirror theory of Tρσ [SU(N )]/Zqt . Gauging a ZNk ⊂ G t . A Tρσ [SU(N )] is a linear quiver theory with gauge/flavour groups specified by a sequence of integers {Ni } and {Mi }, respectively. The partitions determine the integers as detailed in [30] and the quiver becomes M1 Mk M2 Mn−1 Mn Tρσ [SU(N )] : (2.30) ... N1 N2 ... Nk Nn−1 Nn For concreteness, take the node Nk , with Nk > 1, and gauge a ZNk ⊂ U(1)k ⊂ G t inside the Cartan factor of the topological symmetry associated to the k-th node. By the same arguments as in Appendices B.1 and B.2, one straightforwardly derives the resulting theory 1 M1 Tρσ [SU(N )]/ZNt : k Mn ... N1 N2 16 SU(Nk ) (2.31) ... Nn−1 Nn SciPost Phys. 15, 033 (2023) which has a non-trivial ZNk 1-form symmetry. Now, one constructs the mirror theory. Gauging a ZNk ⊂ G ∨f . The mirror quiver gauge theory of (2.30) is given by M1∨ M2∨ Tρσ [SU(N )]∨ = Tσρ [SU(N )] : N1∨ ... N2∨ Mn∨′ −2 Mn∨′ −1 Mn∨′ Nn∨′ −2 Nn∨′ −1 Nn∨′ (2.32) and the integers {Ni∨ } and {Mi∨ } are determined by the partition data ρ, σ. To determine which ZNk ⊂ G ∨f subgroup needs to be gauged, one has two options: one ρ could derive the mirror map of parameters for the specific pair (Tρσ [SU(N )], Tσ [SU(N )]) and f compute which flavours are charged under ZN . In principle, this is straightforward but likely k to be tedious. Alternatively, one can employ the following train of thought: The mirror theory ρ Tσ [SU(N )] can be rewritten in an unframed form M1∨ N1∨ Mn∨′ −2 Mn∨′ −1 M2∨ N2∨ ... Nn∨′ −2 Nn∨′ −1 Mn∨′ ∼ = Nn∨′ 1 M1∨ N1∨ ... N2∨ Mn∨′ Nn∨′ −2 Nn∨′ −1 ///U(1)diag Nn∨′ (2.33) where no explicit flavour group appears. For such a theory, it is implied that an overall U(1)diag subgroup decouples so the two quiver diagrams express the same theory. The next step is to turn the unitary gauge group U(Nk ) in (2.30) into a special unitary gauge group SU(Nk ). This theory still has a trivial 1-form symmetry due to the flavour groups. However, the 3d mirror theory can be found by using the algorithm in [32]. Schematically, one finds Mk M1 Mn M1∨ mirror ←−−→ ... N1 SU(Nk ) ... Nn N1∨ N2∨ 1 1 ... Mn∨′ Nn∨′ −2 Nn∨′ −1 ///U(1)diag Nn∨′ (2.34) Turning U(Nk ) into SU(Nk ) means the 3d mirror has an additional U(1) gauge group. The additional U(1) gauge group in the unframed quiver is the result of gauging the flavour symmetry. Now, there are two U(1) gauge groups connected to the rest of the linear quiver. The number of bonds Mi∨ attached to each U(1) depends precisely on the choice of SU(Nk ), i.e. which k. The splitting can also occur where the same gauge node, for example, U(N2∨ ) is connected to one U(1) gauge group with an edge of multiplicity M2∨ − x and to the other U(1) with an edge of multiplicity x; see for instance Example 2 below. The final step is to gauge the diagonal U(1) flavour symmetry in the left quiver of (2.34) to obtain Tρσ [SU(N )]/ZNt . However, simply introducing a new U(1) gauge node k leads to the ambiguityQof the global form of the product gauge group, which can either be G = U(1) × SU(Nk ) × i̸=k U(Ni ) or G removed by a subgroup of its centre. For G /ZNk , with ZNk embedded into SU(Nk ) as centre and into the diagonal U(1) ⊂ U(Ni ) of each of the other gauge group factors, one obtains back the original theory Tρσ [SU(N )], see [31] or Appendix B. 17 SciPost Phys. 15, 033 (2023) But this theory exhibits a ZNt topological symmetry, see also Section 2.9. In order to generate k the desired Tρσ [SU(N )]/ZNt theory with gauge group G , one needs to gauge the discrete ZNt k k symmetry, which effectively reduces the magnetic lattice to the standard integer lattice. For the 3d mirror, this means that one first gauges a U(1) t topological symmetry, which effectively f removes a U(1) gauge degree of freedom. But one also needs to gauge a ZN in a subsequent k f step. This ZN can be thought of as embedded in the U(1) that one has to be removed. Hence, k the intermediate step is given by M1∨ mirror Tρσ [SU(N )]/ZNt ←−−→ k N1∨ N2∨ 1 1 ... Mn∨′ Nn∨′ −2 Nn∨′ −1 /// U(1)diag ×U(1) ! f (2.35) ZN k Nn∨′ From the unframed quiver on the right, one has to ungauge a U(1)diag × U(1) and also keep a f ZN gauged. The natural choice is to ungauge the two U(1) gauge groups on top; thus, turning k f them into flavour groups up to a choice of ZN . The last step is to choose in which of the two k f U(1)s one embeds the ZN . k This is because, as with the T [SU(N )] theories, one knows the ∨ only difference between Tρσ [SU(N )]∨ and Tρσ [SU(N )]/ZNt should be the splitting of the k flavour groups along with a discrete quotient. Schematically, one finds M1∨ N1∨ N2∨ M1∨ 1 1 ... M2∨ Nn∨′ −2 Nn∨′ −1 ∼ = N1∨ N2∨ Mn∨′ ... /// ! k ZNk Z Nk ∼ = Mn∨′ Nn∨′ −1 (2.36) f ZN Nn∨′ Mn∨′ −2 Nn∨′ −2 U(1)diag ×U(1) Nn∨′ M1∨ N1∨ M2∨ N2∨ ... Mn∨′ −2 Mn∨′ −1 Mn∨′ Nn∨′ −2 Nn∨′ −1 Nn∨′ and the two framed mirrors show that the discrete quotient can be applied diagonally on either one of the two sets of flavour hypermultiplets. This is also clear from Sections 2.1.2 and 2.3, and Appendices C.1 and C.2, as an overall U(1) rotation can be used to shuffle the discrete ZNk charges from one set of fundamental flavours to another. Example 1. One can apply the above procedure to Tρσ [SU(15)] where σ = (33 , 22 , 12 ) and ρ = (6, 4, 3, 12 ) for the example as in Figure 4. The global form of the 0-form symmetry is expected to be G t (bottom left quiver of Figure 4) = G f (bottom right quiver(s) of Figure 4) = SU(2) × U(1)1 × U(1)2 × U(1)3 ∼ = U(2) × U(1)2 × U(1)3 , Z2 (2.37) and one can explicitly verify this structure as demonstrated in (D.47). Alternatively, the Coulomb branch quiver indicates this isometry group as follows: only the leftmost U(1) is balanced, leading to a topological su(2) t because there are monopole operators of U(1) magnetic flux ±1 at R-charge 1 (see also [30]). The remaining U(2) and U(1) gauge nodes provide 18 SciPost Phys. 15, 033 (2023) 3 2 2 2 1 1 2 2 1 mirror 1 2 2 1 2 2 1 1 ≅ 1 U → SU 3 2 2 2 2 ///U(1)diag 2 2 1 1 1 1 ///U(1)diag mirror 1 2 SU(2) 1 2 2 2 2 1 ungauge U(1) gauge U(1) 1 1 1 /// mirror 1 2 SU(2) 1 2 2 2 2 1 2 2 2 Z2 2 Figure 4: 1 2 2 2 ≅ ≅ 1 U(1)diag ×U(1) Z2 1 Z2 2 1 1 2 1 1 1 1 ρ Starting from the mirror pair Tρσ [SU(15)] and Tσ [SU(15)] with σ = (33 , 22 , 12 ) and ρ = (6, 4, 3, 12 ), one can gauge a discrete Z2 0-form symmetry to create a new mirror pair with Z2 1-form symmetry. See also Appendix C.3 for f the choice of Z2 gauging. t one U(1)i=1,2,3 topological symmetry factor each. Let the one associated with U(2) be denoted t by U(1)1 . Since this node is connected to the balanced node, arguments similar to [30] show the existence of a chiral ring operator that transforms as a spinor under su(2) t and has charge ±1 under U(1)1t . Therefore, the Z2 centre action can be absorbed into U(1)1t , resulting in a U(2) t topological symmetry factor. One can also choose the other SU(2) in Tρσ [SU(15)] which gives the mirror pairs displayed in Figure 5. Comparing Figure 4 and 5, one observes that the global form of the 0-form 19 SciPost Phys. 15, 033 (2023) 1 1 1 /// mirror 1 2 SU(2) 1 2 2 2 2 ≅ 2 2 2 2 Z2 1 2 2 U(1)diag ×U(1) Z2 1 1 1 1 Z2 2 ≅ 1 2 1 2 1 1 ρ Figure 5: Again starting from the mirror pair Tρσ [SU(15)] and Tσ [SU(15)] with σ = (33 , 22 , 12 ) and ρ = (6, 4, 3, 12 ), one can gauge a different discrete Z2 0-form symmetry to generate a another mirror pair with Z2 1-form symmetry. See Appendix f C.3 for the choice of Z2 in the mirror. symmetry in Figure 5 is simply PSU(2) × 3 Y U(1)i , (2.38) i=1 which is supported by the explicit calculations in (D.50). This conclusion can also be drawn by examining the Coulomb branch quiver. Since the balanced U(1) gauge node is not directly connected to any of the U(2) or U(1) gauge groups, there is no expectation on a chiral ring operator that transforms non-trivially under the Z2 centre of the SU(2) t topological symmetry. Example 2. Consider the mirror pair Tρσ [SU(9)] with ρ = (3, 23 ) and σ = (32 , 13 ) 3 2 1 3 ←→ 2 2 2 (2.39) 1 2 2 1 whose symmetry algebra is su(3) ⊕ u(1), as apparent from the balanced set of nodes. The global form is evaluated to be G t (LHS (2.39)) = G f (RHS (2.39)) = SU(3) × U(1) ∼ = U(3) , Z3 (2.40) because the U(1) has charge −1 under the Z3 centre symmetry. See (D.51) for details. Alternatively, the left-hand-side quiver in (2.39) allows us to derive this by using the balanced set of nodes. Since the unbalanced gauge nodes connect to the A2 Dynkin diagram (formed by the balanced nodes) on its first node, there exists a chiral ring operator transforming as 20 SciPost Phys. 15, 033 (2023) [1, 0] × (+1) (plus conjugate) under the topological SU(3) t × U(1) t . Thus, the Z3 centre can be compensated by suitable embedding into the U(1) t factor. To create a new mirror pair, we can gauge a Z2 symmetry on both sides of the dual theories. For example, we can gauge the topological Z2t symmetry on w3 . The mirror map, as shown in f (C.24), indicates that gauging the Z2 symmetry leads to the following mirror pair Z2 1 1 ←→ 2 2 SU(2) (2.41) 1 2 2 1 whose symmetry algebra is su(2) ⊕ u(1) ⊕ u(1). The Hilbert series (D.54) then suggests a symmetry group of G t (LHS (2.41)) = G f (RHS (2.41)) = SU(2) × U(1) × U(1) ∼ = U(2) × U(1) , Z2 (2.42) because the centre Z2 acts trivial on one U(1) factor and with charge −1 on the other. This can also be read off from the Coulomb branch quiver. As there is a U(2) node connected to the balanced U(2) node, there exists a chiral ring operator transforming as [1]A1 × (±1) under the associated SU(2) t × U(1) t topological symmetry factors. Therefore, the Z2 centre symmetry then gives rise to a U(2) t isometry factor. The other topological Cartan U(1) t is uncharged under the Z2 centre, as the gauge nodes are not connected to each other. Gauging Zqt on a U(1) node. Analogous to Section 2.3, one can also gauge discrete subgroups of the topological symmetry associated to a U(1) gauge node. From the examples considered, it is clear what the theory after gauge the Zqt is: the same quiver as before, but all hypermultiplets connected to the specific U(1) gauge node have now charge q. The question is then, what the corresponding mirror theory is. This can be determined by utilising the mirror map between the fugacities, as demonstrated in Appendix C. 2.5 T [SO(2N )] theories In a similar vein to T [SU(N )], one can consider the self-mirror theory T [SO(2N )] [30], see Figure 6. For quiver theories composed of alternating SO(n) and Sp(m) gauge nodes, only the Z2 factors of the SO(n) gauge nodes are the discrete parts of the topological symmetry visible in the UV description. If we gauge any of these, we get a T [SO(2N )]-type quiver with a single replacement SO(2k) → Spin(2k).6 The corresponding mirror theory is obtained from T [SO(2N )] by gauging a suitable Z2 inside the flavour symmetry. This leads to a splitting of the flavour node as indicated in Figure 6. Appendix D.5 provides examples and consistency checks for T [SO(6)] and T [SO(8)]. Considering the theory T [SO(2N )]/Z2t obtained from gauging Z2t , the quiver description allows us to use the techniques of [6] to verify the 1-form symmetry and its interplay with the flavour 0-form symmetry. One finds the discrete groups summarised in Table 1 which constitute the short exact sequence 0 → Γ [1] → E → Z → 0 . (2.43) As expected, the flavour 0-form symmetry is always PSO(2N ) since the flavour centre Z is maximal. Moreover, only for T [SO(4N )]/Z2t with a Spin(4l + 2) gauge node does the 1-form 6 This follows as gauging the Z2 topological symmetry of an SO(2n) gauge group leads to an Spin(2k) gauge group. Conversely [21, 33], gauging the Z2 1-form symmetry in Spin(2k) recovers the SO(2k) theory. 21 SciPost Phys. 15, 033 (2023) 2N 2N mirror ... 2 2 ... 2k 2N −2 2N −2 0-form: PSO(2N )f × PSO(2N )t Z2 mirror 2 2 Spin(2k) 2 2 gauge Z2 ⊂ PSO(2N )f 2N ... ... 2k 0-form: PSO(2N )t × PSO(2N )f gauge Z2 ⊂ PSO(2N )t ... ... 2N −2 k 2N −2 0-form: PSO(2N )f × P(O(2k) × O(2N − 2k))t 1-form: Z2 2N − 2k ... ... 2k 2 2 0-form: P(O(2k) × O(2N − 2k))f × PSO(2N )t 1-form: Z2 Figure 6: Gauging of discrete 0-form symmetries in T [SO(2N )] theories. Gauging the Z2t for an SO(2k) gauge leads to a Spin(2k) gauge group. Gauging the mirror f dual Z2 is realised by splitting the fundamental flavours into two sets: one set is unf charged and the other set is charged under Z2 , indicated by an edge with multiplicity k connected to a grey node. symmetry and the flavour 0-form symmetry form a non-trivial extension hinting to a 2-group symmetry. Following [21, 28], it is straightforward to illustrate the 1-form symmetry and 2-group structure via line operators. For the Spin(2l) gauge group, a Wilson line Ws in the spinor representation cannot end on a local operator, because all half-hypermultiplets transform in the vector representation. For l even, the tensor product of the spinor with itself contains a singlet; therefore, Ws2 is equivalent to the identity line without the need for any local operator. The lines that cannot end generate the (Pontryagin dual of the) 1-form symmetry and there is no 2-group structure. For l odd, the tensor product of the spinor with itself contains the vector. Now Ws2 is equivalent to a flavour Wilson line because it can end on a local operator build from the half-hypermultiplets. However, the vector representation is not an allowed representation of PSO(2N ), which means that the Z2 1-form symmetry forms a 2-group with the flavour symmetry. This is consistent with Table 1. f f On the other hand, in the theory T [SO(2N )]/Z2 obtained by gauging the Z2 symmetry, there are two distinct sets of flavour hypermultiplets, each forming half-hypermultiplets H and h in the vector-vector representation of SO(2N −2k) × Sp(N −1) and SO(2k) × Sp(N −1), Table 1: Interplay of 1-form symmetry and the flavour centre for T [SO(2N )]/Z2t theories. theory Γ [1] E Z T [SO(4N )]/Z2t with Spin(4l) T [SO(4N )]/Z2t with Spin(4l + 2) T [SO(4N + 2)]/Z2t with Spin(4l) T [SO(4N + 2)]/Z2t with Spin(4l + 2) Z2 Z2 Z2 Z2 Z2 × Z2 × Z2 Z2 × Z4 Z2 × Z4 Z2 × Z4 Z2 × Z2 Z2 × Z2 Z4 Z4 22 SciPost Phys. 15, 033 (2023) respectively, i.e. Z2 2N −2k k H . h (2.44) ... 2N −2 2 2 The only difference is that h is also charged under Z2 . As in Sections 2.1.2 and 2.3, to study the global form of the flavour symmetry of this theory, one can consider gauge-invariant operators. Using the invariant Sp(N −1) anti-symmetric tensor J, the standard mesons-type invariants are H J H and hJh, both of which then transform in the adjoint representation [0, 1, . . . , 0] D of so(2N −2k) and so(2k), respectively. Likewise, one can consider hJ H, which is Sp(N −1) gauge invariant, but not Z2 invariant due to the Z2 charge of h. Hence, O = Sym2 (hJ H) is indeed a gauge invariant operator transforming as [2, 0, . . . , 0] Dk ⊗ [2, 0, . . . , 0] DN −k . All of these gauge-invariant Higgs branch operators have trivial charges under the so(2k) or so(2N −2k) centre symmetries. This suggests that the global form of the flavour symmetry is PSO(2k) × PSO(2N −2k). 2.6 Sp(k) SQCD and its orthosymplectic mirror The lessons learnt can be readily applied to other orthosymplectic quivers, such as Sp(k) SQCD with N fundamental hypermultiplets and its orthosymplectic mirror quiver [34]. Focusing on N ≥ 2k +1, the SQCD theory admits a manifest flavour symmetry, while there is no topological symmetry for N > 2k + 1 and a U(1) t symmetry for N = 2k + 1. Thus, it is quite natural to consider gauging discrete subgroups of the flavour 0-form symmetry. Conversely, the mirror orthosymplectic quiver does not have a continuous flavour symmetry for N > 2k1 (i.e. no mass parameter) and an SO(2) f symmetry for N = 2k + 1 (i.e. one mass parameter). While the topological symmetry is not manifest in the UV description, certain remnants are: each SO(l) gauge group admits a manifest Z2t symmetry. 1 2N mirror ... 2 2 2k+1 2 gauge Z2 ⊂ PSO(2N )t 1 1 mirror ... ℓ 2 0-form: (SO(2ℓ) × SO(2N − 2ℓ))f × Ht 1-form: Z2 2 2k 0-form: PSO(2N )t × Hf gauge Z2 ⊂ PSO(2N )f 2k ... 2k+1 (N −2k−1) × SO(2k + 1) nodes (N −2k) × Sp(k) nodes 0-form: PSO(2N )f × Ht Z2 ... 2k 2k 2N − 2ℓ 1 ... 2ℓ−2 Spin(2ℓ) 2ℓ ... 2k ... 2k 2 0-form: (SO(2ℓ) × SO(2N − 2ℓ))t × Hf 1-form: Z2 Figure 7: Gauging of discrete 0-form symmetries in Sp(k) SQCD with N fundamentals and its linear orthosymplectic mirror quiver. Here, the isometry group H t/ f is trivial for N > 2k + 1 and U(1) for N = 2k + 1. Again, the split into two sets of flavours can f made symmetric, as a global rotation can shift the Z2 action onto either set. 23 SciPost Phys. 15, 033 (2023) Therefore, one can gauge a Z2 topological symmetry of a specific SO(ℓ) gauge node and inquire about the implications. It is straightforward to observe that this gauging modifies the particular gauge group SO(ℓ) → Spin(ℓ), see for instance [26, 35]. On the mirror side, one gauges a Z2 ⊂ SO(2N ) flavour symmetry, which then leads to a split of the flavour symmetry. This is summarised in Figure 7. Exemplary cases with explicit calculations are provided in Appendix D.6. The interplay of the discrete Z2 1-form symmetry with the continuous 0-form symmetry is simple here. Consider the linear orthosymplectic mirror quiver. For N > 2k + 1, there is no continuous 0-form flavour symmetry that could mix with the 1-form symmetry Z2 . For N = 2k + 1, there exists an enhance U(1) 0-form symmetry, but the 1-form and 0-form symmetry are simply a product of each other. 2.7 Sp(k) SQCD and its unitary D-type mirror quiver It is well-known that Sp(k) SQCD with N fundamental flavours admits a second mirror description [36], based on a DN -type Dynkin quiver: 1 2N ... ←→ 1 2 k ... 2k−1 2k 2k N − 2k − 1 nodes 2k (2.45) 2k k This mirror pair has the advantage that the PSO(2N ) global symmetry is manifest as Higgs branch isometry in the SQCD theory and as Coulomb branch isometry in the D-type Dynkin quiver. It is, hence, natural to study gaugings of discrete Zq symmetries in this manifest 0-form symmetry. Starting with the Dynkin quiver, there are two distinct choices: Firstly, gauging a Zl on a U(l) node which satisfies 2 < l < 2k, one obtains7 1 ... 1 ... l−1 SU(l) l+1 k ... 2k−1 2k 2k (2.46) 2k N − 2k − 1 nodes k which has a Zl 1-form symmetry and the Coulomb branch isometry algebra is su(l) ⊕ u(1)Q ⊕ so(N − l). For the global form, one can study the action of the centre symmetries of the non-abelian factors. One finds G t (2.46) = PSU(l) × U(1)Q × Spin(2N − 2l) Z DN −l , (2.47) where the Z DN −l charges of Q are given by the charges of the congruence class of the j-th fundamental representation [0, . . . , 0, 1, 0, . . . , 0] D with j = 2k−l, see Appendix A.3 for details. For explicit examples including Hilbert series computations see Appendix D.7. 7 The cases l = 1, 2 are addressed separately below. 24 SciPost Phys. 15, 033 (2023) Alternatively, gauging a Zl on a U(l) node which satisfies l ≥ 2k, one obtains 1 ... ... 1 2k−1 k ... 2k 2k SU(2k) 2k (2.48) 2k l − 2k nodes k and the Coulomb branch isometry algebra is the same as in (2.46). However, the “extra” U(1) node is now attached to the balanced A-type Dynkin diagram such that the global form is given by G t (2.48) = PSU(l) × U(1)Q Zl × PSO(2N − 2l) , (2.49) where Q carries Zl charge 2k mod l, i.e. the charges of the congruence class of the 2k-th fundamental representation, see Appendix A.3. Explicit examples for this discrete gauging are given in Appendix D.7. Global form via the mirror. Analogous to the discussion in Sections 2.1.2, 2.3, and 2.5, one can confirm this global symmetry via the Higgs branch of the mirror theory. The starting point is the mirror map (C.27) between the flavour fugacities of Sp(k) SQCD and its unitary D-type Dynkin quiver, see Appendix C.1. This allows us to identify which flavour fugacities f are involved in the discrete Zl gauging on the SQCD side. • l < 2k: The familiar argument then proceeds by splitting the fundamental flavours into two distinct groups: the first l fundamental flavours are grouped as X , transforming 1 as ζl Q− l [1, 0, . . . , 0]Al , and the remaining N − l fundamental flavours, transforming as [1, 0, . . . , 0] DN −l . Building a gauge invariant Higgs branch operator proceeds in two steps: firstly, using the Sp(k) invariant tensors J on constructs operators of the form X J X̃ , which transform as ζl under the discrete symmetry. Secondly, Zl invariance is achieved via O = Syml (X J X̃ ), which transforms as Q−1 [l, 0, . . . , 0]Al−1 ⊗ [l, 0, . . . , 0] DN −l . The Zl centre surely acts trivial on [l, 0, . . . , 0]Al−1 , while the centre charges of [l, 0, . . . , 0] DN −l are (0, l mod 2) if N − l is even or (2l mod 4) if N − l is odd. Thus, the non-trivial transformations under the centres can be compensated if Q transforms as follows: N − l = even Zl × Z2 × Z2 charges of Q: (0, 0, l mod 2) , (2.50a) N − l = odd Zl × Z4 charges of Q: (0, 2l mod 4) , (2.50b) which confirms (2.47). To see this, recall from Appendix A.3 that the congruence class of the j-th fundamental representation of DN −l with j = 2k − l is (0, l mod 2) for N − l even and 2l mod 4 for N − l odd. • l > 2k: The argument is slightly modified: the first set X of flavours transforms as 1 ζ2k Q− 2k [1, 0, . . . , 0]Al−1 , while the second set X̃ transforms as [1, 0, . . . , 0] DN −l . The Higgs branch operator O = Sym2k (X J X̃ ) transforms as Q−1 [2k, 0, . . . , 0]Al−1 ⊗[2k, 0, . . . , 0] DN −l , which has trivial D-type centre charges. To see this, for N − l even, the Z2 × Z2 charges are (0, 2k mod 2) = (0, 0); while for N − l odd, the Z4 charge is 2 · 2k mod 4 = 0. Thus, to compensate potential irreps that are non-trivial under Z2k , one requires that Q has the following charges: N − l = even Zl × Z2 × Z2 charges of Q: N − l = odd Zl × Z4 charges of Q: which then confirms (2.49). 25 (2k mod l, 0, 0) , (2.51a) (2k mod l, 0) , (2.51b) SciPost Phys. 15, 033 (2023) Two special cases. In the l = 2 case of (2.47), a symmetry enhancement is observed in the explicit computations (D.71) and (D.86). These show that there is not only the expected su(2) t , but the topological Cartan symmetry of the “new” U(1) gauge node is also enhanced to a non-abelian su(2) t . These two su(2) t symmetries can both be interpreted as PSO(4) t ∼ = PSO(3) t × PSO(3) t . As in previous sections, one can also gauge a discrete Zqt along the topological fugacity w1 associated to the first U(1) gauge node. The D-type Dynkin quiver is modified in the by now familiar way: the bifundamental of U(1) × U(2) turns into a hypermultiplet that transforms as f fundamental under U(2) but is of U(1) charge q. In the mirror theory, the Zq acts on a single fundamental flavour, as dictated by the mirror map (C.27). In summary, the mirror pair with Zq 1-form symmetry is 2N −2 1 q ←→ 1 2k ... 2 2k−1 2k Zq G= Z DN −1 ¨ , 2k N − 2k − 1 nodes and the global Higgs / Coulomb branch isometry is U(1)Q × Spin(2N − 2) k ... Z DN −1 charges of Q (2.52) 2k k (0, q mod 2) , N = even, 2q mod 4 , N = odd, (2.53) where Q is the topological fugacity of the left-most U(1) gauge node. 2.8 Examples of non-simply laced unitary quivers and their mirrors The last class of quiver theories considered here are non-simply laced unitary quivers,8 whose monopole formula has been proposed in [39]. Consider the following example M1 M2 M3 M4 (2.54) κ N1 N2 N3 N4 with nodes U(N1,2 ) on the “short” side and U(N3,4 ) on the “long” side; wherein the naming is borrowed from Dynkin diagrams. The multiplicity of the non-simply laced edge is denoted by κ. Even though these quiver theories are non-Lagrangian (hence superconformal index and Higgs branch Hilbert series are not computable by the standard methods), we can still study their Coulomb branch using Hilbert series techniques. This allows us to investigate the effects of gauging a discrete Zqt symmetry. Gauging at the long side. To begin with, attempt to gauge a discrete ZNt topological sym3 metry associated to the U(N3 ) node at the long side, with N3 > 1. As a first step, one rewrites (2.54) by expressing U(N3 ) ∼ = (SU(N3 ) × U(1))/ZN3 . By analogous arguments as in Appendix 8 See, for example, [37, 38] for the appearance of such quiver theories via branes and ON planes. 26 SciPost Phys. 15, 033 (2023) B, one arrives at 1 h κ M1 M4 M2 M3 κ N1 N2 SU(N3 ) N4 m1 e Γ := N[ 3 −1 a=0 m2 ℓ Z + κ·a N3 N1 /Z , N3 with (m 1 , m 2 , l, m 4 , h) ∈ e Γ, (2.55) m4 N2 a N3 −1 a N4 a × Z + κ·a × Z + × Z + × Z + N3 N3 N3 N3 , and the Coulomb branch moduli space is the same as that of (2.54). The green edges transform in the fundamental representation of U(N1,2 ) and with charge κ under the U(1). Next, the ZN3 symmetry is gauged. One obtains the following quiver description: h 1 κ M1 M4 M2 N1 N2 m1 m2 Γ := N[ 3 −1 with (m 1 , m 2 , l, m 4 , h) ∈ Γ , M3 κ SU(N3 ) ℓ (2.56) N4 m4 ZN1 × ZN2 × ZN3 −1 × ZN4 × Z , a=0 where Γ is again short-hand for the integer magnetic lattice. This theory exhibits a ZN3 1-form symmetry, by construction. Gauging at the short side. Now, consider gauging a ZNt on the topological fugacity as2 sociated to the U(N2 ) gauge node, with N2 > 1. Again, the first step is to simply rewrite U(N2 ) ∼ = (SU(N2 ) × U(1))/ZN2 . By adopting the arguments of Appendix B, one finds 1 h κ M1 M2 κ N1 SU(N2 ) N3 N4 m3 m4 N2[ ·κ−1 N1 m1 b Γ= ℓ a=0 M4 M3 Z + Na2 /Z N2 with (m 1 , l, m 3 , m 4 , h) ∈ b Γ, (2.57) N2 −1 N3 N4 × Z + Na2 × Z + N2a·κ × Z + N2a·κ × Z + N2a·κ , whose Coulomb branch coincides with that of (2.54). Moreover, the edges highlighted in green transform in the fundamental representation of U(N1,2 ) and with charge κ under the U(1) node. As a next step, gauging the ZN2 results in the following theory: h 1 κ M1 M4 M2 N1 m1 Γ= κ SU(N2 ) κ−1 [ a=0 ℓ with (m 1 , l, m 3 , m 4 , h) ∈ Γ , M3 N3 N4 m3 m4 (Z + a)N1 × (Z + a)N2 −1 × Z + κa 27 N3 × Z + κa N4 × Z + κa , (2.58) SciPost Phys. 15, 033 (2023) where Γ is a short-hand notation for several shifted copies of the standard integer lattice of the magnetic charges. Comment. One could also gauge a discrete Zqt along the topological Cartan U(1) of a U(1) gauge node. In this case, the connected hypermultiplets are modified to have charge q under the U(1), but no other changes to the quiver occur. 2.8.1 C-type quivers A representative example is the mirror pair of O(2k) SQCD with N hypermultiplets in the vector representation and its C-type Dynkin mirror quiver 1 2N ←→ ... 1 (2.59) ... 2 2k 2k 2k 2k O(2k) N gauge nodes which can be realised by a systems of D3-D5-NS5 branes with O5 and ON planes, respectively. The logic is the same as before: Choose a Zqt in the C-type Dynkin quiver, by selecting a gauge node and its associated topological fugacity. Using the mirror map (C.33) for (2.59) f one identifies how the Zq acts on the vectors. For concreteness, consider examples for k = 1 and N = 4: Example: gauging on the long side. Gauging a Z2t on the fourth node yields the mirror pair (i.e. using (C.33) with discrete variable on w4 ) Z2 1 2 ←→ 8 1 O(2) (2.60) 2 2 SU(2) where the ‘new’ U(1) node is connected with a hypermultiplet of charge 2. The global symmetry algebra is su(4) ⊕ u(1), as read off from the balanced set of nodes. Explicit Hilbert series (D.105) show that the global form is G f (LHS (2.60)) = G t (RHS (2.60)) = PSU(4) × U(1)Q , (2.61) because the Z4 centre acts trivial on all appearing representations. Example: gauging on the short side. Gauging a Z2t on the third gauge node results in the new mirror pair (i.e. using (C.33) with discrete variable on w3 ) 1 Z2 2 6 2 ←→ O(2) 1 28 2 (2.62) SU(2) 2 SciPost Phys. 15, 033 (2023) the global symmetry algebra is su(3) ⊕ u(1) ⊕ sp(1), as suggested by the balanced nodes in the unitary quiver. Recalling the maximal subalgebra su(3) ⊕ u(1) ⊂ sp(3), an analysis of the Hilbert series then suggests that the global form is G f (LHS (2.62)) = G t (RHS (2.62)) = PSp(3) × PSp(1) . (2.63) See (D.109) for explicit computations. Example: gauging on the short side. Gauging a Z2t on the second gauge node results in the new mirror pair (i.e. using (C.33) with discrete variable on w2 ) 1 Z2 4 4 2 ←→ O(2) 1 SU(2) (2.64) 2 2 and the balanced set of nodes suggests the symmetry algebra su(2) ⊕ u(1) ⊕ sp(2). A Hilbert series computation (D.113) then indicates the following symmetry group G f (LHS (2.64)) = G t (RHS (2.64)) = PSp(2) × PSp(2) . (2.65) This suggests that the su(2) ⊕ u(1) realise a maximal subalgebra in one sp(2) factor. 2.8.2 B-type quivers Alternatively, we could consider an Sp(k) gauge theory with SO(2n + 1) flavour symmetry. However, to prevent a parity anomaly, we would need to include a suitable Chern-Simons term. The Higgs branch, which is not affected by Chern-Simons levels, is known to be the closure of a B-type nilpotent orbit. Therefore, a natural mirror theory would be a B-type Dynkin quiver, for which analogous arguments apply as above. 2.8.3 A comment on F4 Coulomb branch quivers The reasoning can be also applied to other non-simply laced Coulomb branch quivers, even if there may not exist a known mirror. Such an example is the F4 Coulomb branch quiver of [40]. Table 2 summarises the resulting theories after a suitable Znt is gauged, following the prescriptions (2.58) and (2.56). Here, a few remarks in comparison to the “ungauging scheme” of [41] are in order. The ungauging scheme involves removing a U(1) factor from a selected U(n) gauge group, which in the context of the monopole formula means setting one of the magnetic charges to zero. For simply-laced quivers, this procedure leads to the same consequence as replacing a U(n) gauge group with an SU(n) and quotienting out a diagonal Zn . However, the ungauging scheme becomes problematic when applied to a node on the short side of non-simply laced quivers. If the short node is a U(1) gauge group, then the ungauging simply converts it into a flavour group. In [41], the ungauging of the short U(1) node in the F4 quiver leads to a Coulomb branch that is the next-to-next-to minimal nilpotent orbit closure of so(9). In contrast, if the short node is non-abelian, such as the U(2) node in the F4 quiver, the resulting moduli space cannot be identified with any known space and the procedure has been argued to be “invalid” in [41]. On the other hand, by replacing the short U(2) node with an SU(2) and following the prescriptions in (2.58) and (2.56), one is able to obtain consistent results, as shown in the fourth row of Table 2. The resulting Coulomb branch is the next-to-next-to minimal nilpotent 29 SciPost Phys. 15, 033 (2023) Table 2: The F4 Coulomb branch quiver and its Zq gaugings. The first row is the standard F4 quiver proposed in [40]. Rows 2 - 4 display different choices of gauging a ZNt of a U(N ) node in the Coulomb branch quiver for the minimal nilpotent orbit closure of F4 . The gaugings on the “long” side produce global symmetries given by the balanced set of nodes. For the gauging on the “short” side, the global so(9) symmetry is only visible via the subalgebra su(4) ⊕ su(2). Rows 5 - 7 display the effects of gauging a Zqt inside the topological Cartan factor of the U(1) gauge node. For q = 2, the symmetry algebra is enhance from sp(3) × u(1) to sp(4); while for q > 2, the algebra is simply sp(3) × u(1). In the Hilbert series expressions, χ and φ are characters for the non-abelian symmetry factors and Q is a U(1) fugacity. quiver symmetry Coulomb branch Hilbert series 1 F4 2 3 2 1 1 3 SU(2) 2 A1 × C3 1 + t(χ2,0,0 + φ2 ) + t 2 (1 + χ4,0,0 + χ0,2,0 + χ2,0,0 φ2 + χ0,0,2 φ2 + φ4 ) + . . . = 1 + 24t + 537t 2 + . . . A2 × A2 1 + t(χ1,1 + φ1,1 ) + t 2 (1 + χ1,1 + χ2,2 + χ1,1 φ1,1 + χ2,2 φ1,1 + φ2,2 + φ1,1 ) + . . . = 1 + 16t + 351t 2 + . . . A3 × A1 ⊂ B4 1 + t(χ2 + χ2 φ0,1,0 + φ1,0,1 ) + t 2 (1 + χ4 + χ2 φ2,0,0 + φ0,1,0 + χ2 φ0,1,0 + χ4 φ0,1,0 + φ0,2,0 + χ4 φ0,2,0 +φ1,0,1 +2χ2 φ1,0,1 +χ2 φ1,1,1 +χ2 φ0,0,2 +φ2,2 )+. . . = 1 + 36t + 621t 2 + . . . 1 1 2 SU(3) 2 1 1 2 3 1 + χ1,0,0,0 t + χ2,0,0,0 t 2 + χ3,0,0,0 t 3 + . . . = 1 + 52t + 1053t 2 + 12376t 3 + . . . SU(2) 1 χ C3 × U1 ⊂ C4 2 2 3 2 1 χ χ1,1,0 Q C3 × U1 1+ t(χ0,1,0 +1)+ t 2 (Qχ1,0,1 + 1,0,1 Q +χ0,0,2 +2χ0,1,0 + χ0,2,0 + χ2,0,0 + 1) + . . . = 1 + 22t + 369t 2 + . . . C3 × U1 1+ t(χ0,1,0 +1)+ t 2 (Qχ2,0,0 + 2,0,0 Q +χ0,0,2 +2χ0,1,0 + χ0,2,0 + χ2,0,0 + 1) + . . . = 1 + 22t + 327t 2 + . . . 3 3 2 1 χ 1 4 2 χ 1,0,0 + Qχ0,0,2 + Qχ1,0,0 + Qχ1,1,0 + 0,0,2 Q + Q + + χ0,0,2 + 2χ0,1,0 + χ0,2,0 + χ2,0,0 + 1) + . . . = 1 + 36t + 621t 2 + . . . χ 1 2 χ 2,0,0 2 2 1+ t(Qχ1,0,0 + 1,0,0 Q +χ0,1,0 +1)+ t (Q χ2,0,0 + Q2 1 3 2 1 orbit closure of so(9) as well.9 It is to be noted, that if one uses the prescriptions (2.57) and (2.55), then one recovers the original minimal nilpotent orbit closure of F4 . 9 In general, for non-simply laced quivers, the prescriptions (2.58) do not always provide the same Coulomb branch for all the short nodes. 30 SciPost Phys. 15, 033 (2023) 2.9 Magnetic quivers and gauging discrete topological symmetries Suppose that one is given an unframed unitary magnetic quiver T with only simply-laced edges (i.e. bifundamental hypermultiplets between the unitary gauge nodes). To evaluate the Hilbert series or the index, it is necessary to remove an overall U(1) gauge group factor. In [31], it was emphasised that choosing this U(1) from a U(k) gauge node leads to an SU(k) gauge node, Sk−1 but the magnetic lattice Γ is extended to include shifted versions of the form i=0 Γ + ki . This situation can also be understood from a complementary perspective. Given an unframed unitary magnetic quiver, pick a U(k) gauge node and rewrite it as Sk−1 SU(k)×U(1) i i k−1 ∼ U(k) = , with fluxes (l, h) ∈ , Z+ Z+ . The aim is to remove this k i=0 Zk k U(1) factor. As demonstrated in Appendix B, this rewriting shifts all other magnetic fluxes m by the flux h associated to the U(1); as a result, all magnetic fluxes receive the shifts Γ + ki simultaneously. Now, removing this U(1) means treating it as a background vector multiplet. Nevertheless, all remaining magnetic fluxes are still subject to the shifts Γ + ki . Hence, the Coulomb branch Hilbert series, as well as the index for T , have the form X f (l, m) , (2.66) FT = (l,m)∈ Sk−1 i i=0 (Γ + k ) which is message conveyed in [31]. It turns out that one can refine FT by introducing a Zk -valued fugacity z as follows: the U(1) t topological symmetry of the U(k) node appears in both the monopole formula and the inPk ma . Upon rewriting into magnetic fluxes (l, h) for (SU(k) × U(1)) / Sk−1 Since h ∈ i=0 (Z + ki ), one has wk·n+i for h = n + ki ∈ (Z + ki ) and k some n ∈ Z. This means that one can introduce a discrete fugacity z to keep track of the Zk = z i . This fugacity remains even if the U(1) centre symmetry, setting w k → z such that wk·n+i k is taken to be non-dynamical. One ends up with dex through the factor w k a=1 Zk , this becomes wk·h . k FT (z) = k−1 X i=0 X zi f (l, m) . (2.67) (l,m)∈(Γ + ki ) It is now clear what happens if this discrete Zkt topological symmetry is gauged: the entire range of the summation collapses to the i = 0 sector, i.e., the integer lattice FT /Z t = k k−1 X 1X FT z = (ζk )i = f (l, m) . k i=0 (2.68) (l,m)∈Γ Consequently, the quiver theory, in which U(k) is replaced by an SU(k) and the magnetic lattice is simply the integer lattice, is obtained from the unframed unitary quiver T by gauging SU(k)×U(1) ∼ a discrete Zkt topological symmetry. This Zk distinguishes between = U(k) and Zk t SU(k) × U(1). Additionally, the gauging of the Zk symmetry has introduced a Zk 1-form symmetry into T /Zkt . 2.10 Examples from 5d magnetic quivers One can demonstrate gauging discrete subgroups of the topological symmetry on known magnetic quivers.10 It is most suitable to choose quivers whose Coulomb branches have a known Higgs branch realisation. 10 See also [42–47] for magnetic quivers of theories with 1-form symmetries. 31 SciPost Phys. 15, 033 (2023) E5 quiver. The infinite coupling magnetic quiver for 5d Sp(1) SQCD with 4 flavours realises E5 D5 O ∼ = O , which is also the Higgs branch of Sp(1) with 5 flavours. Thus, one arrives at [48] min min 1 10 /Z2 2 2 4 2 ←→ (2.69) 2 2 It is worth recalling that the magnetic lattice for the left-hand side quiver has the form Γ ∪(Γ + 12 ) with Γ being the standard GNO integer lattice, as can be found in [31] and also see [45, 48– 55] for examples with orthosymplectic quivers. The corresponding discrete Z2t topological symmetry of the magnetic quiver can be gauged in the same vein as before. On the level of the magnetic quiver, this just reduces the relevant magnetic lattice to the integer lattice Γ . f Equivalently, one can gauge a Z2 on the Sp(2) SQCD side, which then gives rise to the following pair of theories 8 1 ←→ (2.70) 2 2 2 4 2 Z2 2 2 and it is straightforward to verify that the Coulomb / Higgs branch Hilbert series reproduce the results of [31, Tab. 9]. The global form of the 0-form symmetry is PSO(8) × U(1). E4 quiver. Similarly, the infinite coupling magnetic quiver for 5d Sp(1) SQCD with 3 flavours E4 A4 realises Omin ∼ = Omin via its Coulomb branch. Of course, this moduli space admits a known Higgs branch realisation and one arrives at 1 2 /Z 2 1 2 2 5 ←→ (2.71) 1 2 The magnetic lattice for the magnetic quiver is of the form Γ ∪ (Γ + 12 ), so the associated Z2t f symmetry can be gauged. The question then becomes what Z2 symmetry is realised on the SQED side. Through explicit calculations, one verifies that 1 4 2 1 ←→ (2.72) 1 2 2 2 reproduces the known Hilbert series [31, Tab. 10]. SO(6) × U(1). 32 Z2 The isometry group in this case is SciPost Phys. 15, 033 (2023) Folded E6 quiver. The infinite coupling magnetic quiver for 5d Sp(1) SQCD with 5 flavours admits a Z2 outer automorphism. Folding the corresponding magnetic quiver leads to E6 D5 Omin → Omin on the Coulomb branch [53]. Since there is a known Higgs branch realisation for D-type minimal nilpotent orbit closures, one arrives at 10 1 4 4 2 2 /Z2 ←→ (2.73) 2 where again the left-hand side quiver has a magnetic lattice of the form Γ ∪ (Γ + 21 ). Gauging this Z2t has a by now clear consequence on the magnetic quiver, as the GNO lattice is reduced f to the integer lattice. On the Sp(1) SQCD side, the corresponding Z2 is realised as follows: Z2 1 4 4 2 2 ←→ (2.74) 2 and one straightforwardly verifies the agreement of the Coulomb branch / Higgs branch Hilbert series, which is given by HS = 1 + 25t + 400t 2 + 3864t 3 + 26600t 4 + 141672t 5 + 621480t 6 + 2337280t 7 + 7763283t 8 + 23265515t 9 + 63954800t 10 + O t 11 , (2.75) and the global symmetry group is PSU(5) × U(1). 3 Discussion and conclusions In this paper, mirror pairs with non-trivial 1-form symmetry have been studied. Starting from known mirror pairs with trivial 1-form symmetry, gauging of discrete Zq subgroups of the 0-form symmetry allowed us to construct new mirror pairs with non-trivial 1-form symmetry. The main results are as follows: 1. It has been shown that theories T /Zqt , obtained by gauging a discrete subgroup Zqt of the topological symmetry, may admit quiver descriptions if the discrete subgroup is suitably chosen. ∨ f 2. The mirror theories T /Zqt can be constructed using T ∨ /Zq , but the precise choice of f Zq in the flavour symmetry of T ∨ can be subtle. This paper provides a simple algorithm f for specifying Zq . f 3. The global form of the 0-form symmetries of (T /Zqt , T ∨ /Zq ) have been derived using both field theory methods and monopole operators (via the balanced set of nodes), and the resulting symmetry groups have been verified through explicit Hilbert series computations. 4. The interplay between continuous 0-form and discrete 1-form symmetries has been studied using established field theory techniques and the equivalence classes of lines. 33 SciPost Phys. 15, 033 (2023) 5. On the technical side, the gauging of discrete subgroups of the topological symmetry on non-simply laced quivers has been proposed and tested on both long and short-side gauge nodes. A comment on the moduli spaces. The maximal branches of the moduli space of vacua in a theory T are the Coulomb branch C (T ) and the Higgs branch H(T ). These are symplectic singularities that can be resolved when the theory T is given either an FI parameter (for the Higgs branch) or a mass parameter (for the Coulomb branch). For instance, consider SQED with N hypermultiplets of charge 1. This theory admits N −1 mass parameters that resolve the C2 /ZN Coulomb branch, and a single FI parameter that resolves the Higgs branch, specifically su(N ) the minimal nilpotent orbit closure Omin . If we gauge a Zqt 0-form symmetry in this theory, the resulting SQED with charge q hypers has the same Higgs branch, but the Coulomb branch is modified to be C2 /ZN ·q . However, there are no additional mass parameters in the theory, which means that the singularity cannot be fully resolved even though a symplectic resolution exists. More generally, one can perform a simple test11 via Hilbert series that shows HSC (T /Z t ) (t) T /Zqt : lim t→1 HS q(t) = 1q , C (T ) or T −→ (3.1) HS f (t) T /Zqf : lim t→1 H(T /Zq ) = 1 , q HS (t) H(T ) and the presence of a 1q fraction in the expression suggests (at least locally) that the Coulomb branch C (T /Zqt ) is a Zq orbifold of C (T ), and a similar relationship holds for the Higgs f branches H(T /Zq ) and H(T ). Again, no additional deformation parameter appears. In contrast, consider T to be U(2) SQCD with 4 fundamental flavours. The maximal su(4) branches are H(T ) = O(22 ) and C (T ) = S(22 ) ∩ Nsu(4) , i.e. the Slodowy slice to the su(4) nilpotent orbit defined by partition (22 ). There are 3 masses resolving the Coulomb branch and 1 FI term resolving the Higgs branch. If we gauge the topological U(1) t symmetry in this theory, the resulting theory is SU(2) SQCD with 4 fundamental flavours. Then, the Coulomb so(8) branch of this theory is C (T /U(1) t ) = C2 /D4 while the Higgs branch is H(T /U(1) t ) = Omin . In this case, the Coulomb branch can be resolved by the 3 + 1 mass parameters, while the minimal orbit closure of so(8) does not admit a symplectic resolution, which is consistent with the absence of an FI parameter in this theory. These symplectic resolutions can also be studied via Hilbert series techniques, see for instance [57, 58]. Generalisations and open questions. In this work, a single Zq factor of the 0-form symmetry has been gauged. One straightforward generalisation is to consider orthosymplectic quivers and gauge several Z2t topological symmetry factors associated to SO(ni ) nodes. The resulting theory is simply obtained by the relevant SO(ni ) → Spin(ni ) and the 1Q replacing t form symmetry is the product group i (Z2 )i . Similarly, one could also entertain the thought of gauging several Zqi inside distinct topological Cartan factors of, say, T [SU(N )]. It is a priori not clear if a simple quiver description exists. Another possibility is to gauge a discrete Zqt group embedded into several topological Cartan U(1) factors of a Coulomb branch unitary quiver. Inspecting the mirror maps for a fully balanced linear quiver (C.7) or (C.15), one observes that the effect on the mirror Higgs branch Following [56], the volume of the Sasakian base S of H or C is evaluated via Vol(S) = lim t→1 (1 − t)d HS(t), where d = dimC (H or C ). 11 34 SciPost Phys. 15, 033 (2023) quiver is as follows: the set of fundamental flavours splits into several sets, with each subset f being acted upon by Zq in a distinct fashion. One might hope to find a simple quiver description on the Coulomb branch side, but this is only possible for very specific q values, similar to the choices in this paper. For instance, gauging a Zqt in two adjacent node U(k) and U(k + 1) in a T [SU(N )] theory, one can expect a quiver-type description with an SU(k), SU(k + 1) node and two “new” U(1)1,2 gauge factors for q = k · (k + 1). While one U(1)1 factor behaves similarly to the discussion in this paper, the second U(1)2 factor is expected to lead to trifundamental hypermultiplets for U(k − 1) × SU(k) × U(1)2 , U(k + 2) × SU(k + 1) × U(1)2 , and SU(k) × SU(k + 1) × U(1)2 . A systematic analysis of these cases is left for future work. Another aspect of 3d mirror symmetry is the exchange of Wilson and vortex line defects [29, 45, 59–61]. Given the central role of line defects in understanding 1-form and 2group symmetries, it would be interesting to systematically analyse the exchange of Wilson and vortex lines under mirror symmetry for the theories with 1-form symmetry. Acknowledgments We would like to thank Fabio Apruzzi, Lakshya Bhardwaj, Mathew Bullimore, Andrea Ferrari, Heeyeon Kim, Noppadol Mekareeya, Matteo Sacchi, and Sakura Schäfer-Nameki for discussions. M.S. is grateful to Ryo Suzuki for use of his computing facilities. M.S. is also grateful to Rudolph Kalveks for invaluable help with Mathematica. Funding information The research of S.N. is supported by the National Science Foundation of China under Grant No. 12050410234 and Shanghai Foreign Expert grant No. 22WZ2502100. The research of Z.Z. is supported by the ERC Consolidator Grant # 864828 “Algebraic Foundations of Supersymmetric Quantum Field Theory” (SCFTAlg). A Notations and conventions A quiver diagram, composed of nodes and edges, encodes a 3d N = 4 theory as follows: • Gauge nodes ⃝ denote dynamical vector multiplets, while flavour nodes □ denote background vector multiplets. The notations are summarised in Table 3a. • An edge between two nodes corresponds to a hypermultiplet H = (X , Y † ), with X , Y two N = 2 chiral multiplets. The notation is summarised in Table 3b. • An exception are so-called non-simply laced edges in a quiver theory. Between unitary gauge node, such an edge has been proposed purely on the level of the conformal dimension of the monopole formula [39] n κ n ←→ k k 1 XX |m1,i − κ · m2, j | , 2 i=1 j=1 (A.1) and it is to stress that this does not correspond to a representation of the gauge groups. For the special case of U(n = 1), such a non-simply laced edge is effectively the same as a U(1) gauge group with a charge κ hypermultiplet. Between orthosymplectic nodes, the conformal dimension has been proposed in [53] X X κ 1 ←→ |ρ(m) − κ · λ(n)| , (A.2) 2·2 n 2k ρ∈[1,0,...,0]B/D λ∈[1,0,...,0]C 35 SciPost Phys. 15, 033 (2023) Table 3: Notation for nodes and links in the quiver diagrams. edge node vector U(n) n SU(n) SU(n) n Spin(n) hyper n k n SU(k) n 2k SO(n) Spin(n) Spin(n) n bifundamental n ⊗ k bifundamental n ⊗ [0, . . . , 0, 1]A half-hyper [1, 0, . . . , 0] D/B ⊗ [1, 0, . . . , 0]C half-hyper in vector × vector 2k N N copies of bifundamental k N Sp(n) 2n Zq Zq n n N copies of fundamental Zq Q fundamental of U(n) but charge Q of U(1) 1 (a) (b) with m, n the magnetic fluxes which are evaluated on the weights ρ, λ, respectively. A.1 Hilbert series A.1.1 Monopole formula The Hilbert series for the 3d N = 4 Coulomb branch is known as the monopole formula [26]. Schematically, the Hilbert series is computed as a sum over magnetic fluxes m valued in the GNO lattice Γ of the gauge group G. X P(t, m)wm t ∆(m) , (A.3) HSC = m∈Γ /W and W denotes the Weyl group of G. A bare monopole operator is characterised by the flux m as well as its conformal dimension ∆(m), which coincides with the third component of the SU(2)R spin. The factors P(t, m) dress a bare monopole operator by gauge invariants formed by the adjoint chiral multiplet of the residual gauge group H(m) . Lastly, w denotes the fugacity of the topological symmetry, assuming that G contains U(1) factors. A.1.2 Higgs branch Hilbert series The Higgs branch Hilbert series [23–25] for the 3d N = 4 quiver gauge theory relevant here is schematically obtained by a Molien-Weyl integral of the form HSH = G PE[χAdj t] Z dµG G 1 G PE[χR · χFF t 2 ] , (A.4) G where the numerator contains the character χAdj of the adjoint representation of the gauge group, while the denominator contains all matter fields characterised by their representations R under the gauge group G and the representations F under the flavour symmetry F . 36 SciPost Phys. 15, 033 (2023) A.1.3 Gauging a discrete 0-form symmetry. Suppose one is given a generating function H(z|t) which is a power series in t with coefficients 2πip that are Laurent polynomials in a U(1) fugacity z. Next, embed a Zq ,→ U(1) via (ζq ) p = e q with p = 0, 1, . . . , q − 1. Gauging this discrete Zq 0-form symmetry is realised in terms of the generating function via a discrete Molien-Weyl sum q−1 1 1X H (ζq ) p · y q |t , q p=0 (A.5) where y is the fugacity for the residual U(1)/Zq ∼ = U(1) symmetry. A.2 Superconformal index The 3d superconformal index can be computed as partition function on S 2 ×S 1 via localisation techniques, see [62–68] for details. Schematically, one arrives at 1 Z= |W m | m X rk(G) Y I dsi Icl · Ivec · Imatter , Trk(G) i=1 2πisi (A.6) where s denotes the gauge fugacities, which are valued in a maximal torus of the gauge group G. The magnetic fluxes m take values in the GNO-lattice of G. A flux m breaks G to the residual gauge group H m (the stabiliser subgroup of m inside G) with Weyl group WH m ≡ Wm . The integration contour is chosen to be the unit circle T for each si . The integrand is composed of classical contributions and the 1-loop determinants of the supermulitpelts. For concreteness, the G = U(N ) case is reviewed: The classical contribution is given by U(N ) Icl (w, m; n) = N Y PN (sa )n w a=1 ma , (A.7) a=1 with w the fugacity of the topological U(1) t symmetry. The N = 2 multiplets have the following 1-loop determinants: • 3d N = 2 Chiral multiplet of R-charge r coupled with unit charge to a gauge field: r (z, m| x) = Ichi x 1−r −1 z ∞ Y |m| 1 − (−1)m z −1 x |m|+2−r+2 j 2 1 − (−1)m z x |m|+r+2 j (−1)m z −1 x |m|+2−r ; x 2 ∞ , (−1)m z x |m|+r ; x 2 ∞ (A.8) j=0 = x 1−r z −1 |m| 2 with a U(1) holonomyQz around S 1 and the Z-valued magnetic flux m on S 2 . Here, the ∞ definition (z; q)∞ = j=0 (1 − zq j ) has been used. • 3d N = 4 Hypermultiplet transforming as bifundamental of U(N ) × U(M ) U(N )×U(M ) Ihyp (s1 , m 1 ; s2 , m 2 | x) = N Y M 1 Y −1 2 Ichi s1,a s2,b , m1,a − m2,b | x a=1 b=1 1 −1 2 · Ichi s1,a s2,b , m2,b − m1,a | x , 37 (A.9) SciPost Phys. 15, 033 (2023) Table 4: Centre symmetries of classical Lie algebras. algebra centre Z Ar Br Z r+1 Z2 Cr Z2 congruence Pr k=1 k · nk mod r + 1 n r mod 2 P r−1 2 n mod 2 j=0 2 j+1 n r−1 + n r mod 2 Dr , r even Z2 × Z2 P r−4 2 j=0 Dr , r odd P r−3 2 Z4 j=0 r n2 j+1 + r−2 2 n r−1 + 2 n r mod 2 2n2 j+1 + (r − 2)n r−1 + r n r mod 4 • 3d N = 2 vector multiplet for a U(N ) gauge group: Y |ma −m b | U(N ) Ivec (s , m| x) = x −|ma −m b | 1 − (−1)ma −m b sa s−1 x b (A.10) a 1 and gauge a discrete subgroup Zd ⊂ U(1) t of the k-th Cartan subgroup of the topological symmetry, provided d|k. One can repeat all steps as above, i.e. rewriting all contributions as U(k) ∼ = (U(1) × SU(k)) /Zk . The only step that requires modifications is (B.4). Recall Sk−1 h ∈ i=0 Z + ki and non-trivial contributions arise for d|(k · h). Since d|k, relevant fluxes h S dk −1 k i need to satisfy d · h ∈ Z, the summation range after Zd gauging becomes h ∈ i=0 Z + k . d Therefore, the resulting theory is given by (B.6), but the magnetic fluxes take values in S dk −1 (h, l, {n a }a̸=k ) ∈ p=0 Γ + dk · p . See also [31] for a related discussion. 41 SciPost Phys. 15, 033 (2023) B.2 Gauging discrete subgroup of topological symmetry revisited In this appendix, the aim is to consider a more general choice of discrete subgroup and to gain further evidence on the resulting theory. Starting from s1 m1 w1 {s2,j } {m2,j } w2 1 2 ... {sk−1,j } {mk−1,j } wk−1 {sk,j } {mk,j } wk {sk+1,j } {mk+1,j } wk+1 k−1 k k+1 ... {sN −1,j } {mN −1,j } wN −1 N −1 {yj } {kj } (B.24) N define the superconformal index of T [SU(N )] as I Y N −1 ℓ Y X dsℓ,a 1 I T [SU(N )] {sℓ }N −1 ; {m ℓ }N −1 , Z T [SU(N )] = ℓ=1 ℓ=1 |Wm ℓ | Tℓ a=1 2πisℓ,a ℓ ℓ=1 (B.25) m ℓ ∈Z I T [SU(N )] = N −2 Y ℓ=1 U(ℓ)×U(ℓ+1) Ihyp · N −1 Y ℓ=1 U(ℓ) Icl U(N −1)×U(N ) (sℓ , m ℓ ; sℓ+1 , m ℓ+1 | x) · Ihyp (s N−1 , m N−1 ; y, k| x) U(ℓ) (wℓ , m ℓ ; nℓ ) Ivec (sℓ , m ℓ | x) , and repeat the analogous step as in the monopole formula. Pick a gauge node U(k) and relabel the magnetic fluxes m k into a (U(1) × SU(k)) /Zk fluxes (h, l), see (B.9) and (B.2). Likewise, the U(k) gauge fugacities sk are transformed into U(1) × SU(k) fugacities (S, σ) via −1 sk,1 = S · σ1 , sk,i s k,k = S −1 · σi · σ−1 i−1 , −1 =S 1