3d $\mathcal{N}=4$ mirror symmetry with 1-form symmetry

The study of 3d mirror symmetry has greatly enhanced our understanding of various aspects of 3d $\mathcal{N}=4$ theories. In this paper, starting with known mirror pairs of 3d $\mathcal{N}=4$ 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 (0-form, 1-form, and 2-group) and the mirror maps between them.


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][4][5][6][7][8][9][10][11][12][13][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 = q 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][18][19][20][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, followed by non-abelian T [SU(N )] and T σ ρ [SU(N )] theories with non-abelian product gauge groups i U(N i ). 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 2 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.  x b x b+1 + x b+1 x b t 1 2 − (N −1)t · PE y 1 2 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 branch operators are unaffected by gauging a t/ f q , 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 q ⊂ U(1) t and the global 0-form symmetry PSU(N ) f for the SQED theory. The centre symmetry Z F = N of su(N ) f is generated by α F = ζ N , while the U(1) gauge group supports a Z G = N ·q centre generated by α G = ζ N ·q . The diagonal α D = (α G , α F ) generates a E = N ·q ⊂ Z G × Z F . The 1-form symmetry Γ [1] = q is generated by α N D = (α N G , α N F ) = (α N G , 1), which acts trivially on the matter content. The short exact sequence 0 → Γ [1] = q → E = q·N → Z F = N → 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 ); q ) [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 ); N ) → H 3 (BPSU(N ); q ). 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] A N −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] A N −1 contains a singlet. Such lines generate the group E = q·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.

SQED with discrete gauge factor
Next, revert the logic: gauge a q subgroup of the PSU(N ) f symmetry of SQED. Conversely, on the mirror side, one gauges a q 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. 3 In brief, lines L 1,2 are equivalent if there exists a local operator O at the junction between them. The set of equivalence classes {L}/ ∼ forms the Pontryagin dual 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: (L 1 , R 1 ) ∼ (L 2 , R 2 ) iff there exists a local operator transforming as R 1 ⊗ R * 2 (or R * 1 ⊗ R 2 ) at the junction of the lines. The equivalence classes give rise to E (Pontryagin dual of E), which encodes the interplay between the centres of gauge symmetry and 0-form symmetry. These groups fit into the short exact sequence 0 → Z → E → 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.      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 q 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 q -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 q 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 q ⊂ U(1) k ⊂ PSU(N ) t subgroup of the topological Cartan U(1) at the k-th node of the abelian quiver 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 where the centre symmetry ℓ ⊂ SU(ℓ) acts on the fundamental representation [1, 0, . . . , 0] A ℓ−1 with charge +1 under ℓ , for ℓ ∈ {k, N −k}, see also Table 4. Moreover, the k × N −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.
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 A k−1 and A N −k−1 Dynkin diagram. Generalising the arguments of [30], there are monopole operators transforming as [0, 0, . . . , q] A k−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 A k−1 Dynkin diagram and the 1-st node of A N −k−1 Dynkin diagram. Compared to the standard case of unit charge bifundamental hypermultiplets, the increased charge q modifies the appearing A ℓ 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. To illuminate this result, it is instructive to also consider the SQED side: where the two distinct sets of fundamentals are denoted as X andX (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 anti-chiral multiplets (outflow from the gauge node). Computationally, gauging a discrete f q is realised via the following flavour fugacities, see Appendix C.1 X a : a = 1, . . . , k : with ζ q ∈ f q . The x a and u d are weight space fugacities for su(k) and su(N −k), respectively. The first observation is that if k|q then the k centre symmetry of SU(k) is gauged, such that a global PSU(k) f factor arises. Similarly, if (N −k)|q the N −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 (2.10) The operator O has k × N −k centre charges (q mod k, −q mod (N −k)). Hence, the k × N −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 · 1 2 = q.

An illustrative example
One of the main messages of this paper is that gauging discrete q subgroups of the topological symmetry for quiver gauge theories T with unitary gauge nodes can result in theories T / t q which admit a simple quiver description. To illustrate this fact, consider U(k) SQCD with N ≥ 2k fundamental flavours k N (2.11) 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. Next, express the gauge group as U(k) ∼ = where k acts as centre on SU(k) and via k charge (k−1) on the U(1) factor. Rewriting U(k) magnetic fluxes m ∈ k into U(1) × SU(k) fluxes (h, l) requires the co-character lattice to be Γ = 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 w → zw 1 k . Next, gauge a discrete q 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:

Gauge a subgroup with q|k
If q|k then only the subgroup q ⊂ k is gauged. The theory becomes with magnetic lattice where the quotient k q signals that this discrete group is not gauged, in the sense of [31].
The resulting theory has a U(1) t × PSU(N ) f 0-form symmetry and a q 1-form symmetry. The potential interplay can be analysed via the action of the centre symmetries: defining (1) = q subgroup that acts trivial on the matter fields. By definition, this establishes the 1-form symmetry. The short exact sequence 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). Taking flavour charges into account, W q is equivalent to a flavour Wilson line transforming as ∧ k [0, . . . , 0, 1] A N −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 E = N ·q and the 1-form symmetry potentially forms a 2-group with the flavour symmetry (depending on the gcd(N , q)).

Gauge a discrete group q with k|q
If k|q then the SU(k) centre k is a subgroup of q and fully gauged. The theory becomes with magnetic lattice k , (2.14) The difference is now that the N hypermultiplets transform as SU(k) fundamental with charge q k ∈ under the U(1). This is indicated by the arrow, cf. Table 3.
In terms of symmetries, the theory T / t q has a U(1) t topological symmetry, PSU(N ) f flavour symmetry, a q 1-form symmetry. Moreover, inspecting the gauge-flavour centre symmetries shows: α G = (ζ k , ζ N ·q ) ∈ k × N ·q and α F = ζ N ∈ N . The diagonal generator α d = (α G , α F ) spans a E = N ·q , and the element N · α d = (ζ N k , ζ N N ·q , 1) generates a Γ (1) = k·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 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 ).  [1] = q . If one also keeps track of the flavour symmetry representations, one finds that O transforms as ∧ k [0, . . . , 0, 1] A N −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.

Symmetries
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?

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 q 0-form symmetry inside, say, the topological symmetry. The mirror of the resulting theory is then obtained by gauging a q 0-form symmetry inside the flavour symmetry. The question is how the q is embedded inside the flavour symmetry, given that the q 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 q ⊂ PSU(N ) t of a T [SU(N )] theory; one inquires about the nature of the resulting theory T [SU(N )]/ t q . Analogous to Section 2.2, see also Appendix B.1, for a specific q embedded in the k-th topological Cartan U(1) factor, the resulting theories T [SU(N )]/ t q 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 q|k or k|q, the resulting theory is given by wherein Γ denotes the standard integer lattice one assigns to the quiver based on [26]. The shifts by i d 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. The by gauging a q ⊂ 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 q symmetry in the overall U(1) flavour symmetry of one of the two sets. For concreteness, consider gauging the q on the set of k fundamental flavours: and Appendix D.2 provides exemplary Hilbert series computations that confirm the mirror symmetry between the theories with non-trivial 1-form symmetry. The (2.18) 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 where the centre symmetries ℓ 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 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 multiplets (outflow from the gauge node). Analogously to (2.9), one can perform the f q gauging by assigning (c.f. Appendix C.2) X a : a = 1, . . . , k : with ζ q ∈ f q . 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 imposing i y i = 1, i.e.
such that the k × N −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 · 1 2 = 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) = q 1-form symmetry. One finds that the centre generators of the combined gauge-flavour symmetry span a E = q·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 2 0-form symmetry leads to a mirror pair: 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 3 subgroup of the 0-form symmetry results in the pair: 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 3 centre symmetry of SU(3); hence, the global form is G = PSU(3) × U(1).

Example 2.
Considering discrete symmetries of the type q with q = a·k allows us to uncover equivalent descriptions. Consider T [SU (5)] and gauge a 6 0-form symmetry. Among the choices considered here, gauging a 6 ⊂ 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 25) or by turing the U(2) node into SU(2) together with charge 3 for the "new" U(1) gauge factor 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 6 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 { y 1 , y 2 , ζ 6 y 3 , ζ 6 y 4 , ζ 6 y 5 } , for (2.25), or {ζ 6 y 1 , ζ 6 y 2 , y 3 , y 4 , y 5 } , for (2.26), (2.27) with ζ 6 ∈ f 6 . But in both cases, the 2 × 3 centre symmetries are gauged by the discrete gauging of the 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 6 action from 2 fundamental flavours to the other 3 fundamental flavours. See also (C. 19)-(C.20).
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 t q 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 (2.28) and the global symmetry becomes where Q has N −1 charge q mod (N −1).

T σ ρ [SU(N )] theories
The class of linear quiver gauge theories with unitary gauge groups and fundamental or bifundamental hypermultiplets is given by the T σ ρ [SU(N )] theories [30], where ρ, σ are two partitions of N . For σ = ρ = (1, . . . , 1) ≡ (1 N ), the corresponding theory is simply . 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 T σ ρ [SU(N )]/ t q .
] is a linear quiver theory with gauge/flavour groups specified by a sequence of integers {N i } and {M i }, respectively. The partitions determine the integers as detailed in [30] and the quiver becomes For concreteness, take the node N k , with N k > 1, and gauge a N k ⊂ 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 . . . which has a non-trivial N k 1-form symmetry. Now, one constructs the mirror theory.
Gauging a N k ⊂ G ∨ f . The mirror quiver gauge theory of (2.30) is given by . In principle, this is straightforward but likely to be tedious. Alternatively, one can employ the following train of thought: The mirror theory T ρ σ [SU(N )] can be rewritten in an unframed form 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(N k ) in (2.30) into a special unitary gauge group SU(N k ). 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 Turning U(N k ) into SU(N k ) 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 M ∨ i attached to each U(1) depends precisely on the choice of SU(N k ), i.e. which k. The splitting can also occur where the same gauge node, for example, U(N ∨ 2 ) is connected to one U(1) gauge group with an edge of multiplicity M ∨ 2 − 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 T σ ρ [SU(N )]/ t N k . However, simply introducing a new U(1) gauge node leads to the ambiguity of the global form of the product gauge group, which can either be But this theory exhibits a t N k topological symmetry, see also Section 2.9. In order to generate the desired T σ ρ [SU(N )]/ t N k theory with gauge group G, one needs to gauge the discrete t N 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 removes a U(1) gauge degree of freedom. But one also needs to gauge a can be thought of as embedded in the U(1) that one has to be removed. Hence, the intermediate step is given by From the unframed quiver on the right, one has to ungauge a U(1) diag × U(1) and also keep a . This is because, as with the T [SU(N )] theories, one knows the should be the splitting of the flavour groups along with a discrete quotient. Schematically, one finds 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 N k charges from one set of fundamental flavours to another.

Example 1.
One can apply the above procedure to T σ ρ [SU (15)] where σ = (3 3 , 2 2 , 1 2 ) and ρ = (6, 4, 3, 1 2 ) 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 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  one U(1) t i=1,2,3 topological symmetry factor each. Let the one associated with U(2) be denoted by U(1) t 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) t 1 . Therefore, the 2 centre action can be absorbed into U(1) t 1 , resulting in a U(2) t topological symmetry factor.
One can also choose the other SU (2) in T σ ρ [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 Figure 5: Again starting from the mirror pair T σ ρ [SU (15)] and T ρ σ [SU (15)] with σ = (3 3 , 2 2 , 1 2 ) and ρ = (6, 4, 3, 1 2 ), one can gauge a different discrete 2 0-form symmetry to generate a another mirror pair with 2 1-form symmetry. See Appendix C.3 for the choice of f 2 in the mirror. symmetry in Figure 5 is simply 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 2 centre of the SU(2) t topological symmetry.

Example 2.
Consider the mirror pair T σ ρ [SU (9)] with ρ = (3, 2 3 ) and σ = (3 2 , 1 3 ) whose symmetry algebra is su(3) ⊕ u(1), as apparent from the balanced set of nodes. The global form is evaluated to be because the U(1) has charge −1 under the 3 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 A 2 Dynkin diagram (formed by the balanced nodes) on its first node, there exists a chiral ring operator transforming as Thus, the 3 centre can be compensated by suitable embedding into the U(1) t factor.
To create a new mirror pair, we can gauge a 2 symmetry on both sides of the dual theories. For example, we can gauge the topological t 2 symmetry on w 3 . The mirror map, as shown in (C.24), indicates that gauging the f 2 symmetry leads to the following mirror pair whose symmetry algebra is su(2) ⊕ u(1) ⊕ u(1). The Hilbert series (D.54) then suggests a symmetry group of because the centre 2 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] A 1 × (±1) under the associated SU(2) t × U(1) t topological symmetry factors. Therefore, the 2 centre symmetry then gives rise to a U(2) t isometry factor. The other topological Cartan U(1) t is uncharged under the 2 centre, as the gauge nodes are not connected to each other.
Gauging t q 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 t q 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.

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 2 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 2 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 )]/ t 2 obtained from gauging t 2 , 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 As expected, the flavour 0-form symmetry is always PSO(2N ) since the flavour centre Z is maximal. Moreover, only for T [SO(4N )]/ t 2 with a Spin(4l + 2) gauge node does the 1-form 2 2 . . .
. . .   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 W s 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, W 2 s 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 W 2 s 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 2 1-form symmetry forms a 2-group with the flavour symmetry. This is consistent with Table 1.
On  theory The only difference is that h is also charged under 2 . As in Sections 2.1.

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 > 2k 1 (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 t 2 symmetry.
. . . 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 made symmetric, as a global rotation can shift the f 2 action onto either set.
Therefore, one can gauge a 2 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 2 ⊂ 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 2 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 2 . 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.

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 D N -type Dynkin quiver: 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 q symmetries in this manifest 0-form symmetry.
Starting with the Dynkin quiver, there are two distinct choices: Firstly, gauging a l on a U(l) node which satisfies 2 < l < 2k, one obtains 7 which has a l 1-form symmetry and the Coulomb branch isometry algebra is For the global form, one can study the action of the centre symmetries of the non-abelian factors. One finds

24
SciPost Phys. 15, 033 (2023) Alternatively, gauging a l on a U(l) node which satisfies l ≥ 2k, one obtains 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 which confirms (2.47). To see this, recall from Appendix A.3 that the congruence class of the j-th fundamental representation of D N −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 which has trivial D-type centre charges. To see this, for N − l even, the 2 × 2 charges are (0, 2k mod 2) = (0, 0); while for N − l odd, the 4 charge is 2 · 2k mod 4 = 0. Thus, to compensate potential irreps that are non-trivial under 2k , one requires that Q has the following charges: which then confirms (2.49).
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 As in previous sections, one can also gauge a discrete t q along the topological fugacity w 1 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 fundamental under U(2) but is of U(1) charge q. In the mirror theory, the f q acts on a single fundamental flavour, as dictated by the mirror map (C.27). In summary, the mirror pair with and the global Higgs / Coulomb branch isometry is where Q is the topological fugacity of the left-most U(1) gauge node.

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 with nodes U(N 1,2 ) on the "short" side and U(N 3,4 ) 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 t q symmetry.
Gauging at the long side. To begin with, attempt to gauge a discrete t N 3 topological symmetry associated to the U(N 3 ) node at the long side, with N 3 > 1. As a first step, one rewrites (2.54) by expressing U(N 3 ) ∼ = (SU(N 3 ) × U(1))/ N 3 . By analogous arguments as in Appendix and the Coulomb branch moduli space is the same as that of (2.54). The green edges transform in the fundamental representation of U(N 1,2 ) and with charge κ under the U(1). Next, the N 3 symmetry is gauged. One obtains the following quiver description: where Γ is again short-hand for the integer magnetic lattice. This theory exhibits a N 3 1-form symmetry, by construction.
Gauging at the short side. Now, consider gauging a t N 2 on the topological fugacity associated to the U(N 2 ) gauge node, with N 2 > 1. Again, the first step is to simply rewrite U(N 2 ) ∼ = (SU(N 2 ) × U(1))/ N 2 . By adopting the arguments of Appendix B, one finds whose Coulomb branch coincides with that of (2.54). Moreover, the edges highlighted in green transform in the fundamental representation of U(N 1,2 ) and with charge κ under the U(1) node. As a next step, gauging the N 2 results in the following theory: 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 t q 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.

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 . . . This suggests that the su(2) ⊕ u(1) realise a maximal subalgebra in one sp(2) factor.

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.

A comment on F 4 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 F 4 Coulomb branch quiver of [40]. Table 2 summarises the resulting theories after a suitable t n 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 n .
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 F 4 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 F 4 quiver, the resulting moduli space cannot be identified with any known space and the procedure has been argued to be "invalid" in [41].
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 F 4 .

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, but the magnetic lattice Γ is extended to include shifted versions of the form 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 The aim is to remove this 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 + i k 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 + i k . Hence, the Coulomb branch Hilbert series, as well as the index for T , have the form which is message conveyed in [31].
It turns out that one can refine F T by introducing a k -valued fugacity z as follows: the U(1) t topological symmetry of the U(k) node appears in both the monopole formula and the index through the factor w k a=1 m a k . Upon rewriting into magnetic fluxes and some n ∈ . This means that one can introduce a discrete fugacity z to keep track of the k centre symmetry, setting w k → z such that w k·n+i k = z i . This fugacity remains even if the U(1) is taken to be non-dynamical. One ends up with It is now clear what happens if this discrete t k topological symmetry is gauged: the entire range of the summation collapses to the i = 0 sector, i.e., the integer lattice 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 a discrete t k topological symmetry. This k distinguishes between SU(k)×U(1) k ∼ = U(k) and SU(k) × U(1). Additionally, the gauging of the t k symmetry has introduced a k 1-form symmetry into T / t k .

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. It is worth recalling that the magnetic lattice for the left-hand side quiver has the form Γ ∪(Γ + 1 2 ) with Γ being the standard GNO integer lattice, as can be found in [31] and also see [45,[48][49][50][51][52][53][54][55] for examples with orthosymplectic quivers. The corresponding discrete t 2 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 . Equivalently, one can gauge a f 2 on the Sp(2) SQCD side, which then gives rise to the following pair of theories where again the left-hand side quiver has a magnetic lattice of the form Γ ∪ (Γ + 1 2 ). Gauging this t 2 has a by now clear consequence on the magnetic quiver, as the GNO lattice is reduced to the integer lattice. On the Sp(1) SQCD side, the corresponding f 2 is realised as follows: 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 (2.75) + 7763283t 8 + 23265515t 9 + 63954800t 10 + O t 11 , and the global symmetry group is PSU(5) × U(1).

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 q 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 / t q , obtained by gauging a discrete subgroup t q of the topological symmetry, may admit quiver descriptions if the discrete subgroup is suitably chosen. 3. The global form of the 0-form symmetries of (T / t q , T ∨ / f q ) 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.

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 2 / N Coulomb branch, and a single FI parameter that resolves the Higgs branch, specifically the minimal nilpotent orbit closure O su(N ) min . If we gauge a t q 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 2 / N ·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 test 11 via Hilbert series that shows 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 q factor of the 0-form symmetry has been gauged. One straightforward generalisation is to consider orthosymplectic quivers and gauge several t 2 topological symmetry factors associated to SO(n i ) nodes. The resulting theory is simply obtained by replacing the relevant SO(n i ) → Spin(n i ) and the 1form symmetry is the product group i ( t 2 ) i . Similarly, one could also entertain the thought of gauging several q i 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 t q 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 quiver is as follows: the set of fundamental flavours splits into several sets, with each subset being acted upon by f q 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 t q 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][60][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.

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 k 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] n 2k  with m, n the magnetic fluxes which are evaluated on the weights ρ, λ, respectively.

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.
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][24][25] for the 3d N = 4 quiver gauge theory relevant here is schematically obtained by a Molien-Weyl integral of the form where the numerator contains the character χ G Adj 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 .

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 that are Laurent polynomials in a U(1) fugacity z. Next, embed a q → U(1) via (ζ q ) p = e 2πip q with p = 0, 1, . . . , q − 1. Gauging this discrete q 0-form symmetry is realised in terms of the generating function via a discrete Molien-Weyl sum where y is the fugacity for the residual U(1)/ q ∼ = U(1) symmetry.

A.3 Centre symmetries of classical Lie algebras
Following [69,70], a representation labelled by Dynkin labels [n 1 , . . . , n r ] lies in a specific congruence class of the centre, as detailed in Table 4.

B Discrete gauging of T [SU(N )]
In this appendix, the monopole formula and the superconformal index are used to determine the theories obtained by gauging a discrete subgroup of the topological symmetry. Before proceeding with the details, the logic of the argument is summarised. Start from a given theory T and consider a function F(w , . . .), like a Hilbert series or index, depending on topological symmetry fugacities w and possibly other variables.
(i) Pick a gauge node U(k) with topological fugacity w and introduce a q -valued variable (ii) Gauging q on the level of F(w , . . .) is realised by (iv) After a change of variables (see below), the q gauging reduces to and restricting to q = a · k with a ∈ , the summation range of h collapses from But this leads to the collapse of the summation range of all other magnetic fluxes onto the integers too.
(v) Lastly, one obtains an expression of F written as a sum over integer-valued magnetic fluxes. One can then read off a Lagrangian description for the theory T / q .

B.1 Gauging discrete subgroup of topological symmetry
Consider the T [SU(N )] quiver and a variation thereof The aim is to show that To begin with, rewrite the U(k) fluxes in (U(1) × SU(k)) / k . Use the following map where 1 is the Cartan generator for the diagonal U(1) ⊂ U(k) and {E i } are the Cartan generators for SU(k). One finds And one finds for contributions of the fundamental U(k) weights with µ i (l) are the weights 12 of the fundamental SU(k) representation evaluated on the fluxes l. The summation range for {m k,i } is k ∩ {m k,1 ≥ m k,2 ≥ . . . ≥ m k,k }, due to the Weyl group action, and translates into with W A k−1 the Weyl group action of SU(k). Moreover, one straightforwardly verifies where Φ + (su(k)) denotes the set of positive roots, here expressed in the weight basis. 13 Next, consider the hypermultiplet with the adjacent gauge nodes where Γ denotes the underlying integer lattice. Next, focus on the fundamental flavour contributions which is exactly the contribution of N copies of U(N −1) × U(1) bifundamentals. Collecting all the results, one ends up with which is indeed the monopole formula for (B.6).

Corollary. Consider the
Therefore, the resulting theory is given by (B.6), but the magnetic fluxes take values in See also [31] for a related discussion.

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 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)) / k fluxes (h, l), see (B.9) and (B.2). Likewise, the U(k) gauge fugacities s k are transformed into U(1) × SU(k) fugacities (S, σ) via Consider the vector multiplet which reduces to the SU(k) vector multiplet contribution. Using the µ(σ) a µ(σ) −1 b ≡ α(σ) a,b yields the root contributions, see also (B.10) and recall the use of weight/Dynkin basis. Note also that the U(1) vector multiplet part receives a fitting classical contribution I ; while such a term is absent for the SU(k) part of the vector. Next, inspect the hypermultiplet contributions wherein the fluxes and fugacities at the k + 1-st node have been redefined as follows: Similarly for the other connected hypermultiplet: using a similar redefinitioñ such that the contour integration transitions nicely into the new variables, again integrated along the unit circle.
Discrete gauging. Next, gauge a q subgroup of the U(1) t Cartan subgroup of the topological symmetry PSU(N ). To do so, one performs a discrete Molien-Weyl sum with f to be determined. Following the rewriting induced on the k-th node U(k) → U(1) × SU(k), the affected terms are Using the same argument as in (B.4) and recalling that h ∈ k−1 j=0 ( + j k ), one might want to consider the case q = a · k with a ∈ . Then (a · k)|(k · h) implies a|h, which in particular requires that h is integer-valued. This implies the collapse of the entire summation range onto the integers. Moreover, the h summation is further restricted to h ∈ a . To be explicit Defining h = a ·h withh ∈ and focusing on the terms involvingh suggests to redefine the U(1) gauge fugacity and topological fugacity as follows: which coincides with (B.21). As a consequence, Putting all pieces together, one obtains which is the superconformal index for wherein the N copies of hypermultiplets between U(N −1) and U(1) transform as (N−1) × (−a), i.e. the U(1) charge is a.

C Mirror maps
Suppose that there is a G-type global symmetry with G being a classical or exceptional Lie algebra. The root space fugacities w i are related to the weight space fugacities η i by the Cartan matrix Most relevant here is the closed formula for the inverse of the A N −1 Cartan matrix given in [73].

C.1.1 Standard mirror map
Suppose that the abelian mirror has topological fugacities w i with i = 1, . . . , N −1, i.e. root space fugacities for su(N ). The mirror theory, SQED with N fundamental flavours, has U(N ) fugacities y f with f = 1, . . . , N . Both sets of fugacities are related to the su(N ) weight space fugacities η i via the following two maps and the inverse Cartan matrix is given in (C.2). The combination can be simplified by observing for f = 1, . . . , N and j = 1, . . . , N −1. Hence, the mirror map becomes for f = 1, . . . , N .

C.1.2 Mirror map after gauging
Suppose that one gauges a q on the U(1) t generated by w k and employs the following parameter map using the Cartan matrices of A k−1 and A N −k−1 , respectively. Using (C.7) for A k−1 and A N −k−1 , the map for w k can be expressed as A straightforward computation yields the new mirror map using the parametrisation (C.8) for the A k−1 fugacities x i and the A N −k−1 fugacities u i .

Remark.
The map (C.10) assigns ζ q charges to each y f ; however, one can remove any overall U(1) phase by a gauge transformation. This leads to two convenient choices: either the first k fundamental flavours are charged under q such that only the last N −k fundamental flavours are non-trivially charged under q . In principle, one could also extend the overall rotation to include Q, but there is no need to do so.

C.2 T [SU(N )] theories
One can construct the mirror map explicitly.

Denote the Coulomb branch root space fugacities of T [SU(N )] by {w
. These can be mapped to the Coulomb branch weight space fugacities via the A N −1 Cartan matrix C i j : (C.12) The Higgs branch U(N ) flavour fugacities are { y a } N a=1 , which are reduced to SU(N ) fugacities The self-mirror property of T [SU(N )] is the reflection in the exchange ω i ↔ η i . The aim is to express the natural flavour fugacities { y a } of the theory in terms of the Coulomb branch fugacities of the mirror. The first step is ( y 1 , y 2 , . . . , y N ) → η 1 , (C.14) This map can be made explicit by using the inverse of the A N −1 Cartan matrix (C.2). Analogous to the abelian case, the combined map reads which is the explicit form of (C.14).

C.2.2 Mirror map after gauging
The next step is utilising the parameter map (B.21) established in Appendix B.1 Lastly, to make contact with the global symmetries, one uses the Cartan matrix for A k−1 and A N −1−k in a standard fashion and one needs to redefine Applying this to (C.15), one finds which displays the split into A k−1 fugacities x j and A N −k−1 fugacities u j .

Remark.
Analogously to the SQED case, one can simplify the ζ q dependence in (C.19) by a suitable overall U(1) rotation. A convenient choice is then given by such that only the first k fundamental flavours are charged under q .
Again, this confirms the two choices in Figure 4.

C.4.1 Standard mirror map
For the balanced D N -type unitary quiver, the mirror map to the flavour symmetry of the Sp(k) SQCD mirror with N fundamental flavours is given by • Let y i denote the U(N ) flavour fugacities.
• Denote by x i the SO(2N ) weight space fugacities. The relation between both is established via (C.25) • Denote by w i the root space fugacities of D N , which are related to the weight space fugacities x i via the D N Cartan matrix (C.26) • Thus, one finds the map between fundamental flavour fugacities y i and root space fugacities w i to be (C.27)

C.4.2 Mirror map after gauging
Suppose that one gauges a discrete q symmetry at the gauge node with topological fugacity w l . Then, analogous to the T [SU(N )] derivation, the fugacity map to the quiver after gauging is simply given by where N i denotes the rank of the i-th node. The remaining topological fugacities w i̸ =l are identified before and after gauging. Lastly, one needs to redefine v such that A l−1 and D N −l representations become manifest. For this, one uses For l ≤ N − 2, the mirror map is given by 14 which clearly displays that the first l fundamental flavours transform under su(l) × U(1), and the remaining N − l fundamental flavours transform under so(2N − 2l).

C.5 O(2k) SQCD and its C-type unitary mirror quiver
Consider the O(2k) SQCD with N fundamental hypermultiplets. The mirror theory is a balanced C-type Dynkin quiver with N gauge nodes.

D Explicit Hilbert series results
In this appendix, some exemplary Hilbert series calculations are presented. Matching the Hilbert series can be used as the stringent test to check the dualities and find the global topological or flavour symmetry groups.

D.1 Linear Abelian quiver
Consider the abelian quiver gauge theory

Explicit character expansions indicate the following global forms
which then confirms the general formula (2.7).
To exemplify, we provide the Hilbert series for N = 5, k = 2, q = 2 here. The perturbative expansion reads with φ m 1 ,m 2 = φ m 1 ,m 2 (u i ) and χ n 1 = χ n 1 (x 1 ) the SU(3) and SU(2) characters of irreps [m 1 , m 2 ] and [n 1 ], respectively. This follows via the Higgs branch fugacity map and the Coulomb branch fugacity map Here, {u i } and x 1 are the corresponding weight space fugacities. Q denotes the U(1) fugacity. In fact, due to the abelian nature, one can even compute the full highest weight generating function (HWG) (D.10) 2,3 , and n 1 the SU(5), SU(4), and SU(2) highest weight fugacities, respectively.

D.2 T [SU(N )] theories
We move on to the examples of T [SU(N )] theories. The employed fugacity maps follow from (2.18) in combination with (2.21).
The symmetry algebra is su (3)   Use the fugacity map C : H : to an x 1 weight space fugacity of A 1 , u i the weight space fugacities of A 3 , and a U(1) variable Q. The Coulomb branch (or Higgs branch) Hilbert series reads The terms in red correspond to the operator O (and its conjugate) in (2.22). The global symmetry is PSU(2) x 1 × SU(4) u 1 ,u 2 ,u 3 × U(1) Q / 4 and Q has 4 charge 2 mod 4. such that x 1 is an A 1 fugacity and { y 1,2 } are A 2 fugacities. The perturbative monopole formula for the left-hand-side quivers reads

D.5 T [SO(2N )] theories
In this appendix, computational details for the T [SO(2N )] theories are provided. For orthosymplectic quivers, the topological symmetries visible in the UV Lagrangian are severely limited. For an SO(k) gauge group, there is only a 2 valued topological 0-form symmetry. For SO (2), there exists a whole U(1) topological 0-form symmetry. Thus, to confirm mirror symmetry after such a 2 is gauged, one needs to identify the 2 in the original mirror pair. Therefore, it is sufficient to provide the 2 -refined Hilbert series of the original mirror pair to demonstrate agreement after gauging. The Hilbert series after gauging the discrete 2 symmetry is simply obtained by averaging over 2 .

D.5.2 T [SO(8)] theories
For T [SO (8)], one can gauge 2 ⊂ PSO(8) t , which corresponds to gauge a 2 factor inside the flavour symmetry for the mirror theory. Now one can choose to gauge this t 2 for the SO(4) or SO(6) gauge node. t 2 of SO(4). Gauging the t 2 of the SO(4) gauge node leads to a "2+2" splitting of the fundamental flavours. Before gauging, the discrete fugacities are attributed as follows: The Coulomb branch Hilbert series of the left-hand side (which equals the Higgs branch Hilbert series of the right theory) reads HS = 1 + (12 + 16z 2 )t + (213 + 192z 2 )t 2 + (1984 + 2048z 2 )t 3 + . . . , with the 2 fugacity z 2 = a. t 2 of SO(6). Gauging the t 2 of the SO(6) gauge node leads to "3+1" splitting of the fundamental flavours in the mirror theory. Before gauging, the discrete fugacities are attributed as follows:

D.6 Sp(k) SQCD and orthosymplectic mirrors
The logic is as in Appendix D.5, to evaluate the Hilbert series of the theories after 2 gauging, one evaluates the 2 -refined Hilbert series of Sp(k) SQCD and its mirror theory. Once agreement is found, the theories after gauging have agreeing the Hilbert series by construction.

D.6.1 Sp(2) SQCD, 5 flavours
Consider Sp(2) with 5 fundamental flavours. . Gauging the t 2 of one of the SO(2) gauge nodes corresponds to a "1+4"splitting of the fundamental flavours in the mirror theory. Before gauging, the discrete fugacities are assigned as follows: The Coulomb branch Hilbert series of the right theory (which agrees with the Higgs branch Hilbert series of the left theory) reads HS 1+4 = 1 + (16a 1 + 29)t + (448a 1 + 532)t 2 + . . . , (D. 60) with the 2 -fugacity a 1 . t 2 inside SO(4). Gauging the t 2 of one of the SO(4) gauge nodes corresponds to a "2+3"splitting of the fundamental flavours in the mirror theory. Before gauging, the discrete fugacities are assigned as follows:   (4). Gauging the t 2 of one of the SO(4) gauge nodes corresponds to a "2+4"splitting of the fundamental flavours in the mirror theory. Before gauging, the discrete fugacities are assigned as follows:   (5). Gauging the t 2 of the SO(5) gauge node corresponds to a "3+3"-splitting of the fundamental flavours in the mirror theory. Before gauging, the discrete fugacities are assigned as follows:

D.7 Sp(k) SQCD and D-type mirrors
In this appendix, computational evidence for the general results of Section 2.7 is provided. The relevant mirror map is discussed in Appendix D.6.