Exploring Seiberg-like N-alities with Eight Supercharges

We show that a large subclass of 3d $\mathcal{N}=4$ quiver gauge theories consisting of unitary and special unitary gauge nodes with only fundamental/bifundamental matter have multiple Seiberg-like IR duals. A generic quiver $\mathcal{T}$ in this subclass has a non-zero number of balanced special unitary gauge nodes and it is a good theory in the Gaiotto-Witten sense. We refer to this phenomenon as"IR N-ality"and the set of mutually IR dual theories as the"N-al set"associated with the quiver $\mathcal{T}$. Starting from $\mathcal{T}$, we construct a sequence of dualities by step-wise implementing a set of quiver mutations which act locally on the gauge nodes. The associated N-al theories can then be read off from this duality sequence. The quiver $\mathcal{T}$ generically has an emergent Coulomb branch global symmetry in the IR, such that the rank of the IR symmetry is always greater than the rank visible in the UV. We show that there exists at least one theory in the N-al set for which the rank of the IR symmetry precisely matches the rank of the UV symmetry. In certain special cases that we discuss in this work, the correct emergent symmetry algebra itself may be read off from this preferred theory (or theories) in addition to the correct rank. Finally, we give a recipe for constructing the 3d mirror associated with a given N-al set and show how the emergent Coulomb branch symmetry of $\mathcal{T}$ is realized as a UV-manifest Higgs branch symmetry of the 3d mirror. This paper is the second in a series of four papers on 3d $\mathcal{N}=4$ Seiberg-like dualities, preceded by the work [arXiv:2210.04921].

1 A brief summary of the paper 1.1 Background UV/IR dualities in d ≤ 6 space-time dimensions provide one of the most powerful tools for probing non-perturbative/strongly-coupled physics in Quantum Field Theories.Seiberg duality [2] is one of the classic examples of an IR duality for a pair of N = 1 gauge theories in four dimensions.There are dualities in lower space-time dimensions that share certain broad features of the Seiberg duality and are often referred to as Seiberg-like dualities.In d = 3, a rich family of such dualities exist for N ≥ 2 gauge theories with Chern-Simons levels [3,4].
Seiberg-like dualities in 3d N = 4 theories without any Chern-Simons terms have been far less studied in the literature.The known examples involve theories which are ugly and bad [5,6] in the Gaiotto-Witten sense [7], and in the latter case the dual should be understood as a low energy effective field theory on a certain singular loci of the moduli space of the original theory [8].
Recently, a family of Seiberg-like dualities were proposed for pairs of 3d N = 4 gauge theories [1] where both theories are good in the Gaiotto-Witten sense.The theory on one side of the duality is a U (N ) SQCD with N f ≥ 2N − 1 fundamental hypermultiplets and P ≥ 1 hypermultiplets in the determinant representation of U (N ).The latter hypermultiplets are referred to as Abelian hypermultiplets, since they are only charged under the central U (1) subgroup of the U (N ) gauge group.These theories will be denoted as T N N f ,P and the IR duality associated with the theory T N N f ,P will be denoted as D N N f ,P .The quiver representation of T N N f ,P is given below (the reader is referred to Section 2.1 for details on the quiver notation): N N f P One of the main results of [1] was to show that the dualities D N N f ,P are non-trivial in the following parameter ranges of N f and P for a fixed N : (N f = 2N + 1, P = 1), (N f = 2N, P ≥ 1) and (N f = 2N − 1, P ≥ 1).These dualities are reviewed in Section 2.2 and summarized in Table 1.Here, we list the gauge groups and the matter content of the dual pairs in each case: The dual pair involves the theory T N 2N +1,1 and an SU (N + 1) theory with N f = 2N + 1 fundamental flavors.
• Duality D N 2N,P : This is given by the self-duality of the theory T = T N 2N,P .
Given the dualities D N N f ,P for the parameter ranges mentioned above, one can use them to construct IR dualities involving more complicated quiver gauge theories.Starting from an appropriate quiver gauge theory T that we will specify momentarily, one can implement these dualities locally at different gauge nodes in a step-wise fashion.This generates a duality sequence and an associated set of quiver gauge theories that are all IR dual to the original theory.We will call this phenomenon an IR N-ality and the set of theories IR dual to each other as an N-al set which includes T .The local operations at the gauge nodes will be referred to as quiver mutations in the spirit of similar operations for Seiberg duality.
In this paper, we will study IR N-ality associated with a quiver T of the following type.The quiver T has both unitary as well as special unitary gauge nodes with only fundamental/bifundamental matter, such that the number of balanced special unitary gauge nodes in the quiver is non-zero 1 .We will demand in addition that the quiver T be good in the Gaiotto-Witten sense.We will refer to this class of quivers as class X .With this quiver T in class X as a starting point, we will implement the allowed quiver mutations to generate the duality sequence and read off the N-al set of quivers.It turns out that a generic theory in the N-al set does not belong to class X .It is still a quiver gauge theory with unitary and special unitary gauge nodes, but the matter sector involves a non-zero number of Abelian hypermultiplets in addition to the fundamental/bifundamental hypermultiplets.We will present a general recipe for constructing the IR N-ality, and explicitly work out the N-al set for a subclass of quivers in X i.e. linear quivers with a single special unitary gauge node.We will also give a recipe for constructing the 3d mirror associated with the N-al set of a generic T ⊂ X and determine it explicitly for the examples studied in this paper.
In principle, one may start from any theory in the N-al set and proceed to generate the others by implementing appropriate quiver mutations.However, the quiver T ⊂ X is a natural starting point in the following sense.A generic quiver in an N-al set containing T has an emergent Coulomb branch global symmetry in the IR such that the rank of the IR symmetry is always greater than the rank manifest in the UV.The quiver T has the minimum UV-manifest rank in the N-al set that it belongs to.It turns out that the duality sequence constructed with T as the starting point leads to at least one theory for which the UV-manifest rank matches the rank of the emergent IR symmetry.For a large class of examples, which includes the quiver families discussed in this paper, the full emergent symmetry algebra can be read off from this special quiver (or quivers) belonging to the N-al set.
The paper is organized in the following fashion.In the rest of this section, we summarize the main results of the paper.In Section 2, we present the general recipe for constructing the N-al set starting from the theory T , after briefly reviewing the quiver notation and the dualities D N N f ,P .In Section 3, we implement this procedure and explicitly work out the N-al set for the case where T is a linear quiver with a single balanced special 1 We will define the balance parameter of an SU (Nα) gauge node as e unitary gauge node and a single unitary gauge node.We extend the construction to the case where T is a linear quiver with three gauge nodes -a single balanced special unitary node and two unitary gauge nodes, in Section 4. In Section 5 we give a recipe for constructing the 3d mirror of a given N-al set and work out several examples.The appendices contain various computational details of the results that appear in the main text.
1.2 Summary of the main results

Systematics of N-ality: Recipe and Examples
One of the main results of this paper is to present a systematic procedure for constructing the N-al set of theories given a quiver T in class X , and implement the procedure in simple examples where T is a linear quiver involving a single balanced special unitary node.The general construction is discussed in Section 2.3.As mentioned earlier, the theories in the N-al set can be read off from a duality sequence generated by the action of certain quiver mutations on T .The allowed quiver mutations, each of which arises from one of the dualities listed above (and reviewed in Section 2.2) are classified in Section 2.3.
The first example, discussed in Section 3, involves the following 3-parameter family of linear quivers: The above is a linear quiver with a U (N 1 ) × SU (N ) gauge group.The SU (N ) gauge node is balanced which implies that the integers (M 1 , M 2 , N 1 , N ) satisfy the constraint N 1 +M 2 = 2N −1, and U (N 1 ) node is either overbalanced or balanced i.e.M 1 +N = 2N 1 +e with e ≥ 0. The N-ality and the associated duality sequence depends crucially on the specific value of e.We consider the two following sub-cases: 1. Overbalanced unitary node (e ̸ = 0): The N-ality and the duality sequence is discussed in Section 3.2.For any e > 2, the N-ality is simply a duality given in Fig. 4. For e = 1, one obtains a triality shown in Fig. 5, while for e = 2 one also obtains a different triality shown in Fig. 6.
2. Balanced unitary node (e = 0): The N-ality, discussed in Section 3.3, is yet another triality, and the associated duality sequence is given in Fig. 7.
The second example, discussed in Section 4, involves the following 4-parameter family of linear quivers: The above is a linear quiver with a U (N 1 ) × SU (N ) × U (N 2 ) gauge group.The SU (N ) gauge node is balanced i.e.N 1 + N 2 = 2N − 1, while the unitary gauge nodes can be either balanced or overbalanced with M 1 + N = 2N 1 + e 1 and M 2 + N = 2N 2 + e 2 such that e 1 , e 2 ≥ 0. The N-ality and the associated duality sequence depends crucially on the specific values of the doublet of in integers (e 1 , e 2 ) as follows: 1. Overbalanced unitary nodes (e 1 ̸ = 0, e 2 ̸ = 0): For e 1 > 2, e 2 > 2, the N-ality is simply a duality as shown in Fig. 8 with details discussed in Section 4.2.The special cases of e 1 , e 2 = 1, 2 can be worked out in a fashion similar to the 2-node quiver.

2.
A single balanced node (e 1 = 0, e 2 ̸ = 0): For e 2 > 2, the N-ality is a triality shown in Fig. 9 with details discussed in Section 4.3.The special cases of e 2 = 1, 2 can again be worked out in a fashion similar to the 2-node quiver.
3. Balanced unitary nodes (e 1 = 0, e 2 = 0): The N-ality in this case, discussed in Section 4.4, is a hexality with the associated duality sequence given in Fig. 10.
In each case listed above, the quiver mutations may be implemented in terms of the sphere partition function.For the 2-node and the 3-node quiver, the relevant computational details can be found in Appendix A and Appendix B respectively.

Emergent global symmetry and N-ality
We mentioned earlier that a quiver T in class X generically has an emergent Coulomb branch symmetry in the IR.The second main result of this paper is to show that there exists at least a single quiver in the N-al set for which the rank of this emergent symmetry of T becomes manifest.In the special case where T is a linear quiver, one can in fact read off the correct symmetry algebra itself from the aforementioned quiver (or quivers).From the examples presented in this paper, we will explicitly confirm this fact.
Consider first the case of the 2-node quiver discussed above.The Coulomb branch symmetry manifest in the UV corresponds to the u(1) topological symmetry associated with the U (N 1 ) gauge node.The IR emergent symmetry of the theory is discussed in Section 3.1 and may be summarized as follows: 1. Overbalanced unitary node (e ̸ = 0): In this case g C = u(1) ⊕ u (1), where the two u(1) factors correspond to the topological symmetry for the U (N 1 ) gauge node and a monopole operator associated with the SU (N ) gauge node.
2. Balanced unitary node (e = 0): In this case , where the origin of the various factors in terms of the monopole operators and the topological symmetry generator is explained in Section 3.1.
For the case of the balanced unitary node i.e. e = 0, the duality sequence in Fig. 7 leads to a triality of quivers.The quivers in the N-al set (with N=3) are given in Fig. 1.

The quiver (T ∨
2 ) is the theory at which the duality sequence terminates.Firstly, note that the rank of the Coulomb branch symmetry for (T ∨ 2 ) manifest in the UV precisely matches that of g C for the theory T .Secondly, one may read off g C itself from the quiver (T ∨ 2 ) using the following argument.Recall that in a good linear quiver with unitary gauge nodes, an array of k balanced nodes contributes an su(k + 1) factor to the Coulomb branch symmetry while every overbalanced node contributes a u(1) factor [7].Given a generic quiver with unitary gauge nodes, which can be constructed using a set of linear quivers glued together Figure 1: Triality of quivers for the case of a balanced unitary gauge node in the 2-node quiver.
by gauging flavor symmetries, the Coulomb branch symmetry is expected to be of the form: where i labels the constituent linear quivers and g i C is the Coulomb branch symmetry of the i-th linear quiver.Applying this argument to (T ∨ 2 ), where the U (1) and the U (N − 1) gauge nodes are balanced while the U (N 1 − 1) node is overbalanced, one can read off the Coulomb branch symmetry as g C = su(2) ⊕ su(2) ⊕ u(1).
For the case e ̸ = 0, the emergent IR symmetry can be similarly read off from the duality sequence associated with the given value of e.We refer the reader to Section 3.2 for further details.
Consider the case of the 3-node quiver next.The Coulomb branch symmetry manifest in the UV corresponds to the u(1) ⊕ u(1) topological symmetry associated with the U (N 1 ) and the U (N 2 ) gauge nodes respectively.The IR emergent symmetry of the theory is discussed in Section 4.1 and may be summarized as follows: 1. Overbalanced unitary nodes (e 1 ̸ = 0, e 2 ̸ = 0): In this case g C = u(1)⊕u(1)⊕u (1), where two of the u(1) factors correspond to the topological symmetries of the unitary gauge nodes and the third arises from a monopole operator charged under the SU (N ) gauge node.

2.
A single balanced node (e 1 = 0, e 2 ̸ = 0): In this case , where the origin of the various factors in terms of monopole operators and topological symmetry generators is discussed in Section 4.1.
For the case where both unitary nodes are balanced i.e. e 1 = 0, e 2 = 0, the duality sequence in Fig. 10 leads to a hexality of quivers which are shown in Fig. 2. Note that the rank of the Coulomb branch symmetry manifest in the UV increases monotonically along the duality sequence in this case.The quiver (T ∨ 5 ) is the theory at which the duality sequence terminates.To begin with, the rank of the Coulomb branch symmetry manifest in the UV for (T ∨ 5 ) precisely matches that of g C for the theory T .The quiver (T ∨ 5 ) involves a pair of balanced unitary nodes -U (N 1 − 1) and U (N 2 − 1), separated by an overbalanced Figure 2: Hexality of quivers for the case where both unitary gauge nodes in the 3-node quiver are balanced.
node U (N − 2) which in turn is attached to a tail of three balanced U (1) nodes.Using our intuition for linear quivers as discussed above, the two isolated balanced nodes contribute an su(2) factor each to the Coulomb branch global symmetry while the overbalanced node gives a u(1) factor.Finally, the tail of three balanced U (1) nodes gives an su(4) factor.
One can therefore read off the Coulomb branch symmetry g C = su(2) ⊕ su(2) ⊕ su(4) ⊕ u(1) from the quiver (T ∨ 5 ), where g C is indeed the emergent symmetry algebra of the theory T as noted above.
For the cases (e 1 ̸ = 0, e 2 ̸ = 0) and (e 1 = 0, e 2 ̸ = 0), the emergent global symmetry can be similarly read off from the duality sequence associated with the quiver T for the given values of e 1 , e 2 .We refer the reader to Section 4.2-Section 4.3 for further details.3d mirror of an N-al sequence: Recipe and Examples N = 4 theories in three space-time dimensions have yet another IR duality which exchanges the Higgs and the Coulomb branches of the moduli space in the deep IR -3d mirror symmetry [9].The quiver gauge theory T and the associated N-al set of quivers will therefore have a common 3d mirror.In Section 5.1, we discuss a general recipe for constructing this 3d mirror, using the technology of S-type operations introduced in [9].For a completely generic quiver gauge theory T in class X , the 3d mirror is not guaranteed to be a Lagrangian theory.However, for the case of T being a linear quiver, one can show that the 3d mirror is indeed Lagrangian and one can explicitly work out the quiver, given the duality sequence associated with T .In Section 5.2 and Section 5.3, we apply the recipe of Section 5.1 to the two-node linear quiver and the three-node linear quiver respectively and construct the corresponding 3d mirrors.The computational details of the construction can be found in Appendix C.
The emergent Coulomb branch symmetry for the linear quivers T is realized in the corresponding 3d mirrors T ′ as a Higgs branch global symmetry which is manifest in the UV.One can readily check this for the two-node and the three-node quivers.Let us consider a specific example of the two-node linear quiver with e = 0, where M 1 = 2, N 1 = 3, N = 4, and M 2 = 4.The corresponding 3d mirror T ′ is shown in the top right corner of Fig. 3. Recall that, g (T ) C = su(2) ⊕ su(2) ⊕ u(1), while from the quiver T ′ one can read off g (1).One can similarly consider the case of a three-node quiver with e 1 = e 2 = 0, where M 1 = 2, N 1 = 3, N = 4, N 2 = 4, and M 2 = 4.The 3d mirror is given by the quiver T ′ in the bottom right corner of Fig. 3.The Higgs branch symmetry of T ′ can be read off as g 1), which evidently matches the emergent Coulomb branch symmetry of the theory T as we discussed above.

IR Dualities involving quivers without Abelian hypermultiplets
As an interesting byproduct of the construction of IR N-ality presented in this paper, one can obtain IR dualities for a pair of theories which both belong to class X .For the subclass of linear quivers that we study in this paper, this implies IR dualities between such linear quivers with unitary and special unitary gauge groups.We obtain an explicit example of such an IR duality from the duality sequence in Fig. 6 associated with the triality of the two-node quiver with e = 2.The dual pair is given as follows: The theory dual to T is also a two-node linear quiver with gauge group SU (N 1 + 1) × U (N − 1).In addition, one can check that the unitary gauge node in the dual theory is also overbalanced with e = 2.The duality holds for generic labels in the quiver T with the constraint e = 2, which implies that the above is a 2-parameter family of dualities.
2 From duality to N-ality: Quivers and mutations In this section, we will present the general construction of the N-al set of theories starting from a given 3d N = 4 quiver gauge theory of class X , after introducing the quiver notation used in this work and briefly reviewing the Seiberg-like dualities discussed in [1].

Quiver notation
Let us begin by discussing the quiver notation that was introduced in [1] and will be used for the rest of this paper.We will consider quiver gauge theories with SU (N ) and U (N ) gauge nodes, hypermultiplets in the fundamental/bifundamental respresentation of both types of gauge nodes, and a number of hypermultiplets in the determinant/anti-determinant representations of the unitary gauge nodes.A generic gauge theory of this type will be denoted by a quiver diagram of the following form: The gauge group and the matter content of the theory can be read off from the quiver diagram using the following dictionary: 1.A node labelled N denotes a U (N ) vector multiplet.

A node
labelled N denotes an SU (N ) vector multiplet.
3. A box labelled F attached to a gauge node denotes F hypermultiplets in the fundamental representation of the gauge group associated to that node.

A thin black line between two gauge nodes represents a bifundamental hypermultiplet.
A collection of M bifundamental hypermultiplets between the same pair of gauge nodes is denoted by a thick black line with the label M .

5.
A box labelled F attached to a unitary gauge node denotes F hypermultiplets in the determinant representation.
6.A thin blue line connecting multiple unitary gauge nodes denotes an Abelian hypermultiplet in the determinant/anti-determinant representation of those gauge nodes.We will show the charges {Q i } of the Abelian hypermultiplet explicitly in the quiver diagram where Q i = ±N i , with N i being the rank (and therefore the label) of the i-th unitary gauge node.A collection of P such hypermultiplets connecting the same set of nodes will be denoted by a thick blue line with the label P .
2.2 Dualities in the T N N f ,P theories: A review In this section, we will briefly review the Seiberg-like dualities for the theories T N N f ,Pa U (N ) SQCD with N f fundamental flavors and P Abelian hypermultiplets.In [1], it was demonstrated that such dualities, denoted as D N N f ,P , exist for the parameter ranges (N f = 2N + 1, P = 1), (N f = 2N, P ≥ 1) and (N f = 2N − 1, P ≥ 1) for a given N .The dual pairs in each case, as well as the 3d mirror theory associated with the pair, are summarized in Table 1.

Duality
Theory T IR dual T ∨ 3d mirror Table 1: Summary of the IR dualities for the T N N f ,P theories, and the 3d mirror associated with each dual pair.For D N 2N,P and D N 2N −1,P , the tail of U (1) gauge nodes in the 3d mirror has precisely P gauge nodes.
The IR dualities listed above can be checked using a variety of supersymmetric observables.In [1], they were checked explicitly using the three-sphere partition function and the Coulomb/Higgs branch Hilbert Series.In this paper, we will make use of the three-sphere partition function extensively to construct the duality sequences 2 .We will therefore also review the partition function identities associated with the dualities below in a form that will be useful later in the paper.For further details on these dualities, we refer the reader to Section 3 and Section 4 of [1].
The dual pair (T , T ∨ ) involves the theory T = T N 2N +1,1 and the theory T ∨ -an SU (N + 1) theory with N f = 2N + 1 fundamental flavors.Let m and m ab be the real masses associated with the fundamental hypers and the Abelian hypermultiplet respectively in the T N 2N +1,1 theory, and η be the FI parameter.Then the duality translates into the following identity: where Trm = 2N +1 i=1 m i .The topological u(1) symmetry of the T N 2N +1,1 theory is mapped to an emergent u(1) IR symmetry in the SU (N + 1) theory, which explains why the FI parameter has to be turned off for the partition functions to agree.The map of the u(1) B symmetry of the SU (N + 1) theory across the duality can be read off from the partition function identity.

Duality D N
2N,P : This is the self-duality of the theory T = T N 2N,P .Let m and m ab denote the real masses for the 2N fundamental hypers and the P Abelian hypers respectively.The associated partition function identity is given as: where m obey the constraint 2N i=1 m i = 0.In particular, for P = 1, the identity can be put into the following form after a simple change of variables: where the masses m are unconstrained and the masses m ′ are related to m as follows: ) An immediate consequence of the duality D N 2N,P is the self-duality of an SU (N ) gauge theory with 2N flavors, which can be seen at the level of the partition function by integrating over the FI parameter η on both sides of the identity (2.2) or the identity (2.3).

Duality D N
2N −1,P : The dual pair (T , T ∨ ) involves the theory T = T N 2N −1,P and the theory T ∨ -a U (1) × U (N − 1) gauge theory with 1 and 2N − 1 fundamental hypers respectively plus P Abelian hypermultiplet with charges (1, −(N − 1)) under the U (1) × U (N − 1) gauge group.Let m and m ab denote the real masses for the 2N − 1 fundamental hypers and the P Abelian hypers respectively.The associated partition function identity is given as: where m, m ′ , m Ab denote the real masses for the hypers in the fundamental representation of the U (N −1) gauge node, the single hyper in the fundamental of U (1) gauge node, and the Abelian hypers charged under both the U (1) and the U (N − 1) nodes respectively.The parameters m and m Ab are unconstrained in the above identity.
For P > 1, the u(1) topological symmetry of the theory T N 2N −1,P is enhanced to u(1) ⊕ u(1) in the IR.On the dual side, this enhanced symmetry is manifest in the UV as topological symmetries of two unitary gauge nodes -U (1) and U (N − 1).For . The rank of the symmetry algebra can be read off from the UV Lagrangian on the dual side as before.In addition, one observes that the quiver has a single balanced gauge node and a single overbalanced gauge node.Using the standard intuition of enhancement of Coulomb branch symmetries for balanced nodes in linear quivers [7], one may therefore infer from the quiver T ∨ that the IR symmetry algebra is enhanced to a su(2) ⊕ u(1).

Constructing the duality sequence: The general recipe
Consider a quiver gauge theory in class X .A theory in this class is specified by a graph of the form discussed in Section 2.1 where the gauge nodes (unitary and special unitary) are connected by fundamental/bifundamental hypermultiplets only.Additionally, at least one of the special unitary nodes must be balanced, while the unitary nodes may be balanced or overbalanced.Finally, we will also impose the condition that the quiver is good in the Gaiotto-Witten sense 3 .
Given the quiver gauge theory T , one can construct a duality sequence by implementing step-wise a set of quiver mutations, which are built out of the dualities summarized in Table 1.We will first discuss the quiver mutations individually and then discuss the construction of the duality sequence using these mutations.

Mutation I
For a given balanced special unitary gauge node in T , one can use the duality D N 2N +1,1 in Table 1 (in reverse) locally to replace the special unitary node with a unitary factor with the same number of fundamental hypers and a single Abelian hypermultiplet.We will refer to this operation as mutation I and denote it as O I : Since SU (N α ) is a balanced node, the labels of the gauge and the flavor nodes obey the constraint: Since a subgroup of the flavor symmetry of the SU (N α ) node is gauged, the Abelian hyper is charged under all the neighboring unitary gauge nodes to which the SU (N α ) node is connected.
In terms of the sphere partition function, the quiver mutation can be realized as follows.The duality D N 2N +1,1 is implemented by using the identity (2.1) locally for the SU (N α ) gauge node in the quiver T : where . . .denotes the terms in the partition function independent of the SU (N α ) node and the U (N α − 1) node respectively, Trm α = Mα a=1 m a α , and η α is the FI parameter of the U (N α − 1) gauge node in T ∨ .The charges for the Abelian hypermultiplet in T ∨ can be simply read off from the partition function -the charge vector Q will have only three non-vanishing entries, i.e.Q = (0, . . ., N α−1 , −(N α − 1), N α+1 , . . ., 0).For a generic (non-linear) quiver, the mutation I can be readily extended as follows: In the quiver T , the SU (N α ) node is balanced, i.e.

Mutation I ′
Consider a gauge node U (N ) with 2N + 1 fundamental/bifundamental hypers plus a single Abelian hyper, where the Abelian hyper can be charged under any of the other unitary gauge node in the linear quiver.We can use the duality D N 2N +1,1 in Table 1 locally at the gauge node to replace the unitary factor with a special unitary node.We will refer to this operation as mutation I ′ and denote it as O I ′ : The labels of the gauge and the flavor nodes obey the constraint: In contrast to the quiver obtained via mutation I above, the Abelian hypermultiplet in (T ) may be charged under any of the unitary gauge nodes and not just the neighboring gauges of U (N α ).On the RHS, the quiver (T ∨ ) is obtained by replacing N α → N α + 1 (keeping all the other gauge/flavor nodes unchanged), and gauging a specific u(1) subalgebra of the topological symmetry algebra, corresponding to the generator: where J i denotes the topological symmetry generator associated with the gauge group U (N i ), and the index a runs over all the unitary gauge nodes in (T ) under which the Abelian hypermultiplet is charged, in addition to U (N α ).The gauging operation amounts to ungauging a U (1) subgroup of the gauge group and is denoted by " U (1)" in the quiver.
In terms of the sphere partition function, the quiver mutation can be realized as follows.Using a simple change of variables, one can rewrite the identity (2.1) in the following fashion: where m denote the real masses for the fundamental hypers with Trm = 2N +1 i=1 m i , and m ab is the real mass for the Abelian hyper.Using this identity locally for the U (N α ) gauge node in the quiver T , we obtain: where Na Trσ a , s α and σ α live in the Cartan subalgebra of U (N α ) and U (N α + 1) respectively.Now, consider the change of variables: (2.10) The partition function identity can then be written as where T ∨ can be readily identified as the quiver above, and the precise u(1) subalgebra to be ungauged can be read off from the argument of the delta function in the matrix model.Note that in the special case where Trσ α−1 + Trσ α+1 + a Qa Na Trσ a = 0, the ungauging operation simply gives an SU (N α + 1) node.In this case, the operation O I ′ simply reduces to the inverse of the O I operation studied above.
For a generic (non-linear) quiver, the mutation I ′ can be readily extended as follows: In the theory T , the labels of the gauge and the flavor nodes obey the constraint: subalgebra to be gauged is given as: where the first sum extends over all the gauge nodes (aside from the U (N α ) node) under which the Abelian hypermultiplet in T is charged, and the second sum extends over all the gauge nodes which are connected to U (N α ) by bifundamental hypers.

Mutation II
Consider a gauge node U (N ) with 2N fundamental/bifundamental hypers plus a single Abelian hyper, where the Abelian hyper can be charged under any of the other unitary gauge node in the linear quiver.We can use the duality D N 2N,1 in Table 1 locally at the gauge node.We will refer to this operation as mutation II and denote it as O II : The labels of the gauge and the flavor nodes obey the constraint: N α−1 +N α+1 +M α = 2N α .On the left, the blue line (solid and dashed) denotes a single Abelian hypermultiplet associated with the charge vector , with L being the number of gauge nodes in the quiver.Note that any number of entries in Q (except for the α-th entry) can be zero.For the quiver on the right, the gauge and flavor nodes remain the same, while the single Abelian hypermultiplet is now associated with a charge vector One can check that this operation squares to an identity operation.
In terms of the sphere partition function, the quiver mutation can be realized as follows.The duality D N 2N,1 is implemented by using the identity (2.3) locally for the U (N α ) gauge node in the quiver T : (2.13) The charge vector Q ′ associated with the quiver T can be read off from the second equality.
The above computation can be readily extended for P > 1.In this case, the mutation preserves the gauge and the flavors nodes as before, while mapping P Abelian hypers of charges {Q l } l=1,...,P to another P Abelian hypers of charges {Q ′l } l=1,...,P .The map of the charges is given as follows: for l = 1, . . ., P, Similar to the P = 1 case, one can check that this operation squares to an identity operation.
The extension of the mutation II to a generic quiver assumes the following form: The labels of the gauge and the flavor nodes obey the constraint: i N α i +M α = 2N α .The map of the charges is given as follows: for l = 1, . . ., P, For the quiver T , the labels of the gauge and the flavor nodes at the node α obey the constraint: N α−1 +N α+1 +M α = 2N α −1, and the blue line (solid and dashed) denotes a single Abelian hypermultiplet associated with the charge vector with L being the number of gauge nodes in the quiver.Note that any number of entries in Q (except for the α-th entry) can be zero.For the quiver T ∨ on the right, the mutation splits the U (N α ) gauge node into a U (N α − 1) and a U (1) gauge node as shown, where the latter node has a single fundamental hyper.The single Abelian hypermultiplet of T ∨ has the charge (1, Q ′ ), where the first entry indicates the charge under the new U (1) gauge node, and Q ′ is an L-dimensional charge vector: Therefore, the charges for the Abelian hypermultiplet transform under the mutation as follows: (2.16) The U (N α − 1) gauge node has N α−1 + N α+1 + M α = 2(N α − 1) + 1 bifundamental/fundamental hypers plus a single Abelian multiplet.Therefore, one can implement the O I ′ operation at the gauge node U (N α − 1) of the quiver T ∨ , and check that one gets back the quiver T .The composition of O I ′ with O III therefore gives the identity operation.
In terms of the sphere partition function, the quiver mutation can be realized by implementing the identity (2.5) locally for the U (N α ) gauge node in the quiver T , we obtain: where for the third equality, we have redefined the integration variable σ ′ by a shift.The charge vector Q ′ associated with the quiver T ∨ can be read off from the third equality.
Now, let us implement O I ′ at the U (N α − 1) gauge node of T ∨ .Let us first make a change of variable: σ ′ → σ ′ − β=α±1 Trσ β + Trσ α .This leads to the following partition function: . (2.18) Using the identity (2.11) for O I ′ , we can replace the σ α -dependent part of the matrix integral: where we have integrated over σ ′ in the second step.The second line can be readily identified as the partition function of the quiver T that we started with above.This shows that the composition of O III and O I ′ gives the identity operation on the quiver T .
The above computation can be readily extended for P > 1.In this case, the mutation gives a quiver where the U (1) node and the U (N α − 1) node are connected by P Abelian hypermultiplets, with all the gauge nodes and flavor nodes remaining the same as the quiver T ∨ above.The P Abelian hypers of charges {Q l } l=1,...,P on the LHS are mapped to another P Abelian hypers of charges (1, {Q ′l } l=1,...,P ) on the RHS.The map of the charges is given as follows: for l = 1, . . ., P, (2.20) One can check that if one implements O I ′ at the gauge node U (N α − 1) of the quiver T ∨ (after an appropriate field redefinition in the theory), one gets back the quiver T .The composition of O I ′ with O III therefore gives the identity operation, as we noted in the P = 1 case.In terms of the partition function, these operations can be implemented in an analogous fashion as the P = 1 case.
The extension of the mutation III to a non-linear quiver assumes the following form: The labels of the gauge and the flavor nodes obey the constraint: The map of the charges is given as follows: for l = 1, . . ., P,

.21)
The Duality Sequence Given the set of four mutations, we can now discuss the construction of a duality sequence starting from a quiver gauge theory T in class X , where T is a good theory with at least one balanced special unitary node.The precise steps for constructing the duality sequence may be summarized as follows: 1. Assuming that the quiver T has L balanced special unitary gauge nodes, the first step involves implementing mutation I at each of the balanced nodes, thereby giving L N-al theories.Note that this quiver mutation will alter the balance condition of the gauge nodes which are connected to the balanced node in question.
2. Each of the L theories obtained in step one will have L − 1 balanced special unitary gauge nodes inherited from T as well as possibly new balanced special unitary gauge nodes which arise because of mutation I.In addition, there will be unitary gauge nodes with various balance parameters and a single Abelian hypermultiplet charged under a subset of the gauge nodes.In the second step, we implement on each such theory all admissible mutations, with each mutation giving an IR dual theory.
3. One continues with this procedure until no new theories can be generated by the set of mutations.The entire set of theories generated in this fashion will be called the N -al set.
In the rest of this paper, we will construct such duality sequences for explicit examples of T .For the sake of simplicity, we will restrict ourselves to examples where the quiver T is given by a linear graph with a single balanced special unitary node and one or more unitary nodes.However, the computation can be readily extended to more complicated quiver gauge theories -for example, ones with more than one special unitary nodes and/or with more involved graphs, following the general procedure outlined above.

N-ality in a two-node quiver
In this section, we will construct an explicit example of the duality sequence starting from a quiver gauge theory T with two gauge nodes of the following form: The SU (N ) node is balanced, i.e.N 1 + M 2 = 2N − 1, while the U (N 1 ) node can either be balanced or overbalanced, i.e.M 1 + N = 2N 1 + e, with e ≥ 0. The above quiver therefore represents a 3-parameter family of quiver gauge theories.We will first discuss the Coulomb branch global symmetries of the quivers for different ranges of e in Section 3.1, pointing out the emergent IR symmetry that arises for the respective ranges of e. Next, we will work out the details of the duality sequence which depend on the specific value of e -we will discuss the e > 0 and e = 0 cases in Section 3.2 and Section 3.3 separately.In particular, we will discuss how the emergent IR symmetry of T (in addition to the rank of the symmetry) becomes manifest in one or more theories of the N-al set.

Monopole operators and global symmetry enhancement
The IR conformal dimensions of the monopole operators of the quiver gauge theory T are given as where a and p denote the GNO charges associated with the U (N 1 ) and the SU (N ) node respectively for a given monopole operator.To begin with, one can check that that the theory is good in the Gaiotto-Witten sense i.e. ∆(a, p) > 1 2 for any a and p.For a good theory, every monopole operator with ∆ = 1 corresponds to a conserved current/global symmetry generator in the IR SCFT.The Lie algebra of the enhanced global symmetry group of the Coulomb branch can therefore be determined by finding the complete set of ∆ = 1 monopole operators.
Consider the monopole operators a = (±1, 0, . . ., 0), p = 0 -their conformal dimensions are ∆(a, For the case of a balanced U (N 1 ) node i.e. e = 0, the conserved currents associated with these monopole operators will combine with the generator of the topological u(1) symmetry algebra to generate an su(2) algebra.This is the standard enhancement that arises from a single balanced unitary gauge node.Next, consider the operator a = 0, p = (1, 0, . . ., 0, −1), for which which generates a u(1) global symmetry for any value of e ≥ 0. This is the generator that enhances the Coulomb branch global symmetry of a balanced SU (N ) gauge theory to a u(1) in the IR.In addition to these, there are global symmetry generators that arise for the case of e = 0. To see these, consider the conformal dimensions of the following monopole operators : For e = 0, the three associated conserved currents will combine to give another su(2) algebra.The appearance of this additional su(2) factor is similar to the enhancement of the Coulomb branch global symmetry studied in [7] for a linear quiver of balanced unitary gauge nodes terminating in an orthsosymplectric node.
To summarize, the Coulomb branch global symmetry algebra of the quiver T is given as 1.Overbalanced U (N 1 ) node : g C = u(1) ⊕ u(1), corresponding to the topological symmetry for the U (N 1 ) gauge node and the monopole operator associated with the SU (N ) gauge node with charge a = 0, p = (1, 0, . . ., 0, −1).
In comparison, the Higgs branch global symmetry algebra can be directly read off from the quiver, i.e. g H = su(M 1 ) ⊕ su(M 2 ) ⊕ u(1) ⊕ u(1), which arise from the fundamental hypermultiplets of the respective gauge nodes.

Overbalanced unitary node: Duality and Trialities
Let us first consider the case where the U (N ) node in T is overbalanced i.e.N + M 1 = 2N 1 + e with e > 0. As discussed in Section 2.3, the first step is to implement mutation I at the balanced SU (N ) node, which leads to the theory T ∨ 1 in Fig. 4.
Figure 4: Duality for the case e > 2. The gauge node at which the mutation I acts is marked by a red cross.
One can realize this mutation in terms of the sphere partition function in the following fashion.The partition function of the quiver T is given as: where s 1 denotes the integration variables associated with the unitary gauge group U (N 1 ), while s 2 denotes the integration variable associated with the SU (N ) gauge group.The mass parameters for the fundamental hypermultiplets at the two gauge nodes are collectively denoted as m = (m 1 , m 2 ), and the FI parameter associated with the U (N 1 ) gauge node is η.Mutation I can be implemented by isolating the s 2 -dependent part of the matrix integral, and using the identity (2.1) (after substituting N → N − 1), which gives: where σ 2 are the integration variables in the Cartan subalgebra of a U (N − 1) group.Substituting the above identity in (3.6), the partition function for T can be written as The matrix integral on the RHS of the above equation can be evidently identified with the sphere partition function of the quiver gauge theory T ∨ 1 in Fig. 4, where Z hyper 1−loop is the contribution of a single Abelian hypermultiplet with charge (N 1 , −(N − 1)) under the gauge group U (N 1 ) × U (N − 1), i.e.
We therefore have the following relation of the two partition functions: Note that the IR enhanced Coulomb branch symmetry u(1) ⊕ u(1) of T is manifest in the dual theory T ∨ 1 as topological symmetries of the two unitary gauge nodes.The emergent nature of the symmetry on one side of the duality is indicated by the fact that a linear combination of FI parameters in T ∨ 1 has to be set to zero for the sphere partition functions to agree.
For a generic integer e > 2, the N-ality is simply a duality, since we cannot implement any of the other mutations on the U (N ) node.In this case, for a fixed e, we have a 2parameter family of dualities.However, for the special cases of e = 1, 2, we can continue further, as we will now demonstrate.
Consider the case of e = 1 to begin with, as shown in Fig. 5.The first step of obtaining the quiver T ∨ 1 from T using mutation I is the same as before.Since e = 1, the U (N 1 ) gauge node of T ∨ 1 has 2N 1 fundamental/bifundamental hypers plus a single Abelian hyper.One can therefore implement mutation II on the U (N 1 ) gauge node which leads to the quiver T ∨ 2 .The mutation can again be realized in terms of the partition function and the details of the computation are given in Appendix A.1.The final result is the partition function relation: Figure 5: Triality for the case e = 1.The gauge node at which a given mutation acts is marked by a red cross at each step.
where T ∨ 2 is the quiver gauge theory in Fig. 5.The duality map for the real masses can be read off from the above identity: Note that the number of independent real mass parameters are M 1 + M 2 which live in the Cartan subalgebra of g H = su(M 1 ) ⊕ su(M 2 ) ⊕ u(1) ⊕ u(1).In addition, the IR enhanced Coulomb branch symmetry u(1) ⊕ u(1) of T is manifest in the dual theory T ∨ 2 , in addition to the theory T ∨ 1 .Mutation II squares to an identity as a quiver operation, and therefore applying the mutation II on the U (N 1 ) gauge node in the T ∨ 2 theory will reproduce the theory T ∨ 1 .Since no further mutations are permitted, the duality sequence ends at T ∨ 2 giving a triality, as shown in Fig. 5.Note that we have a 2-parameter family of trialities in this case.Now consider the case of e = 2, as shown in Fig. 6.The first step again involves implementing mutation I at the SU (N ) gauge node in T which gives the quiver T ∨ 1 .Since e = 2, the U (N 1 ) gauge node of T ∨ 1 now has 2N 1 + 1 fundamental/bifundamental hypers plus a single Abelian hyper.One can then implement the mutation I ′ at this node which replaces the U (N 1 ) node by an SU (N 1 + 1) gauge node, leading to the dual theory Figure 6: Triality for the case e = 2.The gauge node at which a mutation acts is marked by a red cross at each step.The triality implies a duality between a pair of linear quivers with no Abelian multiplet.
The mutation can again be realized in terms of the partition function (the details can be found in Appendix A.1), and lead to the following partition function relation: where the duality map for the real masses can be read off from the above identity: Note that the number of independent real mass parameters are again M 1 + M 2 which live in the Cartan subalgebra of g H = su(M 1 ) ⊕ su(M 2 ) ⊕ u(1) ⊕ u(1).The u(1) ⊕ u(1) Coulomb branch symmetry is emergent in the theory T as well as the theory T ∨ 3 , while it is manifest in the theory T ∨ 1 as a u(1) ⊕ u(1) topological symmetry.This is manifest at the partition function level -the FI parameters of T ∨ 1 have to be set to zero for the above identity to hold.Since no further mutation is allowed on the theory T ∨ 3 , the duality sequence ends here, giving a 2-parameter family of trialities.
Note that the duality sequence in Fig. 6 implies a very interesting IR duality between the theories T and T ∨ 3 .Although the intermediate theory T ∨ 1 has a single Abelian multiplet, the theories T and T ∨ 3 are both linear quivers with a unitary node and a special unitary node.The action of the duality is to simply flip a unitary node to a special unitary node of the same rank and vice-versa.Note that the overbalanced unitary node has e = 2 for both T and T ∨ 3 .

Balanced unitary node: Triality
Let us now consider the case where the U (N 1 ) node in T is balanced i.e.N + M 1 = 2N 1 + e with e = 0.The first step is precisely the same as the duality sequences studied in Section 3.2 -the balanced SU (N ) node is replaced by a U (N − 1) node plus an Abelian hypermultiplet by mutation I, which leads to the theory T ∨ 1 in Fig. 7.The U (N 1 ) gauge node in T ∨ 1 has M 1 + (N − 1) = 2N 1 − 1 fundamental hypers as well as a single Abelian hyper.Therefore, one can implement the mutation III to this node, which leads to the theory T ∨ 2 in Fig. 7.This gives a 2-parameter family of trialities involving the quiver gauge theories T , T ∨ 1 and T ∨ 2 .
Figure 7: Triality for the case e = 0.The gauge node at which a mutation acts is marked by a red cross at every step.
The duality sequence can be realized in terms of the sphere partition function as follows.The first step involving mutation I is exactly the same as in Section 3.2.Starting from the partition function of T ∨ 1 , one can implement mutation III using the identity (2.5) (see Appendix A.2 for details) which leads to the following partition function relation: The duality map for the masses are given as: where m ′3 denotes the mass for the fundamental hyper of the U (1) gauge node in Fig. 7.
As before, the number of independent real mass parameters are M 1 + M 2 living in the Cartan subalgebra of g H = su(M 1 ) ⊕ su(M 2 ) ⊕ u(1) ⊕ u(1).
Let us discuss the matching of the Coulomb branch global symmetry in this case.In Section 3.1, we discussed that the Coulomb branch global symmetry algebra for the theory T with the balanced unitary node gets enhanced to g C = su(2) ⊕ su(2) ⊕ u(1) in the IR, while only a u(1) symmetry is visible in the IR.The first mutation produces the theory T ∨ 2 , for which the symmetry visible in the UV is u(1) ⊕ u(1).In the next step along the duality sequence, the second mutation produces the theory T ∨ 2 .The symmetry manifest in the UV for T ∨ 2 is u(1) ⊕ u(1) ⊕ u(1) which has the same rank as g C .In addition, the enhanced global symmetry algebra can be read off from the quiver gauge theory T ∨ 2 in this particular case.Note that the quiver T ∨ 2 has two balanced gauge nodes -U (1) and U (N − 1), and a single overbalanced U (N 1 − 1) gauge node.The balanced nodes are expected to contribute an su(2) factor each while the overbalanced node contributes a u(1) factor, thereby reproducing the correct global symmetry algebra.One can check this more directly by computing the Coulomb branch Hilbert Series of the theory T ∨ 2 .

N-ality in a three-node quiver
In this section, we will construct another explicit example of the duality sequence starting from a quiver gauge theory T with three gauge nodes of the following form: The SU (N ) gauge node is balanced i.e.N 1 + N 2 = 2N − 1, while the unitary gauge nodes can be either balanced or overbalanced, where M 1 + N = 2N 1 + e 1 and M 2 + N = 2N 2 + e 2 such that e 1 , e 2 ≥ 0. The above quiver therefore represents a 4-parameter family of quiver gauge theories.We will first discuss the Coulomb branch global symmetry of the quivers for different ranges of e 1 and e 2 in Section 4.1, pointing out the emergent symmetry that appears in the IR for the respective ranges.We will then discuss the duality sequence, which depends on the specific values of e 1 , e 2 , for the cases {e 1 , e 2 ̸ = 0}, {e 1 = 0, e 2 ̸ = 0} and {e 1 , e 2 = 0} in Section 4.2, Section 4.3 and Section 4.4 respectively.For each case, we will discuss how the emergent IR symmetry of T becomes manifest in one or more theories of the N-al set.

Monopole operators and global symmetry enhancement
One can show that the quiver T with the constraints mentioned above is a good theory in the Gaiotto-Witten sense, and therefore the Coulomb branch global symmetry algebra can again be read off by enumerating the monopole operators with conformal dimension 1.This can be worked out in a fashion similar to the 2-node quiver in Section 3.1.Below, we summarize the results: 1.If both unitary nodes are overbalanced, the Coulomb branch global symmetry algebra is , where two of the u(1) factors correspond to the topological symmetries of the unitary gauge nodes, while the third one emerges in the IR and is associated to a monopole operator charged only under the SU (N ) gauge node.This case is similar to the case of a two-node quiver with the unitary node overbalanced.
2. If one of the unitary nodes is balanced, the global symmetry algebra is enhanced to . One of the su(2) factors arise from the balanced unitary node in a standard fashion i.e. the generator of the topological symmetry and the conserved currents associated with a pair of monopole operators charged only under the balanced node generate an su(2) algebra.The second is generated by currents associated with three monopole operators which are charged under the balanced unitary node as well as the SU (N ) node but not under the overbalanced unitary node.The two u(1) factors correspond to the generator of the topological symmetry of the overbalanced node and the monopole operator charged only under the SU (N ) node respectively.
3. If both the unitary factors are balanced, the Coulomb branch global symmetry is enhanced further to g C = su(2) ⊕ su(2) ⊕ su(4) ⊕ u(1).The su(2) factors arise from the two balanced unitary nodes in the standard fashion.The currents generating the su(4) factor correspond to the monopole operators that are charged under one of the two balanced unitary nodes in addition to the SU (N ) node, and the monopole operators that are charged under all three gauge nodes.The u(1) factor corresponds to the monopole operator charged only under the SU (N ) gauge node.
The Higgs branch global symmetry algebra can be directly read off from the quiver, i.e. g H = su(M 1 ) ⊕ su(M 2 ) ⊕ u(1) ⊕ u(1), which arise from the fundamental hypermultiplets of the respective gauge nodes.

Overbalanced unitary nodes : a duality
Let us first consider the case where both unitary nodes in T are overbalanced, i.e. e 1 > 0, e 2 > 0. The first step is to apply mutation I to the SU (N ) gauge node which leads to the quiver gauge theory T ∨ 1 , as shown in Fig. 8.
Implementing the mutation in terms of the sphere partition function, one arrives at the following identity (see Appendix B.1 for details): As discussed earlier, the Coulomb branch global symmetry of the IR SCFT in this case is u(1) ⊕ u(1) ⊕ u(1).For the theory T , only a u(1) ⊕ u(1) global symmetry is visible in the UV, and the other u(1) factor appears as an emergent symmetry in the IR.For the theory T ∨ 1 , the full u(1) ⊕ u(1) ⊕ u(1) global symmetry is visible in the UV.
For generic values of the integers e 1 > 2, e 2 > 2, no other mutation can be implemented on the quiver T ∨ 1 .The N-ality in this case is simply a duality.However, for special values of e i = 1, 2, the quiver admits additional mutations, as we found in the case of the 2-node quiver.These cases can be worked in a fashion similar to the 2-node quiver.

A single balanced unitary node : a triality
Let us consider the case where the gauge node U (N 1 ) is balanced, while the node U (N 2 ) is overbalanced -this will lead to a triality of the quiver gauge theories shown in Fig. 9. Starting from the theory T , the first step is the same as in the previous case, i.e. one uses mutation I to replace the SU (N ) gauge node with a U (N − 1) gauge node and an Abelian hypermultiplet, giving the theory T ∨ 1 .The U (N 1 ) gauge node in this theory has flavors plus an Abelian hypermultiplet, and one can therefore implement a mutation III at this node, which gives the theory T ∨ 2 .
The duality sequence can be realized at the partition function level by first implementing mutation I as we did in Section 4.2.Starting from the partition function of T ∨ 1 , one can implement mutation III using the identity (2.5) (see Appendix B.2 for details) which leads to the following partition function relation: The Coulomb branch global symmetry algebra of T is g C = su(2) ⊕ su(2) ⊕ u(1) ⊕ u(1), of which only a u(1) ⊕ u(1) subalgebra is visible in the UV.For the theory T ∨ 1 , a u(1) ⊕ u(1) ⊕ u(1) subalgebra of the full global symmetry algebra is visible.In contrast, the correct rank of g C can be read off from the quiver gauge theory T ∨ 2 .In fact, one can also read off g C by noting that the U (1) and the U (N − 1) gauge nodes are balanced (thereby contributing an su(2) factor each), while the U (N 1 − 1) and the U (N 2 ) nodes are overbalanced (thereby contributing a u(1) factor each).
We can construct the triality in terms of the sphere partition function (see Appendix B.2 for details) leading to the following identity: For a generic integer e 2 > 2, the duality sequence ends with T ∨ 2 .For e 2 = 1, 2, the quiver T ∨ 2 will allow further mutations and these can be worked in a fashion analogous to the treatment for the case of a two-node quiver.

Two balanced unitary nodes : a hexality
Let us now consider the case where both the gauge nodes U (N 1 ) and U (N 2 ) in T are balanced.Starting from the theory T , the first step is to use mutation I to replace the SU (N ) gauge node with a U (N − 1) gauge node and an Abelian hypermultiplet, leading to the theory T ∨ 1 .In the theory T ∨ 1 , the U (N 1 ) node has 2N 1 − 1 fundamental/bifundamental hypers plus an Abelian hypermultiplet.Similarly, the U (N 2 ) node has 2N 2 − 1 fundamental/bifundamental hypers plus an Abelian hypermultiplet.One can either implement mutation III at the U (N 1 ) node in the next step to obtain the quiver T ∨ 2 , or mutation III at the U (N 2 ) node instead to obtain the quiver T ∨ 3 .In the next step, performing mutation III at the U (N 2 ) node of T ∨ 2 , or the U (N 1 ) node of T ∨ 3 leads to the quiver T ∨ 4 .Since the original SU (N ) node was exactly balanced, the central U (N − 1) in T ∨ 4 has N f = 2(N − 1) − 1 fundamental/bifundamental hypers in addition to the Abelian hyper, and therefore admits another mutation III, giving the quiver T ∨ 5 .One therefore arrives at the hexality of quiver gauge theories shown in Fig. 10.It is convenient to perform a field redefinition on T ∨ 5 to recast the quiver in the final form given in the bottom left corner of Fig. 10.The construction of this hexality in terms of the sphere partition function is worked out in detail in Appendix B.3.
The Coulomb branch global symmetry algebra of T is g C = su(2)⊕su(2)⊕su( 4)⊕u(1), of which only a u(1) ⊕ u(1) subalgebra is visible in the UV.One observes that along the duality sequence starting from T , the rank of the global symmetry algebra manifest in the Redef. ( Figure 10: A hexality of quiver gauge theories which arises when e1 = 0 and e2 = 0.The final step involves a field redefinition. UV increases by 1 at every step, until the final mutation leading to the quiver T ∨ 5 , where the full rank of g C is manifest.In addition, one can again read off g C by using the standard balanced node argument.The two su(2) factors arise from the balanced U (N 1 − 1) and the balanced U (N 2 −1) nodes in T ∨ 5 , while the u(1) factor arises from the U (N −2) gauge node.Finally, the su(4) factor arises from the three consecutive balanced U (1) gauge nodes in T ∨ 5 .

Three dimensional mirror for an N-al set
In this section, we discuss the construction of the 3d mirror associated with a given set of N-al theories.After introducing the general procedure in Section 5.1, we construct the 3d mirror explicitly for the case of the two-node quiver in Section 5.2 and the three-node quiver in Section 5.3.

The general recipe : S-type operations
The basic technology for this construction involves the S-type operation [10], which is a generalized version of Witten's S ∈ SL(2, Z) generator acting on a 3d N = 4 theory.Consider a 3d mirror pair (X, Y ) such that both theories admit weakly-coupled descriptions in the UV and the theory X has a global symmetry subalgebra g sub ⊂ g H of the form g sub = ⊕ γ u(M γ ).In addition, we demand that the theories are good in the Gaiotto-Witten sense.Given such a 3d mirror pair (X, Y ), an S-type operation generates a new dual pair (X ′ , Y ′ ) where X ′ has a weakly-coupled description by construction but Y ′ is not in general guaranteed to have a Lagrangian description.There is however an important subclass of S-type operations -the Abelian S-type operations [10,11], for which Y ′ can be shown to always have a Lagrangian and this Lagrangian can be explicitly constructed.We refer the reader to Section 2.2 and Section 2.3 of [11] for a concise proof.
There are two types of Abelian S-type operations that will be relevant for the construction of 3d mirrors in this section -the flavoring-gauging and the identification-flavoringgauging operations, which we review next.The former acts on a generic quiver X of the type mentioned above to give another quiver X ′ in the following fashion: In the quiver diagram for X, we have only shown two flavor nodes labelled M 1 and M 2 manifestly, while the vector multiplets and the remaining hypermultiplets of the theory are collectively represented by the grey blob.The flavoring-gauging operation involves splitting the flavor node labelled M 1 into two flavor nodes labelled M 1 − 1 and 1, introducing F hypermultiplets of charge 1 at the latter node and then promoting it to a gauge node.Similarly, the identification-flavoring-gauging operation acts on the quiver X to give another quiver X ′ as follows: In this case, the operation involves splitting both the flavor nodes labelled M i (i = 1, 2) into flavor nodes M i − 1 and 1, and then identifying the two flavor nodes labelled 1.This is followed by attaching F hypermultiplets of charge 1 to the identified flavor node, which is then promoted to a gauge node.
The corresponding operation on the quiver Y in each case produces a quiver Y ′ which is the 3d mirror of X ′ by construction.As discussed in [11], the quiver Y ′ can be explicitly determined in both cases and has the following general form: The quiver Y ′ is constituted of two subquivers -the quiver Y and a linear chain of F − 1 U (1) gauge nodes connected by bifundamental hypers and a single fundamental hyper at the far end.The subquivers are connected by a single Abelian hypermultiplet which is charged under one of the U (1) gauge nodes of the linear chain (as shown in the quiver diagram above) and various unitary gauge nodes of the quiver Y .The Abelian hypermultiplet has charge 1 under the U (1) node and its charge under the unitary nodes of the quiver Y is collectively denoted by Q.The precise nodes and the charges depend on the quivers (X, Y ) and the details of the S-type operation itself.
We can now write down the general recipe for the construction of the 3d mirror for a given duality sequence generated from a theory in class X : 1.The first step is to identify in the N-al set a quiver which firstly does not have any balanced special unitary nodes and secondly gives a good quiver in the Gaiotto-Witten sense if one deletes the Abelian multiplets as well as the quiver tails introduced by the various mutations.Let us call the N-al theory T ′ and the theory obtained after stripping off the Abelian multiplets and the quiver tails T good .
2. The second step is to find the 3d mirror of T good .A systematic procedure for constructing the mirror theory is to start from a pair of mirror dual linear quivers with unitary gauge nodes (X, Y ) and implement a sequence of S-type operations [10] to arrive at T good and its 3d mirror.
3. If the 3d mirror of T good has a weakly-coupled description T good , then the final step is to perform a certain sequence of Abelian flavoring-gauging and/or identificationflavoring-gauging operations on T good .On the mirror side, such an operation will generically amount to attaching quiver tails with Abelian hypermultiplets to T good , as we have discussed above.The appropriate sequence of Abelian S-type operations on T good is the one that produces the quiver T ′ from T good on the mirror side.The quiver gauge theory T ′ which is obtained from T good by the above sequence of Abelian operations is therefore the mirror dual of T ′ .Since T ′ is IR dual to all the other theories in the N-al set, the theory T ′ is the 3d mirror associated with the duality sequence.
A few comments are in order.Note that the first step specifies a non-trivial set of requirements.For example, given the duality sequence of the 3-node quiver in the e 1 = e 2 = 0 case in Fig. 10, the only quiver that satisfies these requirements is the quiver T ∨ 5 .In this case, the only choice for the quiver T ′ is T ′ = T ∨ 5 .The second step, which involves finding the 3d mirror of T good , is arguably the hardest step if T good is a generic quiver gauge theory.In general, the 3d mirror may not admit a Lagrangian description.However, for the class of quivers discussed in this paper, T good is always a linear quiver with unitary gauge nodes for which the 3d mirror T good is known.The final step involving the Abelian S-type operations can be performed in terms of the sphere partition function or the superconformal index and we refer the reader to the papers [10,11] for the computational details.

The two-node quiver sequence
Let us now implement the recipe given in Section 5.1 to find the 3d mirror for the duality sequence generated from the two-node quiver4 .We will focus on the case e = 0 i.e. the quiver for which the unitary node is balanced.The e ̸ = 0 cases can be treated in an similar fashion.The duality sequence for e = 0 is given by Fig. 7. To begin with, we observe that the only choice for the theory T ′ is T ′ = T ∨ 2 , since in this case the theory obtained after stripping off the Abelian hyper and the attached quiver tail is good in the Gaiotto-Witten sense.The theory T good is then given by the linear quiver: The labels in the quiver obey the constraints: M 1 + N = 2N 1 and N 1 + M 2 = 2N − 1, which implies that the U (N 1 − 1) gauge node is overbalanced while the U (N − 1) gauge node is balanced.The 3d mirror T good is also a linear quiver which can be found using the standard Hanany-Witten construction [13] for generic labels {M 1 , M 2 , N 1 , N } subject to the constraints.One can then implement the third and final step of the construction by performing the Abelian S-type operations on the linear quiver T good .
For the sake of simplifying the discussion of the final step, we will consider the quiver T good with the following labels: M 1 = 2, N 1 = 3, N = 4, and M 2 = 4.The theory T good then has the following form: One can then implement a flavoring-gauging operation O on the quiver T good and its 3d mirror as follows: The operation acts on the flavor node labelled 1 (marked in red) in the quiver T good .On the mirror side, this operation involves attaching to the U (2) gauge node of T good an Abelian hypermultiplet which in turn is connected to a quiver tail of a single U (1) gauge node plus a fundamental hypermultiplet.Note that this is consistent with the form of the quiver Y ′ in Section 5.1 for F = 2. Since the operation correctly reproduces the theory T ′ = T ∨ 2 , the quiver T ′ can be identified as the 3d mirror associated with the duality sequence Fig. 7 for the chosen labels.
One can check that the basic data of the mirror symmetry in the following fashion.Firstly, one can check that the moduli space quaternionic dimensions agree: The theory T ′ has a su(2) ⊕ su(4) ⊕ u(1) ⊕ u(1) Coulomb branch global symmetry algebra which is realized as the Higgs branch global symmetry algebra for the theory T ′ .The Higgs branch symmetry of T ′ is su(2) ⊕ su(2) ⊕ u(1) which is realized as the Coulomb branch symmetry of the theory T ′ .Therefore the emergent Coulomb symmetry of the unitary-special unitary linear quiver T in Fig. 7 is realized as the Higgs branch symmetry of the 3d mirror T ′ , where it is manifest in the UV Lagrangian.

The three-node quiver sequence
Let us now construct the 3d mirror for the duality sequence generated from the three-node quiver.For the sake of concreteness, we will focus on the case e 1 = e 2 = 0 i.e. the quiver for which both unitary nodes are balanced.The duality sequence which gives a hexality of quiver gauge theories is shown by Fig. 10.There is a single choice for the theory T ′ i.e.T ′ = T ∨ 5 , since only in this case the theory obtained after stripping off the Abelian hyper and the attached quiver tail is good in the Gaiotto-Witten sense.The theory T good is then given by the linear quiver: The labels in the quiver obey the constraints: and N + M 2 = 2N 2 , which implies that the U (N 1 − 1) and U (N 2 − 1) gauge nodes are balanced while the U (N − 2) gauge node is overbalanced.The 3d mirror T good is also a linear quiver which can be found using the standard Hanany-Witten construction [13] for generic labels {M 1 , M 2 , N 1 , N 2 , N } subject to the constraints.One can then implement the third and final step of the construction by performing the Abelian S-type operations on the linear quiver T good .
For the sake of simplifying the discussion of the final step, we will consider the quiver T good with the following labels: M 1 = 2, N 1 = 3, N = 4, N 2 = 4, and M 2 = 4.The theory T good then has the following form: Finally, one has to implement an identification-flavoring-gauging operation O on the quiver T good and its 3d mirror as follows: The operation involves splitting the flavor node labelled 2 (attached to the U (2) gauge node of T good ) into two flavor nodes labelled 1 (marked in red), and identifying them into a single flavor node.This is followed by attaching 4 hypermultiplets of charge 1 to the identified flavor node, which is then promoted to a U (1) gauge node.On the mirror side, the operation amounts to attaching to the central U (2) gauge node of T good an Abelian hypermultiplet which in turn is connected to a quiver tail of three U (1) gauge nodes plus a fundamental hypermultiplet at the far end.Note that this is consistent with the form of the quiver Y ′ in Section 5.1 for F = 4. Since the operation correctly reproduces the theory T ′ = T ∨ 5 , the quiver T ′ can be identified as the 3d mirror associated with the duality sequence Fig. 10 for the chosen labels.
One can check that the basic data of the mirror symmetry in the following fashion.Firstly, one can check that the moduli space quaternionic dimensions agree: C Abelian S-type operations and 3d mirror In this section, we perform the Abelian S-type operations discussed in Section 5.2 and Section 5.3 in terms of the sphere partition function and find the explicit forms of the 3d mirrors.
It will be useful for the computations performed below to define a linear quiver of the following form: This a linear quiver of F U (1) gauge nodes connected by bifundamental hypermultiplets and a single fundamental hypermultiplet at one end.The partition function of the quiver is given as: , one can readily check that the above matrix integral is the partition function of the quiver T ′ given above.
The partition function for the 3d mirror of O( T good ) = T ′ can be obtained by using the identity (C.5) to substitute the function Z ( T good ) in the first line of (C.6) and changing the order of integration to perform the integral over u first, i.e.

Figure 3 :
Figure 3: Top row: 3d mirror for a 2-node linear quiver T .Bottom row: 3d mirror for a 3-node linear quiver T .

2 a=1 1 F=F − 2 a=0
cosh π(u − m a f ) Z (T good ) int (σ, t, −u, −m 2 , −m 3 ), (C.7)where O is the operation dual to O, and the function Z (T good ) int can be read off from (C.2)-(C.4).It is useful at this stage to use the following identity: a=1 cosh π(u − m a f ) cosh π(τ a − τ a+1 ) cosh πτ F −1 .(C.8) .15) Mutation III Consider a gauge node U (N ) with 2N − 1 fundamental/bifundamental hypers plus a single Abelian hyper, where the Abelian hyper can be charged under any of the other unitary gauge node in the linear quiver.We can use the duality D N 2N −1,1 in Table1locally at the gauge node.We will refer to this operation as mutation III and denote it as O III :