3d N=2 mirror symmetry, pq-webs and monopole superpotentials

D3 branes stretching between webs of (p,q) 5branes provide an interesting class of 3d N=2 theories. For generic pq-webs however the low energy field theory is not known. We use 3d mirror symmetry and Type IIB S-duality to construct Abelian gauge theories corresponding to D3 branes ending on both sides of a pq-web made of many coincident NS5's intersecting one D5. These theories contain chiral monopole operators in the superpotential and enjoy a non trivial pattern of global symmetry enhancements. In the special case of the pq-web with one D5 and one NS5, the 3d low energy SCFT admits three dual formulations. This triality can be applied locally inside bigger quiver gauge theories. We prove our statements using partial mirror symmetry `a la Kapustin-Strassler, showing the equality of the S^3_b partition functions and studying the quantum chiral rings.


Introduction and summary
Two decades ago Intriligator and Seiberg [1] discovered three dimensional mirror symmetry: a quantum duality acting on theories with 8 supercharges. One salient feature of this duality is the exchange of monopole operators with standard mesonic operators, or of Coulomb branches with Higgs branches. Soon after, 3d mirror symmetry was interpreted as Type IIB S-duality acting on 1 4 BPS brane setups composed of D3, D5 and N S5 branes [2]. These Hanany-Witten setups can be easily generalized to field theories living in a different number of dimensions.
Extending the analysis to models with only 4 supercharges is clearly desirable, but making progress proved quite difficult. 3d N = 2 mirror symmetry is understood in the Abelian case [3,4]. Two important results using brane setups are the brane interpretation of Seiberg dualities and the construction of chiral field theories [5,6] (with SU (N f ) × SU (N f ) chiral global symmetry) from Dp branes ending on D(p + 2)-N S5 branes intersections. In the 3d case we have D3 branes ending on pq-webs [5,7], 1 4 BPS brane setups involving D5 ′ and N S5 branes that give rise to five dimensional SCFT's with minimal supersymmetry.
Our aim is to tackle some problems that to the best of our knowledge are still open. We focus on the set-up consisting of a single D3 brane ending on both sides of a pqweb composed of 1 N S5 brane and K D5 ′ branes, the (1 N S5 , K D5 ′ ) web, and on the S-dual (K N S5 , 1 D5 ′ ) web. We also terminate the D3 branes on two spectator N S5 ′ D5 ′ branes. The low energy theory corresponding to the (1 N S5 , K D5 ′ ) set-up is known, it is the U (1) × U (1) quiver with K flavors for each node [5,6] depicted in fig. 2, with the so-called flavor doubling: the D5 ′ branes are broken into two pieces and each half provides a SU (K) mesonic symmetry. The two sets of flavors talk to each other through a cubic superpotential that preserves a SU (K) × SU (K) chiral global symmetry.
The low description of the S-dual set-up was not known and we derive it by using partial Abelian mirror symmetryà la Kapustin-Strassler [8], it is the quiver theory composed of two U (1) K−1 tails sketched in fig. 2. Each tails supports an U (1) K−1 top topological symmetry and a U (1) K−1 axial axial symmetry which enhances to SU (K) top × SU (K) axial . The two tails talk to each other through a superpotential which is a sum of monopole operators involving gauge groups on both tails. This superpotential has the non-trivial effect of breaking the total enhanced global symmetry to the diagonal SU (K) top × SU (K) axial . We then see that the mirror theory shows gauge doubling: the N S5 branes are broken into two pieces and each half provides a SU (K) topological/axial symmetry.
Our main tools to complement our analysis via partial mirror symmetry are the study of three-sphere partition functions and the analysis of the chiral rings.
The method of localisation applied to SUSY theories defined on compact spaces, pioneered by Pestun in the case of 4d N = 2 theories on S 4 [9] and then widely generalized, allows us to obtain exact results for partition functions and other observables. Path integrals reduce to ordinary matrix integrals which are explicit functions of global symmetry fugacities. Testing dualities amounts to show the equivalence of the matrix integrals under an appropriate map of the fugacities.
To study the chiral rings of mirror theories we rely on the recent progress in under-standing of monopole operators in 3d N = 4 and N = 2 gauge theories [10][11][12][13][14][15][16][17][18]. Having a good control on the monopole operators is crucial in our case since these operators appear in the superpotentials of our theories.
Polyakov showed long time ago that monopole operators enter the effective potential in field theories arising at low energies from spontaneous symmetry breaking [19,20]. It is also known that compactifying 4d supersymmetric field theories on a circle introduces monopole operators in the low energy 3d superpotential [21,22].
Actually since the general paradigm is that mirror symmetry acts on the chiral rings mapping mesonic operators into monopole operators, whenever we have generic mesonic operators in the superpotential we should expect monopole operators in the superpotential of the dual theory. We will see many examples of this. 1 In section 2 we review the Abelian mirror between U (1) with N f flavors and the linear quiver U (1) N f −1 . We detail a proof of the duality, using partial mirror symmetry, and present a complete map between the two quantum chiral rings.
In section 3 we discuss in detail the pq-web (1 N S5 , 1 D5 ′ ), which under S-duality is mapped to itself. We find a proof that the gauge theory associated to the brane setup Along the way we discover a triality with a third theory: a U (1) gauge theory with 3 flavors and a superpotential containing the two basic monopole operators W = M + + M − . The duality, when projected down to the equality of the corresponding S 3 b partition functions, becomes an integral identity which appeared in the mathematical literature as the new pentagon or ultimate integral identity. Here we provide a brane realization and a field theory interpretation of that identity and prove the duality at the level of the full gauge theory. A similar relation for U (1) with 4 flavors and monopole operators in the superpotential has been discussed in [24] and plays an important role in the 3d-3d correspondence.
We then generalize to our main result of D3 branes ending on the (K D5 , 1 N S5 ′ ) ↔ (K N S5 , 1 D5 ′ ) pq-webs in section 4. We also show that the duality discussed above between U (1) with 3 flavors and the XY Z + X ′ Y ′ Z ′ model can be used to provide two different presentation of the theories (two on each side of the duality). Duality patterns for gauge theories associated to S-dual brane setups.
The quivers in the first line are related by the application of the duality T B ↔ T C of figure 1 and describe the (1 N S5 , K D5 ′ )web.
The quivers in the third and forth lines contain many monopole terms in the superpotential, are related by the application of the duality T B ↔ T C and describe D3 branes ending on the Sdual (K N S5 , 1 D5 ′ )-web.
Notice that the symmetry enhancement pattern U (1) k−1 × U (1) k−1 → SU (k) × SU (k) of the theory in the last line of fig. 2 is quite peculiar: on each side one node has 3 flavors, while the other k − 2 nodes have 2 flavors. Nevertheless the monopoles involving the 3flavors-node combine with the other monopoles to form a bifundamental representation of SU (k) × SU (k). This shows that the topological symmetry enhancement in this N = 2 case follows rules that are different from the N = 4 case, where only sub quivers with balanced U (N ) nodes (number of flavors equal to twice the number of colors) are expected to have an enhanced topological symmetry. It would be interesting to investigate these issues further.
In section 5 we study relevant deformations of the dualities. Real mass deformations lead, on the mirror side, to chiral theories with Chern-Simons interactions, both for the same gauge groups (CS couplings) and for different gauge groups (BF couplings).
In section 6 we show how similar techniques can be applied to longer quivers. We consider a circular quiver with flavors at each node that is self-mirror.  [4] focusing on the partial mirror symmetry approachà la Kapustin-Strassler [8]. We complement this derivation with the study of the mapping of the quantum chiral rings of mirror theories.
We begin with the N = 4 case. The most basic example is the duality between the SQED with one hyper and the theory of a free (twisted) hyper. In [11] an exact proof of this duality was given, showing that the monopole operators M ± of the SQED are free fields with scaling dimension ∆ = 1 2, hence equivalent to a free hyper.
In [8] it was shown that all the Abelian mirror duals can be derived from this basic Abelian duality. The key idea is to regard partition functions as functionals of background vector multipletsV associated to the global symmetries of the theory and to prove that the basic Abelian mirror duality is in fact an invariance under Functional Fourier Transform.
The partition function of the N = 4 SQED with one flavor, regarded as a function of the the background vector multipletV associated to the topological symmetry (in a suitable gauge fixing), is given by We refer the reader to [8] for the definition of the kinetic terms for the vector and hypermultiplet S V (V ), S H (V, Q) and record here the contribution of the BF coupling to the action involving the vector and the adjoint scalar multiplets V , Φ and the background linear and scalar multipletsΣ,Φ: The partition function of the free hypermultiplet as a functional of the background vector multipletV associated to the U (1) global symmetry is In the IR the statement of abelian N = 4 mirror symmetry becomes: since the BF coupling action is quadratic it can be regarded as the Functional Fourier Transform kernel and one can interpret this as the statement that the free hyper coincides with its Functional Fourier Transform: By using Functional Fourier Transform and the convolution theorem one can piecewisegenerate all the N = 4 mirror pairs.
More recently thanks to localisation this approach has been revamped in a simpler although less general fashion. Partition functions of N = 4 theories on S 3 can be computed via localisation [25] which reduces them to matrix integrals, functions of real masses associated to the background vector multiplets of the formV ∼ mθθ. The 1-loop exact contribution of an hyper to the S 3 partition function is: , and the the basic abelian N = 4 mirror symmetry duality is expressed by the integral identity The functional integration becomes an ordinary integration over the locus fixed by the localising equation, that is an integration over the zero modes of the adjoint scalar σ taking value in the Cartan generators of the gauge group. The BF coupling becomes the ordinary Fourier transform kernel and the basic abelian mirror symmetry reduces to an ordinary rather than a functional Fourier transform. Using this trick one can prove all the N = 4 mirror dualities.
In the N = 2 case the basic Abelian mirror pair is formed by the U (1) theory with one flavor with W = 0 and the XY Z model: The chiral ring on the U (1) side is generated by the meson operator M and by the two monopole operators with topological charge ±1 that we denote M ±1 or simply M ± . This duality can also be formulated in a form equivalent to the N = 4 case as In [26] it was observed that this duality is the composition of two operations: an Saction which gauges a global symmetry and introduces a new topological symmetry and the introduction of a superpotential coupling.
This latter operation has been later named Flip. If a theory has a chiral operator O with a coupling W = φO to a background chiral φ, the Flip operation adds a coupling to a new background chiral field φ ′ : W = φO → W = φO + φφ ′ and makes φ a dynamical field.
It is also possible to localise N = 2 theories with non trivial R-charges on the squashed three-sphere S 3 b [27] [28]. The contribution to the partition function of a chiral multiplet with a real mass m ′ for a U (1) symmetry and R-charge r is where b is the squashing parameter. For convenience we also introduce the function: that is the contribution of an hyper with axial mass m (using holomorphy [28] we absorbed r in a redefinition of the mass).
The fundamental abelian mirror symmetry duality (2.2) at the level of the partition function becomes: Notice that on the r.h.s. the real masses of the three chirals are consistent with the cubic superpotential, the arguments of the three double sine functions sum up to iQ 2 . The identity (2.6) is known as pentagon identity and plays a key role in the so called the 3d-3d correspondence [26], [29]. This correspondence relates 3d N = 2 theories, obtained from the compactification of M5 branes on hyperbolic 3-manifolds M , to Chern-Simons theories at complex coupling on M . The invariance of Chern-Simons partition functions under changes of triangulations corresponds to 3d mirror symmetry and in particular the pentagon identity represents the the basic 2-3 move.
By a real mass deformation corresponding to integrating out one chiral, we can derive the mirror duality between the U (1) theory with a charge plus chiral and k = + 1 2 CS coupling and a free chiral with k = − 1 2 CS coupling. In this case one of the two basic monopole operators acquires a non zero gauge charge while the gauge invariant monopole is mapped to the free chiral. At the level of partition function this duality is encoded in the identity An example of mirror duality via partial mirror dualisations We will now review how to derive the mirror dual of the U (1) theory with K flavors p i ,p i , i = 1, . . . , K and W = 0, by repeated use of the fundamental Abelian duality via the Kapustin-Strassler piecewise approach [8].
We dualize each flavor p i ,p i , using eq. (2.3) from right to left, into a U (1) q i ,q i , W = q iqi Φ i theory. We end up with a U (1) K+1 gauge theory with 2K charged chiral fields q i ,q i , K gauge-singlets Φ i and a superpotential W = ∑ i q iqi Φ i . There are also 2K BF couplings (Chern-Simons interactions involving different U (1) gauge factors) connecting the original U (1) factor with the 2K new gauge factors.
At this point the original U (1) gauge group has no flavors and we can perform the functional integral over it obtaining a functional delta function. Implementing the delta reduces the number of gauge groups to K − 1 and the mirror duality can be presented as the quiver depicted in fig. 3. Figure 3. The SQED and its quiver dual. On the quiver side in the superpotential W = ∑ i q iqi Φ i each singlet couples to, or flips, the mesonic operator quadratic in the bifundamental field on its left. We represent each flip by slashing the pair of bifundamental fields. Empty circles (squares) indicate a gauge (flavor) U (1) symmetry.

Check of the duality at the level of partition function
We will now retrace the previous piecewise dualisation at the level of the S 3 b partition function. For the SQED we turn on real masses in the Cartan generators of the global symmetry group (2.9) The abelian symmetry U (1) Φ can mix with the R-charge. The partition function reads: (2.10) We now dualize the K hypers by using the fundamental duality (2.6) and we integrate over the original U (1) gauge group which has no flavors and yields a delta function. After changing variables we find: In this expression the K factors s b (−φ a ) are the contributions of the K chirals singlets flipping the bifundamental mesons. We also see that the real masses are compatible with the superpotential W = ∑ K a Φ a q aqa coupling. We remark that the check of the duality at the level of partition functions is for a specific choice of background hence less general than the Functional Fourier Transform approach. However the partition function approach is very convenient since it allows us to keep track of all global symmetries across dualities. For example we see that the real masses in the Cartan generators of the flavor SU (K) ma are mapped into FI parameters on the mirror quiver side, from which we conclude that the topological symmetry on the quiver theory will enhance.

Mapping of the chiral rings
We conclude the discussion of this example by describing the map of the chiral ring generators. We begin by recording the formula that computes the global and R-charges of BPS monopole operators (the scaling dimension of the monopole operators coincides with their R-charge). The charge of the monopole operators under a global symmetry can be computed by summing over the charges of all the fermions in representations R i charged under the gauge symmetries [13], [12], [14], [10], [11]: where ρ are the weights of the representation. Let us write this formula more explicitly in the special cases of interest for this paper. In a quiver U (1) k with bifundamentals B ij going from node i to node j and at the i-the node F i chiral flavors q i,m with gauge charges ±1, the formula above becomes where M (n 1 ,n 2 ,...,n k ) is a monopole operator with topological charges n 1 , n 2 , . . . , n k .
In the theory U (1) with N f flavors, the chiral ring is generated by the N 2 f mesons Q iQj (with ∆ = 2r, where r is the R-charge and the scaling dimension ∆ of the N f flavors chiral fields) and by the two magnetic monopoles with topological charges ±1, M ± . From eq. (2.14) we compute ∆( then we have at level N f (1 − r) These generators satisfy relations of three types: -The N f × N f meson matrix p ipj has rank 1, so every 2 × 2 minor of this matrix is vanishing. These relations are simple on the U (1) side, but are non trivial quantum relations on the mirror side. For example in the N f = 4 case we have: 3 D3 branes ending on the (1 N S5 , 1 D5 ′ ) pq-web: a triality In this section we study the low energy gauge theory description of the theory living on two D3 branes ending a (1 N S5 , 1 D5 ′ ) pq-web, one on each side. In general by (H N S5 , K D5 ′ )-webs we mean K D5 ′ and H N S5 intersecting at a point. Sometimes it is convenient to represent the (H N S5 , K D5 ′ )-web by its toric diagram a rectangle with base K and height H [7].
Let us briefly review the brane setups. The various branes extend in the directions labelled by x in the table below.
The brane setup enjoys a U (1) ⊗ U (1) global symmetry that rotates the 45 and 89 planes. One combination of these symmetries becomes the U (1) R global symmetry of the 3d N = 2 gauge theory, while the other combination becomes another global symmetry that we call U (1) t . We then have the global symmetries inherited from the pq-web. On a (H N S5 , K D5 ′ )-web the 5d gauge theory is SU (K) H−1 linear quiver 2 , and its global symmetries are We choose to terminate the two D3 branes on two NS5', so the brane set up we are interested in is The global symmetry of the gauge theory must be The 3d gauge theory description of this brane setup, which we call T A and show in Fig. 4, is a U (1) 2 quiver theory with 6 chiral fields (A,Ã, Q,Q, P,P ) and two cubic superpotential terms [6]: set-up and its low energy quiver gauge theory description. The bifundamental fields A,Ã come from D3 − D3 strings, the fundamentals Q,Q and P,P come respectively from D3 − D5 ′ strings and from D5 ′ − D3 strings.
There are 6 fields, so 6 possible U (1)'s, but 2 combinations are gauged and 2 combinations are broken by the 2 superpotential terms, then we have 2 topological symmetries associated to the 2 gauge groups, in total we have U (1) 4 global symmetry.
The R-charges of the 6 chiral fields is exactly 2 3 3 . Using eq. (2.13) we can compute the scaling dimension of the supersymmetric chiral monopole operators The Type-IIB S-dual brane setup is and the 3d description of this brane setup, which we call T C depicted in fig. 5, is basically T A without the gauge fields: it's given by 6 chiral fields, which we denote as (a,ã, q,q, p,p), entering in two cubic superpotential terms Also here the global symmetry is clearly U (1) 4 , and the scaling dimensions of the 6 fields is ∆ = 2 3. Figure 5. The S-dual brane set-up and the 3d mirror theories. In T C the bifundamental field a,ã come from D3 − D3 strings, the fundamental q,q and p,p come respectively from D3 − D5 and D5 − D3 strings.
Our first goal is to show that T A and T C are dual. 4 Along the way we will uncover an additional duality with a theory T B , an U (1) gauge theory with 3 flavors and monopole operators in the superpotential, so we will prove the triality shown in Fig. 6. Figure 6. The gauge theory triality.

T A = T B = T C : a gauge theory triality
In the quiver T A each U (1) node has 2 flavors, so we start with the mirror symmetry for U (1) with 2 flavors reviewed in the previous section and sketched in the first line of fig. 7: with the following mapping of the chiral ring generators: The theory T ′ becomes our theory T A if we gauge the symmetry Q acting on the fields (q 1 ,q 1 , q 2 ,q 2 , Φ 1 , Φ 2 ) with charges (+1, 0, 0, −1, −1, +1) as shown in fig. 7. This implies that the chiral ring generators Using the chiral ring mapping (3.8), we see that the symmetry Q acts with Figure 7. Starting from the mirror pair T = T ′ and gauging the symmetry Q yields the we can easily see that this GLSM is actually the product of two copies of U (1) with 1 flavor and W = 0. Using that basic mirror symmetry we conclude that T A is mirror of two copies of the XY Z model, that is T C .
In Appendix A we present a different proof, where the theory remains a quiver at every duality step.
Let us start from the known mirror symmetry for U (1) with 3 flavors and flip the 3 diagonal mesons. We get a duality between theory T , a U (1) 3, linear quiver with W = 0, sketched in the first line of fig. 8. The theory T ′ is our T A except for the two cubic superpotential terms, which are precisely the dual of the two monopole operators in T . So we add the two terms linear in the monopoles to U (1) 3, and call the resulting theory T B as shown in fig. 8. We conclude that T B , i.e. U (1) with 3 flavors p i ,p i and is dual to T A , the linear quiver U (1) 2 with This proves the gauge theory triality in fig. 6.
In T B it is easy to compute the R charges: by the S 3 symmetry permuting the 3 flavors, it is clear that all the 3 flavors have the same R charge r. Setting the R-charges of the monopoles to be 2 we get the R-charges of the fundamental fields In section 3.3 we will discuss the chiral rings of T A , T B , T C and their mapping.
It is interesting to undo the 3 meson flips in T B and consider the SU symmetry, of this theory is obtained flipping the corresponding fields in T C . We obtain a theory of 9 chiral fields, in order to have SU (3) × SU (3) symmetry, these 9 fields must transform in the bifundamental representation, so we call them x i j , the only cubic superpotential compatible with the symmetry has 6 terms: The fields x i j have R-charge 2 3. The duality is The mapping of the chiral ring generators is simply (3.14) The F -terms relations on the right hand side are simply telling us that the 3 × 3 meson matrix q iqj has rank= 1, a fact that is automatic in the left hand side.
If we now flip back the three diagonal mesons in U (1) 3 to get T B , on the right hand side of 3.13 we are left with only 6 fields and two superpotential terms, This is T C . The mapping of the chiral ring at level ∆ = 2 3 reads In this section we will derive the triality at the level of the partition function.
The U (1), q iqi , W = M + + M − and its mirror, or the new pentagon Our starting point is the Aharony duality [30] between theory T , a U (1), p ipi theory with N f = 2 and W = 0 and theory T ′ , a U (1), q iqi theory with N f = 2 and W = S ij q iqj +O + M + +O − M − . In T ′ the 4 mesons are flipped by singlets S ij and the monopoles by singlets O ± . We can prove this duality starting from the partition function of T : and applying 6 times the pentagon identity (2.6) to get: Notice the six singlets flipping the mesons and the monopoles operators. Now we act on both sides with the S element, which gauges the U (1) ξ symmetry and introduces a new FI parameter η: by rearranging we find: We have identified the l.h.s. and r.h.s. of this identity with the partition functions of theories T B and T C . Indeed Z T B has no real mass for the topological symmetry and the masses for the flavor symmetry satisfy a constraint. This is compatible with the breaking of U (1) top × U (1) axial by the a superpotential linear in the monopole operators W = M + + M − . The three flipping singlets are also visible. In Z T C we have 6 chirals with real masses compatible with two cubic superpotentials.
If we now introduce the following parameterization we can rewrite eq. (3.19) as the following identity On the l.h.s. we can shift the integration variable sending a i → a i + ∆ and b Similarly the r.h.s. depends only on the combination a i +b j invariant under the previous shift. By considering also the constraint (3.22) we see that indeed the identity depends only on 4 parameters, which are associated to the Cartan generators of the SU (3) 2 global symmetry. Notice also that on the l.h.s. of eq. (3.21) there is no FI parameter, since the topological symmetry is broken by the monopole superpotential. This is the partition function identity for the duality we found in (3.13).
This integral identity appears in various contexts in the mathematical literature, for example when derived in [31] in was called ultimate integral identity. More recently it has been called new pentagon identity and plays a role in the study of shaped triangulations, [32]. This identity can also be obtained as a particular limit of the elliptic beta-integral discovered by Spiridonov [33] and has been tested also on the 3d index [34].
A similar relation for U (1) with 4 flavors and monopole operators in the superpotential has been discussed in [24] and plays an important role in the 3d-3d correspondence.
Both of these dualities can be generalized to a Aharony-Seiberg duality for U (N c ) with N f flavors, and W = M + + M − [35].

Partition functions triality
We now complete the partition function proof of the triality. We start from theory T A where we turn on real masses for the global symmetry In the table below we record the charges of the fields in fig. 4 and their contribution to the partition function: The trial R-charges under U (1) Ro are consistent with the superpotential couplings indeed one can verify that the real masses of P, Q, A andP ,Q,Ã sum up to iQ 2 . Notice that the abelian symmetry Using this table it is easy to write the partition function of theory T A : where we combined pairs of chirals of opposite gauged charges into hypers and introduce for convenience the parameter 2v = −t − iQ 4. Now we dualize each hyper by using the pentagon identity (2.6): Integrations over x 1 , x 2 give δ(ξ 1 −p+s) δ(ξ 2 −q −s), which we implement by integrating over p, q, in the end we find: In the last step we identified the partition function of theory Z T B , the U (1) theory with 3 flipped flavors and a superpotential This proves the first part of the triality, which follow by piecewise mirror symmetry. The triality can be summarised as follows: As we have just shown Z T A = Z T B follows from the pentagon identity (2.6) while the equality Z T B = Z T C is due to the new pentagon identity. Finally Z T A = Z T C gives a new interesting identity: (3.27) We can introduce the function: which satisfies: and This Fourier-transform-like identity describing the S-duality of the brane setup in fig. 5 (given that the (1, 1) pq-web is self dual) is very reminiscent of the S-transform action on codimension-two defects discussed in [36].

The full structure of the three chiral rings
In the previous sections we proved that the three theories T A , T B , T C are dual, using Kapustin-Strassler mirror symmetry arguments, this implies in particular that the chiral rings of the three theories must be isomorphic. It is however instructive to study the relations in three chiral rings separately. This section is not necessary for the following and it can be skipped.
The chiral ring of T C , which is Wess-Zumino model apq+ãpq with no gauge interactions, is very simple to compute. It is generated by the 6 fields a, p, q,ã,p,q with ∆ = 2 3.
The T A chiral ring contains monopole operators with charges a, b under the topological We can use the two T A topological charges to organize the structure of the chiral rings. For example at level ∆ = 2 3 the mapping of the T A ↔ T C ↔ T B chiral ring generators is conveniently written as In the next sections we will discuss the relations in the three chiral rings.

T C chiral ring
In theory T C the 6 generators obey 6 quadratic F-terms relations coming from the cubic superpotential W T C = apq +ãpq.
At the second level, ∆ = 4 3, we can construct 6⋅5 2 = 21 operators, but 6 of these are vanishing, we are left with 15 = 12 + 3 operators, that can be organized as follows: The operators in the chiral rings can be thought of as the holomorphic functions on the moduli space of vacua, which is simply the product of the moduli space of vacua of two XY Z models: It's also easy to compute the Hilbert Series [37] of the chiral ring. The (unrefined) Hilbert Series for a single XY Z model is where we need to subtract 2 to avoid overcounting the identity operator. The Hilbert Series for the product of 2 XY Z models is thus which admits the small t expansion From the last equation we can infer that at level k there are 6k operators living on the edges of an hexagon in the Z × Z lattice with vertices ±(k, 0), ±(k, k), ±(0, k), and, for k > 0, 3k − 3 operators living inside this hexagon, along the 3 main diagonals of the hexagon.

T B chiral ring
In T B there are 3 basic relations The T B chiral ring is generated by 6 ∆ = 2 3 operators and by 3 ∆ = 4 3 operators.
As we have seen the 6 ∆ = 2 3 operators obey 9 quadratic relations. Considering the F-term of the ∑ i Φ i p ipi of the form Φ i p i = Φ ipi = 0, we find additional relations at level ∆ = 5 3: Φ i M ij = 0. Hence, even if at first it looks like that the T B and T C chiral rings are different, they are just two different presentation of the same algebraic structure.
For instance let us look at the level ∆ = 4 3 operators, there are 15 = 12 + 3 operators, that can be organized as: 42) and immediately matched to the corresponding generators in T C (3.35).
T A chiral ring The chiral ring structure is more involved to uncover in the case of T A . The 6 quadratic relations must be quantum relations involving monopole operators, it turns out these relations are setting to zero the 6 monopole operators M x,y with x + x − y + y = 2, for instance It would be nice to apply the formalism developed in [16-18, 38, 39] and perform a direct computation of the Hilbert Series.

Comments on the triality
In this section we gave a proof that the 3 gauge theories T A , T C , T B are dual. The theory T A is associated to the brane setup 1 N S5 ′ − 1 D3 − (1 N S5 , 1 D5 ′ ) − 1 D3 − 1 N S5 ′ while we argued that the mirror dual theory T C is associated to the S-dual brane setup A natural question is the relation between theory T B and the brane setups. In the next section we will generalize this result to the setups We will find that we can associate to each brane setup two slightly different gauge theories, and the difference consists precisely in replacing a subquiver T B with a subquiver T C . So in a sense the theory T B is associated to both brane setups The fact that we can do this replacement without introducing BF couplings is nontrivial, since in general if we perform a partial dualization we do get BF couplings. For example if in a quiver theory we dualize a pair of chiral fields to a U (1) theory with 1 flavor, we also need to add BF couplings between the new U (1) node and all the nodes under which the 2 chirals were charged.

D3 branes ending on the
In this section we will extend the previous results to the following brane setup: The global symmetry of the first brane set up is and the low energy description on the D3 branes is the well known quiver gauge theory T A,K which we discuss in the next section. The gauge theory corresponding to the second brane setup is not known. We will find it using partial mirror symmetry, i.e. dualizing all the 4K + 2 chiral fields of T A,K .
We leave the discussion on the low energy description of D3 branes ending on the (K D5 ′ , H N S5 ) pq-web for the future.

T A,K : global symmetries and chiral ring
We begin with the brane set-up 4.1 and study its low energy description T A,K . This the U (1) × U (1) gauge theory sketched in fig. 9, with a bifundamental flavor A,Ã and K flavors for each U (1) node, that we denote P i ,P i and Q i ,Q i . There is also a cubic superpotential with 2K terms: The global symmetries of the setup are the non Abelian SU (K) 2 acting on the flavors, the two topological U (1) symmetries, and two additional U (1) flavor symmetries. The last U (1) 2 flavor symmetry acts exactly as U (1) t and U (1) φ in T A .
The exact superconformal R-symmetry mixes with U (1) t so it has to be obtained from Z-minimization. We parameterize the R-symmetry in terms of the R-charge of the 4K flavor fields, that we denote r and leave r as an unknown parameter. All the flavor fields have the same R-charge as a consequence of the SU (K) 2 global symmetry and of an additional global Z 2 × Z 2 symmetry.
From the exact marginality of the superpotential (4.4), it follows that the 2 bifundamental chirals A,Ã have R-charge 2 − 2r.
The global charges of the monopole operators can be obtained by means of (2.14). Denoting by M a,b T A,K the monopole with topological charges (a, b), we find As usual, a generic monopole operator in the chiral ring is equivalent to a product of the basic monopole operators have the same R charge. This suggests an enhanced symmetry. Indeed the 5d CFT associated to the pq-web (2 D5 ′ , 1 N S5 ) is U (1) with 2 flavors, which is actually a free theory of 4 hypers, [2]− [2], and supports the tri-fundamental symmetry SU (2) 3 .

Dualization to T B,K
We now group the chiral fields of theory T A,K in 2K + 1 pairs of chirals with opposite gauge charges, and dualize each pair using the standard abelian duality where the new fields q i ,q i , p i ,p i and a,ã have R charge 1 − r and 1 − (2 − 2r) respectively. In this new dual frame we findT A,K a U (1) 2K+3 theory with 4K +2 charged chiral fields, 2K + 1 singlet chiral fields and BF couplings depicted in fig. 10. The superpotential contains the 2K + 1 flipping terms, plus the original 2K terms of (4.4), which are now written in terms of the monopole operators M ± a for the U (1) a , a,ã node, M ± i,L for the U (1) L,i , p i ,p i nodes and M ± i,R for the U (1) R,i , q i ,q i nodes: At this point we can integrate over the two original gauge U (1)'s, that have zero flavors, and produce a functional Dirac delta. We can implement the Dirac delta to present our gauge theory as the linear quiver T B,K depicted in fig. 11, with U (1) 2K−1 gauge symmetry. This theory provides a low energy description of the brane set up (4.2).
Let us compute the R charge of some basic monopole operators, recalling that the 4K 'horizontal' bifundamentals have R charges 1 − r and the 2 'vertical' bifundamentals a,ã have R charges 2r − 1. We find that there are 2K monopole operators with topological charge given by a string of ±1 extending symmetrically from the central node, with R charge 2: ...,0,1,...,1 1 1,...,1,0 The part of the superpotential involving the monopoles, which is the sum of 2K terms, is precisely the sum of the 2K operator in eq. (4.9). These operators are not in the chiral ring.
There also 4 K 2 = 2K(K − 1) monopole operators with topological charge given by a string of ±1 extending to the right or to the left of the central node, with R charge 2r: (4.10) These operators combine with the 2K singlets Φ i,L and Φ i,R to form 2 SU (K)×SU (K) bifundamental representation of R charge 2r. These are dual to the 2K 2 mesons P iPj and Q iQj of theory M a,b T A . The chiral ring of T B,K is generated by -K 2 + K 2 monopole and singlets Φ i,L R operators, with R charge 2r.
and precisely maps to the generators of the chiral ring of T A,K .

Dualization to T C,K
We can also use the triality discussed in detail in Section 3 to give a different presentation of the theory associated to the brane setup (4.2). Inside T B,K we can see a subquiver isomorphic to T B : U (1) with 3 flavors (a,ã with R = 2r −1 and p K ,p K ; q K ,q K with R = 1 − r) plus 3 flipping fields (Φ a , Φ K,L , Φ K,R ) and the monopole superpotential M (0,...,0 +1 0,...,0) + M (0,...,0 −1 0,...,0) ). Using the duality between T B and T C we can locally modify the quiver of fig. 11 to the quiver of fig. 12, replacing T B with a copy of T C , i.e. a pair of XYZ models with fields that we label A,Ã with R = 2 − 2r and B,B; CC with R = r. Theory T C,K has 2K − 2 nodes and the superpotential now reads where both  which is equal to the scaling dimension of the monopoles associated to the other nodes of the quiver with 2 flavors.

S-duality and gauge theory tetrality
We can also apply the T C → T B replacement inside theory T A,K . We get a new theory T D,K , a U (1) 3 quiver gauge theory with K − 1 flavors for the external nodes and 1 flavor for the central node, depicted in the top right corner of fig. 13. Here the obvious global symmetry is only SU (K − 1) 2 × U (1) 6 , but there must be an enhancement to Naming the bifundamental fields a,ã, b,b and the central flavor c,c, the superpotential now reads In conclusion we have shown that there are four dual gauge theories descriptions for the two S-dual brane set-up as summarized in fig. 13.

S 3 partition function derivation
Finally we check the duality web at the level of partition functions. We start from T A,K with real masses m a , ∑ To obtain the mirror dual we first use eq. (2.6) to dualize the bifundamentals, we then collect all the fields connected to the x 1 node and use eq. (2.12) to dualize into the mirror linear quiver: (4.17) We do the same for the x 2 integration (4.18) and we find We can rewrite this by shifting w a → w a + ξ 1 and u a → u a + ξ 2 and sending u a → −u a as: We observe that the real masses φ a on the l.h.s. and on the r.h.s. are identified and that there is no real mass turned on for the central U (1) top,s . This is consistent with W T B,K breaking the global symmetries on the two tails to the diagonal and the central node topological symmetry. The 2K + 1 flipping terms are also visible. This allows us to identify (4.20) with the partition function of T B,K .
We can integrate over the central node by using the new pentagon identity (3.21): Again we observe that the real masses on the left and right tails are identified which is consistent with W T C,K . There are also the 2K − 2 flipping terms hence we can identify (4.21) as the partition function of T C,K .
Finally we find the last dual presentation to complete the diagram in fig. 13. In (4.16) we implement the T C = T B duality to replace: If substitute this into eq. (4.16) we find which is the partition function of T D,K . From this expression we see how the real masses are mapped and conclude that indeed as anticipated the symmetry enhancement involves only mesonic symmetries.

Massive deformations
Complex mass deformations We are extracting h D5 from the pq-web and pushing them to the right. This deformation corresponds in the 3d low energy theory to turning on a complex mass for h mesons in T A,K : Integrating out the 2h massive fields P i ,P i , i = 1, . . . , h we obtain at low energies the theory T def A,K depicted in fig. 14, a U (1) 2 quiver gauge theory with 4(K − h) chirals entering a cubic superpotential and 2h chirals entering a quartic superpotential: This superpotential supports a global symmetry SU (K − h) 2 × U (1) × U (h) × U (1) 2 top × U (1) t , as expected from the global symmetry of the brane system. Notice that the global symmetry associated to the stack of h D5 branes is now U (h) rather than SU (h) 2 × U (1), since the k D5's are not broken into two parts. The complex mass parameter m is related to the distance between the (K − h D5 ′ , 1 N S ) pq-web and the h-D5's stack. When this distance goes to zero, there is a divergence in the Lagrangian that must be cured integrating in the 2h chiral fields P i ,P i . 6 As an aside, if we push h R D5 ′ s to the right and h L D5 ′ s to the left of the pq-web, while Let us go back to the theory T def A,K and try to determine the mirror dual. We can start from the dual (1 D5 ′ , K N S ) web and extract h N S5 branes. This leads to the low energy theory T def C,K which is given by T C,K−h , with the monopole superpotential mirror of the cubic part of (5.2), joined to an additional U (1) h tail as shown in fig. 14. The Coulomb branch operators (adjoint singlets Φ i and monopoles) of these additional h nodes enter the superpotential through the mirror of the quartic part in (5.2), AÃ(∑ i Q iQi ). Recalling the map (2.16), we see that whole superpotential is given by The adjoint singlets couple to the central bifundamentals a,ã through the coupling aã(∑ h i=1 Φ i ) which breaks all the axial symmetries of the U (1) h tail, the U (1) h topological symmetries instead enhance to U (h). Notice also that the coupling aã(∑ h i=1 Φ i ) is non-local in the quiver: the h singlets Φ i enter the superpotential both with a bifundamental in the right U (1) h tail, and with the bifundamental connecting the two U (1) K−h−1 tails. 6 Notice that in the brane setup In the latter case the quartic superpotential term involving the h flavors and the bifundamentals breaks SU (h) 2 → SU (h). In the former case it's impossible to write such a superpotential term so there is an accidental chiral symmetry in the gauge theory. We are grateful to Amihay Hanany for discussions about this point. Figure 14. The effect of a complex mass deformation.
We can check the duality at the level of partition function. For T def A,K we have: where the quartic superpotential imposes ζ = 2v.
It is then easy to check that the piecewise dualisation leads to: Notice that the effect of the coupling aã(∑ h i=1 Φ i ) which breaks the axial symmetries of the joint U (1) h tail is visible in the last line where all the axial masses are set equal to −2v.

Real mass deformations
A different type of deformations of our brane set-ups correspond to turning on real masses in the 3d low energy the theories. For example, in the brane set-up for T A in fig. 4, we consider the deformation where we move toward infinity along x 7 one of the two D5 ′ halves. The (1 N S5 , 1 D5 ′ )-web is deformed to a junction of three 5-branes. The toric diagram associated to the junction is the triangle with unit base and hight shown fig. 15. This deformation corresponds to the following limit on the real masses: 7 The key property we use is the asymptotic of the double sine function: In Z T A we need to "follow the vacuum" by shifting x 1,2 → x 1,2 + s to find The effect of the limit is to remove one chiral on each node and to replace it with half CS units of opposite sign as shown in the first quiver from the left in fig. 15. The superpotential contains only the cubic termPÃQ.
In T B the limit (5.7) yields: 10) in this case the limit removes two chirals and two singlets and creates some background CS units as shown in the centre of fig. 15. The superpotential consists of a cubic term flipping the hypermultiplet and the linear monopole term M + to whichPÃQ in T A is mapped.
Finally on in T C the limit (5.7) gives: With different choices of signs we can integrate out chirals of opposite charge generating CS terms with opposite levels, see for example [40]. Here we are only interested in a qualitative analysis of the effect of these real mass deformations on the dual theories.
Again two chiral multiplets are removed and replaced by two background half CS units of opposite sign as shown on last quiver on the right in fig. 15. Figure 15. The effect of a real mass deformation on the triality.
It is easy to generalize this mass deformation to the case (1 N S5 , K D5 ′ ) case. When we send to infinity a D5 ′ half the (1 N S5 , K D5 ′ )-web is deformed into the configuration corresponding to a toric diagram given by a right trapezoid with bases of length K and K − 1 and unit height shown in fig. 16. To implement this limit we define a new set of Figure 16. The effect of a chiral mass deformation on T A,K and on its mirror T C,K .

vector masses and axial masses
(5.12) In T A,K we shift the integration variables x 1,2 → x 1,2 + s and use the asymptotics (5.8) to find (we also absorb some linear terms in a renormalisation of the FI parameters): 13) The effect of this limit is to remove one chiral on the left and one on the right and to introduce two opposite half CS units as shown on the top of fig. 16. The superpotential coupling P i AQ i is removed.
The mirror dual can be easily obtained via piece-wise dualisation. We apply the pentagon identity (2.6) to dualize the 2(K − 1) hypers connected to the left and right nodes and the bifundamental hyper. The chirals with half CS unit are instead dualized by means of eq. (2.7). After implementing the delta functions and applying the new pentagon identity (3.21) we can present the result as the quiver on the bottom of fig. 16. On each tail the i-th bifundamental chiral is removed and a BF coupling with half CS unit is created. Finally in the superpotential we need to remove the term M −(0,...,0,1,...1,1 1,1,...,1,0...,0) dual to the cubic coupling P i AQ i containing the chirals that we have integrated out.
One can iterate this deformation and move to infinity all the K D5 ′ halves on the same side of the N S5 and reach a configuration described by a toric diagram given by a right triangle of base K and unit hight. It should be clear that on the gauge theory side this deformation corresponds to integrate out all the fundamental chirals in the lower triangle in T A,K . On the dual side instead we are left with a deformed version of T C,K where the quiver tails contain chiral bifundamentals going only in one direction plus BF couplings.

Vector mass deformation
Another interesting non-minimal deformation corresponds to moving towards infinity two D5 ′ halves on opposite side of the N S5 along the x 7 direction with the same sign. This has the simple effect of removing a D5 ′ from the pq-web: To implement this deformation in Z T A,K we define a new set of vector masses we shift the integration variables x 1,2 → x 1,2 + s and take the s → ∞ limit. Only the i th left and right hypers are affected by this limit which we evaluate by using the asymptotics (5.8): . (5.14) We see that this limit has the trivial effect of removing two hypers and contributing a finite shift to the FI parameters. The remaining part of the partition function will have no m i and φ i dependence hence it will be a function of 2K real masses only. We conclude that as expected this vector mass deformation reduces T A,K → T A,K−1 . The mirror duals are obviously T B,K−1 and T C,K−1 .

Axial mass deformation
We can also send two D5 ′ -halves, on opposite side of the N S5, to infinity along x 7 with opposite signs. In this case the (1 N S5 ′ , K D5 ′ )-web is deformed into the configuration corresponding to a toric diagram given by a parallelogram with bases of length K − 1 and unit height shown in fig. 17. To implement this deformation we take φ i = s → ∞. In Z T A,K this limit yields: The i th hypers are removed and replaced by two Chern-Simons terms with k = ±1 as shown in fig. 17. Notice that actually there is no-dependence on the mass m i as it can be absorbed in a shift in the FI parameters. Also in this case the deformed theory depends on 2K real masses only.
In this case the cubic superpotential couplings P i AQ i +P iÃQi are removed.
We can now find the mirror of this configuration by piecewise dualisation. We first dualize all the hypers by means of the pentagon identity (2.6). We then dualize the Gaussian CS couplings by writing them as inverse Fourier transforms. At this point, after few manipulation we can apply the new pentagon identity (3.21) to present the dual theory as the quiver on the bottom of fig. 17 with partition function: From this expression we see that the axial mass deformation removes a bifundamental hyper on each tail and replaces it with a BF coupling at level k = ±1.
(1, K) and (K, 1) pq-5branes. By iterating the real mass deformation one can describe more general configurations. In particular the case where we integrate out all the chirals corresponding to moving to infinity in opposite directions the two stacks of K D5 ′ -halves on the two sides of the N S5, deforms the (1 N S5 , K D5 ′ )-web into a (1, K) pq-5brane. Recall that a D5 ′ is a (1, 0) pq-5brane and an N S5 is a (0, 1) pq-5brane.
The low energy theory describing the configuration 1 N S5 ′ −1 (1,K)5 −1 N S5 ′ is then obtained by taking the limit φ i = s → ∞ for i = 1, ⋯K in T A,K and it is given by a U (1) 2 quiver with a bifundamental, and ±K Chern-Simons units for each node. The low energy description of the S-dual set-up 1 DS5 ′ − 1 (K,1) − 1 D5 ′ can be obtained by taking the same limit in T C,K , which has the effect of removing all the bifundamentals in the two tails and replacing them by BF couplings. Since all the nodes in the tails have no matter we can easily perform all the Gaussian integrations and present the theory as a U (1) 2 quiver with a bifundamental, and ±1 K Chern-Simons units for each node. The inversion of the Chern-Simon level is indeed what is expected by dualisation.
Actually since the low energy theory of the set-up 1 N S5 ′ − 1 (1,K) − 1 N S5 ′ has only a bifundamental we can perform all the integrations and find:

Abelian necklace quivers
In this section we consider a circular, or necklace, quiver U (1) K , with K pairs of bifundamental chiral fields A i ,Ã i and two pairs of chiral flavors P i ,P i , Q i ,Q i at each node as shown in Fig. 18. The superpotential is given by: Figure 18. The necklace quiver.
This quiver theory should be related to the low energy theory on D3 branes stretched between K pq-web of the basic form brane direction is a circle. 8 The brane set up is invariant under the action of IIB S-duality and we expect the necklace quiver to be self mirror. We will show that this is indeed the case first by implementing the piecewise dualisation on the S 3 b partition function, then by working out the map of the chiral ring generators. 8 The quiver enjoys a U (1) 3K+1 global symmetry, while the brane setup should support U (1) t × U (1) K global symmetry (one U (1) for each (1 D5 ′ , 1 N S ) pq-web). The discrepancy should be due to an accidental symmetry in the field theory description, similarly to what happens for the brane setup N S − D5 ′ − D5 ′ − . . . − D5 ′ − N S: the low energy 3d description is U (1) with k flavors and W = 0. The global symmetry of the U (1) gauge theory includes an SU (k) 2 , while on the brane there is only one SU (k) factor.
(Higher rank versions) of these type of quivers, known as Hand-Saw quivers, appear in a variety of contexts. For example they appear in connection with Laumon spaces and W-algebras [41], [42] providing a mathematical proof of a finite analog of the AGT correspondence in presence of surface operators.
Hand-Saw quivers play also a central role in the work of [43] where they have three avatars. They appear as the theories on the vortices on the Higgs branch of 5d N = 1 A n quiver gauge theories. They also describe the 5d gauge theories when the Coulomb branch parameters are tuned to special values and fluxes are turned on. The last incarnation is as q-deformed free field correlators of q-deformed W N primaries.
Here we focus on the self-mirror property of the abelian case leaving the discussion of the higher rank for a future work.

S 3 b partition function
The S 3 b partition function with K real masses ξ i for the topological symmetry, K vector masses µ i (with ∑ µ i = 0), K axial masses φ i , bifundamental mass B and the usual masses t and 2v = −t − iQ 4 is given by: (6.2) To find the piecewise mirror we begin by dualizing all fields by means of the pentagon identity (2.6): Integrations over We then use the new pentagon identity (3.21) to perform the integrals ds i . Alternatively one can apply K-times the T C = T A duality, which reduces to the identity (3.30), in (6.2) and then implement the delta functions.
j=1 ξ j so that for example p i+1 − p i → p i+1 − p i − ξ i and obtain: Finally we define F = ∑ K j=1 ξ j , shift p i → p i + i F K and, after few manipulations we arrive at the following result for the mirror dual necklace: 9 By comparing the partition functions (6.2) and (6.7) it is easy to work out the mirror map: (6.8) As expected mirror symmetry acts by swapping the real masses associated to the topological symmetries with those associated to the flavor symmetries while axial masses change sign. This is similar to what happen in the mass deformed T SU (N ).

Chiral ring generators map
The full moduli space is quite complicated and has many branches. Here we focus on the two extreme ones the Coulomb (where all charged fields are set to zero) and the Higgs branch (where monopole and singlets Φ i are set to zero).

Higgs branch operators
The Higgs branch component of chiral ring is generated by the 2K mesons M Q,i , M P,i = Q iQi , P iPi , and by 2K(K − 1) length-L "extended mesons" operators of the form A i+kPi+L+1 , L = 0, ⋯, K − 2 , (6.9) 9 We omit a mirror-map invariant prefactor containing background Chern-Simons terms.
where for L = 0 we have M i 0 = Q iPi+1 andM i 0 =Q i P i+1 . These operators start at node i and end at node i + L going in the right directions. 0 ≤ L < K, so the mesons don't wrap around the whole necklace. Actually also the longest mesons (for L = K − 1) vanish since Q i P i+K = Q i P i andQ iPi+K =Q iPi vanish on the Higgs branch where Φ i = 0.
We could also consider operators likeQ i ∏ L k=1 A i+k Q i+L but they are zero in the chiral ring because of F-terms.
There are also 3 mesonic operators constructed with bifundamental fields only: Coulomb branch operators The Coulomb branch generators are the K singlets Φ i with scaling dimension R[Φ i ] = 4r − 2.
Then we have 2K(K − 1) basic monopole operators with topological charges given by strings of +1's or −1's starting at node i and ending at node i + L with L < K. Their scaling dimension is given by: the two wrapping monopoles are mapped to the two wrapping mesons L andL. We can also see this by looking at effect of the mirror map (6.8) on the real masses for of wrapping mesons K( iQ 4 + t ± 2B) → K( iQ 4 + t ± F K ) . which suggests that under mirror symmetry the 2K(K −1) basic monopoles are mapped to the extended mesons M i L ,M i L . For example we can see how the masses of the mesons Q iPi+1 , Q i P i+1 are transformed by the mirror map (6.8): Finally the K singlets Φ i 's, the bifundamental meson M B ≃ A iÃi and the 2K mesons Q iQi , P iPi are mapped into themselves. Indeed they respectively have real masses equal to −2t, 2t and iQ 2 + 2v ± φ i hence they are left unchanged by the mirror map (6.8).
We close this section by noticing that while the dual quiver we have been discussing in this section is obtain by applying K times the identity T A = T C , we can find many dual presentations of the necklace quiver by applying the T C = T B duality as shown Fig. 19. Figure 19. We can apply the T C = T B to generate many dual presentation of the necklace quiver.
The two U (1) nodes in the quiver both have 2 flavors, so we can use the mirror symmetry for U (1) with 2 flavors reviewed earlier: The mapping of the chiral ring generators is Now let us flip the two mesons p 1p2 and p 2p1 of U (1) 2, W = 0 and call the resulting theory T . Flipping accordingly on the right hand side, and calling the resulting theory T ′ , we obtain a new version of the duality: 3) The mapping of the chiral ring generators between T and T ′ reads Now we use Aharony duality for U (1) with 2 flavors in the form with (q 1q1 , q 2q2 ; q 1q2 , q 2q1 ; φ u , φ d ) ↔ (Φ 11 , Φ 22 ; Φ 12 , Φ 21 ; M + , M − ) . (A.6) Flipping q 1q1 and q 2q2 on the l.h.s. of (A.5) we obtain a duality between T ′ and a third theory U (1) 2 with only the two off-diagonal mesons flipped, that we call T ′′ : T and T ′′ are the same theory, but following the mapping of the chiral rings we obtain a non trivial duality that exchanges the two chiral singlets Φ u , Φ d with the two monopole operators M + , M − : The symmetry we need to gauge, in order to go from T to T A , acts on the fields of T (p 1 ,p 1 , p 2 ,p 2 , Φ u , Φ d ) with charges (+1, −1, 0, 0, −1, +1), so it acts on the chiral ring generators (p 1p1 , p 2p2 ; Φ u , Φ d ; M + , M − ) with charges (0, 0; +1, −1; 0, 0). The dual symmetry must have the same charges on the corresponding chiral ring generators: all the mesons in T ′′ have to be neutral, while the monopoles have to be charged. On T ′′ the symmetry is the topological symmetry, so gauging the symmetry means to ungauge the gauge symmetry of T ′′ , and we get precisely T C .
We end up with a duality between T A = S ⋅ T and T C = S ⋅ T ′′ .