New N=1 Dualities from M5-branes and Outer-automorphism Twists

We generalize recent construction of four-dimensional $\mathcal{N}=1$ SCFT from wrapping six-dimensional $\mathcal{N}=(2,0)$ theory on a Riemann surface to the case of $D$-type with outer-automorphism twists. This construction allows us to build various dual theories for a class of $\mathcal{N}=1$ quiver theories of $SO-USp$ type. In particular, we find there are five dual frames to $SO(2N)/USp(2N-2)/G_2$ gauge theories with $(4N-4)/4N/8$ fundamental flavors, where three of them are non-Lagrangian. We check the dualities by computing the anomaly coefficients and the superconformal indices. In the process we verify that the index of $D_4$ theory on a certain three punctured sphere with $Z_2$ and $Z_3$ twist lines exhibits the expected symmetry enhancement from $G_2 \times USp(6)$ to $E_7$.


Introduction
Supersymmetric gauge theories have been extremely fruitful in our endeavor to uncover the rich structure of quantum field theory. One of the most remarkable phenomenon discovered in supersymmetric gauge theory is Seiberg duality [1] where two different UV gauge theories flow to the same fixed-point in the IR . The original example studied by Seiberg was N = 1 SQCD with SU (N ) gauge group, which was subsequently generalized to SO(N ) gauge groups by Intriligator-Seiberg [2] and to U Sp(2N ) gauge groups 1 by Intriligator-Pouliot [3].
One can generalize this construction to N = 1 theory. The simplest way is to give mass to the chiral adjoints in the N = 2 vector multiplets. In the IR, the massive chiral adjoints will be integrated out and we land on a SCFT [19,20]. One can construct more general theories by requiring non-baryonic U (1) F to be conserved. This gives rise to a new class of N = 1 SCFTs generically not the same as the mass-deformed N = 2 theories in class S [21,22]. This class of theories are subsequently generalized to include the Riemann surface with punctures in [4,23,24] so that the theory can have larger global symmetries. Further studies of N = 1 class S theories have been done in [25][26][27][28][29]. In this paper, we generalize this construction to the case of Γ = D N series with outer-automorphism twists.
This construction requires extra data beyond the choice of the Riemann surface, namely the degree of the normal bundles L(p) ⊕ L(q) → C g,n with p + q = 2g − 2 + n. This stems from the fact that we have one parameter ways to twist the 6d N = (2, 0) theory while preserving N = 1 SUSY in 4-dimensions. 3 The punctures also have to be more general than the N = 2 counterpart. In our case, we put Z 2 valued 'color' to the punctures in addition to the usual N = 2 data. In order to realize SQCD of gauge group SO(2N )/U Sp(2N − 2) with (4N − 4)/(4N ) fundamental quarks, 4 we put the Γ = D N theory on a 4-punctured sphere with two twisted full punctures with each color and two twisted null punctures with each color and choose the normal bundle to be (p, q) = (1, 1). For the case of G 2 theory, start with Γ = D 4 with 4 punctures, but also with Z 3 twist line running between two U Sp(4) punctures of each color. We also need two twisted null punctures of each color as well.  The notion of pair-of-pants decomposition needs an extra ingredient because of the normal bundles. It can be realized by putting colors to the pair-of-pants itself. It turns out there are five different colored pair-of-pants decompositions for our setup, thereby giving five dual frames to the SQCD. 5 The list of dual theories we find are summarized as follows. For the SO(2N ) theory, the five dual frames are: • U SO : SO(2N ) with 4N − 4 fundamentals (vectors) • U SO c1 : SO(2N ) with 4N − 4 fundamentals and mesons [2] • Three non-Lagrangian duals: U SO s , U SO as , U SO c2 and for the U Sp(2N − 2) theory: • U Sp : U Sp(2N − 2) with 4N fundamentals • U Sp c1 : U Sp(2N − 2) with 4N fundamentals and mesons [3] • Three non-Lagrangian duals: U Sp s , U Sp as , U Sp c2 and for the G 2 theory: • U G 2 : G 2 with 8 fundamentals • U G 2 c1 : Spin(8) with 6 quarks in 8 V and 8 S and mesons • Three non-Lagrangian duals: Three out of five dual theories are non-Lagrangian. We will explain these non-Lagrangian duals in detail in later sections. We provide evidence to these dualities through computing the anomaly coefficients and the superconformal indices. In order to compute the superconformal index of G 2 theory, we also discuss N = 2 index with Z 3 twist line and G 2 puncture. Especially, we find that the theory with U V curve given by three punctured sphere with U Sp(6), G 2 and twisted null punctures has enhanced E 7 flavor symmetry as expected in [18] where it was identified as the theory of Minahan-Nemeschansky [30].
The paper is organized as follows. In section 2, we review construction of the N = 1 theories of class S from which we construct our dual theories. We will also discuss the effect of outer-automorphism twist in the setup. In section 3, we propose dualities of SO(2N ) gauge theories and check the 't Hooft anomaly coefficients. In section 4, we discuss the dualities of U Sp(2N − 2) gauge theories. In the section 5, we discuss the dualities of G 2 gauge theory. Finally, in section 6, we check our duality proposals by computing the superconformal index. In the appendix, we derive certain chiral ring relations for the T SO(2N ) and the twistedT SO(2N ) blocks, which are necessary in other sections.

Setup and Data
In order to obtain an N = 1 SCFT from M5-branes dubbed the theories of class S, we need the following data: • Choice of the 'gauge' group Γ = A n , D n , E 6,7,8 .
• Riemann surface C g,n of genus g and n punctures. We call it a UV-curve.
• Choice of two normal bundles L 1 (p), L 2 (q) of degree p, q over C g,n such that p + q = 2g − 2 + n.
• The choice of appropriate boundary condition on each punctures.
The choice of Γ labels the 6-dimensional N = (2, 0) theory and we compactify the 6d theory on C g,n to obtain the N = 1 theory in 4-dimension. When compactifying the theory, we have to perform partial topological twist in order to preserve any supersymmetry. It turns out that there is an integer parameter family of different ways to twist the theory while preserving 4 supercharges. This can be understood as the choice of the normal bundles L 1 (p) ⊕ L 2 (q) → C g,n . The total space of this rank-2 bundle becomes Calabi-Yau 3-fold if it satisfies p + q = 2g − 2 + n. The data on a puncture is specified by the following conditions which are all equivalent: • 1 4 -BPS boundary condition of N = 4, d = 4 SYM theory.
• Choice of the singular boundary condition of a generalized Hitchin equation on C g,n DzΦ 1 = DzΦ 2 = 0 , • Choice of the singular boundary condition of a generalized Nahm's equation.
When one of p or q is zero, then we go back to the N = 2 theories of class S [6,12]. In this case, the data on the puncture is specified by a 1 2 -BPS boundary condition of N = 4, d = 4 SYM theory, or the embedding of SU (2) group to Γ. Equivalently, one of the Higgs field Φ 1,2 vanishes and we get the ordinary Hitchin equation. When Γ = A n−1 it is labeled by a Young tableau with n boxes. Colored N = 2 punctures Generally, N = 1 puncture will involve both Φ 1 and Φ 2 in (2.1) developing singularities at the same point where the punctures sits. Throughout the paper we restrict ourselves to the case where only one of them develops a singularity at a given point. In a sense this makes our system N = 2-like near the puncture. We will label each puncture by a color σ = ± along with the choice of embedding ρ : SU (2) → Γ. We will call them as colored N = 2 punctures.
When the group Γ admits an outer-automorphism (when the corresponding Dynkin diagram is symmetric under a discrete action o), we can twist the punctures accordingly [17,18,31]. Once we twist the puncture, the punctures are no longer labeled by the SU (2) embedding into Γ but into G, see Table 1. Another thing to notice here is that the number of twisted punctures cannot be arbitrary, but is required to be such that the product of monodromies around the punctures should be equal to one. 6 For example, we need to have even numbers of Z 2 -twisted punctures. In the case with Z 3 punctures, we could also have odd number of Z 3 -twisted punctures as in the figure 23.

Colored pair-of-pants decomposition
For a given such configuration, we can have various different dual frames by considering different pair-of-pants decompositions. On each pair of pants, we also label it by a color σ = ±. The number of the pair-of-pants labelled by + and by − are given by the degree of line bundles p and q respectively. Now, for a given pair of pants, we have the following data:  1. The choice of color σ p of the pair of pants itself.
When we glue two pair of pants, we gauge the flavor symmetry associated to punctures we glue. When the σ p of two pair of pants are the same, we gauge it using the N = 2 vector multiplets, and when the σ p are different, we glue it through N = 1 vector multiplet. Note that when we glue two punctures, we can always choose the coloring of the punctures as the same as the pair of pants that we are gluing. See figure 4 for an illustration of the construction. Now, for a given colored pair of pants with color σ p , we identify the building block as follows. When all the punctures have the same color as the pair of pants itself, we identify the theory as the same one as N = 2 theory. For example, when all the punctures are (untwisted) full punctures, then we get T Γ theory. 7 When a full puncture has a different sign from the pair of pants, we add a 'meson' field that transforms as an adjoint of Γ associated to the puncture. Moreover we add a superpotential term for the meson field: W = Tr(M µ), where µ is an operator associated to the puncture. The operator µ transforms under the adjoint representation of Γ and has the conformal weight ∆ = 2.
For a theory in class S, we have U (1) F global symmetry in addition to the N = 1 superconformal symmetry and the global symmetry labeled by the punctures. Suppose we have only maximal punctures meaning ρ is given by the trivial embedding and has the full global symmetry G. We define the U (1) F global charge to be  Here red/blue means σ = ± respectively. Three red punctures and two blue punctures with p = 1, q = 2. Grey tube denotes N = 1 vector, white tube denotes N = 2 vector multiplet. We have 3 mesons associated to the blue puncture on the right and two red punctures on the left.
where J i are the global U (1) charge at each pair of pants. Note that each pair-of-pants or three punctured sphere describes N = 2 superconformal theory. It has SU (2) R × U (1) R R-symmetry which is broken down to U (1) R × U (1) J i upon coupling to N = 1 vectors. The coloring σ i labels the choice of the sign of U (1) J i charge we can make. The color σ p i of a puncture tells us the charges of the operator µ p i . We assign U (1) F of µ to be 2σ p i . When we have a meson field M p i , the U (1) F charge for the µ p i is reversed to −2σ p i and the meson has charge 2σ p i instead. In addition to the operators corresponding to the punctures, we also have 'internal' operators µ i associated to the punctures glued via cylinders in the pair-of-pants decompositions. The U (1) F charge for µ i is given by 2σ i . When the gluing is done through N = 2 vectors, we also have an adjoint chiral multiplet φ. The U (1) F charge for φ is −2σ i , so that the N = 2 superpotential term W = Tr(φµ+φμ) preserves the U (1) F whereμ is the operator corresponds to the other glued puncture. For the N = 1 gluing, we can have a superpotential term W = Tr(µμ) which is exactly marginal.
This global symmetry is not anomalous and in general not baryonic. The true R-charge in the IR will mix with U (1) F charge in the UV. Therefore one needs to perform a-maximization [32] to obtain the correct R-charge.

Non-maximal punctures via Higgsing
If the labeling of the punctures ρ is non-maximal, we 'Higgs' a maximal puncture down to a non-maximal one in the following ways: For the puncture with the same color as the color of the pair-of-pants σ p , we give vev to the moment map µ = ρ(σ + ), and for the puncture with different color σ p , we give vev to the meson M = ρ(σ + ) where ρ is the embedding of SU (2) into G which labels the puncture itself. For the latter case, this yields the superpotential W = trρ(σ + )µ + trM µ where µ are the components of µ which commute with ρ T and M are the mesonic fluctuations around its vev. Higgsing breaks the global symmetry from G down to the commutant G F of ρ(SU (2)) in G.
When some of the punctures are non-maximal, it shifts the U (1) F of (2.2) by a certain amount, if the color of the puncture is different from the pair-of-pants. The shifted U (1) F is given by where (ρ p i , σ p i ) labels the punctures and their colors.
Dualities from colored pair-of-pants decompositions As we discussed above, the different pair of pants decomposition describes different dual frames. Additional ingredient here is the assignment of color σ p i for each pair of pants. This adds another choices on the top of the pair of pants decomposition and it makes the duality structure richer than the N = 2 counterpart. We call it colored pair-of-pants decomposition. Figure 5: Colored pair-of-pants decompositions for a 4-punctured sphere with two twisted full punctures and two twisted null punctures of each color. The degrees of normal bundles are (p, q) = (1, 1). Each subscript stands for: crossing-type 1, swap, Argyres-Seiberg type, crossing-type 2. The first two dual frames have Lagrangian descriptions. The theory U SO c1 turns out to be identical to the dual theory of [2]. The latter three theories are all non-Lagrangian theories. The theory U SO s is an SO version of [4].
The SQCD with SO(2N ) gauge group and 4N − 4 vectors can be realized by choosing the normal bundles and the UV curve to be L(1) ⊕ L(1) → C g=0,n=4 . Two of the punctures are twisted maximal ones having U Sp(2N − 2) flavor symmetries with each color, and we also put two twisted punctures with no flavor symmetry with each color. Since we have 4 distinct punctures and two distinct pair-of-pants, there are many more dual frames compared to the case of N = 2 theory. See figure 5.
One can also consider having other type of punctures to realize U Sp(2N − 2) gauge theories or G 2 gauge theory. The colored pair-of-pants decompositions will be almost the same as this example. There are five dual frames, one of them being the electric gauge theory. There is one Lagrangian dual which we denote as crossing 1 and three non-Lagrangian theories which we name as swap, Argyres-Seiberg type and the crossing 2 type. This fact will be universal regardless of the choice of the gauge group, as it can be easily read off from the geometry. In the later sections, we study each theories in more detail.

Dualities for SO(2N ) gauge theory
In this section, we study dualities for the SO(2N ) gauge theory with 4N − 4 vectors.

T SO(2N ) andT SO(2N ) theory and Higgsing
For a class S theory of type Γ, the most basic building block is T Γ which is given by wrapping the 6d theory on a three punctured sphere with 3 maximal punctures. The theory has Γ A × Γ B × Γ C global symmetry, and has special dimension 2 operators µ A,B,C that transform under the adjoint of Γ A,B,C respectively. These operators satisfy a chiral ring relation This relation is proved in [20] for the Γ = SU (N ) where the theory is usually called as T N . We will mainly use the twistedT SO(2N ) theory to construct various theories of interest. It has We prove the chiral ring relation The number of effective vector multiplets n v and hypermultiplets n h for T Dn andT Dn can be computed using the equations (3.16) and (3.19) of [31]. Each puncture contributes by There is also a contribution from the bulk from which we can compute the n v , n h for T Dn andT Dn to get We will use these formula in later sections to compute the anomaly coefficients.

Higgsing theT SO(2N ) theory
From theT SO(2N ) theory, we can obtain other building blocks by partially closing the full puncture to a one with smaller global symmetries. The SU (2) embedding ρ : induces a decomposition of adjoint representations into the representations of SU (2) and its commutant where V j is the spin-j representation of SU(2) and R j are the representations of the flavor symmetry G F associated to the puncture. For example, when we close one of the twisted puncture having U Sp(2N −2) completely to have no global symmetry, we obtain a free theory with bifundamental of SO(2N )-U Sp(2N − 2). More concretely, we give vev to the operator µ associated to the puncture as where α are the simple roots of U Sp(2N − 2) and E + α are the corresponding raising operators. 8 The ρ ∅ denotes the principal embedding of SU (2) into U Sp(2N − 2), and σ + = σ 1 + iσ 2 where σ i are the Pauli matrices. This embedding leaves no flavor symmetry at all. Under this embedding the adjoint representation of U Sp(2N − 2) decomposes as where V j is the spin-j representation of SU (2). The dimension of the nilpotent orbit of ρ(σ + ) then gives us the number of free half-hyper multiplets produced in the process. Thus we find that after Higgsing, the theory flows to an SO(2N )-U Sp(2N − 2) bifundamental along with 2(N − 1) 2 free half-hypermultiplets. See for example section 2 of [31].

Dualities for SO(2N )-coupledT SO(2N ) theories
Before going into the SQCD, let us consider the theory that does not have a known Lagrangian description. Consider a theory realized by the UV curve given by 4 punctured sphere with two red and blue colors each. Choose all the punctures to be the twisted maximal ones having U Sp(2N − 2) flavor symmetries. We decompose it as two pair-of-pants with red and blue colors and arrange all the punctures to lie in the same color as the pair-of-pants. Each pairof-pants givesT SO(2N ) block. Let us call the red punctures to be A, B and blue punctures to be C, D. Here µ is the dimension 2 operator transforming in the adjoint representation of the SO(2N ) flavor symmetry ofT SO(2N ) whileμ is its counterpart coming from the otherT SO(2N ) block. The U (1) F charge for µ is +2 whileμ has −2. The U (1) F charges of the operators are determined by the color choice σ for each punctures as described in section 2. Diagrammatically we can represent this theory as in figure 8a. We will call this theory as T SO . This theory can also be obtained by starting from twoT SO(2N ) blocks coupled along with their SO(2N ) flavor symmetry by an N = 2 vector multiplet and then integrating out the adjoint chiral in the vector multiplet by giving it mass and then flowing to the IR. Since the operators µ andμ both have R-charge 1, the operator µμ is marginal.
A dual of this theory can be obtained by exchanging the punctures labeled B and C. We will also have to integrate in mesons M B and M C that transform in the adjoint representation of U Sp(2N − 2) B and U Sp(2N − 2) C respectively [4]. The superpotential in the dual theory is given by W =ĉtrμμ + tr ΩM B Ωμ B + tr ΩM C Ωμ C , (3.14) where Ω is the U Sp(2N − 2) invariant antisymmetric form. We now have the dual operatorŝ µ B ,μ C for the punctures B, C which have their U (1) F charges reversed, and also meson operators M B , M C which has the same U (1) F charges as µ B , µ C . We depict this theory by figure 8b. Figure 8: Dual frames of the twoT SO blocks coupled by SO gauge group. The red/blue color means σ = +/− respectively.
We can further exchange punctures A and D to obtain a third theory which is dual to the previous two. The superpotential now becomes with extra meson fields M A and M D . See the figure 8c. One can also derive these dualities starting from N = 2 S-duality and giving mass to the adjoint chiral multiplet in the N = 2 vector multiplet and integrating it out and then flowing to the IR. Then by using the chiral ring relation derived in the appendix A and integrating in the mesons, one can reproduce the superpotentials (3.14), (3.15). We refer to the section 2.2.4 of [4] for details.
Following the nomenclature used in [4], we refer to the dual theories obtained above as being in the "crossing frame" T SO c and the "swapped frame" T SO s respectively. These three duality frames will be the basis of the dualities discussed in this section.

Dualities for SO(2N ) SQCD
Now, let us move on to discuss dualities for the theory with UV Lagrangian descriptions.

Intriligator-Seiberg duality
By partially closing the punctures A and D in the electric theory T SO , we reduce it to SQCD with gauge group SO(2N ) and N f = 4N − 4 fundamental (vector) flavors. Partial closing of the puncture is implemented by giving appropriate vevs as in (3.11) to µ A and µ D . Closing the punctures changes the dual theories as well. Upon Higgsing, the two copies ofT SO(2N ) become free bifundamentals of SO(2N ) × U Sp(2N − 2) B and SO(2N ) × U Sp(2N − 2) C . Therefore, the original theory T becomes SO(2N ) gauge group with 4N − 4 fundamental(vector) flavors Figure 9: global symmetry is manifest. This is nothing but the usual SQCD. We also have the marginal superpotential We will denote this theory as U SO . Now, let us look at the theory obtained by closing the punctures A and B of crossing frame, T SO c . The theory so obtained has two meson fields M B , M C each transforming under the adjoint of U Sp(2N − 2) B and U Sp(2N − 2) C . Also we get superpotential terms as We can writeμ B andμ C in terms of the fundamental dual quarksQ as µ B =Q BQB and µ C =Q CQC which are in the adjoint (=symmetric) representations of U Sp(2N − 2) B,C . Thê µ andμ are given by the dual quark bilinears asμ =Q B ΩQ B andμ =Q C ΩQ C which are in the adjoint of SO(2N ). The duality frames obtained through this procedure are depicted in figure 9. These two duality frames are related to each other by the Intriligator-Seiberg duality [2]. Applying Intriligator-Seiberg duality to the SO(2N ) gauge theory with 4N − 4 fundamentals we find that the magnetic dual is given by the theory with SO(2N ) gauge group and 4N − 4 dual quarksQ along with mesons and the superpotential term W = trMQQ. In the absence of any other superpotential the global symmetry of this theory would be SU (4N − 4) with the mesons transforming in the symmetric representation of while the meson splits into the following irreducible representations.
Note that M ij is dual to the meson of the electric theory formed by Q Bαi Q Cαj . The electric superpotential trµμ induces a mass term for M ij . The dual superpotential of the magnetic theory can be written as Integrating out the massive mesons M ij then gives us the superpotential of (3.17). We will denote this theory as U SO c1 since it arises from exchanging the two punctures in the electric theory.

Non-Lagrangian dual 1: Swap
An interesting non-Lagrangian dual (figure 10) to the SO(2N ) SQCD is obtained by the Higgsing the swapped theory T SO s of figure 8c. In this frame the Higgsing of U Sp(2N − 2) A and U Sp(2N − 2) D is implemented through a vev ρ ∅ (σ + ) to the meson fields M A and M D . The low energy dynamics of this theory can be obtained as follows. With a little abuse of notation, let M A now represent the fluctuations around the vev ρ A (σ + ). The deformed superpotential now becomes where we rewrite the components of (µ A ) ij and (M A ) ij by decomposing into SU (2) representations as in (3.10). The indices j, m with m = −j, −j + 1, . . . , j − 1, j labels the spin-j representations of SU (2) and k = 1, . . . , dimR k . Since there is no flavor symmetry left here, we do not have any k dependence. Since the first term of (3.19) break the U (1) F , we should shift its charge appropriately. Also we want our superpotential term to have U (1) R charge 2. In order to achieve this, we shift the U (1) F flavor symmetry and R-symmetry to where F 0 and R 0 are the U (1) F and R-charges of the fields before Higgsing. The U Sp(2N − 2) A flavor symmetry gets broken and the resulting non-conservation of the associated global currents can be expressed as (3.21) The right-hand side vanishes only if m = −j. This implies that the operators (μ A ) j,m−1 are no longer BPS and hence the superpotential terms that couples them to mesonic fields become Repeating the same analysis for U Sp(2N − 2) D then leads to the following superpotential for our proposed dual (3.23) We will denote this theory as U SO s .

Non-Lagrangian dual 2: Argyres-Seiberg type dual
Another type of dual theory to the SQCD can be obtained from Higgsing punctures B and D of the duality frames in figure 8. This is possible since the punctures with the same colors are indistinguishable in the non-Lagrangian theory of figure 8 and therefore their labels can be interchanged. In the present case we relabel A ↔ B.
Higgsing the frames T SO and T SO s give us the theories U SO and U SO s respectively. However an Argyres-Seiberg type dual, U SO as , is obtained upon closing the afore mentioned punctures in T SO c (see figure 11). Firstly, Higgsing the puncture D will make the theoryT SO(2N ) on the upper sphere to be the theory of bifundamentals. Therefore we haveT SO(2N ) theory with SO(2N ) flavor symmetry gauged and coupled to 2N − 2 fundamentals (vectors).
The punctures A and C have different colors from their pair of pants. Therefore we will have meson field M A and M C coupled through where we replaced the operatorμ B by the product of the quarks (Q αi Q βi ) A . In order to Higgs the puncture A, we give vev to the meson field M B = ρ ∅ . We can now consider low The R-and F-charges are shifted to One interesting aspect of this dual description compared to the N = 2 counterpart is that this dual theory has the same gauge group as the electric one. In the N = 2 case, this type of duality changes the gauge group to be SU (2) subgroup of T Γ [11,13], whereas in the present case the gauge group is still SO(2N ) unbroken.

Non-Lagrangian dual 3: Crossing type
One more dual frame can be obtained from Higgsing B and C punctures of T SO c of figure 8b and relabeling A ↔ B and C ↔ D. We call this the crossing type dual and denote it by U SO c2 (see figure 12). It consists of twoT SO(2N ) blocks coupled to each other along their SO(2N ) flavor symmetry. Also there will be mesons M A and M D with a vev M A = M D = ρ ∅ . The low energy superpotential for the theory becomes and shifted R-and F-charges

't Hooft anomaly matching
Now we test our dualities by computing the anomaly coefficients in different dual frames.

Non-Lagrangian duals
Upon giving a mass to the adjoint chiral superfield in the vector multiplet of an N = 2 theory and hence reducing SUSY down to N = 1, the residual U (1) R symmetry that is preserved by this deformation is given by where I 3 is the Cartan of SU (2) R in the parent theory. Thus we can write the trR N =1 and trR 3 N =1 anomalies in terms of the anomalies of the parent N = 2 theory as and where n v is the effective number of vector multiplets and n h is the effective number of hypermultiplets in the parent theory.
It is now straight-forward to check that the trR and trR 3 anomalies of the duality frames shown in figure 8 match. This is because the mesons have R-charge 1 and hence do not contribute to the R-anomalies. Thus all the R-anomalies of these theories are destined to match as a direct consequence of their matching in the N = 2 parent theories. This also implies that the flavor central charge given by Kδ ab = −3trRT a T b will only get contributions from the coupledT SO(2N ) blocks and hence match in all the three duality frames.
Let us now consider the matching of trFT a T b across the various duality frames. The global current F = i σ i J i is given by the sum of J i where the global symmetry J is given by for the each building blockT SO(2N ) . Note that if the correspondingT SO(2N ) block has a U (1) F -charge σ then To begin with, consider the anomaly coefficient for is 4N as can be checked by comparing the dual theories of figure 28. Thus for the electric theory, T SO (figure 8a) we have This matches trivially to the anomaly coefficient of the theory, T SO c (figure 8b). It is much more interesting to compare this with the anomaly coefficient of T SO s (figure 8c) which, after taking the contributions of the meson M A into account, becomes Note that on the dual side, after giving a vev to the meson M A , the R-charge gets shifted: . This will not affect the contribution of theT SO(2N ) block, since its trρ(σ 3 ) = 0. For the mesons, we will only consider the contributions of (M A ) j,−j since the others decouple. This implies The meson M B does not contribute to the R-anomalies since its R-charge is not shifted and is equal to 1. Putting these together, we find that in this frame  Moving on, we now compare the trR 3 anomalies on the two sides and find Thus, due to the shift in R-charges theT SO(2N ) now contributes Adding the contributions of theT SO(2N ) block and the mesons we find that the trR 3 anomalies match with those of the bifundamental. The trRT a B T b B and trFT a B T b B anomalies for the bifundamental are given by (− 1 2 )(2N ) and (−1)(2N ) respectively. On the dual side these have the same values as in the scenario before Higgsing. This is because trρT a B T b B = 0 for theT SO(2N ) block. We therefore conclude that these anomalies have the same value in the electric and the swapped theory.
Similarly, we can match the anomaly coefficients of U SO and U SO as . In U SO as we have (up to the gaugino-contributions) which match with those in the electric theory. The coefficients of trRT a and trFT a C T b C are not affected by Higgsing and therefore match with their electric counterparts.
The anomaly coefficients in U SO c2 can also be matched to those in the other duality frames. This follows from the matching between the anomalies of the bifundamental and theT SO(2N ) block (with mesons) shown in figure 15. We now repeat the same procedure as the previous section for U Sp(2N − 2) gauge theory with 4N fundamentals.

Dualities for
We begin by considering twoT SO(2N ) blocks coupled to each other at a U Sp(2N −2) puncture via an N = 1 vector multiplet, giving the electric theory of figure 16a. The superpotential for this theory is We will henceforth denote this theory by T Sp . The frames dual to T Sp can be obtained by using the rules of section 2 to move the punctures around. This gives us the set of theories shown in figure 16.
We will call the theory in figure 16b as 'crossing frame 1' and denote it by T Sp c1 . It is obtained by exchanging punctures B and C. Since these punctures will no longer have the same color as their pants, we will therefore have to integrate in mesons M B and M C transforming as the adjoints of the respective symmetries. The superpotential of the theory becomes Similarly when we exchange the puncture A and D, we end up with the theory in 'crossing frame 2' (figure 16c) which is denoted by T Sp c2 . Once again we will have to couple mesons M A and M D via the superpotential 3) The theory in 'crossing frame 3' (figure 16d) is obtained by exchanging punctures B and D. This will correspond to a pair of pants decomposition where one of the pants has no outer automorphism twists. In other words it consists of anT SO(2N ) block coupled to a  Interestingly this gives us a duality between an N = 1 theory with a U Sp(2N − 2) gauge group and a theory with SO(2N ) gauge group. We will denote the theory in this duality frame by T Sp c3 .
The theory in figure 16e will be called the 'swapped' theory and we will denote it by T Sp s . It is obtained by moving the 4 punctures around such that none of them have the same color as the pants in which they reside. This will require us to integrate in mesons at each puncture. The superpotential will now become W =ĉtrμΩμΩ + trM A Ωμ A Ω + trM BμB + trM CμC + trM D Ωμ D Ω .  figure 16a and 16b), with a vev to their adjoint representation operators, we obtain the usual pair of Intriligator-Pouliot dual theories [3]. The electric theory is given by figure 17a. We will use the short-hand notation U Sp to denote this theory. It is a U Sp(2N − 2) gauge theory with 4N fundamental quarks. It has an SO(2N ) B × SO(2N ) C ⊂ SU (4N ) global symmetry. Its superpotential is given by where now µ ij = (Q αi Q αj ) B andμ ij = (Q αi Q αj ) C . Here (Q B ) αi is the quark transforming as the bifundamental of SO (2N . Applying Intriligator-Pouliot duality to the above electric theory, we get a theory with 4N quarksQ transforming under a U Sp(2N − 2) gauge group. In the absence of any superpotential this theory will enjoy SU (4N ) global symmetry. The spectrum of the theory will also include mesons transforming in the anti-symmetric representation of SU (4N ). In terms of the SO (2N  Integrating out the massive mesons M αβ then gives us the theory of figure 17b. We will use U Sp c1 to denote this theory.
Non-Lagrangian dual 1: Swap As usual the R-and F-charges get shifted to: (4.9) We will denote this theory by U Sp s .

Non-Lagrangian dual 2: Argyres-Seiberg type dual
A more interesting non-Lagrangian dual is obtained if one considers Higgsing the A and D punctures of T Sp c3 (figure 16d). Closing the puncture for U Sp(2N − 2) A reduces the corre-spondingT SO(2N ) block to a bifundamental of U Sp(2N − 2) × SO(2N ). Giving vev to M D then gives mass to one of the quarks. We therefore end up with a theory of 2N − 3 fundamentals of SO(2N ) coupled to a T SO(2N ) block as shown in figure 19. The dual superpotential Figure 19: Argyres-Seiberg type dual U Sp as to U Sp gauge theory.
where (μ D ) αβ = (Q mα Ω mlQ lβ ) D and the new U (1) F and U (1) R charges are We will use the short-hand notation U Sp as for this theory.

Non-Lagrangian dual 3: Crossing type dual
The crossing type dual of U SO can be obtained by exchanging its (closed) punctures A and D. This will bring A (and similarly D) into a pair pants whose color is opposite to that of A. The statement that these punctures are closed in U Sp is then equivalent to saying that the puntures are Higgsed by giving a vev to the mesons that we had to couple to the pants. We will denote this theory by U Sp c2 . The quiver diagram for U Sp c2 is shown in figure 20. Its superpotential is while R-and F-charges are where R 0 and F 0 are the charges in the theory without a vev for the mesons i.e. T Sp c2 (see figure 16c).

't Hooft anomaly matching
Let us go on to put the dualities to test.

Non-Lagrangian duals
It is a simple exercise to check that the trR and trR 3 anomalies of the electric theory, the theories in the crossing frames 1 and 2, and the theory in the swapped frame match since the mesons have R-charge 1 and hence do not contribute to the R-anomalies. This also implies that the flavor central charge given by Kδ ab = −3trRT a T b will also only get contributions from theT SO(2N ) blocks and hence will match in all the these frames. It is instructive to match the trR and trR 3 anomalies of T Sp and T Sp c3 . Thus in the electric frame these anomalies get contributions from the twoT SO(2N ) blocks and the gauginos in the U Sp(2N − 2), N = 1 vector multiplet. EachT SO(2N ) block contributes while the gauginos give trR gaugino = 1(N − 1)(2N − 1) .  Similarly, Let us calculate the above anomalies in T Sp c3 . The mesons will not contribute since they have R-charge 1. Thus the contributions come from a T SO(2N ) block, aT SO(2N ) block and the SO(2N ) gauginos. For the T SO(2N ) block we find that and The anomalies of T Sp c3 can now be computed:  Thus we see the anomalies of the duality frames proposed here match perfectly.
Let us now consider the matching of trFT a T b across the various duality frames. For thẽ To begin with, consider the anomaly coefficient for T a ∈ sp(2N − 2) A . Note that k sp(2N −2) for T SO(2N ) is 4N as can be checked by comparing the dual theories of figure 28. In the electric frame T Sp , we find trFT a A T b A = −2N δ ab (4.23) This matches trivially to the anomaly coefficient of T Sp c1 and T Sp c3 . It is much more interesting to compare this with the anomaly coefficient of T Sp c2 and T Sp s . After taking the contributions of the meson M A into account, the anomaly evaluates to This agrees with the original theory. We can analogously match the anomaly coefficient when We now consider the case when T a ∈ so(2N ) B . This time, by comparing the dual theories of figure 29, we find that the contribution ofT SO(2N ) to k so(2N ) is 4N − 4 and hence in T Sp , the requisite coefficient is After adding the contributions of the meson M B in the theories corresponding to T Sp c1 and T Sp s respectively , it is simple to check that their coefficients match the original theory. The above discussion also applies when comparing the anomaly coefficients with T a ∈ so(2N ) C or T a ∈ sp(2N − 2) D . The matching of these coefficient between T Sp and T Sp c2 is trivial.
It is much more non-trivial and interesting to match the anomalies of SO(2N ) B and SO(2N ) C in T Sp and T Sp c3 . Let us start by comparing the SO(2N ) C anomalies. In the electric theory we find that Using the linear quiver to evaluate trR N =2 T a C T b C in the T SO(2N ) block we find that the anomaly in the magnetic theory matches that in the electric theory. We can then immediately see that the anomalies of SO (2N   Also M B does not contribute to the R-anomalies since their R-charge is not shifted and is equal to 1. Putting these together, we find that in this frame which is identical to the corresponding anomaly of the bifundamental. Moving on, we now compare the trR 3 anomalies on the two sides and find On the dual side, since R = R 0 − ρ(σ 3 ), where R 0 = 1 2 R N =2 + I 3 , therefore  On the dual side these have the same values as in the scenario before Higgsing. This is because trρT a B T b B = 0 for theT SO(2N ) block. We therefore conclude that these anomalies have the same value in U Sp and U Sp s . The anomalies of U Sp and U Sp c2 can also be matched in a similar manner by comparing the contributions made by their building blocks shown in figure 22a and 22c.
We now compare the anomalies of the Argyres-Seiberg type dual, U Sp as . Note that in U Sp and (4.33) In the U Sp as , the R-charges are shifted to R = R 0 − ρ(σ 3 ). Also, the meson, M B , does not get a vev. It therefore has an R-charge 1 and hence does not contribute. In the T SO(2N ) block, trρ(σ 3 ) = 0, which implies The contribution from those components of M D which continue to stay coupled to the theory after giving a vev is For the purpose of anomaly matching we can consider the 2N − 3 fundamentals coupled to the T SO(2N ) block as a bifundamental of SO(2N ) × U Sp(2N − 2) with shifted R-charges. As usual the shift will correspond to the embedding of SU (2) in U Sp(2N − 2). The shift in the R-charge of the bifundamental does not change its contribution to trR, since trρ = 0 for the bifundamental. Thus we find that trR in U Sp as is given by  As was mentioned before, the meson, M B will contribute trivially while the contribution from those modes of M D that are still coupled to the theory becomes (4.38) The contribution of the bifundamental is given by Combining all these contributions we find hence providing a nontrivial check of our proposal. It can also be checked, via a pretty direct calculation, that the trRT a T b and trFT a T b anomalies also match in these theories.

Dualities for the G 2 gauge theory
In this section, we study a G 2 gauge theory and its dual frames. The G 2 gauge group can be obtained from Γ = D 4 theory with Z 3 outer-automorphism twist. Since the D 4 theory allows both Z 2 -twisting σ 2 and Z 3 -twisting σ 3 . We should take the twist lines with slightly more care to go to various different dual frames. We study the G 2 gauge theory with 8 fundamental quarks in the 7 dimensional representation of G 2 . A dual theory for the G 2 gauge theory was first proposed in [10] where the dual theory is given by SU gauge group with anti-symmetric tensors. We find new dual descriptions for the G 2 gauge theory flowing to the same fixed point in the IR. We test the duality via anomaly matching and comparison of superconformal indices.

G 2 gauge theory and its dual from coupled E 7 blocks
To obtain the G 2 -dual we propose the following procedure: start with the strongly coupled block of [18] given by D 4 theory on a three punctured sphere with a twisted null puncture, a U Sp(6) puncture and a G 2 puncture as in figure 23. Even though the E 7 flavor symmetry is not manifest, the theory exhibits enhanced E 7 symmetry which is the theory of Minahan-Nemeschansky [30]. We will demonstrate in section 6 that the superconformal index of the theory of figure 23 agrees with the E 7 theory. Now prepare two copies of this theory. By gauging the G 2 symmetry common to the two blocks we obtain an N = 2 SCFT with a G 2 gauge group which can be represented by figure 24a. We can obtain its S-dual by exchanging the punctures. One of its S-dual can be obtained by exchanging two null punctures. It is given by an N = 2 SCFT with Spin(8) Figure 23: Three punctured sphere with U Sp(6) and G 2 punctures. gauge symmetry along with three hypermultiplets in 8 V and three hypermultiplets in 8 S representations which can be represented as in the figure 24b. This duality was first found in [33]. Another frame can be found by colliding two null punctures and two U Sp (6) punctures. This is similar to the Argyres-Seiberg duality, where in this case we partially gauge the theory with U Sp(6) 2 × G 2 flavor symmetry. Let us go to the N = 1 construction. It can be done by giving colors to the punctures and the pair of pants. In figure 24a, let's color the two punctures on the bottom to be red, and the other two punctures to be blue. Also color the pair of pants on the bottom to be red and the other to be blue. Since the color of the punctures and the pants are the same, we can identify the 'matter content' to be the same two E 7 blocks as before. Then we glue two G 2 punctures by N = 1 vector multiplet with the superpotential where µ andμ transform in the adjoint representation of G 2 . A dual frame is described by a Spin(8) gauge theory, with quarks in the 8 V × 6 of Spin(8)×U Sp(6) A and another in the 8 S ×6 representations of Spin(8)×U Sp(6) B . There are also mesons transforming in the adjoint representations of U Sp(6) A and U Sp(6) B respectively. One can also prove the duality starting from N = 2 construction and then giving mass to the chiral adjoint in the vector multiplet if we assume the chiral ring relation 2) and then following the procedure of [4]. The dual superpotential is given by Figure 25: Lagrangian duals to the G 2 gauge theory with 8 fundamentals Upon Higgsing the U Sp(6) flavor symmetries, in the electric frame, down to U Sp(4), we obtain two copies of bifundamentals of G 2 × U Sp(4) with the G 2 being gauged. Higgsing is achieved by giving a vev to the adjoint of U Sp(6) along the partition: 6 = [2, 1 4 ]. We will use the short-hand notation U G 2 to denote this theory. In the dual frame we will have to give the same vev to the mesons M A and M B . This will generate a mass for the dual quarks with SU (2) quantum numbers (j = 1 2 , m = − 1 2 ). We integrate these out and obtain the low energy theory which is described by 5 vectors and 5 spinors of the Spin(8) gauge group and transforming as 4 ⊕ 1 of their respective U Sp(4) flavor symmetries. The low energy superpotential in the dual frame becomes withμ Aj,j being quadratics Spin (8) invariants. The R-charge in magnetic frame is shifted by , where as usual ρ specifies the SU (2) embedding in U Sp (6).
. Some of the mesons decouple.and we are left with the mesons M j,j,k coupled to the magnetic theory. This theory will be denoted by the symbol U G 2 c1 .

Non-Lagrangian duals
We can also get several non-Lagrangian duals to the G 2 theory using different colored pairof-pants decompositions. See the figure 26.

Non-Lagrangian dual 1: Argyres-Seiberg type
The Argyres-Seiberg type dual of figure 26a is obtained by colliding the punctures A and B on the Riemann surface. This will land us upon a theory consisting of an E 7 block (with Figure 26: Non-Lagrangian dual theories for the N = 1 G 2 gauge theory with 8 fundamentals (6) × U Sp(4) × G 2 block via an N = 1, G 2 vector multiplet. We will also have to integrate in mesons (with appropriate vevs) to compensate for the mismatch between the colors of the punctures and the pair of pants. Its superpotential is with the shifted charges being We will use the symbol U G 2 as to denote this theory.

Non-Lagrangian dual 2: Swapped G 2
We arrive at the swapped G 2 frame by permuting all the four punctures such that we exchange A with B and C with D. This is equivalent to coupling two U Sp(6) 2 × G 2 theories along their G 2 puncture. We will have to integrate in 4 mesons M A , M B , M C and M D . We give vevs to these mesons such that U Sp(6) C and U Sp(6) D get completely Higgsed while U Sp(6) A and U Sp(6) B get Higgsed down to their respective U Sp (4). This theory will henceforth be denoted by U G 2 s . Its superpotential becomes while the charges get shifted such that

Non-Lagrangian dual 3: Crossing-type
The crossing-type frame is shown in figure 26c. It consists of two blocks with U Sp(6) × U Sp(4) × SO(8) flavor symmetry glued along their SO(8) puncture. The spectrum of the theory also includes mesons M C and M D as the punctures C and D lie in pants that are colored oppositely to their own color. We will give a vev to the mesons such that the U Sp(6) flavor symmetry of these punctures gets completely Higgsed. The superpotential then becomes 11) and the new charges are given by We will use the symbol U G 2 c2 to represent this theory.

Anomaly matching
We now show that the anomalies of our proposed dual frames match.
trR and trR 3 In the G 2 electric theory, we find that which matches with the electric theory.
The effective number of hypers and vectors in a block with U Sp(6) × U Sp(4) × SO(8) flavor symmetry is 102 and 72 respectively [16]. Using this result we find that the trR anomaly in the U G 2 c2 is trR = 2(72 − 102) + 28 + 2 × 9 = −14 , here the first term on the RHS corresponds to the contribution of the non-Lagrangian blocks to trR, the second term is the contribution from SO(8) gauginos while the last term is the contribution of the mesons used to Higgs the U Sp(6) flavor symmetry of the blocks. Using As before the various terms on the RHS are obtained from the contribution of the non-Lagrangian blocks, the SO(8) gauginos and the mesons respectively. The effective number of hypers and vectors in the block with U Sp(6) 2 ×G 2 symmetries can be obtained by comparing the N = 2 theory obtained by gluing two such blocks along their G 2 puncture and its S-dual corresponding to two copies of the block with U Sp(6) 2 × SO(8) punctures glued along their SO(8) puncture with a Z 3 twist around the cylinder. This will also provide us with the central charges of the various flavor symmetries. Following this procedure we find that in the U Sp(6) 2 × G 2 block, n v = 86 and n h = 112. Using this and including the contribution of the mesons that stay coupled to the theory (after Higgsing one of the U Sp(6) down to U Sp(4) and completely Higgsing the other U Sp(6)), we find that in U G 2 s trR = 2 × (86 − 112) + 14 + 2 × 9 + 2 × 3 = −14 . (5.20) If we now calculate the trR 3 anomaly in this theory, we find This is in agreement with our proposal. We can use our knowledge of the number of hypers and vectors and central charges in the U Sp(6) × U Sp(4) × SO(8) block to evaluate this data for the U Sp(6) × U Sp(4) × G 2 block which are: n v = 79 and n h = 102. The trR anomaly can now be calculated in U G 2 as and is found to match with that in the other duality frames: here the first term on the RHS is the contribution from the U Sp(6) × U Sp(4) × G 2 block while second term is the contribution from the E 7 theory. The third term is the contribution of G 2 gauginos while the last two terms are the contributions of the mesonic excitations. The coefficient of trR 3 in this theory is In the G 2 theory, each block contributes In the SO(8) theory, the F charges are shifted such that F → F − 2ρ A (σ 3 ) + 2ρ B (σ 3 ). The contribution of the pants with color 'σ' is therefore given by This shows a perfect match with the G 2 theory. In the non-Lagrangian duals, the shifted charges R = R 0 − ρ(σ 3 ) and F = F − 2σρ(σ 3 ), (for pants with color 'σ') give rise to the following expression for trRF 2 : where we have used the SU (2) embedding index I to evaluate trRρ 2 and trFρ 2 . The final expression in (5.26) is independent of the color of pants, as should be the case. Also, on each pair-of-pants trR 0 F 2 0 = −n h . Using this and taking the contribution of mesons into account, it can be verified that each pair-of-pants in the decomposition of U G 2 c2 and U G 2 s , contributes a −14 to the anomaly, thereby establishing the match with the electric frame. In U G 2 as , since the pair-of-pants decomposition is not symmetric thus the pants contribute different amounts to the total anomaly. The pants with U Sp(6) × U Sp(4) × SO(8) punctures contributes −92 while the other pant contributes 64, thereby bringing the total to −28 which is same as in the electric theory.
trRT a T b and trFT a T b After Higgsing the U Sp(6) punctures in the G 2 -frame of figure 24a, we are left with a U Sp(4) A × U Sp(4) B flavor symmetry which is enhanced to U Sp (8) in the electric theory when there is no superpotential. We now match the 'tHooft anomalies of these flavor symmetries in the electric and the magnetic frames. In the G 2 theory we find that It is straight forward to check that these match with those in the SO(8) theory, once we use the shifted R and F charges. Thus in the SO(8) theory we have and which is same as the corresponding anomalies of the G 2 theory. The same discussion will also apply in the case of anomalies for the U Sp(4) B flavor symmetries.
In U G 2 c2 the anomaly coefficients can be obtained from the flavor central charges: Since k sp(4) = 7, we find that the anomaly coefficients match those in the electric frame. The same holds for the anomalies of U Sp(4) B .
The anomalies of in U G 2 s can be obtained from the embedding index of U Sp(4) in U Sp (6). Thus for the pair-of-pants containing the puncture A we find Since the 6 of U Sp(6) becomes 4 ⊕ 1 ⊕ 1 of U Sp(4), therefore I = 1. We will also have to add the contribution of the mesons. Thus The anomalies of U Sp(4) B match those in the electric frame in an analogous manner.
The anomalies of U G 2 as can also be shown to match after using the fact that k sp(4) B = 7 and proceeding in the same way as in U G 2 s for the anomalies of U Sp(4) A .

Superconformal index
In this section, we put our new dualities to test by comparing the superconformal indices for the dual theories. We first review superconformal indices for the N = 2 theories of class S studied in [34][35][36][37] which was extended to the case of type D by [38]. In the process, we close some of the loose ends regarding the Z 2,3 -twisted punctures of D n theories. Then we compute the superconformal indices for the N = 1 theories studied in section 3, 4, 5 using a similar formalism developed in [4,39].

N = 2 index
The N = 2 superconformal index is defined as where (j 1 , j 2 ) are the Cartans of the Lorentz group SU (2) 1 × SU (2) 2 , r and R are the U (1) R and SU (2) R generators respectively. The f i denote the Cartans for the flavor symmetry group. For any class S theories, the indices can be thought of as a correlation function for a topological field theory. It turns out that the indices for a class S theory defined by a Riemann surface C with genus g and n twisted or untwisted punctures labeled by ρ 1,··· ,n can be written as where the summation is over the representations λ of Γ. Let us explain the meaning of various symbols.
• The function P λ is some special function defined by requiring the function f λ (a) = K full (a)P λ (a) to be orthonormal under the measure given by the vector multiplet index I V (a): The function P λ can be Schur function or Macdonald polynomial or related to the wave function of elliptic Ruijsenaars-Schneider model depending on the number of fugacities (p, q, t) we want to keep. The P λ also depend on the choice of twisted/untwisted puncture.
• The K-factor K ρ is labeled by a embedding ρ of SU (2) into G, where G = Γ for the untwisted puncture and G is the group formed by folding the Dynkin diagram with the choice of outer-automorphism as in the table 1. The embedding ρ induces a decomposition of adjoint into the form ⊕ j R j ⊗ V j where V j is the spin-j irrep of SU (2) and R j are representations for the flavor symmetry group associated to the puncture. For the case of the Macdonald index (p = 0), the K-factor can be written as [40] K Λ (a) = PE where PE stands for the plethystic exponential. For example, for the full puncture, it is simply given by For the null puncture ∅, it is given by where d i are the degrees of invariants of G and (x; q) = ∞ i=0 (1−xq i ) is the Pochhammer symbol. The general form of K Λ (a; p, q, t) has been conjectured in [4] to be • The argument a ρ I can be determined by looking at the embedding of ρ(SU (2)) × G F into G where ρ(SU (2)) is image under the map ρ and G F is the flavor symmetry group associated to the puncture. The fundamental of G can be decomposed in terms of spin-j irreps of SU (2) as fund G = ⊕ j R F j ⊗ V j . One can match the fugacities by using characters. First write down the character for the fundamental of G. And then compare it with the characters of the representations of SU (2) × G F . By comparing the two, one can map the fugacities for the G F to the fugacities of G appear in P λ (a). See the section 4.2.1 of [38] for more details.
Now, let us focus on the examples of twisted D n -type theories. We will restrict our discussion to the case of Macdonald index p = 0.
We implemented computation of Macdonald polynomials using the procedure outlined in appendix B of [40] through direct Gram-Schmidt process using Mathematica and LieART [41]. There is more efficient method of computing Macdonald polynomials for A, B, C, D, E 6,7 through determinantal construction [42]. We refer to appendix A of [38] for a nice review on the construction of Macdonald and Hall-Littlewood polynomials.
D n -type theories with Z 2 -twist The function P λ in our case becomes the normalized Macdonald polynomial of type G where G is either Γ = D n or G = C n−1 depending on the choice of untwisted and twisted puncture.
where P M,G is the Macdonald polynomial given by the root-system of G. 9 The N λ (q, t) is a normalization factor given by inner product of two Macdonald polynomials where [dx] G stands for the Haar measure of the group G. For the D 4 case, we have two different choice of twisting, namely Z 2 and Z 3 which gives C 3 and G 2 . We will treat this special case later in this section.
The superconformal index for the T SO(2n) theory is given by , (6.10) where the P λ is given by the SO(2n) Macdonald polynomial. One can start from this theory and then by partially closing or Higgsing the punctures, to obtain general theory corresponding to a 3 punctured sphere. In more extreme limit, one can consider completely closing the punctures. Then the index should be trivial, which completely fixes the factor in the denominator which is the structure constant of the TQFT. More generally, when we have twisted punctures, the structure constant can be fixed by requiring it to become trivial when we close all the three punctures. Therefore we can write the index for theT (SO(2n)) theory as , (6.11) where the sum is over the representations of U Sp(2n − 2) not SO(2n). For the P SO(2n) , we restrict the sum to the case of outer-automorphism invariant representations. In terms of Dynkin labels, they are of the form [λ 1 , λ 2 , · · · , λ n−1 , λ n−1 ]. One can completely close one of the U Sp puncture to obtain the free theory of SO(2n) × U Sp(2n − 2) bifundamental half-hypermultiplets. It is given by .

(6.12)
We have checked this relation up to n = 5 and to a few orders in q.
When we glue three punctured spheres, we integrate with a vector multiplet measure. From the orthonormality condition (6.3), we arrive at the same result of (6.2). One interesting aspect here is that whenever there is a twisted puncture, summation over the representations of Γ reduces to that of G.
The Γ = D 4 theory can be twisted in two different ways because the outer-automorphism group is generated by Z 2 and also Z 3 . The Z 2 twisting gives C 3 = U Sp(6) puncture and the Z 3 twisting gives G 2 puncture. Consider the three punctured sphere given by one U Sp(6) puncture and one G 2 puncture with twisted null puncture as in the figure 23. From the TQFT structure, we can write its index as .

(6.13)
Here the sum is over the representations of G 2 . For the SO(8) and U Sp (6) punctures, this means summing over the representations invariant under the Z 3 action. In terms of the Dynkin labels, they are [λ 1 , λ 2 , λ 1 , λ 1 ] and [λ 1 , λ 2 , λ 1 ] for the G 2 representation [λ 2 , λ 1 ]. The TQFT structure requires S-duality invariance of the index. In our case, it translates to the condition that the indices for the first two frames of G 2 -coupled two E 7 theories as in figure 24 being equal. We should have where I G vecl is the vector multiplet index for the gauge group G and I bifund denotes the index of the SO(8) × U Sp(6) bifundamentals (6.12). We represent the G 2 fugacities with ω while the SO(8) fugacities are given by z = (z 1 , z 2 , z 3 , z 4 ) andz = (z 4 , z 2 , z 3 , z 1 ). The transformation of SO(8) fugacities from z toz implements the Z 3 twist around the SO(8) cylinder in figure  24b. Orthogonality of the SO(8) wave-functions upon integration with respect to the SO(8) vector multiplets implies that only those representations that are of the Z 3 invariant form mentioned before, contribute to the RHS of (6.14). This is enough to show the identity of (6.14) As a remark, we find that the index for the E 7 theory can also be written as We can get this form from the identity where the representations λ are now restricted to belong to the Z 3 invariant form discussed above. We do not have an analytic proof of the identity (6.16), but we were able to check this relation for several low-dimensional representations. From the form (6.15), the index becomes 1 upon closing all the punctures. For the case of UV curves without twisted punctures, we always get 1 upon closing all the punctures. It is not clear whether it should be the case with twisted punctures, because even after closing a twisted puncture it still carries non-trivial information. Nevertheless, it turns out that the superconformal index is unity for the theory having a UV curve with only null punctures (with or without twist) of type A n , D n .
Enhancement of Global symmetry U Sp(6) × G 2 to E 7 As we have discussed in section 5, the theory given by U Sp (6) and G 2 punctures is expected to have enhanced E 7 global symmetry [18]. Here we check this explicitly through the computation of index. We find that the index of this theory computed by (6.13) can be indeed written in terms of the characters of E 7 .
The product algebra G 2 × U Sp(6) is embedded into E 7 such that [43] 56 → (7, 6) ⊕ (1, 14) , We find that the index of the U Sp(6)×G 2 theory can be written in terns of the E 7 characters. For example, the Schur index (p = 0, q = t) can be written as where we used the above decompositions to write as (6)  21 (b) and so on. Especially, the Hall-Littlewood index (p = 0, q = 0) is known to reproduce the Hilbert Series of the Higgs branch when the UV curve has genus 0 [36]. The Higgs branch of Minahan-Nemeschansky E 7 theory is known to be the moduli space of E 7 instantons with instanton number 1. The Hilbert series of 1 instanton moduli space is entirely given in terms of the characters for the symmetric product of adjoint representations: This relation for the exceptional group was proven in [44,45] and studied in the physics literatures by [46][47][48]. We verified that the Hall-Littlewood index for the U Sp where we used the Dynkin label here.
Bifundamentals of G 2 × U Sp(4) through Higgsing the E 7 theory As we have discussed in section 5, we can obtain a free theory of G 2 × U Sp(4) bifundamentals by partially Higgsing the U Sp(6) global symmetry down to U Sp(4) of the E 7 theory. We obtain the K-factor from decomposing the adjoint of U Sp (6) to the representations of SU (2)× U Sp(4) which is where we used Dynkin labels to write the representation of U Sp(4). The fugacities for the U Sp(4) puncture is (t, t 1/2 b 1 , t 1/2 b 2 ) in the α-basis meaning all the weights are given as a linear combination of the simple roots. The fugacities for the null puncture is (t, t −1 ) for the G 2 and (t 5/2 , t 4 , t 9/2 ) for the U Sp(6) in the α-basis.

N = 1 index
Now, let us move on to the discussion of the superconformal indices of N = 1 class S theories. The N = 1 superconformal index is defined as I(z; p, q, ξ) = Tr(−1) F p j 1 −j 2 +R/2 q j 1 +j 2 +R/2 ξ −F /2 z Q , (6.24) where F is the U (1) F global symmetry preserved in the class S theory. The N = 1 index of theories constructed in the present paper can be obtained from the N = 2 index of their building blocks. These building blocks can be classified into the colored T σ N blocks (σ = ±) and the N = 1 and N = 2 vector multiplets that couple them together. Their contribution to the N = 1 index is given by I N =1 = I N =2 (p, q, t = ξ σ √ pq), where ξ σ gives their charge with respect to the U (1) F flavor symmetry. 10 As mentioned previously, the underlying TQFT structure implies that the N = 2 superconformal index of class S theories can be written in terms of orthogonal functions f λ (a; p, q, t). It is expected that in general f λ (a; p, q, t) are related to the wave-functions of elliptic Ruijsenaars-Schneider model. There is some evidence that for theories of type A N these functions satisfy the identity [37] f λ (a; p, q, t) We will henceforth assume that this identity continues to hold for theories of type D N and their outer-automorphism twists. This identity implies that the functions P λ (p, q, t) are invariant under t ↔ pq/t. Upon reducing this to the case of N = 1 index, it ensures the invariance of P λ (p, q, ξ σ √ pq) under ξ ↔ ξ −1 . The superconformal index of two T σ N blocks coupled by an N = 1 vector multiplet can be written as where a, b, c, d are the fugacities for the flavor symmetries of the theory while z are the fugacities for the gauge group. I T σ N is the N = 1 index of T σ N theory obtained from its N = 2 index. Due to orthonormality of the wavefunctions, the index in (6.26) formally simplifies to .

(6.27)
Here f σ λ (a) is short-hand for f λ (a; p, q, t = ξ σ √ pq) and has to be chosen appropriately according to the flavor symmetry of puncture "a". f ± λ (∅) correspond to the structure constants in the N = 2 index. The sum in (6.27) is over the set of representations whose Dynkin labels are of the form explained earlier in the paper.
In the Intriligator-Seiberg (magnetic) frame U SO c1 , the superconformal index is which matches with the index of the electric theory upon using (6.31).
In the swapped frame it is the mesons that get a vev, leading to a shift in the R-and F-charges. The shift of the charges can be accommodated into the index by the following substitution: in theT σ N block of the swapped theory, replace the fugacities for U Sp(2N − 2) with those for SU (2) × G F ⊂ U Sp(2N − 2) using ξ σ / √ pq as the fugacity for SU (2). The index of the swapped theory, U SO s , is therefore given by where M σ ∅ is the contribution from the mesonic excitations M Aj,−j and M Dj,−j that stay coupled to the theory: Similarly the index for the theory, U SO as , in the Argyres-Seiberg frame can be written as while the index for the theory, U c2 , in the crossing frame is given by The equality of the indices in the various duality frames can be established by using the identity along with (6.31) and the invariance of P U Sp λ under the Weyl symmetries of U Sp(2N − 2). 11

U Sp dualities
Following a similar procedure as in the case of the SQCD with SO(2N ) gauge group, we can now write down the index of the various duality frames of SQCD with U Sp(2N − 2) gauge group. Before Higgsing some the punctures, we compare the indices of the unHiggsed theories in the various duality frames we obtain by moving the punctures around. The index for the electric theory, T Sp is , (6.40) 11 More specifically we use the fact that P U Sp λ (a) = P U Sp λ (a −1 ).
where the sum now is over the representations of U Sp (2N − 2), as was explained earlier. In the duality frame T Sp c1 obtained by exchanging punctures B and C, we find    Note that in the dual frame the SO(2N − 1) sub-group of one of the three SO(2N ) flavor symmetries of the T N block is gauged while the other two SO(2N ) flavor symmetries are in one to one correspondence with flavor symmetries at the ends of the linear quiver. We thus expect the operator µ 1αβ transforming in the adjoint representation of SO(2N ) 1 to be identified with Ω ij q i 1α q j 1β in the linear quiver.
Here Ω is the invariant anti-symmetric form of the U Sp(2N − 2) group. Similarly we can also identify the operator that corresponds to the dual of µ 2αβ . We now want to establish the chiral ring relation To see this note that the F -term equation of motion of the linear quiver are given by Using these relations we find that = Ω ij Ω lm q i 3α q j 3β q l 3β q m

3α
= Ω ij Ω lm q i 4α q j 4β q l 4β q m 4α = trμ 2 , whereμ αβ is the operator transforming in the adjoint of the SO(2N ) gauge group in the linear quiver. Propagating this relation across the quiver we then establish that trµ 2 1 = trµ 2 2 . By symmetry we thus expect that in the strongly coupled T SO(2N ) block the following chiral ring relation holds trµ 2 1 = trµ 2 2 = trµ 2 3 . We can also consider the superconformal linear quiver of 2N −2 nodes given by SO(2N )× U Sp(2N − 2) × · · · × U Sp (2N − 2). The quiver then ends in a U Sp(2N − 2) flavor symmetry on the left and a SO(2N ) symmetry on the right. This theory can be shown to be S-dual to aT SO(2N ) block coupled to a superconformal tail whose nodes are U Sp(2N − 2) × SO(2N − 1) × U Sp(2N − 4) × · · · × SO(3). The U Sp(2N − 2) node of the tail is obtained by gauging one of the two U Sp(2N − 2) flavor symmetries of theT SO(2N ) block. We will also need to couple a half-hyper to this node in order to ensure that its β-function vanishes. These theories can be visualized as in figure 29. Now if µ ij 1 is the dimension 2 operator ofT SO(2N ) theory transforming in adjoint repre-sentation of U Sp(2N − 2) flavor symmetry while µ 3αβ is the dim. 2 operator transforming as the adjoint of the SO(2N ) flavor symmetry then we identify their duals in the linear quiver to be such that The F -term relations of the linear quiver are Ω ij (q i 1α q j 1β + q i 2α q j 2β ) = 0 , q i 2α q j 2α + q i 3β q j 3β = 0 , (A.8) Ω ij (q i 3α q j 3β + q i 4α q j 4β ) = 0 , . . .