Infinitely many N=1 dualities from $m+1-m=1$

We discuss two infinite classes of 4d supersymmetric theories, ${T}_N^{(m)}$ and ${\cal U}_N^{(m)}$, labelled by an arbitrary non-negative integer, $m$. The ${T}_N^{(m)}$ theory arises from the 6d, $A_{N-1}$ type ${\cal N}=(2,0)$ theory reduced on a 3-punctured sphere, with normal bundle given by line bundles of degree $(m+1, -m)$; the $m=0$ case is the ${\cal N}=2$ supersymmetric $T_N$ theory. The novelty is the negative-degree line bundle. The ${\cal U}_N^{(m)}$ theories likewise arise from the 6d ${\cal N}=(2,0)$ theory on a 4-punctured sphere, and can be regarded as gluing together two (partially Higgsed) ${T}_N^{(m)}$ theories. The ${T}_N^{(m)}$ and ${\cal U}_N^{(m)}$ theories can be represented, in various duality frames, as quiver gauge theories, built from $T_N$ components via gauging and nilpotent Higgsing. We analyze the RG flow of the ${\cal U}_N^{(m)}$ theories, and find that, for all integer $m>0$, they end up at the same IR SCFT as $SU(N)$ SQCD with $2N$ flavors and quartic superpotential. The ${\cal U}_N^{(m)}$ theories can thus be regarded as an infinite set of UV completions, dual to SQCD with $N_f=2N_c$. The ${\cal U}_N^{(m)}$ duals have different duality frame quiver representations, with $2m+1$ gauge nodes.

and U

Introduction
Different 4d N = 1 supersymmetric theories can RG flow to the same IR SCFT [1]. Such dual descriptions are not merely two similar UV completions of the same IR physics, but rather encode the IR physics quite differently, exchanging strong and weak coupling effects such as Higgsing and mass terms. The original duality of [1] relates the electric SU (N c ) SQCD theory, with N f flavors, to a magnetic SU (N f − N c ) theory, with N f flavors and added meson singlets and superpotential.
We will be focussing on SU (N c ) SQCD with N f = 2N c , where the gauge group is selfdual 1 . In [4], a new dual of N f = 2N c SQCD was found, involving two copies of the T N theory 1 Upon adding a quartic Wtree on the electric side, the theory is completely self-dual, as the meson singlets of the magnetic theory get a mass and can be integrated out. This theory can be obtained from the self-dual N = 2 SQCD superconformal field theory with N f = 2Nc, upon breaking N = 2 to N = 1 by an added mass term for the adjoint chiral superfield; see [2,3] for discussion of the N = 1 duality from this perspective. of [5] (see [6] for a nice, recent review), along with 2N 2 + 2N gauge singlets and a specific superpotential. In [7], another new dual of N f = 2N c SQCD was found, involving a single T N theory, two quarks/anti-quarks, N 2 + N gauge singlets, and an intricate superpotential. For N = 2, the T 2 theory reduces to eight free chiral multiplets, the gauging can then be written as a standard Lagrangian, and the duals in this case reduces to ones analyzed in [8,9].
In this paper, we argue for the existence of two infinite classes of 4d N = 1 theories, T where M ij = Q iQj , and λ ij;k˜ are chosen to preserve a SU ( this is a one-complex dimensional conformal manifold of SCFTs. The U (m) N is a quiver gauge theory consisting of 2m + 1 gauge nodes and components constructed from T N , along with a specific superpotential. The m = 2 case is illustrated in the the generalized quiver diagram of figure 1.
2 . The edges connecting the nodes denote bifundamental chiral multiplets. A small box with an 'x'-mark denotes a singlet chiral multiplet coupled to the bifundamental.
(b) Quiver diagram for U (2) N . The triangle refers to the T N theory. Here a small box with 'x'-mark refers to a certain deformation or Higgsing of the theory which breaks one of the SU (N ) ⊂ SU (N ) 3 global symmetries in T N . There are gauge/flavor singlets as well.  the N = 2 T N theories only have a Lagrangian description in the N = 2 case. Nevertheless, for all N , results can be obtained via holomorphy [10,11], much as in [4,12] for the T N case. Also, a-maximization [13] enables us to determine exact R-charges of the chiral operators and the central charges. We thus compute the exact R charges, the anomaly coefficients, and the superconformal index [14,15] of the T (m) N and the U (m) N theories. All of these theories have a natural description as being of class S: the low-energy limit of the six-dimensional N = (2, 0) theory of type Γ = A N −1 , compactified on punctured Riemann surfaces C g,n , generalizing the 4d N = 2 theories of [5,16]. For the 4d N = 1 theories, in addition to C g,n (called the UV curve) we need to assign a pair of integers (p, q) C (p,q) g,n ≡ L(p) ⊕ L(q) → C g,n , with p + q = −χ(C g,n ) = 2g − 2 + n, (1.2) where p ≡ c 1 (L(p)) and q ≡ c 1 (L(q)) and the condition is to preserve N = 1 supersymmetry [17][18][19][20] (as discussed in these references, there are more general possibilities). From the 6d perspective, various dualities can be understood as arising from different choices of the (generalized) pair-of-pants decompositions of the same Riemann surface [7,17,[19][20][21][22][23][24][25][26]. For N = 1 theories, when we decompose C g,n into pants, the (p, q) integers are also decomposed into sums over the pants, with each pair of pants also satisfying (1.2), with g = 0 and n = 3. Previous works on class S field theories restricted to (p, q) ≥ 0, whereas here we consider cases with negative degree. In particular, our T (m) N theory arises from reducing the 6d A N −1 N = (2, 0) theory on the three-punctured sphere C 0,3 , with the line bundle degrees Some perspectives or expressions that are compatible with negative degree include gravity duals [18,19,[27][28][29], superconformal indices [23] and generalilzed Hitchin system associated to the UV curve [20,[30][31][32][33]. A possible objection to combining positive and negative degree pairs of bundles as in (1.3) is that they are unstable 2 to transitions m → m − 1, eventually reducing down to m = 0. We find that the T (m) N theories are stable, but the U (m) N exhibit m → m − 1 cascade processes, via renormalization group flows in the associated 4d QFTs.
The 6d A N −1 , N = (2, 0) theory on a 4-punctured sphere (with punctures being appropriately decorated) gives into two pairs-of-pants, one can assign degrees as in (1.3), (m+1, −m) to one and (−m, m+1) to the other. This suggests N f = 2N c SQCD is dual to theories labeled by general m, with a RG flow down to m = 0, leading to an infinite set of duals. We will flesh out this relation, and provide a number of checks. Among the checks is a matching of the superconformal index [34], which can be seen easily via the generalized TQFT structure studied in [23] and in [4,7]. The outline of this paper is as follows. In section 2, we review the 4d N = 1 SCFT in class S and show how to obtain the theories corresponding to general (p, q) through the nilpotent Higgsing. In section 3, we will discuss the construction of T (m) 2 theory in detail. For the case of Γ = A 1 , we always get a Lagrangian theory with SU (2) gauge groups. From these building blocks, we show how to obtain the dual theories of SU (2) SQCD. In section 4, we generalize the construction to T (m) N which involves multiple copies of T N theory. Using these building blocks, we construct dual theories of SU (N ) SQCD. In section 5, we compute the superconformal indices of the T  In this section, we briefly review the N = 1 class S theories, and our particular constructions.
Data The N = 1 class S theories we consider are labelled by: 1. The choice of a 'gauge group' Γ ∈ ADE of the 6d, N = (2, 0) theory.
2. The choice of a Riemann surface C g,n (UV curve) of genus g and n punctures.
3. The choice of the degree of line bundles (p, q) over C g,n satisfying (1.2). 4. We decorate each of the punctures i = 1, · · · n with an SU (2) embedding ρ i into Γ and a Z 2 -valued color σ i .
We will here focus on Γ = A N −1 , though much of the discussion is valid for general Γ. The total space C (p,q) g,n ≡ L(p) ⊕ L(q) → C g,n in (1.2) is a local Calabi-Yau 3-fold, so M5-branes wrapped on the base C g,n preserves 4 supercharges in the 11-dimensional M-theory. The fourth data labels the punctures that specify the global symmetry of the theory. Here we restrict to the class of punctures that we call the 'colored N = 2 punctures', since locally they are of the same type that appear in N = 2 class S theories [5,16]. For Γ = A N −1 , the choice of ρ i is in one-to-one correspondence with the choice of a partition of N , or equivalently a Young diagram of N boxes. The commutant of the SU (2) embedding ρ i gives the flavor symmetry associated with the i-th puncture.
Such N = 1 class S theories admit a U (1) + ×U (1) − global symmetry [18], with generators (J + , J − ), from those Cartans of the SO(5) R-symmetry of the N = (2, 0) theory that can be preserved after a partial topological twist on the UV curve. Defining R 0 is a U (1) R symmetry and F is a non-R global U (1) symmetry. The exact superconformal R-symmetry is a linear combination where is fixed by a-maximization [13]. For the case p = q, this gives = 0.
Pair-of-pants decomposition and duality The pair-of-pants decomposition of (hyperbolic) C g,n yields a way to build the theory, and find duals. One decomposes the total space C (p,q) g,n , including the normal bundle degrees, with p+q = 1 for each pant (g = 0, n = 3). If one restricts to (p, q) both non-negative, the two options for each pant are (1, 0) or (0, 1), which are denoted by a coloring σ = ±, with C (p,q) g,n then decomposed into p pants of color σ = + and q pants with σ = −. Two pants of same color are glued with an N = 2 vector multiplet, while pants of opposite colors are glued with an N = 1 vector multiplet. See figure 3 for an illustration of the construction. Figure 4 gives the theory corresponding to the pair-of-pants decomposition in figure 3. Different pair-of-pants decompositions of C g,n give IR dual theories.
Each puncture has a SU (N ) symmetry, which is unbroken if the puncture is maximal. In addition to the N = 1 SU (N ) current multiplet, there is a SU (N ) adjoint-valued chiral superfield multiplet, µ (often called the "moment-map" operator). The N = 1 current multiplet and µ combine to form the N = 2 SU (N ) current multiplet when N = 2 supersymmetry is preserved. When the two pants of the same color are glued, the diagonal combination of these N = 2 SU (N ) currents is gauged. When there is an oppositely colored puncture on the pants, we also have extra chiral multiplet M in the adjoint of SU (N ), with a superpotential coupling W = TrM µ, so M effectively replaces the role of µ via a Legendre transform.
Non-maximal punctures are labelled by an SU (2) embedding ρ. We then partially close, or Higgs, the puncture by giving a nilpotent vev ρ(σ + ) to µ if the color of puncture is the same as the pants, and to M if the puncture has the opposite color. This breaks the global + -+ --+ + + - Figure 3: An example of colored pair-of-pants decomposition. Here red/blue means σ = ± respectively. Three red punctures and two blue punctures with p = 2, q = 1. Grey tube denotes N = 1 vector, white tube denotes N = 2 vector multiplet. There are 3 punctures of opposite color. There is an adjoint chiral multiplet attached to each of them. Here we assumed all punctures to be maximal. symmetry associated to the puncture from SU (N ) to the commutant of the ρ(SU (2)) inside SU (N ). The building blocks corresponding to a sphere with generic three punctures can be identified from the previous works [35,36] for the case of same colored puncture, and [4,7] for the oppositely colored puncture.

General (p, q) class S theories from nilpotent Higgsing
We aim to find N = 1 class S theories corresponding to C (p,q) g,n satisfying (1.2), here allowing for negative p or q. The idea is to start with a theory with positive degrees, (p , q ) ≥ 0, and obtain negative degrees via nilpotent Higgsing of the puncture. Following the prescription in [4,7], for the case Γ = A n−1 , we can identify the Higgsed theory. For example, to get the three punctured sphere with degree (m+1, −m), we start with a sphere with m+3 punctures, and line bundles of degree (m + 1, 0), with 3 + punctures and m − punctures. If we Higgs all m of the − punctures, we are left with three + punctures with degrees (m + 1, −m). This procedure allows us to identify the theory corresponding to non-positive (p, q). In the following, we mainly focus on the three (+ colored) maximal punctured sphere with normal bundle degrees (m + 1, −m), which yields the N = 1 theories that we denote by T (m) N , The m = 0 case reduces to the T N theory of [5]. As we discuss, the T (m) N , theory can be constructed from gluing m + 1 copies of the T N theory with a number of singlet chiral multiplets and then Higgsing/closing the punctures. The closure of the puncture is implemented via giving a nilpotent vev to associated chiral adjoints M . This can thought of as a nilpotent mass deformation when Γ = A 1 , i.e. for N = 2. We will discuss this in detail in later sections.

SU (2) theories
Let us start with the SU (2) case, coming from the 6d Γ = A 1 theory, and recall that the T 2 theory of [5] reduces to 8 free chiral multiplets. Likewise, there is a Lagrangian description for every (p, q). We first consider the T To obtain the 3-punctured sphere with normal bundle degrees (m + 1, −m) = (2, −1), we start with the UV curve C (2,0) 0,4 with (n + , n − ) = (3, 1) where n ± denotes the number of ± punctures. Upon closing the − puncture, we will obtain the UV curve C (2,−1) 0,3 with all + punctures. Before closing the puncture, the Lagrangian description of the 4d N = 1 theory is given as in figure 6. The field content of the theory is given as in the table below: Here J ± are combinations of R 0 , F defined so that R 0 = 1 2 (J + + J − ) and F = 1 2 (J + − J − ). They are the 'candidate R-charges' which were used in [7]. The exact R-charge is given by a   0,4 , with (n + , n − ) = (3, 1) and its corresponding quiver diagram, see also [5]. Each node denotes SU (2) global/gauge symmetries.
linear combination of the two, which is determined by a-maximization [13]. In terms of the quiver diagram 6b, SU (2) A,B refers to the blue flavor nodes on the left, and SU (2) C refers to the blue flavor node on the right, and SU (2) D corresponds to the red flavor node on the right. The theory has a superpotential W = Trφ(q 1 q 1 + q 2 q 2 ) + TrM q 2 q 2 .
We now close the red puncture corresponding to SU (2) D by giving a nilpotent vev, M ∼ σ + . This triggers a relevant RG flow, giving a mass to some components of the q 2 matter multiplet. Upon integrating them out, we obtain an IR SCFT described by the quiver diagram of figure 7. It can also be understood as the Fan corresponding to the partition 2 → 2 [7]. The matter content is given as in the table below: 3 theory. The 'x'-marked box denotes a closed puncture. It also means there is a singlet coupled to the quarks connected.
The remaining theory has superpotential which is generic for the global symmetry with (J + , J − ) = (2, 2) charges. 4 The charged matter is that of N = 2 SU (2) with N f = 3, but the theory is N = 1 supersymmetric because one of the flavors does not couple to the adjoint, instead coupling to the gauge singlet M . This theory has a quantum moduli space of vacua, with several branches. The M field can have arbitrary expectation value, and M gives a mass to the q 2 field. The low-energy theory for M = 0 thus has an accidental N = 2 supersymmetry, given by N = 2 with N f = 2 flavors, with global symmetry SU (2) A ×SU (2) B ×SU (2) R ×U (1) R . That theory has [37] a Coulomb branch, with modulus u = Trφ 2 , and two Higgs branches, emanating from the massless monopole and dyon points on the Coulomb branch, at u ∼ ±Λ 2 L ∼ ±M Λ. Each Higgs branch is a copy of C 2 /Z 2 , and either SU (2) A or SU (2) B is spontaneously broken, depending on which branch. For M → 0, the two Higgs branches meet at the origin of the Coulomb branch, with additional moduli from q 2 , subject to the F-term Trq 2 q 2 = 0. It would be interesting to interpret this moduli space via geometric construction.
The IR theory at the origin of the moduli space is an N = 1 interacting SCFT. It has a manifest SU (2) 3 flavor symmetry, with three (J + , J − ) = (2, 0) moment map chiral operators, in the adjoint representations of SU (2) A,B,C , given by The operator µ C is dressed with the adjoint chiral multiplet φ to have the correct charges, (J + , J − ) = (2, 0). Despite the apparent difference between µ A,B vs µ C , the IR SCFT is expected to be S 3 permutation symmetric under permutation of the SU (2) A,B,C symmetries. Because the theory is N = 1 supersymmetric and not N = 2, these chiral operators are not in the SU (2) A,B,C current multiplets, and they receive anomalous dimension. The exact superconformal R-charge is as in (2.2), R = R 0 + F, and then chiral scalar operator dimensions are given by ∆(O) = 3 2 R(O), e.g. ∆(µ A,B,C ) = 3 2 (1+ ), ∆(Trφ 2 ) = 3(1− ), ∆(M ) = 3(1− ), with determined via a-maximization to be 5 0.52. We find that the superconformal index computed from this gauge theory description agrees with the TQFT prediction of [23]. The index is compatible with the S 3 permutation symmetry.

T (m=2) 2
We start from the theory corresponding C . There are three different ways to do this, starting from the three dual frames of the unhiggsed theory as in the figure 8. The unHiggsed theory has SU (2) × SU (2) gauge group with bifundamental hypermultiplets and two more fundamentals attached to each of the gauge groups. The blue parts of the quiver are N = 2 supersymmetric, with chiral adjoints φ for each gauge group and N = 2 matter couplings. and (n + , n − ) = (3,2) where n ± denotes the number of ± punctures respectively.
The red nodes are N = 1 supersymetric, given by two chiral multiplets transforming as adjoints of the flavor groups, coupled via a superpotential of the form where µ a is the gauge invariant bilinear of chiral multiplets, in the adjoint of the SU (2) a global symmetry. We then close the − punctures by giving nilpotent vevs to the two chiral multiplets M a attached to the − punctures. This triggers a relevant deformation of the theory which leads to a new SCFT in the IR. Since the three different quivers are dual to each other before Higgsing, they all flow to the same SCFT in the IR. The nilpotent M a vev in quivers 1 and 2 gives rise to mass terms for some of the quarks, which we integrate out. Figure 9 describes the quiver after Higgsing. In the figure, an 'x'-marked box denotes the remnant of a closed puncture, where a gauge / flavor singlet component of M a remains, with coupling to the remaining quarks in the theory. Quiver 3 requires a special treatment since the second nilpotent vev does not introduce a mass term.
Consider first quiver 1. The nilpotent M a on the right/left-hand side gives the same type of the matter content as in the figure 7, with matter and charges as in the table below: The singlet field attached to the 'x'-marked box couples to the neighboring quarks, which gives rise to a cubic superpotential term similar to that in (3.3). In addition, there is a quintic coupling between the quarks and the adjoint chiral multiplets: Quiver 2 can be understood by considering a decoupling limit of the SU (2) gauge group corresponding to the rightmost gauge node. The left-hand side of the quiver is then the same as the T (1) 2 theory. We list the matter content and charges of the theory in the table below: The superpotential for the quiver 2 is where we suppress gauge and flavor indices, which are as determined by the symmetry. The superpotential is generic given the (J + , J − ) = (2, 2) or R 0 = 2 and F = 0 symmetry.
Non-mass deformation Let us consider quiver 3. When we close one of the − punctures, we get a similar description as quiver 1 and 2. Now, we need to further close the −(red) SU (2) puncture by giving a vev to the chiral flavor adjoint of say SU (2) 0 . Before closing the last puncture, we have a superpotential term TrM 0 φ 1 (q 0 q 0 ) where q 0 is the quark transforming as a fundamental of SU (2) 0 , and φ 1 is the chiral adjoint of SU (2) 0 . The nilpotent vev M 0 = σ + then gives the deformation term Trσ + φ 1 (q 0 q 0 ). Though not a mass term for the quarks, it nevertheless turns out to be a relevant deformation, breaking the SU (2) 0 global symmetry. To see that Trσ + φ 1 (q 0 q 0 ) is relevant, note that it has charge (J + , J − ) = (2, 0) which means the exact R-charge (before the deformation) is R = 1 + , which is relevant, R < 2, since a-maximization gives 0.46. This gives a 1.55 before the deformation. The SU (2) 0 breaking M 0 = σ + yields a superpotential with terms where µ m=−1,0,1 = Trσ m φ 1 q 0 q 0 is in the adjoint of SU (2) 0 . Much as in [4], the first term in (3.10) leads to SU (2) 0 current non-conservation for the m = 0, 1 components: so, for m = 0, 1, J m and µ m−1 pair up to become long multiplets. The remaining superpotential is The charges (J + , J − ) must be shifted to be conserved and unbroken The matter content after Higgsing is as in Figure 9, with charges: We will consider similar type of deformations in section 4.

't Hooft Anomalies
The anomaly coefficients of T 2 , in all three dual frames, are: a-maximization yields 0.534 and a 1.45 for the T 2 theory in all three dual frames. Here SU (2) 0 is the flavor symmetry SU (2) A .

T (m) 2
We can generalize previous subsection to construct a general T (m) 2 theory. Start with the UV curve C (m+1,0) 0,m+3 with (n + , n − ) = (3, m). By closing all the − punctures, we arrive at the sphere with 3 + punctures and normal bundle degree (m + 1, −m). We can consider a number of different dual frames, but let us consider the analog of quiver 2 in figure 9. The resulting theory will be a quiver gauge theory, with SU (2) m gauge symmetry, bifundamental chiral multiplets for the neighboring nodes, and 2 fundamental chirals at the end nodes. In addition, we have adjoint chiral multiplets for each gauge nodes, and m gauge/flavor singlet chiral multiplets. We summarize the matter contents and their charges in the table 1. The superpotential is (with indices, and their contractions, suppressed) The 't Hooft anomaly coefficients for this theory are The value of is fixed, by maximizing a( ), to be As a check, (m = 0) = 1 3 which is the value of the free field theory T 2 . The central charge a( (m)) grows linearly in m, which is not surprising from the quiver gauge theory perspective.
The T (m) 2 theories do not have any exactly marginal deformations: there are m+(m−1)+ 1 + m = 3m couplings from the terms in the superpotential (3.15), and the gauge couplings, and there is no linear relation among their beta functions. The conformal manifold is an isolated point; this is consistent with geometric construction, since the three punctured sphere has no complex structure modulus. with (n + , n − ) = (2, 2). The theory enjoys multiple dualities [8,9] which also has a class S interpretation [4]. Moreover, this theory is known to have 72 dual frames [38,39]. We now argue that gluing two copies of T e.g. (m = 0) = 1/3 gives k = 2; the gauged SU (2) will be asymptotically free if 2k < 3N c = 6, which is satisfied for all m in (3.18).
There are several, dual descriptions of the resulting theory, corresponding to the dual descriptions of each pair-of-pants discussed in section 3.2. Let us pick the dual frame referred   where one copy has flipped (J + , J − ) charges, as listed in the table. In addition to the added gauge multiplet, we have a superpotential term where µ σ=± = q σ 1 q σ 1 is the operator, with (J + , J − ) = (2, 0) or (0, 2), associated to the glued punctures and superpotential (with gauge indices contracted and coupling constants λ) We argue that the U (m) 2 theories RG flow to the same IR fixed point as N f = 4 SU (2) SQCD, which is the m = 0 case of U (m) 2 . As a first check, we find that the 't Hooft anomaly coefficients of the U (m) 2 quiver theory are m-independent: Matching of operators Among the single trace, gauge invariant operators of U However, there initially appears to be a mismatch in our proposed duality between U seems to contribute extra gauge singlet operators, M ± i and u i = Tr(φ ± i ) 2 , for i = 1 . . . m. Classically, these would lead to a mismatch with N f = 4 SU (2) SQCD, not only in the spectrum of operators, but also in the moduli space of vacua. Actually, as we now discuss, the quantum theory does not have the M i and u i classical moduli. They are quantum-lifted in a way similar to what happens in magnetic SQCD, where the classical electric condition rank(M ) ≤ N arises from non-perturbative dynamics in the dual [1]. A vev of the would-be moduli would induce a dynamically generated superpotential, which is inconsistent with the F -term constraints.
To see this in our setup, suppose first that some M − n−1 has a non-zero vev, which spontaneously breaks J + and gives a mass to the quarks q − n from the first term of (3.21). This effectively decouples the side of the U consistent with the symmetries for all j. This would lead to a q − j runaway that is incompatible with F M − i = 0, so the apparent M − n−1 flat direction is actually lifted. Likewise, if u − n gets an expectation value, the associated non-zero φ − n spontaneously breaks J + and gives a relevant deformation from the second term of (3.21) (since q − i φ − i−1 q − i has R-charge 1 or (J + , J − ) = (0, 2)). In order to preserve J + symmetry in the IR, the charge of q − n becomes (J + , J − ) = (0, 1) and the SU (2) n−1 instanton factor gets J + charged, (Λ − n−1,L ) b L ∼ u − n so the theory admits which has a runaway for q ± i that is incompatible with F M ± j , so the u n flat direction is lifted.  theory via the method of [2]. The exact NSVZ beta functions for the gauge couplings of SU (2) 0 and SU (2) ± i are (with g σ i the gauge couplings for SU (2) σ i ) , (i = 2, · · · , m) .
The exact beta functions for the superpotential couplings are where the anomalous dimension γ O is given by ∆ theory has a one complex dimensional conformal manifold. This can also be seen via the the method of [40]. There are 6m + 2 couplings, which break U (1) (6m+2)−1 global symmetries (the −1 is because we preserve U (1) F ), so there is a one-complex dimensional conformal manifold that preserves the SU (2) 4 × U (1) F × U (1) R global symmetry.
Cascading RG flow to SQCD The duality frame of figure 12b is the U (m) 2 theory, which we claim is dual to the U (m) 2 theory, giving another description of the theory obtained by gluing two copies of T (m) 2 . The U m theory has superpotential term where µ σ=± = φ σ 1 q σ 1 q σ 1 is the operator with (J + , J − ) = (2, 0) or (0, 2) associated to the punctures that we are gluing and (with implicit gauge index contractions) In this dual frame, the SU (2) 0 gauge group has N f = N c and no adjoint, so it confines, with a quantum deformed moduli space constraint as in [10]. At energies below the SU (2) 0 dynamical scale, the SU (2) 0 node is eliminated, and its adjoining fundamentals are replaced with the SU (2) 0 neutral composites where V + and V − (the SU (2) analog of baryons) are gauge singlets, while the mesons V +− transform as a bifundamental of SU (2) +1 × SU (2) −1 , with the constraint [10] det The superpotential (3.31) becomes (with implicit trace over gauge and flavor indices) We see that V ± combine with M ± 1 to become massive, so they can all be integrated out, setting V ± = M ± 1 = 0. The quantum constraint on the moduli space (3.31) then implies that V +− = 0. The non-zero V ± bifundamental vev Higgses SU (2) +1 × SU (2) −1 to the diagonal SU (2) subgroup. It follows from the superpotential (3.35) that φ ± 1 become massive, and are integrated out. The resulting low-energy theory is thus similar to the original theory (shown in figure 14 . The above analysis applies to that theory, again reducing m, giving a cascading RG flow that eventually ends up at the m = 0 theory, U 2 , which is simply SU (2) SQCD with N f = 4. thus also flows to the same IR SCFT as SQCD.

SU (N ) theories
We here generalize the discussion in section 3 to N = 1 SU (N ) SQCD with 2N flavors. The new element is that we have to replace each bifundamental or trifundamental chiral multiplet, in the links of the quiver, by the T N theory and its deformations. We first construct the N = 1 T

Review of the T N theory
Recall that the T N theory is an N = 2 SCFT with SU (N ) A × SU (N ) B × SU (N ) C flavor symmetry. The theory also has ∆ = 2 "moment-map" chiral operators, µ A,B,C , in the adjoint of the SU (N ) A,B,C respectively. These operators satisfy the chiral ring relation [12] Trµ k A = Trµ k B = Trµ k C , (4.1) for k = 2, 3, · · · N . There are also operators Q ijk ,Q ijk which transform as the trifundamental and anti-trifundamental of SU (N ) A × SU (N ) B × SU (N ) C with scaling dimension N − 1. The T N theory has a Coulomb branch of complex dimension (N − 2)(N − 3)/2, and a Higgs branch, which meet at the origin. See [6,12] for more detailed discussion on the chiral ring operators and their relations of the T N theory.
Since the T N theory at the origin is a N = 2 SCFT, it has U (1) R N =2 × SU (2) R symmetry. When we couple this theory to an N = 1 theory, we preserve (J + , J − ) = (2I 3 , R N =2 ), where I 3 is the Cartan generator of SU (2) R . As in the previous section, one linear combination of J + , J − will become exact R-charge, and F = 1 2 (J + − J − ) will be a charge of the global symmetry of the theory. The µ A,B,C operators have the charge (J + , J − ) = (2, 0), and Q ijk ,Q ijk have (J + , J − ) = (N − 1, 0). The 't Hooft anomaly coefficients of the T N theory are: We start with a m + 3-punctured sphere with 3 + punctures and m − punctures and degrees (p, q) = (m + 1, 0). Here we assume all the punctures to be the maximal one carrying SU (N ) global symmetry. Let us choose the colored pair-of-pants decomposition so that we get the quiver as described in the figure 15a.
The theory is composed of m+1 copies of T N theory that are connected via N = 2 vector multiplets and m extra chiral multiplets M (i) (i = 1, · · · , m) transforming under the adjoint of the SU (N ) i global symmetry associated to the − punctures. We denote the moment map operators of the + colored operators by µ A,B,C and those of − colored operators by µ (i) (i = 1, · · · m). We use φ i for the adjoint chiral multiplets in the N = 2 vector multiplet and µ k ,μ k for the moment map operators for the symmetry group SU (N ) k that are being gauged. The superpotential is where ρ is the principal embedding of SU (2) into SU (N ). This will induce a relevant deformation to the theory which we name as T (m) N . Here we closely follow the discussion of [4]. We can decompose the adjoint representation of SU (N ) in terms of sum of the spin-j irreducible representation V j of SU (2) as adj = N −1 j=1 V j . Using this, one can write each components of the adjoint of SU (N ) in terms of (j, m) with m = −j, −j + 1, · · · , j − 1, j. After giving the vev, the superpotential can be written as This superpotential preserves (J + , J − ) = (2, 2) upon the shift where m (i) are the weights of the SU (2) representations or the image of J 3 = σ 3 /2 under ρ i associated to each puncture (i) being closed. The vev breaks the original SU (N ) global symmetry, with the non-conservation of the current given by The semi-short multiplet (J (i) ) j,m and the chiral multiplet µ (4.8) We summarize the 'matter content' of the theory in the table 4.
Anomaly coefficients To compute the 't Hooft anomaly coefficients of the T (m) N theory, we need to compute effect of the Higgsed T N block, with the nilpotent vev. Accounting for the above shifts, we find that we simply need to add the contributions from M j,−j to that of the T N theory. This gives, for the single puncture Higgsed T N or equivalently the theory corresponding to the UV curve C Combining this with the known results of the T N theory and the quiver description depicted in figure 15 and the charges of the singlets as given in (4), we obtain the anomaly coefficients of the T (m) N as follows: Note that the anomalies involving the SU (N ) A,B,C are the same as that of T N theory. These coefficients can also be obtained from the formula given in the section 5.2 of [7] by extrapolating all the formulas to the negative p or q.
The trial a-function is 11) and the value of is fixed by a-maximization to be For m = 0, we find = 1 3 , which is the expected value for the N = 2 T N theory. The value of a increases linearly with respect to m and grows cubically with respect to N . We can also determine the SU (N ) flavor central charge k SU (N ) [41,42] to be for m > 0, we see the flavor central charge is less than N for m > 0. In many respect, the T N theory behaves as N fundamental flavors [12] since it contributes the same amount to the beta function of the gauge coupling. For the T (m) N case, it contributes to the beta function as that of N f < N .

Infinitely many N = 1 duals for gauged T N theories
As a preparation of the SQCD, let us first consider the theory obtained by gluing two copies of T N theory by gauging one of the SU (N ) flavor groups on each of T N . It can be obtained from choosing the UV curve to be the 4-punctured (all maximal, 2 +, and 2 − colored) sphere with (p, q) = (1, 1). See the figure 16a. This theory and its dualities have been studied in [4,17] which we review here. This theory has SU (N ) symmetry with the 'matter content' as given in the table 5.
For this theory, the superconformal R-charge is given by R 0 = 1 2 (J + +J − ). The µ A,B,C,D 's are the operators present in the T N theory, which are associated to the punctures on the   UV curve. The operators µ ± are the operators corresponding to the punctures that we are gluing/gauging. We can write a superpotential term which preserves all the global symmetries of the theory. Now let us describe the dual theories of the coupled T N . We couple two copies of T has to be flipped in order to write the superpotential term (4.13). See figure 16. The 'matter content' of the theory is given in the table 6.  The theory has a superpotential where Since the coupled theory for any m comes from the same UV curve, we expect they all flow to the same SCFT in the IR. Let us compute the anomaly coefficients of the quiver theory. We can use the anomaly coefficients we computed for the T We see that the anomaly coefficients are independent of m, therefore it agrees with the gauged T N which corresponds to the case with m = 0. We will match the set of supersymmetric operators by computing the superconformal index in section 5.
Cascading RG flows to the gauged T N theory In section 3.4 we saw that in the dual frame of the form figure 12b, the central gauge node SU (2) 0 confines and we get a cascade of RG flows which ultimately reduces the whole system to SU (2) SQCD with 4 flavors. Here, we will argue that a similar mechanism occurs when two T (m) N blocks are glued to each other to give the duality frame of figure 17a. Guided by the SU (2) case, we claim that the N = 1 node in the sub-quiver shown in figure 17b undergoes confinement with a quantum deformed moduli space. At energies below confinement-scale, the spectrum of the quiver will include operators that transform as bifundamentals of the ±1-th nodes of the original quiver. The quantum deformation of the moduli space will imply that these bifundamentals have a nonzero expectation value, breaking the product gauge group SU (N ) +1 × SU (N ) −1 down to the diagonal SU (N ). The expectation value will also make the adjoint chiral fields coupled to the ±1-th nodes massive, which will therefore get integrated out. The upshot will be a reduction of m → m − 1: at low energies, the quiver shown in figure 17b reduces to that shown in figure  17c. This process triggers a cascade of RG flows which reduces the quiver of figure 17a down to that shown in figure 16a.
(a) Another quiver description obtained by gluing two copies of the T    As an evidence to support our claim about figure 17b, we consider the theory obtained by gluing two T (1) N blocks via an N = 1 vector multiplet along one of their full punctures. The other full puncture of each block is glued (via an N = 2 vector) to an N = 2 quiver tail corresponding to the minimal puncture, giving the quiver in figure 18. If our claim is correct then the central N = 1 node of this quiver should also exhibit confinement, and the theory will then flow to the quiver of figure 19. We now argue that this is indeed the case. Note that the quiver of figure 18 is dual to the linear quiver shown in figure 21. When the 'x'-marked punctures of the figure 18 are not closed, as in figure 20a, the theory is dual to the linear quiver of figure 20b [43]. The only difference here is that we added gauge singlets to the punctures. From here, we close the punctures at each ends by a nilpotent Higgsing to get the linear quiver theory as given in the figure 21 [7]. We have also shown the (J + , J − ) charges of the various fields in the same figure. The superpotential terms of this quiver are given by all the single trace gauge singlet local operators with charges (J + , J − ) = (2, 2).
Let us now dualize the central N = 1 node of figure 21, followed by dualizing the ±1-st nodes, then dualize the ±2-nd nodes and so on until we finally dualize the ±(N − 2)-th nodes of the quiver. This will land us on a linear quiver which has an N = 1 vector multiplet at the 0-th, ±(N − 2)-th and ±(N − 1)-th nodes while the rest of the nodes have an N = 2 vector multiplet as shown in figure 22. Notice that the N = 1 node at either ends of the quiver in the current duality frame is equivalent to an SQCD with N f = N c + 1 flavors. These nodes will therefore undergo s-confinement. The low energy theory of this quiver will then be given  by fields describing the mesonic and baryonic fluctuations of the end nodes. Equivalently, we can Seiberg dualize this node to get the theory of free chiral multiplets. This corresponds to the quiver of figure 23. Once again the superpotential of this quiver can be written down Figure 23: The low energy theory of the quiver in figure 22 obtained by noticing that the nodes at its left and the right ends undergo s-confinement. Here j = 1, · · · , N − 2.
by considering all the chiral gauge invariant operators which have charges (J + , J − ) = (2, 2). This will include the low energy superpotential of N f = N c + 1 SQCD that is expected to be there after s-confinement of the edge nodes in figure 22. Figure 24: The duality frame of the theory in figure 23 obtained by dualizing its 0-th node, followed by the ±1-th nodes and so on until we dualize the ±(N − 3)-th nodes.
In order to proceed we will first have to go through the following series of dualities: dualize the 0-th node in the quiver of figure 23 followed by the ±1st nodes, then the ±2nd nodes and so on until we finally dualize ±(N − 3)-th nodes. This series of dualities will produce a quiver whose central and last two nodes on either sides are gauged using an N = 1 vector multiplet while the rest of the nodes are gauged using an N = 2 vector multiplet. This quiver is depicted in figure 24.
If we now dualize the nodes at the left and the right ends of the quiver in figure 24, we obtain the quiver of figure 25. Here j = 1, · · · , N − 3.
We will now have to again go through the series of dualities mentioned in the previous paragraph, this time stopping when we dualize the ±(N − 4)-th nodes. This gives us the quiver of figure 26. Dualizing the penultimate nodes on either sides of this quiver gives the quiver that can be represented by figure 27. We can now repeat the series of dualities outlined earlier (starting by dualizing the 0-th node, followed by dualizing the (±1)-st node and so on) multiple times such that we ultimately land on a linear quiver that corresponds to figure 28. Dualizing the 0-th node of this quiver then lands us on the duality frame of figure 19 which is the result we sought. where

18)
A C (a) A quiver description dual to the SU (N ) SQCD with 2N flavors.
N obtained by gluing the two copies of T   with Anomaly coeffecients As an intermediate step, let us consider the Higgsed T Here A and G are the two maximal punctures while B is the name we used for the minimal puncture. The anomalies of the T (m) N theory with all its colors inverted can be obtained by interchanging the roles of J + and J − in the above table.
We now compare the anomaly coefficients of our proposed dual theories. For U (m) N , we find: As before we find that these coefficients are independent of m and match perfectly with those of SU (N ) SQCD with 2N flavors.
Cascading RG flows to SQCD As in the case of the section 4.3, let us consider a dual description for the T N theory itself to show that it flows to the same theory as the SU (N ) SQCD with 2N flavors. The 'matter content' of the theory U N . We call this as U  The set of chiral operators in the T N theory contains (anti-)trifundamental operator Q ijk andQ ijk . When an oppositely colored puncture of the T N block is closed, the operators , and the corresponding charges being (J + , J − ) = (N − 1, −2 ) or (−2 , N − 1) depending on the choice of color. These operators will be important to our analysis and we will label those coming from the i-th block in figure 30 as Q σ,(i) ,Q σ,(i) suppressing indices.
The superpotential for the theory is given as where with Here we formed the gauge invariant operators µ σ m so as to transform as the adjoint of SU (N ) ±m according to whether σ = ± whileμ σ k is constructed such that it transforms as the adjoint of SU (N ) 0 .
By applying a sequence of dualities, we have showed earlier that the central SU (N ) 0 -node confines. From this, we conjecture that the SU (N ) 0 -node undergoes confinement with N 2 mesonic operatorsQ ±,(1) and quantum deformed moduli space given by det Q ±,(1) where Λ b 0 is the SU (N ) 0 instanton factor, with the exponent b determined by The scaling dimensions of the two sides of (4.26) agree, upon using ∆ = 3 2 R U V , where R U V is the superconformal R-charge before gauging SU (N ) 0 . Gauging SU (N 0 ) breaks the separate which is consistent with the U (1) A charge of the product of operators on the LHS of (4.26). The operators on the LHS of (4.26) carry U (1) R IR charge zero, as required for a quantum deformed chiral ring relation (and that is why other Q ±,(i) ,Q ±,(i) do not appear in (4.26)).
The first and second term in the LHS of (4.26) are analogs of detM and BB in SQCD with N f = N c . We put the second term in quotes because we have not fully determined the dependence on the µ ± j,j beyond what is fixed by the symmetries. In any case, the F terms of superpotential (4.22) sets the operators µ ±,(i) j,j to zero, setting the terms in quotes to zero in (4.26). On the deformed space (4.26), the Q ±,(1) thus have non-zero expectation value. Then φ + 1 and φ − 1 will become massive via the last term of (4.22) with k = 1. Moreover, the SU (N ) +1 × SU (N ) −1 gauge symmetry is broken down to the diagonal SU (N ), which will again undergo confinement. This is an iterative cascade of RG flows, reducing m in each step, eventually flowing to SU (N ) SQCD with 2N flavors with a quartic superpotential in the IR.

Superconformal index
The superconformal index for a N = 1 superconformal field theory is defined as where we introduced the fugacity ξ for the U (1) F which is present for generic class S theories. For the theory having a Lagrangian description in the UV, the index can be simply computed by multiplying the contributions from each matter multiplets in the UV and then by integrating over the gauge group. The contribution of each matter multiplets is calculated using the exact R-charge in the IR [14]. In our case, the only possible non-anomalous U (1) symmetry that can mix with R-symmetry in the IR is U (1) F . Therefore we can obtain the index using the UV R-charge as long as we keep the fugacity ξ turned on. Once we know the exact R-charge R = R 0 + F, we can simply redefine ξ → ξ(pq) /2 to obtain the true superconformal index.

Topological field theory and superconformal index
For an N = 1 SCFT in class S, the superconformal index can be written in terms of a correlation function of the 2d (generalized) topological field theory living on the UV curve. This topological field theory is related to a deformation of 2d Yang-Mills theory [23,[44][45][46][47][48].
The index can be written as where (p, q) are the degrees of the line bundles and n is the number of punctures, which should satisfy the relation p + q = 2g − 2 + n. Here we suppressed the p, q, ξ dependence and the sum is over the representations λ of Γ labelling the six-dimensional (2, 0) theory. The basis function ψ ρ,σ λ (a) corresponding to the puncture labelled by the embedding ρ : SU (2) → Γ and color σ can be written in the following form where t σ = ξ σ √ pq and we suppressed the p, q dependence. The K-factor does not depend on λ, but the form of the function depends on the type of puncture. P λ is a symmetric function of a which in certain limit reduces to the Macdonald polynomial. The argument at ρ σ is determined by the embedding ρ of SU (2) into Γ labelling the puncture (see [49]). The structure constant can be written as C σ λ = (ψ ∅,σ λ ) −1 in terms of the basis function ψ's. Let us compute the index of the T (m) N starting from the theory given by the UV curve C (m+1,0) 0,m+3 with (n + , n − ) = (3, m) where we know how to write the index from the TQFT: Now, we want to Higgs all the − punctures. Complete Higgsing or closing of a puncture is implemented via replacing the wave function ψ ρ,σ λ (b) corresponding to the puncture to close by ψ ∅,σ λ (t ρ σ ). From the relation C σ λ = (ψ ∅,σ λ ) −1 , we see that the degree of the normal bundle corresponding to the color σ reduces upon Higgsing. We get where we suppressed ρ i to denote full punctures. One can also flip all the colors ± in the components to get the same index with ξ → ξ −1 . This is of the same form as the equation (5.2), from which we can plug in (p, q) = (m + 1, −m) with 3 + colored punctures.
Once we have the equation (5.5), it is a piece of cake to show that the index is the same for the dual theories, independent of m. Gluing two copies of T (m) N with opposite color by a cylinder to form the theory corresponding to the 4-punctured sphere with (p, q) = (1, 1), the index can be written as We here used the fact that wave functions are orthonormal: [dz]I vec (z)ψ + λ (z)ψ − µ (z) = δ λµ , (5.7) where I vec (z) is the contribution to the index from a N = 1 vector multiplet. Therefore for any choice of m ∈ Z the gluing gives us the same index as that of the theory described by 2 full + punctures and 2 full -punctures and (p, q) = (1, 1). It describes the two copies of T N theory glued by N = 1 vector multiplet. The same argument goes through when we Higgs or partially close the full punctures of each color to minimal punctures to get the SQCD. In the paper [23], the superconformal index for the generic (p, q) was proposed from the structure of the (generalized) topological field theory, initially without concrete SCFTs that realize the indices. The SCFT that we discuss here gives such a concrete realization.

Direct computation for the SU (2) theories
The proof of the previous section holds as long as the index of the T N theory can be written in terms of the basis wave function ψ λ (a). Here, we confirm the TQFT formula for T where v are the weight vectors of the representation R of the symmetry group the chiral multiplet is charged under. Here the notation z v is a short-hand for i z v i i . Here, we used the elliptic gamma function which is defined as Γ(z; p, q) = ∞ m,n=0 to write the index in a concise form. We will suppress the p, q dependence of Γ(z; p, q) whenever possible. The vector multiplet contribution to the index is given by chi (z ± a ± b ± )I (0,1) chi (z ± c ± d ± ) , (5.14) where we glued two T (m) 2 with opposite F charges. We have verified this identity to hold for m = 1, 2 at the leading orders in p and q. where we also refined the index for the SQCD. One can easily check the index preserves SU (8) flavor symmetry by relabelling the fugacities. We should keep in mind thatĨ (m) in (5.15) is not a genuine index of the theory, since T (m) 2 itself does not have the SU (4) symmetry. There is a cubic coupling which breaks SU (4) → SU (2) 2 , and this coupling cannot be tuned to zero as we have discussed in section 3.3. But after gluing two copies of T

Conclusion and outlook
Guided by the construction of 4d QFTs from M5 branes wrapping Riemann surfaces, we constructed an infinite set of dual theories of 4d N = 1 SU (N ) SQCD with 2N flavors. These theories are parametrized by an integer m ∈ Z ≥0 and involve 2m copies of the T N theory of [5], 2N quarks/anti-quarks along with 2m(N − 1) singlet chiral superfields as their building blocks. As a check of the dualities we compared their central charges, anomaly coefficients and superconformal indices. Along the way, we constructed a family of new N = 1 SCFTs with SU (N ) 3 flavor symmetries, which generalize the N = 2 T N theory.
The dual theories discussed here can be used to construct more duals, for example by applying them to the magnetic dual of [1]. This will result in adding extra chiral multiplets transforming as adjoints of global symmetries SU (N ) A,C and cubic superpotential terms. We can also consider the swapped dual of [4], and also Argyres-Seiberg type duals of [7,25]. Moreover, as we have discussed in the section 3.2, even the building block T (m) N itself has many different dual descriptions, so the number of duals grows rapidly with m.
One question is how to generalize our dualities to N f = 2N . This may be possible e.g. by considering a mass deformation of the T (m) N theory, as was done in the T N case [50]. From the class S perspective, this involves understanding dualities in the presence of irregular punctures. Another direction would be a more detailed study of phase structure and chiral ring of the new theories. The spectral curve of the generalized Hitchin system associated to the N = 1 theories [30][31][32][33]51] will be useful. It will be also interesting to generalize our construction of T (m) N to D and E type theories and also with outer-automorphism twists using the N = 2 results [52][53][54][55][56][57][58], as well as possible generalizations using the theories of [59,60], which will provide analogous infinitely many duals for other gauge groups.