Infinitely many \( \mathcal{N}=1 \) dualities from m + 1 − m = 1

Journal of High Energy Physics, Oct 2015

We discuss two infinite classes of 4d supersymmetric theories, T N (m) and \( {\mathcal{U}}_N^{(m)} \), labelled by an arbitrary non-negative integer, m. The T N (m) theory arises from the 6d, A N − 1 type \( \mathcal{N}=\left(2,0\right) \) 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 \( \mathcal{N}=2 \) supersymmetric T N theory. The novelty is the negative-degree line bundle. The \( {\mathcal{U}}_N^{(m)} \) theories likewise arise from the 6d \( \mathcal{N}=\left(2,0\right) \) 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 \( {\mathcal{U}}_N^{(m)} \) 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 \( {\mathcal{U}}_N^{(m)} \) 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 \( {\mathcal{U}}_N^{(m)} \) theories can thus be regarded as an infinite set of UV completions, dual to SQCD with N f = 2N c . The \( {\mathcal{U}}_N^{(m)} \) duals have different duality frame quiver representations, with 2m + 1 gauge nodes.

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Infinitely many \( \mathcal{N}=1 \) dualities from m + 1 − m = 1

Received: July = 1 dualities from m two (partially Higgsed) T 0 1 2 theories. The T 0 1 2 UN N N 0 1 2 0 Open Access , c The Authors 1 San Diego , La Jolla, CA 92093 , U.S.A 2 Department of Physics, University of California We discuss two infinite classes of 4d supersymmetric theories, T N(m) and U N(m), labelled by an arbitrary non-negative integer, m. The T N(m) theory arises from the 6d, AN−1 type 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 N = 2 supersymmetric TN theory. The novelty is the negative-degree line bundle. The UN 6d N = (2, 0) theory on a 4-punctured sphere, and can be regarded as gluing together various duality frames, as quiver gauge theories, built from TN components via gauging (m) theories likewise arise from the and nilpotent Higgsing. We analyze the RG flow of the UN integer m > 0, they end up at the same IR SCFT as SU(N ) SQCD with 2N flavors and (m) theories, and find that, for all (m) theories can thus be regarded as an infinite set of UV quartic superpotential. The UN completions, dual to SQCD with Nf = 2Nc. The UN (m) theories can be represented; in - = 1 1 Introduction 2 3 5 SU(N ) theories Review of the TN theory Conclusion and outlook Review of class S theories General (p, q) class S theories from nilpotent Higgsing SU(2) theories T (m=2) The simplest example: T2(m=1) 4.4 Infinitely many N = 1 duals for SU(N ) SQCD with 2N flavors Superconformal index Topological field theory and superconformal index Direct computation for the SU(2) theories 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(Nc) added meson singlets and superpotential. TN theory of [5] (see [6] for a nice, recent review), along with 2N 2 + 2N gauge singlets 1Upon 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 (a) Quiver diagram for U2(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 N(2). The triangle refers to the TN theory. Here a small box with ‘x’-mark global symmetries in TN . There are gauge/flavor singlets as well. symmetries. The small subscripts label the node and symmetry group. involving a single TN theory, two quarks/anti-quarks, N 2 + N gauge singlets, and an the gauging can then be written as a standard Lagrangian, and the duals in this case reduces to ones analyzed in [8, 9]. T (m) and U N(m), labelled by an arbitrary integer m ≥ 0. T (m) theories are superconformal N N theories that have several duality frame representations. We argue that, for all m, UN and quartic superpotential conformal manifold of SCFTs. The UN gauge nodes and components constructed from TN , along with a specific superpotential. (m) can be obtained by gluing (via gauging) two copies of the T N(m) theories (when N > 2, we glue partially Higgsed T (m)). The T (m) theories are new N = 1 SCFTs, which N N theless, for all N , results can be obtained via holomorphy [10, 11], much as in [4, 12] for the TN case. Also, a-maximization [13] enables us to determine exact R-charges of the chiral (m) is a quiver gauge theory consisting of 2m + 1 (a) A quiver diagram describing the T2(3) theory. (b) A quiver diagram describing the T N(3) theory. a number of dual descriptions for the T N(m) theory itself. 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) and the UN N All of these theories have a natural description as being of class S: the low-energy limit theories, in addition to Cg,n (called the UV curve) we need to assign a pair of integers (p, q) try [17–20] (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, 19–26]. For we consider cases with negative degree. Some discussion of negative degree bundles, and related aspects, appear in the context of gravity duals [18, 19, 27–29], superconformal indices [23] and generalized Hitchin system associated to the UV curve [20, 30–33]. The novelty here is that we consider the field theory realization associated with negative degree bundles. In particular, our T (m) theory arises from reducing the 6d AN−1 N = (2, 0) theory N on the three-punctured sphere C0,3, with the line bundle degrees L(p) ⊕ L(q) → Cg=0,n=3, (p, q) = (m + 1, −m) A possible objection to combining positive and negative degree pairs of bundles as in (1.3) is 2We thank Edward Witten for this remark. The 6d AN−1, N appropriately decorated) gives find that the T (m) theories are stable, but the UN N via renormalization group flows in the associated 4d QFTs. SU(N ) SQCD with Nf = 2Nc via L(1) ⊕ L(1) → Cg=0,n=4 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]. 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 T2(m) theory in detail. 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) which involves multiple copies of TN theory. Using N these building blocks, we construct dual theories of SU(N ) SQCD. In section 5, we compute the superconformal indices of the T (m) theory as further checks of our proposed dualities. Four-dimensional N In this section, we briefly review the N = 1 class S theories, and our particular Review of class S theories For more detail, we refer to the papers [4, 7, 18–20, 25]. 1. The choice of a ‘gauge group’ Γ ∈ ADE of the 6d, N = (2, 0) theory. 2. The choice of a Riemann surface Cg,n (UV curve) of genus g and n punctures. 3. The choice of the degree of line bundles (p, q) over Cg,n satisfying (1.2). total space Cg,n wrapped on the base Cg,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 gives the flavor symmetry associated with the i-th puncture. that can be preserved after a partial topological twist on the UV curve. Defining R0 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 R0 ≡ 2 (J+ + J−) , F ≡ 2 (J+ − J−) RN =1 = R0 + F 1 − = 0. Pair-of-pants decomposition and duality. The pair-of-pants decomposition of (hyperbolic) Cg,n yields a way to build the theory, and find duals. One decomposes the total , with Cg(p,n,q) then decomposed into p pants of color σ = + and q pants with σ = − . Two pants of same color are glued with an N = 2 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 Cg,n give IR dual theories. Each puncture has a SU(N ) symmetry, which is unbroken if the puncture is maximal. try is preserved. When the two pants of the same color are glued, the diagonal combination the pants, we also have extra chiral multiplet M in the adjoint of SU(N ), with a superpotenis the same as the pants, and to M if the puncture has the opposite color. This breaks the 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. There is an adjoint chiral multiplet attached to each of them. Here we assumed all punctures to be maximal. General (p, q) class S theories from nilpotent Higgsing obtain negative degrees via nilpotent Higgsing of the puncture. Following the prescription This procedure should be regarded as a conjecture, which fits with the computation of central charges and superconformal indices, and our analysis provides some additional 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 (b) The quiver diagram corresponding to the (a) A colored pair-of-pants decomposition for UV curve and the colored pair-of-pant decomthe 4-punctured sphere. position on the left. sponding quiver diagram, see also [5]. Each node denotes SU(2) global/gauge symmetries. can be constructed from gluing m + 1 copies of the TN 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 detail in later sections. SU(2) theories theory of [5] reduces to 8 free chiral multiplets. Likewise, there is a Lagrangian description for every (p, q). We first consider the T (m) theories, and then obtaining duals of N = 1 2 The simplest example: T (m=1) 2 start with the UV curve C0,4 (2,0) with (n+, n−) = (3, 1) where n ± denotes the number of ± is given as in figure 6. The field content of the theory is given as in the table below: (2,−1) with all + It also means there is a singlet coupled to the quarks connected. −1 −1 (J+, J−) They are the ‘candidate R-charges’ which were used in [7]. The exact R-charge is given by a 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 We now close the red puncture corresponding to SU(2)D by giving a nilpotent vev, 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 −1 −2 (J+, J−) The remaining theory has superpotential symmetric because one of the flavors does not couple to the adjoint, instead coupling to the the charge assignments of 3.2. is not in the chiral ring due to the F -term for M . gauge singlet M . This theory has a quantum moduli space of vacua, with several branches. The M field can have arbitrary expectation value, and hM i gives a mass to the q2 field. The Each Higgs branch is a copy of C2/Z2, 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 It would be interesting to interpret this moduli space via geometric construction. operators, in the adjoint representations of SU(2)A,B,C , given by expected to be S3 permutation symmetric under permutation of the SU(2)A,B,C symmeare not in the SU(2)A,B,C current multiplets, and they receive anomalous dimension. The determined via a-maximization to be5 ' 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 S3 permutation symmetry. We start from the theory corresponding C0,5 and then close the two − punctures to obtain C0,3 (3,0) with (n+, n−) = (3, 2) (unhiggsed theory) (3,−2). 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 transforming as adjoints of the flavor groups, coupled via a superpotential of the form Wm = a∈red nodes (a) Quiver 1. (b) Quiver 2. (c) Quiver 3. Figure 8. Three dual frames corresponding to the UV curve C0,5 n± denotes the number of ± punctures respectively. (3,0) and (n+, n−) = (3, 2) where (a) Quiver 1. (c) Quiver 3. (b) Quiver 2. 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 Ma 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 Ma 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 Ma 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 T2(1) theory. We list the matter content and charges of the theory in the table The superpotential for the quiver 2 is where we suppress gauge and flavor indices, which are as determined by the symmetry. Non-mass deformation. tures, we get a similar description as quiver 1 and 2. Now, we need to further close the a mass term for the quarks, it nevertheless turns out to be a relevant deformation, breaking which is relevant, R < 2, since a-maximization gives the deformation. ' 0.46. This gives a ' 1.55 before (J+, J−) (J+, J−) m=−1,0,1 The matter content after Higgsing is as in figure 9, with charges: − → J − − 2m . − 2 (J+, J−) We will consider similar type of deformations in section 4. ’t Hooft anomalies. The anomaly coefficients of T2(2), in all three dual frames, are: TrJ+, TrJ +3 TrJ−, J −3 TrJ +2J− TrJ+J −2 −2 −6 −18 a-maximization yields We can generalize previous subsection to construct a general T2(m) theory. Start with the UV curve C0,m+3 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 ' 0.534 and a ' 1.45 for the T2(2) theory in all three dual frames. Figure 10. One of the dual frames describing the T2(3) theory. qi (1 ≤ i ≤ m) qm+1 (i = m) SU(2)i−1 (J+, J−) 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) m−1 W = The ’t Hooft anomaly coefficients for this theory are J−, J −3 J +2J− J+J −2 J+SU(2)2A,B,C J−SU(2)2A,B,C −m m − 8 −9m −2 a( ) = (3TrR3 − TrR) = 3 + 3(19m + 5) − 27 2 + (9 − 63m) 3 . The value of is fixed, by maximizing a( ), to be (m) = −3 + √133m2 + 16m + 4 . 0.5492 . 21m − 3 charge a( (m)) grows linearly in m, which is not surprising from the quiver gauge theory The T (m) theories do not have any exactly marginal deformations: there are m + 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. S interpretation [4]. We now argue that gluing two copies of T (m) with an N = 1 vector multiplet, for any 2 S language, we have chosen two pairs-of-pants labelled by an integer m which gives the Moreover, this theory is known to have 72 dual frames [38, 39]. same 4-punctured sphere. of flavors contributed by each T (m) theory upon gauging SU(2)X=A,B,C global symmetries 2 is given by the ’t Hooft anomaly k = −3TrRSU(2)2X = 3(1 − ) 3Nc = 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 to there as quiver 2. As we claimed in section 3.2, there is a non-manifest S3 permutation symmetry among the SU(2)A,B,C global symmetries. Correspondingly, there are two dual ways to gauge the the SU(2) flavor group; see figure 12. Let us pick the dual frame shown in figure 12a. We will label duality frames of this type as U2 matter content and their charges are given by two copies of T2(m) where one copy has flipped a superpotential term (a) The U2(2) quiver, obtained by gauging the SU(2) flavor group on the left-hand side of figure 9c. SU(2)i±−1 (b) The Ub2 (2) quiver, obtained by gauging the SU(2) flavor group on the right-hand side of figure 9c. flow to the same SCFT as SU(2) SQCD with 4 flavors. 1+ (i = 1) qi+ (2 ≤ i ≤ m) qi− (i = 1) qi− (2 ≤ i ≤ m) qm−+1 (i = m) Mi− (J+, J−) m−1 J +2J−, J+J −2 J+SU(2)2A,B, J−SU(2)2C,D J−SU(2)2A,B, J+SU(2)2C,D −2 −5 SQCD, which is the m = 0 case of U2 coefficients of the U2 (m) quiver theory are m-independent: We argue that the U2 (m). As a first check, we find that the ’t Hooft anomaly J−). Matching of operators. Among the single trace, gauge invariant operators of U2 an SU(8) global symmetry (though it is broken by (1.1) to SU(2)4) with meson / baryon operators in the 82 and the remaining meson/baryon operators are in the (2, 2, 2, 2) of the qm−+1qm− . . . q2−q1−q1+q2+ . . . qm+qm++1 However, there initially appears to be a mismatch in our proposed duality between U2 (m) seems to contribute but also in the moduli space of vacua. Actually, as we now discuss, the quantum theory does not have the Mi and ui classical moduli. They are quantum-lifted in a way similar to what 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. This effectively decouples the side of the U2 in the figure 13. This gives TrJ+(SU(2)n−−1 (m) quiver in with gauge group SU(2)i−≥n, as )2 6= 0, so the low-energy SU(2)n−−1 instanton Mn−−1 the theory admits u− which has a runaway for qi± that is incompatible with FMj± , so the un flat direction is lifted. The superpotentials (3.25), (3.26) involves only the quarks on the other (+) side of vector multiplet, but not in the T (m) theory itself or when they are coupled via N = 2 N vector multiplet. We give a refined check of operator matching through computing the superconformal index in section 5. The index of the U2 provides a strong check of the duality. Therefore we conjecture that for every choice of m, (m) theory flow to the same SCFT as SQCD in the IR. (m) theory agrees with that of the SQCD, which Exactly marginal deformations. formal manifold of exactly marginal deformations N = 1 SU(2) SQCD with 4 flavors has a large coni, j = 1 . . . 8, including a one-complex dimensional line of fixed points which preserve SU(2)4 flavor symmetry. This line of fixed points can also be seen in the U2 The exact NSVZ beta functions for the gauge couplings of SU(2)0 and SU(2)i± are (with (m) theory via the method of [2]. The exact beta functions for the superpotential couplings are βg0 ∝ −(2 + 2γq+ + 2γq− ) , 1 1 βgiσ ∝ −(2 + 2γφiσ + γqiσ + γqiσ+1 ) , (i = 2, · · · , m) . βλ0 ∝ 1 + γq+ + γq− , 1 1 (m) theory has a one complex dimensional conformal manifold. This can also be seen conformal manifold that preserves the SU(2)4 × U(1)F × U(1)R global symmetry. i+ (1 ≤ i ≤ m) qi− (1 ≤ i ≤ m) qm−+1 (i = m) Mi− SU(2)i±−1 (m) theory; SU(2)0± is the shaded node in figure 12. by gluing two copies of T2(m). The Ubm theory has superpotential term Cascading RG flow to SQCD. which we claim is dual to the U2 The duality frame of figure 12b is the Ub2 (m) theory, giving another description of the theory obtained punctures that we are gluing and (with implicit gauge index contractions) m−1 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 V +− = q1+q1− V − = q1−q1− , The superpotential (3.31) becomes (with implicit trace over gauge and flavor indices) We see that V ± combine with M1± to become massive, so they can all be integrated det(V +−) − V +V − = Λ04 . m−1 (m) thus also flows to the same IR SCFT as SQCD. massive, and are integrated out. The resulting low-energy theory is thus similar to the applies to that theory, again reducing m, giving a cascading RG flow that eventually ends (m−1). The above analysis SU(N ) theories new element is that we have to replace each bifundamental or trifundamental chiral multiplet, in the links of the quiver, by the TN theory and its deformations. We first construct the argue that this flows to the same theory as obtained from gluing two TN theories. Then we we have SU(N )2 × U(1) flavor symmetry. We then glue two such theories to obtain U N(m), and other dual versions, which give new dual descriptions of SU(N ) SQCD with 2N flavors. Review of the TN theory adjoint of the SU(N )A,B,C respectively. These operators satisfy the chiral ring relation [12] mental and anti-trifundamental of SU(N )A × SU(N )B × SU(N )C with scaling dimension 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 TN theory. N =2), where I3 is the Cartan generator of SU(2)R. As in the previous section, one linear combi(a) A quiver before Higgsing given by the UV curve C3,3 −(N − 1)(3N + 2) 13 (N − 1)(N − 2)(4N + 3) J−, J −3 J +2J− J+J −2 J+SU(N )2A,B,C J−SU(N )2A,B,C 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. that are being gauged. The superpotential is W = X Trφk(μk − μ˜k) + X Trμ(i)M (i) . deformation to the theory which we name as T N(m). 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 After giving the vev, the superpotential can be written as − → J − − W = X Trφk(μˆk − μˆ0k) + X μ1,−1 + X μj(,im) Mj,−m . (i) (i) Now, we close the punctures by giving a nilpotent vev to Mi’s as symmetry, with the non-conservation of the current given by Finally, the remaining superpotential is j,m−1 combine into a longW = X Trφk(μˆk − μˆ0k) + X m N−1 i=1 j=1 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) theory, we need to compute effect of the Higgsed TN block, with the nilpotent vev. Accounting that of the TN theory. This gives, for the single puncture Higgsed TN or equivalently the Mj(,i−)j (1 ≤ j ≤ N − 1) theory corresponding to the UV curve C0(1,2,−1): J−, J −3 J +2J− J+J −2 J+SU(N )2Z,Z0 J−SU(N )2Z,Z0 1 − N (1 − N )(2N + 1) 13 (N − 1)(4N 2 − 2N − 3) 13 (1 − N )(4N 2 + 4N + 3) −N Combining this with the known results of the TN 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) as follows: J−, J −3 J +2J− J+J −2 J+SU(N )2A,B,C J−SU(N )2A,B,C 13 (N − 1)(4N 2 − 5N − 6 + m(4N 2 + 4N + 3)) m(1 − N ) (N − 1)(m − 3N − 2) 13 m(3 + N − 4N 3) −N The trial a-function is − 1 + 2N − 3 2 + 1 , Note that the anomalies involving the SU(N )A,B,C are the same as that of TN 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. a( ) = (N − 1)(1 − ) 3N 2( + 1)2 − 3N 2 2 + + 1 − 2 3 2 + 3 + 2 and the value of is fixed by a-maximization to be 3(2m(N 2 + N + 1) + N 2 − 2N − 2) (J+, J−) (2, −2j) (0, 2j + 2) the gauge group. For m = 0, we find 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 kSU(N) [41, 42] to be kSU(N)δab = −3TrRT aT b = the TN 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) case, it contributes to the beta function as that of Nf < N . As a preparation of the SQCD, let us first consider the theory obtained by gluing two copies of TN theory by gauging one of the SU(N ) flavor groups on each of TN . It can be obtained studied in [4, 17] which we review here. This theory has SU(N )A × SU(N )B × SU(N )C × SU(N )D ×U(1)F ×U(1)R global symmetry with the ‘matter content’ as given in the table 5. 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 TN . We couple two copies of T N(m) with an N = 1 vector, the (J+, J−) charge assignment of one of the T (m) has to be flipped N in order to write the superpotential term (4.13). See figure 16. The ‘matter content’ of the theory is given in the table 6. (b) A quiver description obtained by gauging the SU(N ) flavor group of two copies of the T (2) N = 2 vector multiplets. (2, −2j) (−2j, 2) (J+, J−) i+ (0 ≤ i ≤ m) i+ (1 ≤ i ≤ m) Mj+,−,(ji) (2 ≤ i ≤ m + 1) The theory has a superpotential Wσ = X Trφkσ(μkσ − μ˜kσ) + X m+1 N−1 i=2 j=1 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 (m) and add up with that of T (m) with flipped J+ and N N J+, J +3, J−, J −3 J +2J−, J+J −2 J+SU(N )2A,B, J−SU(N )2C,D J−SU(N )2A,B, J+SU(N )2C,D (2N + 1)(1 − N ) 13 (N − 1)(4N 2 − 2N − 3) −N We see that the anomaly coefficients are independent of m, therefore it agrees with the We will match the set of supersymmetric operators by computing the superconformal index in section 5. Cascading RG flows to the gauged TN 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) blocks are glued to each other to give the duality frame of figure 17a. Guided by the SU(2) case, we claim that the 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 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 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. As an evidence to support our claim about figure 17b, we consider the theory obtained by gluing two T (1) blocks via an N = 1 vector multiplet along one of their full punctures. N tail corresponding to the minimal puncture, giving the quiver in figure 18. If our claim is theory will then flow to the quiver of figure 19. We now argue that this is indeed the case. cascade of RG flows which reduces it to the quiver of figure 16a in the IR. figure 17a. The dynamics that lead to this behavior are local to this section of the quiver and do not depend upon the rest of the quiver. here at low energies. node reducing it to the sub-quiver of figure 17c. This process triggers a cascade of RG flows in figure 17a reducing it to the quiver of figure 16a. to the minimal puncture. The T (1) blocks are glued to each other via an N = 1 vector multiplet N (a) Gauged TN theory with quiver tails attached. (b) Linear quiver dual of the above quiver. theory with quiver tails attached. attached to each ends. Here j = 1, · · · , N − 1. 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 charges of the various fields in the same figure. The superpotential terms of this quiver are 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. j = 1, · · · , N − 3. This corresponds to the quiver of figure 23. Once again the superpotential of this quiver can be written down by considering all the chiral gauge invariant operators which have SQCD that is expected to be there after s-confinement of the edge nodes in figure 22. 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 produce a quiver whose central and last two nodes on either sides are gauged using an 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. We will now have to again go through the series of dualities mentioned in the previous 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 in figure 26. Here j = 1, · · · , N − 4. quiver shown here. All the singlets become massive and integrated out. 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. The matter content for the theory UN The superpotential is given by Let us now consider the case of SQCD with SU(N ) gauge group and 2N flavors. From the class S point of view, what we need to do is to start with 4-punctured (all maximal, 2 + and maximal punctures of each color. This will result in replacing the TN block we glued to the end of the quivers by bifundamental hypermultiplets of SU(N )×SU(N ). See the figure 29. (m) similar to the figure 29b is given in the table 7. W = W +0 + W −0 + Trμ0+μ0− , m+1 N−1 i=2 j=1 (a) A quiver description dual to the SU(N ) SQCD with 2N flavors. Here we have maximal punctures of each color and minimal punctures of each color. 1, −1 1, −1 (J+, J−) (1, −N + 1) (2, −2j) (−N + 1, 1) (−2j, 2) q+, q˜+ (i = m) i+ (0 ≤ i ≤ m) Mj+,−,(ji) (2 ≤ i ≤ m + 1) q−, q˜− (i = m) Mj−,−,(ji) (2 ≤ i ≤ m + 1) (m) theory. Here 1 ≤ j ≤ N − 1. σ,(m+1) = Trq˜σqσ(φσm)N−j−1 . Anomaly coeffecients. As an intermediate step, let us consider the Higgsed T (m) theory by Higgsing one of the punctures. Let us call it TeN(m). This theory is given by the UV curve quiver diagram of the theory is the left half of figure 30 with central gauge group ungauged. J−, J −3 J +2J− J+J −2 J−SU(N )2A, J−SU(N )2G J+SU(N )2A, J+SU(N )2G J−U(1)2B J +2U(1)B, J −2U(1)B m(1 − N ) m(N − 1) − 2N 2 − 31 (4N 3 − N − 3) − N − 3) −N −2N 2 J+, J +3, J−, J −3 J +2J−, J+J −2 J+SU(N )2A, J−SU(N )2C J−SU(N )2A, J+SU(N )2C J+U(1)2B, J−U(1)2D J+U(1)2D, J−U(1)2B J +2U(1)B,D, J −2U(1)B,D −N 2 − 1 − 1 −N −2N 2 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) theory with all its colors inverted can be obtained by We now compare the anomaly coefficients of our proposed dual theories. For U N(m), we The theory will undergo cascading RG glow to the SQCD. with U(1)B baryonic symmetry. The anomalies of this theory are given as: 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 TeN 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 UN similar as in section 4.3, but we get SU(N )A × U(1)B × SU(N )C × U(1)D × U(1)R × U(1)F (m) (figure 30) is quite global symmetry instead. It is described in the table 8. The set of chiral operators in the TN theory contains (anti-)trifundamental operator Qijk and Q˜ijk. When an oppositely colored puncture of the TN block is closed, the operators N−1 ≤ ` ≤ N2−1 , and q+, q˜+ (i = m) i+ (0 ≤ i ≤ m) i+ (1 ≤ i ≤ m) Mj+,−,(ji) (1 ≤ i ≤ m) q−, q˜− (i = m) choice of color. These operators will be important to our analysis and we will label those The superpotential for the theory is given as W = W +0 + W −0 + Trμ0+μ0− + X Trμˆk+μˆk− , X Trφkσ(μkσ − μ˜kσ) + X m N−1 i=1 j=1 the adjoint of SU(N )0. By applying a sequence of dualities, we have showed earlier that the central SU(N )0node confines. From this, we conjecture that the SU(N )0-node undergoes confinement with det Q˜±N,−(11)Q∓N,−(11) − “(μj+,j,=(11)μ−,(1)) 12 N(N−1)” = Λb(N−1) , j,j=1 0 b = 3N − 2k = 3 UV N, where k = −3TrRUV SU(N )02 = (1 − UV )N. (2, −2j) (0, 2j + 2) (−2j, 2) (2j + 2, 0) RUV is the superconformal R-charge before gauging SU(N )0. Gauging SU(N0) breaks the separate U(1)F± is consistent with the U(1)A charge of the product of operators on the l.h.s. of (4.26). The operators on the l.h.s. of (4.26) carry U(1)RIR 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 l.h.s. of (4.26) are analogs of detM and BB˜ in SQCD 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 I(p, q, ξ; x) = Tr(−1)F pj1+j2+ R20 qj2−j1+ R20 ξF Y xiFi . 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 nonanomalous U(1) symmetry that can mix with R-symmetry in the IR is U(1)F . Therefore to obtain the true superconformal index. Topological field theory and superconformal index 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, 44–48]. The index can be written as X(Cλ+)p(Cλ−)q Y ψρi,σi (ai) , λ where (p, q) are the degrees of the line bundles and n is the number of punctures, which function of a which in certain limit reduces to the Macdonald polynomial. The argument Let us compute the index of the T (m) starting from the theory given by the UV curve I[C0(m,m++13,0)] = X(Cλ+)m+1 Y ψλ+(ai) Y ψλ−(bi) . close by ψ∅,σ(tρσ). From the relation Cλσ = (ψλ∅,σ)−1, we see that the degree of the normal λ I[T N(m)](p, q, ξ; ai) = X (Cλ+)m+1 (Cλ−)m ψλ+(a1)ψλ+(a2)ψλ+(a3) , 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) with opposite color by the index can be written as I(a, b, c, d)=X (Cλ+)m+1 λ,μ (Cλ−)m ψλ+(a)ψλ+(b) = X Cλ+Cλ−ψλ+(a)ψλ+(b)ψλ−(c)ψλ−(d) . We here used the fact that wave functions are orthonormal: 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 TN theory can be written theories (5.5) by directly computing the index using the matter content of section 3. v∈R v∈R where v are the weight vectors of the representation R of the symmetry group the chiral multiplet is charged under. Here the notation zv is a short-hand for Qi zivi . Here, we used the elliptic gamma function which is defined as 1 − z−1pm+1qn+1 1 − zpmqn Ivec(p, q; z) = whenever possible. The vector multiplet contribution to the index is given by the Haar measure for the gauge group G to the vector multiplet index for convenience. For the SU(N ) gauge group, we get Ivec(p, q; z) = (p; p)N−1(q; q)N−1 Y i6=j defined to be (z; q) = Qm∞=0(1 − zqm). T (m) theory. Let us compute the superconformal index of the T2(1) theory discussed in 2 section 3.1. We would like to compute the index in the UV using the description given as in figure 7 and show that it agrees with the TQFT formula. The index on the electric side can be written as 2πiz Ivec(z)Ic(h0i,2)(z±2,0)Ic(h1i,0)(z±a±b±)Ic(h1i,−1)(z±c±)Ic(h0i,4)(1) I(m) = means f (z2)f (z−2)f (z0). One tricky part here is choosing the correct contour for this integral. Usually, one so that we pick up the poles only inside the unit circle. This works as long as there is no chiral multiplet with R0 or R charge less than equal to zero. But if there is a chiral being products of the fugacities corresponding to the gauge/flavor symmetries. The index for the T2(m) can be written as dzi Ivec(zi)Ic(h0i,2)(z±2,0)Ic(h1i,−1)(zi±−1zi±)Ic(h0i,4)(1) Ichi where z0 = c. We confirmed that this indeed gives us the same index as the TQFT hold, we have the identity 2πiz Ivec(z)I(m)(ξ)I(m)(ξ−1) = where we glued two T2(m) with opposite F charges. We have verified this identity to hold for m = 1, 2 at the leading orders in p and q. SQCD vs UbN (m) theory. Let us compute the index in the dual frame UbN(m). In this frame, we should be able to see SU(8) flavor symmetry since it cascades to the SQCD in the IR. In order to see this from the index, first we refine the index 5.13 as I˜(m)(a) = ai=1,2,3. And then we find 2πiz Ivec(z)I˜(m)(a, ξ)I˜(m)(b, ξ−1) = 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 that I˜(m) in (5.15) is not a genuine index of the theory, since T (m) 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 T2(m), we have exactly marginal deformations which includes the point with enhanced symmetry. Conclusion and outlook Guided by the construction of 4d QFTs from M5 branes wrapping Riemann surfaces, we 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 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) itself has many different dual descriptions, so the number of duals grows rapidly with m. by considering a mass deformation of the T (m) theory, as was done in the TN case [50]. From N 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 our construction of T (m) to D and E type theories and also with outer-automorphism of [59, 60], which will provide analogous infinitely many duals for other gauge groups. Acknowledgments JS would like to thank Dan Xie for motivating him to understand the class S theories with general twists and pointing out the possibility of an infinite number of duals. PA and JS thank Ibrahima Bah and Kazunobu Maruyoshi for the related collaborations and discussions. 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Prarit Agarwal, Kenneth Intriligator, Jaewon Song. Infinitely many \( \mathcal{N}=1 \) dualities from m + 1 − m = 1, Journal of High Energy Physics, 2015, 35, DOI: 10.1007/JHEP10(2015)035