Gaiotto duality for the twisted A 2N −1 series

Journal of High Energy Physics, May 2015

We study 4D \( \mathcal{N} \) = 2 superconformal theories that arise from the compactification of 6D \( \mathcal{N} \) = (2, 0) theories of type A 2N −1 on a Riemann surface C, in the presence of punctures twisted by a ℤ2 outer automorphism. We describe how to do a complete classification of these SCFTs in terms of three-punctured spheres and cylinders, which we do explicitly for A 3, and provide tables of properties of twisted defects up through A 9. We find atypical degenerations of Riemann surfaces that do not lead to weakly-coupled gauge groups, but to a gauge coupling pinned at a point in the interior of moduli space. As applications, we study: i) 6D representations of 4D superconformal quivers in the shape of an affine/non-affine D n Dynkin diagram, ii) S-duality of SU(4) and Sp(2) gauge theories with various combinations of fundamental and antisymmetric matter, and iii) realizations of all rank-one SCFTs predicted by Argyres and Wittig.

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Gaiotto duality for the twisted A 2N −1 series

Received: January Gaiotto duality for the twisted A2N 1 series Oscar Chacaltana 3 Jacques Distler 1 Yuji Tachikawa 0 2 0 Institute for the Physics and Mathematics of the Universe, University of Tokyo , Kashiwa, Chiba 277-8583 , Japan 1 Theory Group and Texas Cosmology Center, Department of Physics, University of Texas at Austin , Austin, TX 78712 , U.S.A 2 Department of Physics, Faculty of Science, University of Tokyo , Bunkyo-ku, Tokyo 133-0022 , Japan 3 ICTP South American Institute for Fundamental Research, Instituto de F sica Te orica, Universidade Estadual Paulista , 01140-070 S ao Paulo, SP , Brazil We study 4D N = 2 superconformal theories that arise from the compactification of 6D N = (2, 0) theories of type A2N1 on a Riemann surface C, in the presence of punctures twisted by a Z2 outer automorphism. We describe how to do a complete classification of these SCFTs in terms of three-punctured spheres and cylinders, which we do explicitly for A3, and provide tables of properties of twisted defects up through A9. We find atypical degenerations of Riemann surfaces that do not lead to weakly-coupled gauge groups, but to a gauge coupling pinned at a point in the interior of moduli space. As applications, we study: i) 6D representations of 4D superconformal quivers in the shape of an affine/non-affine Dn Dynkin diagram, ii) S-duality of SU(4) and Sp(2) gauge theories with various combinations of fundamental and antisymmetric matter, and iii) realizations of all rank-one SCFTs predicted by Argyres and Wittig. ArXiv ePrint: 1212.3952 Open Access, c The Authors. Article funded by SCOAP3. Supersymmetric gauge theory; Supersymmetry and Duality; Extended Supersymmetry; Duality in Gauge Field Theories Contents 1 Introduction and summary 2 4D theories and punctures 2.1 Punctures, the fields k(z) and the Hitchin field (z) 2.2 Local properties of punctures 2.3 Graded Coulomb branch dimensions 2.4 Constraints 2.4.1 General structure of constraints 2.4.2 Number of constraints 2.4.3 Explicit form of constraints Collisions of punctures 3.1 OPE of punctures on a plane 3.2 Degeneration of a curve via the OPE 3.2.1 Example 1 3.2.2 Example 2 3.2.3 Example 3 3.3 Determining the OPE via the Higgs field 3.3.1 Untwisted-untwisted 3.3.2 Twisted-twisted 3.3.3 Twisted-untwisted 3.3.4 Degenerating k-differentials 3.3.5 Example 1 3.3.6 Example 2 3 4 5 Atypical degenerations 4.1 Gauge theories on a fixture 4.1.1 Fixtures with [2N 1, 12] 4.1.2 Separating [2N 1, 12] to a pair of [2N + 1] and [2N 1, 1] 4.1.3 Derivation 4.2 SU(N ) SU(N ) cylinder 4.2.1 How it arises 4.3 SU(2) SU(2) cylinder D-shaped quivers 5.1 Affine Dn-shaped quivers 5.1.1 Degenerations of the 5-punctured sphere 5.1.2 The moduli space of coupling constants vs. the complex structure moduli space 5.2 Comparison to Kapustins work i 1 3 Non-affine Dn-shaped quivers 6 SU(4) and Sp(2) gauge theories 6.1 SU(4) gauge theory 6.1.1 8(4) 6.1.2 1(6) + 6(4) 6.1.3 2(6) + 4(4) 6.1.4 3(6) + 2(4) 6.1.5 4(6) 6.2 Sp(2) gauge theory 6.2.1 6(4) 6.2.2 1(5) + 4(4) 6.2.3 2(5) + 2(4) 6.2.4 3(5) 6.3 A family of SU(2) Sp(2) gauge theories Rank-1 SCFTs 7.1 Summary of rank-1 SCFTs 7.2 On a new rank-1 SCFT with (u) = 3 A Tables of properties of twisted sectors A.1 A3 twisted sector A.1.1 Punctures A.1.2 Cylinders A.1.3 Free-field fixtures A.1.4 Gauge-theory fixtures A.1.5 Interacting SCFTs A.1.6 Mixed fixtures A.2 A5 twisted sector A.3 A7 twisted sector A.4 A9 twisted sector Introduction and summary Considerable progress has been made recently in the program of understanding the 4D theories that arise from the compactification of 6D N = (2, 0) theories on a Riemann surface, C, possibly in the presence of codimension-two defects of the (2,0) theories, which correspond to punctures on C [15].1 Much of the richness of the construction stems from the variety of available defects. When an N = (2, 0) theory of type J = A, D, or E has a 1To avoid confusion, we note that, in this paper, [1] denotes a reference, [5] stands for a B-partition (an embedding sl(2) so(2N + 1)), and [3] is a generic partition. nontrivial outer-automorphism group, there exists, in addition to untwisted defects, a sector of twisted defects equipped with an action of a element of the outer-automorphism group of J as one goes around the defect. The general local properties of twisted and untwisted defects were studied in [6]. In particular, the A2N1 series has a sector of defects twisted by the Z2 outer automorphism of A2N1. In this paper, we study the global properties of theories of type A2N1 in the presence of such Z2-twisted defects. Just as untwisted defects of the A2N1 series are classified by embeddings of sl(2) in sl(2N ), twisted defects in this series are classified by embeddings of sl(2) in so(2N + 1). Equivalently, untwisted defects are classified by partitions of 2N , while twisted defects are classified by certain partitions of 2N + 1, called B-partitions. So, for instance, the twisted sector contains a maximal twisted puncture, denoted by the B-partition [12N+1] and with flavour group SO(2N + 1), and a minimal twisted puncture, denoted [2N + 1] and with trivial flavour group. The local properties of these and other twisted punctures can be computed following [6]. In this paper we will provide some new, explicit algorithms to make these calculations easier. One especially interesting twisted defect is the one with B-partition [2N 1, 12], which arises from the collision of a minimal untwisted and a minimal twisted defect. Such defect is unique in that it can be continuously deformed into a pair of defects [2N + 1] and [2N 1, 1]. In the global picture, this property leads to a number of elements that were absent in the untwisted story: three-punctured spheres that correspond to gauge theories fixed at a point in the interior (not a cusp) of their moduli space. cylinders whose pinching decouples a semisimple gauge group. It is well known that Lagrangian 4D N = 2 superconformal quivers whose gauge group is a product of special-unitary groups can be constructed only in the shapes of (affine and non-affine) Dynkin diagrams of type A-D-E [7]. The An-shaped quivers were used originally by Gaiotto to deduce local properties of untwisted defects in the AN1-series, and are realized as compactifications on untwisted spheres. In this paper we find a realization of the affine and non-affine Dn-shaped quivers as compactifications of the A2N1 series on twisted spheres. An equivalent expression for the Seiberg-Witten curve for the affine Dn-shaped quivers was found long ago by Kapustin [8] from a IIA brane construction. (Quite recently, a uniform way to derive the Seiberg-Witten solutions for these ADE quiver theories were found by [9, 10] using instanton calculus.) We present a full tinkertoy representation of the twisted A3 theory, and, as an application, study the S-dual frames of SU(4) and Sp(2) gauge theories with matter in the fundamental and antisymmetric representations. We show how to construct the rank-one 4D SCFTs studied by Argyres and Wittig in [11, 12].2 These are SCFTs whose only Coulomb branch operator has scaling dimension = 3, 4, 6 respectively, and which are not the more familiar MinahanNemeschansky theories with E6,7,8 flavour symmetry. The = 3 theory will be found in the context of the twisted A2 theory, although we leave a systematic analysis of the twisted A2N series for later, due to the subtle issues pointed out in [13]. We will be able to pin down numerical invariants of this = 3 theory left undetermined in [11, 12]. The rest of the paper is organized into two parts. The first part consists of section 2, 3 and 4. In section 2, after recalling the general method for obtaining a 4D theory from a 6D N = (2, 0) theory on a Riemann surface, we describe the algorithms to compute the local properties of twisted punctures of type A2N1, elaborating on [6]. In section 3, we develop the method to identify the behaviour of the theory when two defects are brought together. In section 4, we study atypical degenerations in detail, where the degeneration of the Riemann surface does not correspond to the emergence of a weakly-coupled gauge group. The second part of the paper deals with applications. In section 5 we show how a Dn-shaped quiver gauge theory can be realized in terms of the 6D N = (2, 0) theory of type A2N1 on a sphere with twisted punctures. In section 6 we study the S-duality properties of all superconformal SU(4) and Sp(2) gauge theories. In section 7 we discuss rank-one SCFTs and their realizations in terms of 6D N = (2, 0) theory. In appendix A we list all twisted fixtures and cylinders for A3, and tabulate the properties of twisted punctures for A5,7,9. 4D theories and punctures In section 2.1 we recall the construction of 4D theories from the compactification of the 6D N = (2, 0) theory of type A2N1 on Riemann surfaces with punctures. Section 2.2 through 2.4 detail algorithms to compute local properties of punctures. We show extensive tables of local properties in appendix A. After going over section 2.1, a busy reader can skip the rest of this section, and continue directly to section 3. Consider the 6D (2,0) theory of type A2N1, compactified on a Riemann surface C with a partial twist to preserve supersymmetry [1, 14]. We allow for the possibility of having 2These were called non-maximal mass-deformations of the E6,7,8 theories in [11, 12], but we prefer to call them just new rank-one theories, since physically they are not deformations of the E6,7,8 theories, although their Seiberg-Witten curves are. codimension-two defects of the (2,0) theory, localized at punctures on C. This construction leads at low energies to a 4D N = 2 SCFT. We are interested in classifying and characterizing the 4D SCFTs that arise for various choices of C and defects on it. Usually, the moduli space of the 4D SCFT can be identified with the complex-structure moduli space of C, so that cusps in the latter correspond to weakly-coupled limits of the theory, where a certain gauge group almost decouples. We will see in section 4 that there exist counterexamples to this statement when twisted punctures are included. The Seiberg-Witten curve of the theory can be realized as a ramified cover of C. To describe explicitly, we should consider the VEVs of certain protected operators in the 6D theory, which, upon compactification on C, give rise to meromorphic k-differentials k on C, where the k are the dimensions of the Casimirs of A2N1, i.e., k = 2, 3, . . . , N . The k have poles at the positions of the punctures on C. We then have the following equation for : Then, the sector of untwisted punctures is the one corresponding to the identity element, while the twisted sector corresponds to o. As one goes around a twisted puncture on C, (2.2) tells us that the k-differentials of odd degree k must change sign. Now, untwisted punctures are classified by sl(2) embeddings in sl(2N ), whereas twisted punctures are classified by sl(2) embeddings in so(2N + 1). More practically, recall that sl(2)-embeddings in sl(2N ) are in bijection with partitions of 2N . Similarly, sl(2)embeddings in so(2N + 1) are in bijection with B-partitions of 2N + 1, which are defined as partitions of 2N + 1 where every even part has even multiplicity. For example, [42, 33, 26] is a B-partition, but [6, 5, 42] is not. If z is a local coordinate on C such that the puncture is at z = 0, the k-differentials near z = 0 have the behaviour: We call the set {pk}, for k = 2, . . . , 2N , the pole structure of the puncture. For an untwisted puncture, all the pk should be integer, while for a twisted one, the pk for odd (even) k must be half-integer (integer) because of (2.2). Let us now relate the discussion to the Hitchin system. Following [14], the classical integrable system associated to our 4D N = 2 theories is a Hitchin system on C with gauge group sl(2N ). Let be the Higgs field for the Hitchin system, i.e., is an sl(2N )-valued meromorphic 1-form on C in the adjoint representation of sl(2N ). Then, the SeibergWitten curve of this system, (2.1), is given by the spectral curve for the Hitchin system, where X is an element in sl(2N ) specifying the puncture, and the ellipsis denotes a generic element of sl(2N ). Since is not gauge invariant, the defect is actually characterized by the conjugacy class of X, known as a (co)adjoint orbit in sl(2N ). When the mass parameters of the puncture are set to zero, X is nilpotent, and the orbit is called a nilpotent orbit. Nilpotent orbits in sl(2N ) are classified by sl(2) embeddings in sl(2N ), or, equivalently, by partitions of 2N . If a puncture is labeled by a partition , the nilpotent orbit that defines its boundary condition (2.5) is the one corresponding to the transpose partition T of 2N . The analogous boundary condition for a twisted puncture was given in [6]. First, decompose the sl(2N ) Lie algebra as a direct sum of eigenspaces of the Z2 outer automorphism, j = j1 + j1, where j1 sp(N ) is the invariant subalgebra. Then, if the twisted defect is at z = 0, the local boundary condition for the Higgs field is (z) = Xz + zA1/2 + A. Here, X is an element of a nilpotent orbit in sp(N ), and A and A are generic elements of j1 and sp(N ), respectively. As before, nilpotent orbits in sp(N ) are classified by sl(2) embeddings in sp(N ), or, equivalently, by C-partitions of 2N , which are defined as partitions of 2N where every odd part has even multiplicity. (For example, [62, 34, 2] is a C-partition, but [52, 3, 1] is not.) Then, a twisted puncture in the A2N1 series is labeled by a B-partition of 2N + 1, but its Higgs-field boundary condition is given by a C-partition of 2N . There is a map, called the Sommers-Achar map [1517], which is a generalization of the Spaltenstein map on nilpotent orbits, which gives us the Hitchin-system data associated to : Here, C() is a discrete group. Then, X is the nilpotent element (+), seeing as an sl(2) embedding in Sp(N ). In [6], was called the Hitchin pole of the puncture labeled by the Nahm pole . is given by the C-collapse of the reduction of the transpose of (a 2N + 1 partition) to a 2N partition; see section 2.2 and section 3.4.4 of [6]. Local properties of punctures The local properties of a twisted puncture that we can compute are: 1. the pole structure {pk}, 3. the constraints on the leading coefficients c(pkk), 4. the local flavour symmetry group, 5. the contribution to the conformal-anomaly central charges, (a, c), or, equivalently, to the effective number of hypermultiplets and vector multiplets, (nh, nv). Let us briefly explain the terminology. A constraint refers to a relation among leading coefficients c(pkk), or to the fact that a leading coefficient can be expressed in terms of more basic gauge-invariants, as we will see in section 2.4. (In principle, subleading coefficients may have been constrained too, but it turns out that this does not occur.) Once the local form of the Higgs field for a specific puncture is known, as in (2.5) and (2.6), one can find the local form of the k-differentials from (2.1) and (2.4), read off the the pole structure {pk}, find the constraints, and compute the {nk}. However, carrying out this honest procedure is quite tedious in practice. In what follows, we describe algorithms to compute these properties directly from the B-partition, which we found after looking at a large number of examples. First, in section 2.3, we explain how to calculate the {nk}, and then, in section 2.4, how to compute the constraints. Once these are known, the pole structure {pk} can be easily reconstructed. We will see that the only twisted defect that gives rise to a Coulomb branch operator of dimension two is the one with B-partition [2N 1, 12]. This occurs through a constraint of the form c(34) = (a(32/)2)2. This particular puncture will play an important role in section 4. For untwisted punctures in the A series, it is well known that there are no constraints at all, and so the pole orders {pk} are exactly the same as the {nk}, for each k. (By contrast, untwisted punctures in the D series generically do exhibit constraints [5].) The Lie algebra of the global symmetry group Gflavour of a twisted puncture labelled by the embedding : su(2) so(2N + 1) is the commutant of (su(2)) in so(2N + 1). It is easier to give a formula for Gflavour in terms of the B-partition p corresponding to : Gflavour = S Y O(nl) Y Sp(nl/2) where l runs over the parts of the partition p, and nl is the multiplicity of l in p. For even l, nl must be even because p is a B-partition, so the formula above makes sense. (In our notation, Sp(1) SU(2).) As for the contributions to nh and nv (and thus a and c), these can be easily computed from the embedding : su(2) so(2N + 1). The formulas were given in [6], Here, A and B are the Weyl vectors of A2N1 and BN , respectively; h = (3) is the Cartan of the embedded su(2), and we have decomposed g = so(2N + 1) = PrZ+1/2 gr, where gr is the eigenspace of h with eigenvalue r. The contributions to nh and nv for the twisted sectors of the A3,5,7,9 theories are given in appendix A. Graded Coulomb branch dimensions Consider a twisted puncture in the A2N1 theory, specified by a B-partition p of 2N + 1. We want to compute the contributions {nk} to the dimensions of the graded Coulomb branch. The formula for the {nk} is most easily expressed in terms of a number of auxiliary sequences, which we now define. Let q = pt be the transpose partition to p. First, let us define a sequence by Next, let s be the sequence of partial sums of q, Finally, define a sequence r of corrections by = (1, . . . , 1, 2, . . . , 2, . . .). | {q1z } | {q2z } rk = Then, the contribution nk for the twisted puncture with B-partition p is nk(p) = nk([12N+1]) = 3k k 1. 2 2 General structure of constraints The structure of constraints for twisted punctures in the A series is relatively simple. These constraints satisfy some properties: All terms in a constraint should have the same total scaling dimension and pole order. (The cl(k) and a(k) have scaling dimension k and pole order l.) Hence, we can talk l about the scaling dimension and pole order of a constraint. A constraint of scaling dimension k, if it exists, should be linear in cl(k), i.e., it will be of the form cl(k) = fl(k)(c, a), where fl(k)(c, a) stands for a polynomial (in other coefficients cl(k) and parameters al(k)) of scaling dimension k and pole order l = pk. The polynomials fl(k) always have the form of squares or cross-terms (as in the expansion of the square of a sum), or sums of these. Let us look at some representative examples of constraints: i) c(712)= 41 c(76/)2 2 , iii) c(610)= ii) c(77/)2= 12 c(24)c(33/)2, iv) c(59)= a(35)c(24). The first and third examples are squares, while the second and fourth are cross-terms. Also, the first and second examples involve only the cl(k), while the third and fourth involve also new parameters al(k). We call constraints that do not introduce any new parameters, such as the first two examples above, c-constraints. A c-constraint of scaling dimension k (which necessarily has pole order l = pk) tells us that the leading coefficient cl(k) is dependent on others, and so the local contribution to nk should be reduced by one. (2k) = (al(k))2, tells us that c(22lk) is the By contrast, the third example, of the form c2l square of another, more basic gauge-invariant parameter, al(k). Thus, it effectively trades a parameter of scaling dimension 2k by a parameter of scaling dimension k; in other words, the contribution to n2k is reduced by one, while the contribution to nk is raised by one. We call this type of constraint, which introduces a new parameter, an a-constraint. Finally, the fourth example is a cross-term involving the parameter al(k). However, the a(k) will already have been introduced by an a-constraint as in the previous paragraph. l Hence, this cross-term constraint should be taken to be a c-constraint, not an a-constraint. Generically, for every al(k), there will be exactly one a-constraint (a square in al(k)) that introduces it, and (possibly) many cross-term c-constraints (linear in al(k)). Number of constraints Now, for a given twisted puncture, let us explain the rule to find at which scaling dimensions k there exists a constraint. Denote by p the B-partition of 2N + 1 that labels our twisted puncture. Consider, as before, the transpose partition, q = pt, and let s be the sequence of partial sums of q, as in (2.11) above. We will see that s contains all the information about constraints. Let us first note that a B-partition always has an odd number of parts, so suppose our B-partition p has 2l + 1 parts, and let p2l+1 be the last part of p. Then, an a-constraint of scaling dimension k exists if and only if: 1. k belongs to s. 2. k is even. 3. k is not a multiple of 2l + 1. If k = 2m satisfies these conditions, the local contribution to n2m should be reduced by one, and the contribution to nm should be raised by one. Similarly, a c-constraint of scaling dimension k exists if and only if: 1. k belongs to s. 2. If k is odd, it must satisfy a cross-term condition. Let j be the unique index such that k = sj . Then: 1) sj must not be the last element of s; 2) both sj1 and sj+1 must be even, sj1 = 2u and sj+1 = 2v; and 3) sj = u + v. 3. If k is even, it must be a multiple of 2l + 1. p2l2+1 + 1 If k satisfies these conditions, the local contribution to nk should be reduced by one. Example. For the B-partition p = [15, 132, 94], we have 2l + 1 = 7, p2l+1 = 9, q = [79, 34, 12], s = [7, 14, 21, 28, 35, 42, 49, 56, 63, 66, 69, 72, 75, 76, 77]. For c-constraints whose dimensions are of the form k = r(2l + 1) = 7r, we should allow only 5 r 8. Thus, we have a-constraints at the dimensions k = 66, 72, 76, and c-constraints at the dimensions k = 35, 42, 49, 56, 69. We can also compute the pole structure {pk}. For instance, we first find n35 = 63/2, and, since we just found that at k = 35 we have a c-constraint, we must have p35 = 65/2. Also, we can compute n66 = 60 and n33 = 63/2, and we know that at k = 66 we have an a-constraint; hence, p66 = 61 and p33 = 61/2. Notice that, although the k = 66 a-constraint introduces a new parameter, a(6313/)2, with the same dimension and pole order as the leading coefficient c(6313/)2, these are independent. (By construction, the entry to the right of p2l+1(2l + 1) cannot be a multiple of 2l + 1.) This block of multiples of 2l + 1 in s will be important, since it gives rise to a particular set of c-constraints for p. So, let us look at it in detail. Consider the first 2 p2l2+1 multiples of 2l + 1 in s, split into two groups: For completeness, let us call s the set of entries of s that are not in s or s, so s = s s s is a disjoint union. Notice that if p2l+1 is odd, the term p2l+1(2l + 1) is in s. This term never gives rise to a constraint. Entries in s. None of the entries in s correspond to constraints. Rather, they can be used to define certain quantities that make c-constraints more transparent; see the example of the minimal puncture, p = [2N + 1], below. Entries in s. Each entry in s can be interpreted as a dimension for a c-constraint. Let us look at these constraints in more detail. We study first the even entries. Let 2k be in s. The corresponding c-constraint is, schematically, a square: Explicit form of constraints Now, the rules described above (to compute the dimensions at which a- and c-constraints occur) are sufficient for most purposes, but if we want to know what the constraints look like more specifically, which we need to compute explicit Seiberg-Witten curves, we should study the constraint structure of twisted punctures a little more systematically. We will do so below. Recall our B-partition p has 2l + 1 parts, p = {p1, . . . , p2l+1}. Then, q must be of the form q = [(2l + 1)p2l+1 , . . . ]. Such a term would arise if and only if there exist c-constraints of dimensions 2k and 2k, and if k + k = 2k and l + l = 2l. On the other hand, the odd entries, say 2k + 1, of s always yield c-constraints that are sums of cross-terms, as (2.20), but with k + k = 2k + 1. where fl(/k2)(c) is a polynomial in leading coefficients, of total dimension k and total pole order l = p2k (e.g., f2(4) = c(24) 41 c(12) 2). The ellipsis above (and in the rest of this subsection) stands for possible cross-terms, which are of the form Entries in s. Let us now study the constraints in s. Again, let us look at even and odd entries separately. Each even entry, 2k, in s is the dimension of an a-constraint, cl(2k) = al(/k2) 2 + . . . Finally, let us look at the odd entries, 2k + 1, in s. If 2k + 1 satisfies the requirements of section 2.4.2, it yields a cross-term c-constraint involving parameters introduced by a-constraints, where u + v = 2k + 1 and l = l + l. Also, if the first c-constraint dimension in s is odd, the c-constraint will include a mixed cross-term of the form Examples. Consider the minimal twisted puncture, p = [2N + 1]. Then, q = [12N+1] and s = [1, 2, . . . , 2N + 1]. So, we have only c-constraints, which, since l = 0, are of scaling dimension k = N + 1, N + 2, . . . , 2N . To write the c-constraints specifically, we define auxiliary quantities rk, for 0 k N , by r0 1, r1 0, and the rest by rk 21 (c(pkk) Rp(kk)), where Rp(kk) is the sum of all terms of the form (rj )2 or 2rj rj , with j, j < k, of total scaling dimension k and total pole order pk. Then, expressing the c(pkk) back in terms of the rj , for 0 j k N , reveals a nice pattern of squares of cross-terms that should be completed, in a unique way, by the sought c-constraints. For instance, for N = 5, we define r0 1, r1 0, c(55/)2 21 c(33/)2c(12) . r2 21 c(12), Then, we can write: c(33/)2 = 2r3r0 + 2r1r2, c(4) = 2r4r0 +(r2)2 +2r1r3, c(55/)2 = 2r5r0 +2r3r2 +2r1r4, 2 c(6) = 2r2r4 +(r3)2 +2r1r5, c(77/)2 = 2r3r4 + 2r2r5, 3 c(99/)2 = 2r4r5, Here, the expressions of scaling dimension k for 0 k 5 are equivalent to (2.24), while those with 6 k 10 are the actual c-constraints. Thus, introducing the rk makes clear what the c-constraints should be. Now, let us discuss the puncture p = [2N 1, 12]. We find that there are a-constraints (c-constraints) for every even (odd) scaling dimension k in the range 4, 5, . . . , 2N . These constraints follow a pattern of squares and cross-terms in al(j) parameters; see appendix A for examples. More importantly, our rules of section 2.4.2 tell us that the [2N 1, 12] puncture is the only one with an a-constraint of scaling dimension four, that is, the only one that gives rise to an independent parameter a(2) of scaling dimension two. Finally, we study the puncture of type p = [93, 52]. The scaling dimensions of the c-constraints are 15, 20, 31, while those of the a-constraints are 28, 34. We find: c(34) = (a(17))2. Collisions of punctures In this section we study what happens when two or more punctures collide. We call this process the operator product expansion (OPE) of punctures. In section 3.1 and section 3.2, we discuss the overall strategy for analyzing the OPE, by first considering the OPE on an infinite plane, and then on a compact curve. We then describe an explicit algorithm to compute the OPE in section 3.3. OPE of punctures on a plane So far we have studied how to compute the properties of a single puncture. Let us now see what happens if two or more punctures come close together. First, we would like to study the simpler case of a non-compact Riemann surface, the complex plane. Consider a sixdimensional space of the form R4 C. We denote by z the coordinate on C, and consider k punctures of types p1, p2, . . . , pk to be localized, respectively, at z = z1, z2, . . . , zk. Now, at very large |z|, the system looks as if consisting of: a 4D N = 2 superconformal theory X, which depends on the types and positions of the k punctures, and such that a certain subgroup H of its global symmetry group, GX , can be identified with a subgroup of F , and a dynamical gauge multiplet for H, with coupling constant depending on the types and positions of the k punctures, which couples X to q. We call this process the operator product expansion (OPE) of the k punctures, and we call X the coefficient of the OPE. We schematically represent the outcome of the OPE as X H q. If q is the full puncture and H = G, the theory X is the same as the 4D theory obtained by compactifying the 6D theory on a sphere, with punctures of type pi at z = zi, and a full puncture at z = . Otherwise, we say that the theory X is the 4D theory obtained by compactifying the 6D theory on a sphere, S, with punctures of type pi at z = zi, and an irregular puncture at z = , determined by the choice of pi, and say that the gauge group H arises from the cylinder connecting the irregular puncture with the regular puncture of type q.3 We denote such irregular puncture by the pair (q, H), and, if there are inequivalent embeddings H F , we add a label to distinguish which embedding we mean; see section 3.2.3 for an example. We call q the regular puncture conjugate to the irregular puncture (q, H). While the detailed properties of the theory, X, depend on the punctures, pi (and the various cross-ratios of their positions), certain features are encoded purely in the pair (q, H). For instance, H, seen as a subgroup of the global symmetry group GX of the theory X, has some level k 0. (k = 0 if and only if X is the empty theory.) This level is strictly determined [5] by demanding that the H gauge theory on the cylinder (q, H) H q has vanishing -function. Similarly, the local contribution of the irregular puncture to nh, nv and to the graded Coulomb branch dimensions of X are determined by the pair (q, H) [5]. Degeneration of a curve via the OPE Let us now consider the OPE on a compact curve. Let C be a sphere, with k + k regular punctures of types p1, . . . , pk; p1, . . . , pk . We assume that the punctures are such that all the graded Coulomb branch dimensions are non-negative. (Otherwise, the theory is bad, and taking the 4D limit is a more delicate issue.) Now consider the limit where C degenerates into two spheres, C1 C2, with p1, . . . , pk on C1 and p1, . . . , pk on C2. We would like to understand the behaviour of the 4D theory in this limit. We proceed as follows: Replace the punctures p1, . . . , pk with their OPE, as in section 3.1, obtaining a regular puncture q, a gauged subgroup H of the flavour symmetry of q, and the 4D theory X, which is the 4D limit of a sphere with p1, . . . , pk plus an (ir)regular puncture (q, H). Then we have the following system consisting of X H I(q, q) H X where I(q, q) is a sphere with two regular punctures of type q and q, respectively. As explained in [18], a sphere with two regular punctures is a supersymmetric hyperKahler non-linear sigma model with global symmetry F F , where, in our case, we gauge the subgroup H H F F . Any point on the target space of the 3Conventionally, in the Hitchin system literature, a regular puncture is a point on C where the Higgs field, , is allowed to have at worst a simple pole, while an irregular puncture is a point where may have higher-order poles. Our use of the term irregular puncture (as also used in [4, 5]) differs from such conventional use. In particular, in our nomenclature, the full puncture p is a limiting case of an irregular puncture, p = (p, G). No other regular puncture can be thought of as an irregular puncture in this manner. non-linear sigma model breaks F F to the stabilizer subgroup, F and hence the gauge symmetry H H is always Higgsed to H F . Sometimes, the D-term and the F-term constraints for H force the theory X coupled to q via H to be Higgsed to a theory Y . Similarly, the theory X coupled to q may be Higgsed to a theory Y . In the end, we have a 4D system of the form: Y H Y Now, a sphere with k regular punctures and an irregular puncture has a degeneration where we consecutively collide two punctures, so that the resulting 4D theory consists of several three-punctured spheres coupled to each other. These three-punctured spheres, which we call fixtures [4, 5], contain either three regular punctures, or two regular punctures and one irregular puncture. A table of all possible fixtures makes finding the 4D description of an arbitrary degeneration a simple task. Let us illustrate these ideas with a few examples, all with untwisted punctures for simplicity. Example 1 Consider the A2N1 theory compactified on a 4-punctured sphere, with punctures (p1, p2, p3, p4) = ([N 2], [N 2], [2, 12N2], [2N 1, 1]). The OPE of p1 with p2 is a full puncture, [12N ], coupled via H = Sp(N ) to the theory, X, which is 4N free hypermultiplets transforming as 2 copies of the fundamental 2N dimensional representation of Sp(N ). The OPE of p3 with p4 yields a full puncture, [12N ], coupled via H = SU(2N 1) to the theory, X, which is (2N 1)(2N 2) free hypermultiplets transforming as (2N 2) copies of fundamental representation of SU(2N 1). The 2-punctured sphere, with two full punctures is T SU(2N, C), which Higgses H H = Sp(N ) SU(2N 1) down to H = Sp(N 1). So, in the end, the 4D theory looks like Sp(N 1) [12N ], Sp(N ) [12N ], SU(2N 1) [2, 12N2] [2N 1, 1] 2( ) + 4(1) (2N 2)( ) + (2N 2)(1) i.e. an Sp(N 1) gauge theory with 2N fundamentals plus 2(N + 1) free hypermultiplets. Here the symbol stands for the 2(N 1)-dimensional fundamental representation of Sp(N 1). Note the cylinder [12N ], Sp(N ) Sp(N1) [12N ], SU(2N 1) connecting two irregular punctures. Example 2 As a second example. consider the A4 theory, compactified on the 4-punctured sphere with punctures (p1, p2, p3, p4) = ([4, 1], [3, 2], [22, 1], [22, 1]). The OPE of p3 and p4 yields the full puncture, q = [15], coupled to the theory X = R2,5, via H = G = SU(5). Here R2,5 is a non-Lagrangian SCFT discussed in [4]. Note that, since q is the full puncture and H is G, the 3-punctured sphere corresponding to X contains three regular punctures, ([22, 1], [22, 1], [14]). In between, we have the 2-punctured sphere with one full puncture and one [2, 13] puncture. This Higgses H H = SU(2) SU(5) down to H = SU(2). However, in order to satisfy the F-term equations, the theory X is Higgsed down to the theory Y , where the [15] puncture is replaced by a [2, 13]. The end result is (E6)6 + 1(2) + 5(1) an SU(2) gauging of the (E6)6 SCFT, with an additional doublet and 5 free hypermultiplets. Note that, in this case, the cylinder connects an irregular puncture with its conjugate regular puncture. Example 3 Now, let us turn to an example from the D4 theory. Consider the 4-punctured sphere Here, each very even partition (e.g., [24]) corresponds to two nilpotent orbits in so(8), and our sphere includes one of each type (indicated by the red/blue colour); see [5]. When we take the OPE of p1 with p2, we obtain the full puncture, q = , coupled to X = the (E7)8 SCFT, via H = Spin(7) (and similarly for the OPE of p3 with p4). However, there are three inequivalent embeddings of Spin(7) Spin(8), depending on which of the three 8-dimensional irreducible representation of Spin(8) decomposes as 7 + 1. We can indicate this choice by putting a subscript on H, or (in the notation of [5]) by colouring the Young diagram corresponding to q: , Spin(7)) = ( , Spin(7)) = ( , Spin(7)) = ( , Spin(7)8v ) , Spin(7)8s ) , Spin(7)8c ) where, as in section 3.2.1, we have a cylinder connecting two irregular punctures. Determining the OPE via the Higgs field In light of section 3.1 and 3.2, we would like to study the basic problem of two punctures p1 and p2 colliding on a plane. We have seen that in the collision limit, an irregular puncture (q, H) arises, which is connected to a regular puncture q by a cylinder with gauge group H. Let us discuss how to find q and H. To determine q, we construct a solution to the Higgs field on the plane that includes p1 and p2, and compute the residue that arises in the collision limit. This residue provides the Higgs-field boundary condition for q. Thus, one can determine the Nahm pole for q, e.g., by looking at the degeneracy of the mass deformations in the residue. Also, the number of independent mass deformations is equal to the rank of H. To gather more information about H, we consider the k-differentials k, and take the limit where the punctures collide, which reveals the scaling dimensions of the Casimirs of H. Knowing these usually suffices to identify the gauge group. Only in a handful of cases, often to distinguish Sp(k) from SO(2k + 1), must one do further consistency checks, such as computing the matter representation for the fixture that arises in the degeneration limit, and corroborating that it provides the right contribution to the beta function of H. Because of these observations, in the next subsections we will study the Higgs field on a plane with two punctures, in the limit where these collide. Later, in section 3.3.4, we will do the same for k-differentials. But before doing this, let us briefly discuss a situation that will arise often. Consider C to be the complex plane or a sphere, with complex coordinate z, and put k punctures, p1, . . . , pk, on C. Let the positions of the punctures be z1, . . . , zm, zm+1, . . . zk, so that we can collide the first m of them by taking the limit 0. Now consider a meromorphic k-differential on C of the form Qim=1(z zi)ri Qjk=m+1(z zj )rj where , s, r1, . . . , rk are rational numbers; A is a coefficient. This is a typical term in a k-differential, including the case of the Higgs field (k = 1). In the 0 limit, we get C in the presence of the k m punctures pm+1, . . . , pk, plus a new puncture, q, at z = 0. The 0 limit may also be represented by the conformally equivalent picture of a sphere C that bubbles off C, containing the m punctures p1, . . . , pm, plus the irregular puncture (q, H). Such picture is obtained through the change of variables z = /z. Then, requiring that (3.4) have a finite limit as 0 in the z coordinates puts a lower bound on , We will use this result quite often. If the bound is not saturated, the k-differential simply vanishes, in the 0 limit, on both C and C. On the other hand, if the bound is saturated, we have three possibilities when 0: 1. If Pim=1 ri > s + k, (3.4) vanishes on C, but not on C. 2. If Pim=1 ri < s + k, (3.4) vanishes on C, but not on C. 3. If Pim=1 ri = s + k, (3.4) does not vanish on C nor C. In most cases (such as in the following subsections), the coefficient A in (3.4) will represent a physical degree of freedom, and so it should not be lost in the 0 limit; therefore, it will be desirable that the bound be saturated. Case 1 (2) corresponds to A being a degree of freedom for the theory on C (C), whereas case 3 corresponds to A being a degree of freedom of the gauge group on the cylinder, which in the 0 limit looks like a mass deformation on both C and C. However, in a few cases where the coefficient A carries redundant information (because of local constraints), consistency will require that the bound not be saturated. We will see such cases in section 4. Consider now the Higgs field on a plane with two untwisted punctures of type p1 and p2 at positions z = 0 and z = , respectively, where z is the coordinate on the plane. Let A and B be representatives of the (massless or mass-deformed) adjoint orbits in sl(2N ) corresponding to p1 and p2, respectively. Then, we can write an ansatz, (z) = A(z ) +zB(zz +z)(z )P (z) dz, where P (z) is a power series in z whose coefficients are generic elements of sl(2N ). P (z) simply represents the infinite degrees of freedom contained in the plane. At finite , the expansions of (z) near p1 and p2 are, respectively, z B + generic in sl(2N ) dz, (z) = Az(z(z1)1+) B dz, where we used z = /z in (3.6) and took the 0 limit. The punctures q, p1, p2 are at z = 0, , 1, respectively. Notice that the Riemann-Roch theorem for a 3-punctured sphere requires only two coefficients, not three, for (z). In other words, in a fixture, the choice of representatives, A and B, for the adjoint orbits for two punctures, p1 and p2, completely determines the adjoint orbit (that is, its mass deformations), plus a representative of such orbit, for the third puncture, q. Twisted-twisted Let us now take two twisted punctures of type p1 and p2 at positions z = 0 and z = on a complex plane with coordinate z. Let A and B be representatives of the (massless or massdeformed) adjoint orbits in Sp(N ) corresponding to p1 and p2, respectively. Recall the decomposition of sl(2N ) in eigenspaces of the Z2-outer automorphism, sl(2N ) sp(N ) o1. Then, we can write the following ansatz for the Higgs field, 3(4) The remaining degeneration limit, , 1(5) + 1(4) The S-dual theory 1(5) + 1(4) 1(5) + 1(4) Sp(3)5 SU(2)8 SCFT + 12 (2) is an SU(2) gauge theory, with three half-hypermultiplets in the 2, coupled to the Sp(3)5 SU(2)8 SCFT. An alternative realization of the same family of theories (with the addition of one free hypermultiplet) is given by 1(5) + 2(4) + 1(1) The two other degenerations both give rise to an SU(2) gauge theory, with matter in the three half-hypermultiplets in 2, coupled to the Sp(3)5 SU(2)8 SCFT. Sp(3)5 SU(2)8 SCFT + 12 (2) + 1(1) Sp(3)5 SU(2)8 SCFT + 1(1) Note that this S-duality is the Example 13 of Argyres-Wittig [11] and the Sp(3)5 SU(2)8 SCFT is one of the new rank-1 SCFT, to which we will come back in section 7. The best we can do, to capture Sp(2) with matter in the 3(5), is 2(5) + 2(1) Sp(3)5 SU(2)8 SCFT + 2(1) is an SU(2) gauging of the Sp(3)5 SU(2)8 SCFT plus two free hypermultiplets. Note that this S-duality is the Example 12 of Argyres-Wittig [11]. A family of SU(2) Sp(2) gauge theories In this subsection, we wish to discuss a 5-punctured sphere which interpolates between two of the gauge theories discussed in the previous section, namely Sp(2) with 6(4), Sp(2) with 1(5) + 4(4) + 3(1). We achieve this by gauging an SU(2) subgroup of the flavour-symmetry group of one of these theories. The first has flavour symmetry Spin(12)k=8; the second has flavour symmetry group SU(2)5 Spin(8)8 3 free hypers. In each case, gauging an SU(2) (where, in the latter, we treat the 3 free hypermultiplets as 3 half-hypers in the fundamental of SU(2)) breaks the flavour-symmetry group to F = SU(2)8 Spin(8)8. Taking the SU(2) from weak coupling to strong coupling, it again decouples, leaving us with the other Sp(2) gauge theory. Along the way, a crucial role is played by the gauge theory fixture, which is an SU(2) gauge theory, with matter in the 4(2) + 4(1), at its Z2 symmetric point. Only an SU(2) Sp(2) subgroup of the Spin(8) global symmetry of the gauge theory is manifestly realized by the punctures. Being at the Z2 symmetric point means that a Z2 subgroup of the outer-automorphisms of Spin(8) acts an automorphism of the SCFT. In particular, it exchanges two 8-dimensional representations of Spin(8) which transform, respectively, as (3, 1) + (1, 5) and as (2, 4) under SU(2) Sp(2) Spin(8). Thus, the matter can be interpreted either as transforming as the 12 (3, 1; 2) + 12 (1, 5; 2) or as the (4, 2; 2) of SU(2) Sp(2) SU(2)gauge. There are nine distinct degenerations of the 5-punctured sphere. The first three correspond to weakly-coupled descriptions with an SU(2) Sp(2) gauge group and matter in the 23 (2, 1) + 21 (2, 5) + 4(1, 4). Degeneration A. Degeneration B. Sp(2) + 4(4) + 21 (2, 5) , Degeneration C. (2, 4) + (1, 4) Degeneration E. The next two correspond to weakly-coupled descriptions with an SU(2)Sp(2) gauge group and matter in the (2, 4) + 4(1, 4). Degeneration D. Three more degenerations correspond to the S-dual description when the latter Sp(2) gauge theory is strongly-coupled: an SU(2) SU(2) gauging of the (E7)8 SCFT. Degeneration F. Degeneration G. Degeneration H. , SU(2) SU(2) SU(2) SU(2) , ! , SU(2) SU(2) SU(2) SU(2) , ! 32 (2, 1) + 21 (2, 5) + (1, 4) SU(2) + or (2, 4) + (1, 4) Rank-1 SCFTs In this section, we summarize our current knowledge of rank-1 theories and their realizations, including constructions of new such theories using the twisted A series. Summary of rank-1 SCFTs Recall that a rank-1 SCFT has, by definition, only one Coulomb branch operator, u. When all mass deformations are turned off, the scaling dimension (u) of u may only take the values 65 , 43 , 23 , 2, 3, 4, or 6, as reviewed in, e.g., [20]. Examples of such SCFTs can be constructed from a D3-brane probing an F-theory 7-brane: The SCFTs with (u) = 65 , 43 , or 32 , obtained in this way, can also be realized as the superconformal points of Argyres-Douglas type [21, 22]. The SCFTs with (u) = 3, 4, or 6 are the interacting theories found in [23, 24], with flavour symmetries E6,7,8, respectively. We call these SCFTs the old rank-1 theories. For some time, these were thought to exhaust the list of rank-1 SCFTs. However, three new ones were found in [11]: We call these the new rank-1 SCFTs. Let us now see that both old and new rank-1 theories can be realized by compactifying a 6D N = (2, 0) theory on a punctured curve. The constructions below are not necessarily unique; there are often distinct configurations that yield the same isolated SCFTs (possibly plus free hypermultiplets) in the 4D limit. The old SCFTs with (u) = 65 , 34 , or 32 can be obtained from the untwisted A1 theory on a sphere with an irregular puncture and, possibly, a regular puncture; see, e.g., [2527]. (Here, we mean irregular in the sense of these papers, not in ours.) The old SCFT with (u) = 4 is obtained from the untwisted A3 theory on a sphere with two full punctures and a puncture of type [22] [28]. The old SCFT with (u) = 6 is obtained from the untwisted A5 theory on a sphere with three punctures of types [16], [23] and [32] [28]. The new SCFT with (u) = 6 is obtained from the untwisted D4 theory on a sphere with two punctures of type [3, 22, 1] and one puncture of type [22, 14]. This realization includes three free hypers [5]. The new SCFT with (u) = 4 is obtained from the A3 theory on a sphere with an untwisted puncture of type [2, 12] and two twisted punctures of type [22, 1]. This realization comes with a free half-hypermultiplet in the fundamental of the SU(2) flavour symmetry. This is the example we studied in section 6.2.3. To obtain the new SCFT with (u) = 3, we need to extend our analysis to the twisted A2n theory. The Z2 twist of the A2n theory is particularly subtle, as emphasized in [13]. Hence, we prefer to postpone a systematic analysis of the twisted A2n theory. Still, it is possible to show how to obtain the missing new SCFT from a 6D construction. In [11], the new theory with (u) = 3 is introduced in the following way. Consider the SU(3) gauge theory with one hyper in the fundamental and one hyper in the symmetric tensor representation. The S-dual theory is an SU(2) gauging of the new SCFT, coupled to n half-hypermultiplets in the doublet. The field-theory arguments in [12] constrain n to be 0 or 2, and require the flavour symmetry h of the SCFT to satisfy kh = (8 n)/I, where I is the index of the embedding of su(2) in h. Now, the tensor product of two fundamentals of SU(3) decomposes as the direct sum of a fundamental plus a symmetric representation. The tensor product can in turn be obtained from the bifundamental of a product group SU(3)1 SU(3)2, by taking a diagonal subgroup SU(3)diag, such that SU(3)diag is embedded in SU(3)1 in the standard way, but embedded in SU(3)2 with the action of complex conjugation, i.e., the nontrivial outer automorphism. So, consider the A2 theory on a fixture with a simple puncture and two full punctures, which by itself simply gives rise to a bifundamental. Then, the SU(3) gauge theory with matter in the 1(3) + 1(6) can be realized by connecting the full punctures in the fixture by a cylinder with a Z2 twist line looping around it. In other words, we have the A2 theory on a torus with one simple puncture and a Z2 twist loop. See the left side of the figure below. In the S-dual frame, shown on the right side of the figure, we have a fixture with an untwisted simple puncture and two twisted full punctures (denoted by ), and the full punctures are connected by a cylinder with a Z2-twist line along the cylinder. Clearly, this gives a weakly-coupled SU(2) gauge field coupled to the fixture. Also, the flavour symmetry group of the fixture must contain the explicit SU(2)2 U(1) as a subgroup. Thus, based on our findings, the fixture may be: an interacting SCFT with flavour symmetry group H SU(2)2 U(1), plus free hypers in the (2, 1) + (1, 2) of SU(2)2 and neutral under U(1) (if n = 2). To see which of these two possibilities is the right one, recall [18] that when the Riemann surface has two twisted (or untwisted) full punctures with flavour symmetry G, the holomorphic moment maps 1, 2 of the two G-actions on the Higgs branch must be equal, Let us see what happens if n = 2. In this case, the Higgs branch is X H1 H2, where X is the Higgs branch of the interacting SCFT with an action of SU(2)1 SU(2)2, and H1,2 is the Higgs branch for the SU(2)1,2 free hypers, respectively. Then we have tr 12 = tr 22 tr i2 = tr X,i2 + tr 2Hi Acknowledgments The authors would like to thank Andrew Neitzke for very helpful conversations on the treatment of Hitchin fields. The authors would also like to thank Philip Argyres and Alfred Shapere for suggesting to YT the possibility of introduction of the twist fields in the type A theories in the first place, during the conference Quantum Theory and Symmetries held in July 2009. The work of O. C. is supported in part by the INCT-Matematica and the ICTP-SAIFR in Brazil through a Capes postdoctoral fellowship. The research of J. D. is based on work supported by the National Science Foundation under Grant No. PHY-0969020, and by the United States-Israel Binational Science Foundation under Grant #2006157. The work of Y. T. is supported in part by World Premier International Research Center Initiative (WPI Initiative), MEXT, Japan through the Institute for the Physics and Mathematics of the Universe, the University of Tokyo. Tables of properties of twisted sectors A3 twisted sector Pole structure Since the A3 and D3 (2,0) theories are isomorphic, the defects have labels in both descriptions. In D3 notation, an untwisted puncture is labeled by a D-partition of 6, whereas a twisted one is labeled by a C-partition of 4. To facilitate comparison with the tables in [2], we list below the labels for the punctures in both descriptions. untwisted Partition of 4 D-partition of 6 (A3) (D3) [14] [16] [2, 12] [22, 12] [22] [3, 13] [3, 1] [32] [4] [5, 1] , SU(2) SU(2) A.1.3 Free-field fixtures Untwisted fixture # of Hypers Representation , SU(2) SU(2) , 2 of SU(2) (2, 3) of SU(2) SU(3) (4, 4) of SU(4) SU(4) 12 (2, 2, 4) of SU(2) SU(2) Sp(2) Twisted fixture # of Hypers Representation , ) (2, 1, 4) + 12 (1, 2, 6) of SU(2) SU(2) SU(4) 12 (3, 2) of SU(2) SU(2) (1, 2) of SU(2) SU(2) 12 (6, 4) of SU(4) Sp(2) , SU(2) SU(2)) , SU(2) SU(2)) (1, 2) + (2, 1) of SU(2) SU(2) 12 (2, 5) of SU(2) Sp(2) 12 (2, 5) + (1, 4) of SU(2) Sp(2) # of Hypers Representation SU(2) SU(2) 12 (2, 1; 2) + 21 (2, 2; 1) + 21 (2, 1; 1) +(1, 2; 2) + (1, 1; 2) of SU(2)2 G 12 (3, 1; 2) + 12 (1, 5; 2) + (1, 4; 1) or 12 (2, 4; 2) + (1, 4; 1) of SU(2) Sp(2) G (4; 2, 1) + (4; 1, 2) of SU(4) G (4, 1; 3) + (1, 2; 3) of Sp(2) SU(2) G (4, 1; 4) + 12 (1, 2; 5) of SU(4) SU(2) G (4, 1; 4) + 12 (1, 4; 6) of SU(4) Sp(2) G A.1.5 Interacting SCFTs Untwisted fixture (d2,d3,d4) (nh,nv) (0,0,1) (24,7) (0,1,1) (30,12) SU(2)6 SU(8)8 (0,1,2) (40,19) Twisted fixture (d2,d3,d4) (nh,nv) (0,1,1) (23,12) SU(2)5 Sp(3)6 U(1) (0,1,2) (33,19) SU(2)5 SU(4)8 Sp(2)6 (0,0,2) (26,14) SU(2)52 SO(7)8 Sp(4)6 SU(2)8 Sp(2)62 SU(2)6 U(1) Sp(2)62 SU(4)8 Mixed fixtures Untwisted fixture Theory (E6)6 + (1, 1, 4) of SU(2) SU(2) SU(4) Sp(3)5 SU(2)8 SCFT + 1 (1, 1, 4) of SU(2) SU(2) Sp(2) 2 Sp(3)5 SU(2)8 SCFT + 1 (1, 2, 1) of SU(2) SU(2) SU(2) 2 A5 twisted sector Flavour Hitchin B-partition C-partition Pole structure ([4, 2], Z2) A7 twisted sector Flavour Hitchin B-partition C-partition Pole structure c(56) = c(56) = 41 c(34) = c(75/)2 = 2a(32/)2a(23) c(46) = c(24) = 1 c(55/)2 = 21 c(12)c(33/)2 c(36) = 1 Sp(2) SU(2) (317, 6127 ) SU(2) U(1) (296, 5823 ) SU(2) U(1) (288, 5629 ) c(171)/2 = 2a(53/)2a(34) c(171)/2 = c(53/)2a(34) c(75/)2 = 2a(32/)2a(23) c(6) = 2a(32/)2a(4) 4 5/2 c(77/)2 = 12 c(33/)2c2(4) Pole structure {1, 5 , 3, 9 , 5, 13 , 7, 17 , 9} 2 2 2 2 {1, 5 , 3, 9 , 5, 13 , 7, 15 , 9} 2 2 2 2 {1, 5 , 3, 9 , 5, 13 , 7, 15 , 9} 2 2 2 2 {1, 5 , 3, 9 , 5, 11 , 7, 15 , 9} 2 2 2 2 {1, 5 , 3, 9 , 5, 11 , 7, 15 , 9} 2 2 2 2 {1, 5 , 3, 9 , 5, 11 , 7, 15 , 9} 2 2 2 2 {1, 5 , 3, 9 , 5, 11 , 7, 15 , 8} 2 2 2 2 {1, 5 , 3, 7 , 5, 11 , 7, 15 , 8} 2 2 2 2 {1, 5 , 3, 7 , 5, 11 , 6, 13 , 8} 2 2 2 2 {1, 5 , 3, 7 , 5, 11 , 6, 13 , 8} 2 2 2 2 ([42, 2], Z2) {1, 5 , 3, 7 , 5, 11 , 7, 15 , 8} 2 2 2 2 ([32, 22], Z2) {1, 5 , 3, 7 , 5, 11 , 6, 13 , 8} 2 2 2 2 {1, 5 , 3, 9 , 5, 11 , 7, 15 , 8} 2 2 2 2 {1, 5 , 3, 7 , 5, 11 , 7, 15 , 8} 2 2 2 2 c(910) = (a(95/)2)2 c(910) = (a(95/)2)2 = 41 (c(95/)2)2 c(78) = (a(74/)2)2 c(810) = (a(45))2 = 14 (c(53/)2)2 c(78) = (a(74/)2)2 c(810) = (a(45))2 c(78) = (a(74/)2)2 c(810) = (a(45))2 c(810) = (a(45))2 c(56) = (a(53/)2)2 c(8) = 2a(3) a(5) 6 5/2 7/2 c(710) = (a(75/)2)2 SO(7) SU(2) (635,620) Sp(2) SU(2) (618,607) Sp(2) U(1) SU(2) U(1) SU(2) U(1) SU(2) U(1) c(171)/2 = c(53/)2a(34) c(8) = c(53/)2a(5) 6 7/2 c(191)/2 = 2a(54/)2a(35) c5(/52) c(55/)2 12 c(33/)2c(12) c(77/)2 = 21 c(33/)2c2(4) Open Access. 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Oscar Chacaltana, Jacques Distler, Yuji Tachikawa. Gaiotto duality for the twisted A 2N −1 series, Journal of High Energy Physics, 2015, 75, DOI: 10.1007/JHEP05(2015)075