Tinkertoys for the E 6 theory

Journal of High Energy Physics, Sep 2015

Compactifying the 6-dimensional (2,0) superconformal field theory, of type ADE, on a Riemann surface, C, with codimension-2 defect operators at points on C, yields a 4-dimensional \( \mathcal{N}=2 \) superconformal field theory. An outstanding problem is to classify the 4D theories one obtains, in this way, and to understand their properties. In this paper, we turn our attention to the E 6 (2,0) theory, which (unlike the A- and D-series) has no realization in terms of M5-branes. Classifying the 4D theories amounts to classifying all of the 3-punctured spheres (“fixtures”), and the cylinders that connect them, that can occur in a pants-decomposition of C. We find 904 fixtures: 19 corresponding to free hypermultiplets, 825 corresponding to isolated interacting SCFTs (with no known Lagrangian description) and 60 “mixed fixtures”, corresponding to a combination of free hypermultiplets and an interacting SCFT. Of the 825 interacting fixtures, we list only the 139 “interesting” ones. As an application, we study the strong coupling limits of the Lagrangian field theories: E 6 with 4 hypermultiplets in the 27 and F 4 with 3 hypermultiplets in the 26.

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Tinkertoys for the E 6 theory

Received: May E6 theory Oscar Chacaltana 0 1 2 4 5 Jacques Distler 0 1 2 3 5 Anderson Trimm 0 1 2 3 5 0 Department of Physics, University of Texas at Austin 1 01140-070 S ̃ao Paulo , SP , Brazil 2 Instituto de F ́ısica Te ́orica, Universidade Estadual Paulista 3 Theory Group and Texas Cosmology Center 4 ICTP South American Institute for Fundamental Research 5 Bala-Carter f Projection Matrix Compactifying the 6-dimensional (2,0) superconformal field theory, of type ADE, on a Riemann surface, C, with codimension-2 defect operators at points on C, yields a 4-dimensional N = 2 superconformal field theory. An outstanding problem is to classify the 4D theories one obtains, in this way, and to understand their properties. In this paper, we turn our attention to the E6 (2,0) theory, which (unlike the A- and D-series) has no realization in terms of M5-branes. Classifying the 4D theories amounts to classifying all of the 3-punctured spheres (“fixtures”), and the cylinders that connect them, that can occur in a pants-decomposition of C. We find 904 fixtures: 19 corresponding to free hypermultiplets, 825 corresponding to isolated interacting SCFTs (with no known Lagrangian description) and 60 “mixed fixtures”, corresponding to a combination of free hypermultiplets and an interacting SCFT. Of the 825 interacting fixtures, we list only the 139 “interesting” ones. As an application, we study the strong coupling limits of the Lagrangian field theories: E6 with 4 hypermultiplets in the 27 and F4 with 3 hypermultiplets in the 26. Supersymmetric gauge theory; Supersymmetry and Duality; Extended Su- 1 Introduction 2 The E6 theory The Hitchin system ALE geometry Puncture properties Flavour groups Pole structures Puncture collisions Global symmetries and the superconformal index Levels of enhanced global symmetry groups Cataloguing fixtures using the superconformal index Computing the expansion of the index Regular punctures Free-field fixtures 3.3 Interacting fixtures with one irregular puncture 3.4 Interacting fixtures with enhanced global symmetry 3.5 Mixed fixtures A detour through the twisted sector E6 and F4 gauge theory Adding (E8)12 SCFTs E6 + 4(27) F4 + 3(26) n = 3 n = 2 n = 1 n = 0 Connections with F-theory 6 Isomorphic theories A Bala-Carter labels – 1 – 2 B Projection matrices superconformal theories, realizing them as compactifications of the 6D (2,0) theories on a punctured Riemann surface. Because of its six-dimensional origin, this class of theories (sometimes called class “S”) is endowed with a powerful set of tools [1–11] for studying its physical properties. For a theory in this class, it is trivial to write its low-energy Seibergglobal symmetry group, graded dimensions of the Coulomb branch, etc. Equally interesting, the understanding of the theories in this class expands our knowlbut also isolated interacting fixed points of the renormalization group with no known Lagrangian description. Gaiotto’s construction generates infinitely many new examples of these interacting theories. Thus, while it seems unlikely that this construction covers the the construction can themselves be classified. The (2,0) theories are classified by a choice of simply-laced Lie algebra j [14]; (a class of) punctures on the Riemann surface are labeled by nilpotent orbits in j [15, 16], and degenerating Riemann surfaces can be decomposed into a collection of three-punctured spheres connected by cylinders. The set of theories becomes larger if we allow for the (2,0) theory to be “twisted” by an outer-automorphism of j, when traversing a nontrivial cycle of the punctured Riemann surface, C. In particular, a twist when circling a puncture introduces a new class of defects, called “twisted punctures,” which are classified by nilpotent orbits in non-simply-laced Lie algebras obtained by dividing j by the action of the outer automorphism [9, 17]. In [4], we extended the classification of [3] to the (untwisted) DN series, and in [10] and [11] we incorporated Z2 In this paper, we extend our classification program to the (2,0) theory of type E6. We leave the study of this theory in the presence of Z2 outer automorphism twists for another publication. There is no known construction of the E6 theory as a low-energy theory of a stack of M5 branes, as was the case for the A- and D-series. Rather, the only known construction is as a compactification of IIB string theory on a K3 manifold at an E6 of the E6 theory is controlled by a Hitchin system [2] with gauge group E6. As a byproduct, we realize E6 gauge theory with matter in the 4(27), as well as F4 gauge theory with matter in the 3(26), as compactifications of the E6 (2,0) theory on a 27s, appeared first in [18]. Our solution to the superconformal F4 gauge theory is new. The E6 theory The Hitchin system 6D (2,0) theory of type E6 on a Riemann surface C is described by the Hitchin equations on C with complexified gauge group E6 [2]. We may also include codimension-two defects of the (2,0) theory localized at points on C; we refer to these as “punctures”. A class of punctures is classified by nilpotent orbits (or, equivalently, by embeddings of sl(2)) in the complexified Lie algebra e6 [9]. One of the main points of the construction is that a properties of the nilpotent orbits that label the punctures on C, without any detailed knowledge of the (2,0) theory. a local boundary condition for the Higgs field, representation of the gauge group, X is a representative of the nilpotent orbit d(O) in e6, and . . . represents a generic regular function of z taking values in e6. Here, d(O) is the image of O under the Lusztig-Spaltenstein map d [4, 7, 9]. Representatives of all nilpotent orbits in e6 can be found in [19], and a diagram specifying the action of d, as well as other properties of the e6 orbits, are collected in appendix C of [9] (taken from [20, 21]) When d is not injective, we distinguish different punctures with the same d(O) by their Sommersunder the action of C(O). As in our previous papers, we call O, which labels the puncture, the Nahm pole, and d(O), which appears in the Hitchin system boundary condition, the Hitchin pole. The physical properties of a puncture labeled by O will be directly related to geometric properties of the orbits O and d(O), and the discrete group C(O). Unlike classical Lie algebras, there is no natural parameterization of the nilpotent orbits of exceptional Lie algebras in terms of partitions or Young diagrams. Instead, the notation due to Bala and Carter is standard in the representation theory literature. This notation has been briefly discussed in previous works [9, 16, 22], but, for completeness, we review it in appendix A, and discuss how to extract relevant information from it. which is given by the spectral curve of the Hitchin system, i.e., by the characteristic polyextract from them is the same. (s0 and s1 are trivial — they are 1 and 0, respectively.) The sk(z) are holomorphic kdifferentials on C (with poles at the punctures), and can be expressed as polynomials in the representation R. Lie algebra g, the number of Casimirs is equal to the rank of g, and their scaling dimensions are the exponents (minus 1) of g. Unlike the sk or the Pk, the Casimirs encode the non In our previous papers [3, 4, 10, 11], the Lie algebra was of classical type, and R was the vector for DN ). In such cases, the coefficients sk directly provide a basis for the Casimirs These dimensions match precisely those of the non-trivial coefficients sk if R is chosen to be the fundamental representation. Thus, the sk can be taken to be the Casimirs of is the vector representation, the sk with k odd vanish, and the non-trivial coefficients are for large enough N , the sk would not have given directly the Casimirs, but instead a lot of redundant information. For example, for A5, there are five Casimirs, with dimensions 2, 3, 4, 5, 6. However, we have 34 non-trivial coefficients sk, with dimensions 2, 3, 4, . . . , 35. These sk are polynomials in the five Casimirs. R to be the adjoint representation of e6, as it is readily available in the form of structure constants; we used those from the computer algebra system GAP 4 [24]. Instead of trying to compute the 78 coefficients sk, we focus directly on the trace invariants Pk for values of k only as large as needed to extract the Casimirs. For the adjoint representation of e6, the Pk From the Pk, one can construct a less-redundant basis, P6 − 4608 P8 − 9 P6P2 + 32256 P8(P2)2 − 108 P14 − 14880 P12P2 + (P6)2P2 − 44431441920 P6(P2)4 + 409480168734720 This basis was constructed so that it reduces the constraints in our punctures to a min497664 P6(P2)3 − 9172942848 flips under the Z2 outer automorphism of E6). So, we can declare the k-differentials As for the Seiberg-Witten curve, to write it explicitly, we need to know how the 78 using the Casimirs [25, 26]. Let us briefly review that construction. ALE geometry J (where J is of A-D-E type) on the Riemann surface C can also be obtained, in a dual ALE fibration over C of type J [25, 26]. For e6, the threefold is realized as the hypersurface ⊂ tot(KC6 ⊕ KC4 ⊕ KC3 ) 2 = 5 = 6 = 8 = 9 = 12 = 144 (−3φ24 + 4φ2φ6 − φ8) (4φ12 + 6φ26 − 12φ23φ6 + 4φ62 + 3φ22φ8) where the k(z) are k-differentials on C [22] (in the “Katz-Morrison basis” [27]), related to The Seiberg-Witten solution is obtained by computing the periods of the holomorphic of a 2-sphere in the fiber times a curve on C. In the conformal case (which will be our focus in this paper), many of these cycles will necessarily be noncompact (the curve on C being a open curve, stretching between punctures). But, precisely for the parabolic case slightly singular nature of X~u itself. Puncture properties We describe below how to compute the properties of a puncture. There is a systematic way to compute every property of the puncture, except for the constraints, so it is easiest to compute the other properties first, and use them to guess the constraints. Below, let O be the Nahm nilpotent orbit that labels a given puncture, and su(2)O the associated su(2) embedding in e6. Flavour groups the centralizer of su(2)O in e6. A list of the centralizers for each O can be found in table 14 of [9], taken originally from [28]. Let fi be the simple, nonabelian factors in f. The level ki of fi is the central charge of the fi current algebra. (See, e.g., [12] for a review.) Following [9], the ki can be computed from the decomposition of the adjoint, 78, of e6 under the subalgebra su(2)O × f. These decompositions can, in turn, be deduced from the Bala-Carter label for O, and are summarized in the table in appendix A. R1 = 8 + 3(1), R2 = 2(8), R3 = 8 + 1, R4 = 2(1), Let the decomposition of the 78 be e6 = M Vn ⊗ Rn,i , where Vn is the n-dimensional irrep of su(2) (denoted by “n” in the table) and Rn,i is the corresponding (reducible) representation of fi. Let ln,i be the index of Rn,i. Then, the level of fi is ki = Pn ln,i. in appendix A, we have, for f1 = su(3), R1 = 3 + 8(1), R2 = 8(2), R3 = 9(1), R4 = 2, The conformal anomaly central charges of the 4D theory, a and c, can be conveniently written in terms of effective numbers of hyper- and vector multiplets, nh and nv, as the contributions from the punctures plus certain global terms. The formulas in eq. eigenspaces of the Cartan element of su(2)O. Here, let us recast those formulas in terms of the weighted Dynkin diagram for O, which can be found in table 14 of [9]. Let ~x be the six-dimensional vector consisting of roots. The “Weyl vector” is W~ = 12 P (The dot product is Euclidean.) In this notation, the formulas in eq. (3.19) of [9] are: h∨(E6) dim(E6) − 2 h∨(E6) dim(E6) − 2 1 W~ · ~x 1 W~ · ~x − 2 n0 rank(E6), and n1/2(~0) = 0. Thus, 2 h∨(E6) dim(E6) = 624 (dim(E6) − rank(E6)) = 588 (manifold) dimension dimC(O) of O: (dim(E6) − rank(E6)) − 2 dimC(O). The dimensions of the nilpotent orbits of E6 are listed in table 14 of [9]. For a non-trivial example, consider the puncture 2A1, which has weighted Dynkin Pole structures The “pole structure” is the set of leading pole orders {p2, p5, p6, p8, p9, p12} in the expansion To compute the pole structure, we need a representative of the Hitchin nilpotent orbit d(O). A table of representatives of all nilpotent orbits of E6 can be found in table 2 of [19]. In this table, a nilpotent representative is given by a sum of weighted Dynkin diagrams, and each weighted Dynkin diagram represents an element in the root vector space of e6 for by the labels of the Dynkin diagram. The nilpotent representative is the sum of these rootvector space elements. This procedure is most easily understood in terms of an example. sum of five elements [19], root vector spaces for the positive roots Fortunately, GAP4 provides a Chevalley basis for the adjoint representation of e6, so it is pole structure {1, 3, 4, 6, 6, 9} for the D4(a1) puncture. (Actually, there are three orbits, D4(a1), A3 + A1 and 2A2 + A1, that map under Spaltenstein to D4(a1), so we have three are different.) The constraints for some E6 punctures are, in some cases, much less obvious than those and ungraded local contributions to the Coulomb branch. Let us be specific. Let z be a local coordinate on C centered at the puncture, and let c(k) be the coefficient of z−l in the expansion of φk = φk(z) in z. Recall that, in the notation l of our previous papers, a “c-constraint” is a polynomial relation among coefficients cl(k) (of homogeneous bi-degree in both k and l). On the other hand, an “a-constraint” is a relation that defines a new quantity, al (k), of dimension k, in terms of the cl(k). Only the cl(k) with l > 0 parameterize the Hitchin nilpotent orbit [16]. In the absence of constraints, all the c(k) with 0 < l ≤ pk are independent, so their total number, P pk, should be equal to l A c-constraint reduces the total number of independent parameters by one, whereas an a-constraint does not affect this number. So, one should have: X pk − (number of c-constraints) = d Hence, d tells us how many c-constraints exist. On the other hand, the graded sum of the presence of “a”-constraints, k is not restricted to the degrees of the Casimirs), should be equal to nv. An a-constraint replaces a parameter of a certain degree k by another one of a different degree k0 < k. So, to get precisely nv, one must take into account all a-constraints and c-constraints. For example, consider the Higgs field near a D5 puncture, and the corresponding We find the pole structure {1, 2, 3, 4, 4, 6}, and thus, cube of a single quantity, which we call a (24), of scaling dimension 4 and pole order 2: c(8) = 3 a(4) 2 c(12) = Since there is no Casimir of E6 of dimension four, a(4) cannot be a coefficient in one 2 parameter. It is not hard to discover additional relations by inspection, and we see that they, too, involve the a(24) parameter: c(36) = c(49) = − 41 a(24)c(25), We can take one of the relations linear in a(24), say the one for c(36), as the a-constraint (that is, the relation that defines a(24) unambiguously in terms of the cl(k)), and the other the formula in section 2.4.2. We conclude that we have found the correct set of constraints for this puncture. As we will see next, when two punctures collide, we get a new puncture. Thus, working with the k-differentials, we must be able to reproduce the pole structure and constraints of the new puncture from those of the colliding punctures. We can regard this expectation as another test of the correctness of our expressions. Puncture collisions Suppose we have two punctures on a plane, so the Higgs field has two simple poles with X1 +X2 (by the residue theorem applied to the sphere that bubbles off), which corresponds to a new puncture. Generically, X will be mass deformed. The mass deformations are interpreted as VEVs of the scalars in the gauge multiplet associate to the factor in the gauge group which becomes weakly coupled in the collision limit. One can also study this degen Alternatively, one can bypass the Higgs field, and study the collision directly with the (given by their pole structures), and imposing at each pole the constraints of the corresponding puncture. Then, taking the collision limit, the pole structure and constraints of the resulting puncture on the plane arise naturally. As an example, let us see that the collision of two D5 punctures on a plane produces an Sp(2) gauge group, gauged off an A3 puncture. Let us write generic Casimirs for the u2 + zv2 + z(z − x)P2(z) z(z − x) u5 + zv5 + z(z − x)w5 + z2(z − x)P5(z) u6 + zv6 + z(z − x)w6 + z2(z − x)P6(z) z2(z − x)2 z3(z − x)3 z4(z − x)4 z4(z − x)4 z6(z − x)6 where P2(z), P5(z), . . . , P12(z) denote regular functions in z. To solve the constraints at each D5 puncture, we introduce new parameters s4 and t4 of dimension four, and write: u6 = u8 = 3s42, u9 = − 4 u12 = w12 = (3s4t42 + s4w8 + 2t43x), v6 = 2 (t4u2 + s4v2 + t4v2x), v8 = 3(2s4t4 + t42x), v9 = − 4 (t4u5 + s4v5 + t4v5x), v12 = 2 t4(3s42 + 3s4t4x + t42x2), the Casimirs in this limit is: 3(3s4t24 + s4w8) 3(t34 − t4w8 − s4y8) + . . . , where the . . . indicate less singular terms in z. So, u2 and s4 can be interpreted as the VEVs of Coulomb branch parameters (of degree two and four) of the gauge group (which, with a little more work, can be checked to be Sp(2)). In the limit u2, s4 → 0, we obtain the Casimirs for the massless puncture, with pole orders {1, 4, 4, 6, 7, 9}, and with constraints c(79) = Cataloguing fixtures using the superconformal index For the E6 theory, we find 880 fixtures with three regular punctures which correspond SCFTs has a manifest global symmetry group, which is given by the product of the flavor symmetry groups of the three punctures. This global symmetry group may, in general, become enhanced to a larger group. To determine the global symmetry group and number of free hypermultiplets for each of these fixtures, we use the superconformal index [29–33]. The superconformal index of E-type class S theories has not yet been systematically studied. However, since the methods used for A- and D-type theories generalize to any root system, we assume the superconformal index1 for a fixture in the E6 theory takes the usual form resentations of e6. w∈W where R+ denotes the set of positive roots, W the Weyl group, and `(w) the length of the Weyl group element w. flavor symmetry of the ith puncture. atriv denotes the set of fugacities dual to the Cartan of the embedded su(2) of the trivial puncture. • The K-factors are discussed in [32–35]. We will not need their detailed form for our Consider a fixture corresponding to an interacting SCFT, with global symmetry Gglobal, plus free hypermultiplets transforming in a representation R of a flavor symmetry F . Let 1In what follows we will consider the Hall-Littlewood limit of the index [32], which depends on one number of free hypers in the fixture and the global symmetry of the fixture can be read off from the first two non-trivial terms in the Taylor expansion of the index. Schematically, this is given by I = 1 + χFRτ + χaGdfijxt τ 2 + . . . of Gfixt, where both of these representations are viewed as reducible representations of the the Plethystic exponential) and removing the contribution of the free hypermultiplets in (2.5), we arrive at ISCFT = I/Ifree = 1 + χaGdgjlobal τ 2 + . . . from which we can read off the global symmetry of the interacting SCFT. Computing the expansion of the index In (2.4) the term in the sum coming from the trivial representation of e6 gives, to second I = 1 + χaGdmj anifest τ 2 + · · · encoding the manifest global symmetry group. The global symmetry group of the fixture I = 1 + χaGdmj anifest τ 2 + To compute (2.6), we consider each e6 representation in the sum to be a reducible representation of su(2) × f and plug in the corresponding character expansion, where the embedded representations can be obtained using the projection matrices listed in appendix B. Of the 881 fixtures involving three regular punctures, we find that 1 is a free-field fixture, 60 are mixed fixtures and another 134 are interacting fixtures with an enhanced fixtures, the global symmetry group is the manifest one. used the fact that P λ = χλ + O(τ 2). 2Since the theories considered here are all “good” or “ugly” (in the sense of [8]), the lowest possible As an example, consider the fixture come from the sum over the 27, 27, 78, 351, 351, 3510, 3510, and 650 of e6. The expansion of the superconformal index is given by3 in the fundamental representation of E6. The index of these free hypers is given by The index of the underlying SCFT is then ISCFT = I/Ifree Computing the other numerical invariants of the fixture, we find that this is the (E8)12 theory of Minahan and Nemeschansky [36] with 27 additional free hypermultiplets. Levels of enhanced global symmetry groups Since the superconformal index gives the branching rule for the adjoint representation of Gglobal under the subgroup Gmanifest, it most cases it is straightforward to determine the level of each factor in Gglobal from those of Gmanifest: if Hk0 is a subgroup of Gk, then k is given by [12] where IH,→G is the index of the embedding of H in G. There are two cases which require a little more work. The first is when a manifest U(1) becomes enhanced to SU(2). Since we do not know how to assign a level to a U(1) flavor 3For simplicity, we write the dimension to stand for the character of the corresponding representation. The subscript is the U(1) weight. k = symmetry (which would require a precise understanding of how the generator is normalized), we cannot immediately determine the level of the enhanced SU(2) from the index. The second case is when some factor Hk in Gmanifest is embedded diagonally as Hk ,→ Hk1 × Hk2 . Since the only embedding of H in itself has index one, in this case, all we know is that k1 + k2 = k. If any of these remain as factors in Gglobal (that is, if they do not combine with some other factor, with known level, to enhance Gglobal), we cannot determine their levels from sphere for which the SCFT appears in some degeneration, with Hki in the centralizer of subgroup of Gglobal being weakly gauged. Unfortunately, there are a few such fixtures for which no puncture can be gauged (some of these can still be gauged in the twisted sector, which will be discussed in section 4). For these, we do not have a way to determine the levels. In the end, there are two interacting fixtures whose levels we cannot completely determine. Tinkertoys Regular punctures terms in the Laurent expansion of the k-differentials. {1, 4, 5, 7, 8, 11} {1, 4, 5, 7, 8, 10} {1, 4, 5, 7, 7, 10} (E6(a3), Z2) {1, 4, 5, 6, 7, 10} SU(3)24 × SU(2)13 (549, 533) {1, 4, 5, 6, 7, 10} c(1102) = − c(56) 2 + a(56) 2 Sp(2)10 × U(1) c(79) = Spin(7)16 × U(1) SU(3)12 × U(1) SU(2)54 × U(1) c(68) = 3 a(34) 2 c(69) = c(58) = − 4c(46)c(2) + 4c(35)a(3) 1 2 {1,3,4,5,6,8} 2A2+A1 (ns) (D4(a1),S3) {1,3,4,6,6,9} A3 + A1 (ns) (D4(a1),Z2) {1,3,4,6,6,9} c(12) = a(34) 16 a(4) 2 − c(68) 9 9 3 SU(2)9 × U(1) (465,457) {1,3,4,6,6,9} {1,3,4,6,6,9} SU(2)8 × U(1) (408,402) A2 + 2A1 {1,3,4,5,5,7} {1,2,4,4,4,6} c(46) = a(23) 2 c(46) = −8 {1, 2, 3, 4, 4, 6} c(36) = c(48) = 3 a(24) 2 Note that there is a special piece, consisting of three punctures: 2A2 + A1, A3 + A1 and the special puncture D4(a1). For 2A2 + A1, the Sommers-Achar group is the nonabelian group, S3. It acts on a(4), a0(4) as 0 −1 −1 −3 1 −1 −1 3 ! −1 −1 −1 −3 −1 1 −1 3 For A3 + A1, the Sommers-Achar group is the Z2 subgroup of S3, generated by a0(4) → survive as Coulomb branch parameters. Free-field fixtures three punctures. For the free-field fixtures, one of the punctures is an irregular puncture4 (in the sense used in our previous papers), which we denote5 by the pair, (O, Gk), where O is the regular puncture obtained as the OPE of the two regular punctures which collide, and this fixture is attached to the rest of the surface via a cylinder (O, Gk) ←−−−→ O 4Or, in the case of fixture 13, a full puncture, corresponding to the trivial orbit, 0. 5For brevity, we will often omit the level, k, when denoting an irregular puncture. A4 + A1 A3 + A1 2A2 + A1 A2 + 2A1 (A5, SU(2)1) (A4, SU(2)0) (2A2, SU(3)0) (2A2, (G2)4) (A2 + A1, SU(3)0) (2A1, (G2)0) (A2, SU(3)0) (2A1, Spin(7)4) (0, Spin(8)0) (0, Spin(9)4) (0, (F4)12) (0, Spin(10)8) 54 (2, 27) (2A1, SU(4)0) Interacting fixtures with one irregular puncture In the tables below, nd is the number of Coulomb branch parameters of degree d. The total Coulomb branch dimension is P nv = Pd(2d − 1)nd. d nd and the effective number of vector multiplets is (n2,n3,n4,n5,n6,n8,n9,n12) (nh, nv) (0, (F4)12) (0, 0, 0, 0, 1, 0, 0, 0) (0, Spin(10)8) (0, 0, 1, 0, 0, 0, 0, 0) (0, (F4)12) (0, (F4)12) (0, 2, 0, 0, 0, 0, 0, 0) (0, 1, 0, 0, 1, 0, 0, 0) (0, (F4)12) (0, 0, 0, 0, 2, 0, 0, 0) (E8)12 SCFT (E7)8 SCFT [(E6)6 SCFT]2 (E6)12 × SU(2)7 SCFT (F4)12 × SU(2)72 SCFT The (E6)12 × SU(2)7 and (F4)12 × SU(2)72 first appeared in [4], as fixtures in the untwisted D4 theory. Interacting fixtures with enhanced global symmetry (n2,n3,n4,n5,n6,n8,n9,n12) (nh, nv) A4 + A1 (A1, SU(5)2) (0, Spin(10)8) (A1, SU(6)6) (0, 0, 0, 0, 2, 0, 0, 0) [(E8)12 SCFT]2 (0, 0, 0, 0, 1, 0, 0, 1) (E8)24 × SU(2)13 (0, 0, 0, 0, 1, 1, 0, 1) (112, 49) (E7)24 × Spin(7)16 (0, 0, 0, 0, 1, 1, 1, 0) (100, 43) SU(12)18 the SU(2)6 is manifest). The 4-punctured sphere has global symmetry The S-dual (0, SU(6)) SU(6) (E6)24−k × (E6)k × SU(3)122 SCFT F = SU(3)122 × SU(2)24−k × SU(2)k (E6)6 SCFT (G2)10 × SU(3)12 × SU(2)18 × SU(2)6 SCFT Similarly, for (0, SU(6)) SU(6) the global symmetry group is Now there are two S-dual presentations of the theory: empty (E6)18 × (E6)6 × SU(3)12 × U(1) SCFT F = SU(3)12 × SU(2)24−k × SU(2)k × U(1) (G2)10 × SU(2)18 × SU(2)6 × U(1) SCFT ( A˜2, SU(3)) SU(3) (G2)10 × SU(3)12 × SU(2)18 × SU(2)6 SCFT Again, the fact that one of the SU(2) levels is manifest suffices to determine the other. As another example, consider the pair of fixtures In each case, only the diagonal SU(2)54 subgroup, of the indicated SU(2)s, is manifest. Moreover, these fixtures are not gaugeable within the untwisted theory. So there is no obvious way to determine the individual SU(2) levels. Fortunately, the twisted sector provides the empty fixture (D4, SU(3)) which allows us to gauge the SU(3)12 symmetry of each of these fixtures: From the S-duals (D4, SU(3)) SU(3) (D4, SU(3)) SU(3) (E7)18 × U(1) SCFT (E6)6 SCFT Spin(8) (0, Spin(8)) Spin(9) (0, Spin(9)) (E7)18 × U(1) SCFT 1(9) + (E6)6 SCFT and the Lie-algebra embeddings (e7)k ⊃ (f4)k ⊕ su(2)3k (e7)k ⊃ so(9)k ⊕ su(2)2k ⊕ su(2)k (e7)k ⊃ so(8)k ⊕ su(2)k ⊕ su(2)k ⊕ su(2)k Finally, let us turn to the mixed fixture 1(1, 2) + SU(3)12 symmetry of the D4 puncture, as before, we find that the S-dual is a Spin(8) gauge theory, with matter in the 1(8v) + 1(8s) + 1(8c) + 2(1), coupled to two copies of the (E6)6 (0, Spin(8)) Spin(8) (E6)6 SCFT 1(26) + (E6)6 SCFT Applications E6 and F4 gauge theory E6 + 4(27) as the 4-punctured sphere E6 gauge theory, with four fundamental hypermultiplets, is superconformal. It is realized The S-dual theory is an SU(2) gauging of the SU(4)54 × SU(2)7 × U(1) SCFT, with an additional half-hypermultiplet in the fundamental. (A5, SU(2)) SU(4)54 × SU(2)7 × U(1) SCFT The k-differentials, which determine the Seiberg-Witten solution, are (z − z1)(z − z2)(z − z3)(z − z4) (z − z1)(z − z2)(z − z3)4(z − z4)4 u6 z122z344 (dz)6 u8 z122z364 (dz)8 u9 z122z374 (dz)9 (z − z1)2(z − z2)2(z − z3)4(z − z4)4 (z − z1)2(z − z2)2(z − z3)6(z − z4)6 (z − z1)2(z − z2)2(z − z3)7(z − z4)7 u12 z132z394 (dz)12 (z − z1)3(z − z2)3(z − z3)9(z − z4)9 and, for calculational purposes, it is usually convenient to use SL(2, C) to fix (z1, z2, z3, z4) = (0, ∞, f (τ ), 1) in (5.1). found in [18]. (A5, SU(2)) matter in the 12 (2) + 2(1). The S-dual theory is an SU(2) gauging of the Sp(3)26 × SU(2)7 SCFT, with additional 2(26) + 2(1) Sp(3)26 × SU(2)7 SCFT + 2(1) F4 gauge theory, with three fundamentals, is also superconformal. It is realized as of the E6 + 4(27). Adding (E8)12 SCFTs this theory is obtained by Higgsing E6 → F4, using one of the hypermultiplets in the 27. In practice, given the solution to E6 +4(27), the solution to F4 +3(26)+2(1) is obtained by noting that • The Coulomb branch geometry of F4 + 3(26) + 2(1) is the geometry of the fixed-locus Starting with the E6 +4(27) Lagrangian field theory, we can start replacing hypermultiplets the flavour symmetry group of the theory is F = SU(3)142−n × SU(n)54 × SU(2)7 × U(1) SCFT, with an additional half-hypermultiplet in the fundamental (the U(1) is absent for n = 0). 5.2.1 n = 3 With one copy of the (E8)12 SCFT, is dual to (A5, SU(2)) (A5, SU(2)) 1(27) + (E8)12 SCFT SU(3)54 × SU(3)12 × SU(2)7 × U(1) SCFT 5.2.2 n = 2 With two copies of the (E8)12 SCFT, there are two possible realizations. Either 1(27) + (E8)12 SCFT 1(27) + (E8)12 SCFT SU(3)122 × SU(2)54 × SU(2)7 × U(1) SCFT (A5, SU(2)) SU(3)122 × SU(2)54 × SU(2)7 × U(1) SCFT These give two, apparently distinct, realizations of the SU(3)122 × SU(2)54 × SU(2)7 × U(1) n = 1 With three copies of the (E8)12 SCFT, we have 1(27) + (E8)12 SCFT (A5, SU(2)) SU(3)132 × SU(2)7 × U(1) SCFT Finally, the E6 gauging of four copies of the (E8)12 SCFT, is dual to (A5, SU(2)) SU(3)142 × SU(2)7 SCFT Connections with F-theory are, respectively, the (E6)6, (E7)8 and (E8)12 superconformal field theories of Minahan and Nemenschansky [36]. For higher n, the properties of these SCFTs were computed in [39]. The results may be summarized as follows III∗ II∗ Flavour symmetry Graded Coulomb branch dimensions (E6)6n × SU(2)(n−1)(3n+1) n3l = 1, l = 1, 2, . . . , n 3n2 + 14n − 1, n(3n + 2) (E7)8n × SU(2)(n−1)(4n+1) n4l = 1, l = 1, 2, . . . , n 4n2 + 21n − 1, n(4n + 3) (E8)12n × SU(2)(n−1)(6n+1) n6l = 1, l = 1, 2, . . . , n 6n2 + 35n − 1, n(6n + 5) Manifest flavour symmetry III∗ II∗ [n2, n − 1, 1] [n3, n − 1, 1] [(3n)2] [(2n)3] [n5, n − 1, 1] SU(3)62n × SU(2)6n × U(1)2 (E6)6n × SU(2)k + 12 (2) SU(2)8n × SU(4)8n × SU(3)8n × U(1)2 (E7)8n × SU(2)k + 12 (2) SU(2)12n × SU(3)12n × SU(5)12n × U(1)2 (E8)12n × SU(2)k + 12 (2) the predicted level (given that the hypermultiplet transforms as 12 (2) under the SU(2)). of the SU(2). examples of the higher-n theories product SCFT in fixture 39 of section 3.4. It also appeared as an interacting fixture in the D4 theory in [4]. product SCFT in fixture 6 of section 3.4. Further examples can be found in the Z2-twisted sector. Notably, the fixtures Isomorphic theories In our table of interacting fixtures with enhanced global symmetry in section 3.4, we find can be checked by various dualities. Some, however, cannot and we list them below. SU(3)12×SU(2)36×SU(2)18×SU(2)9×U(1)2 SCFT SU(3)122 × SU(2)9 × U(1)3 SCFT SU(3)122 × Sp(2)10 × U(1)3 SCFT It would be nice to check these conjectured isomorphisms by comparing the expansions Acknowledgments We would like to thank S. Katz, D. Morrison, A. Neitzke, R. Plesser and Y. Tachikawa for helpful discussions. J. D. and O. C. would like to thank the Aspen Center for Physics (supported, in part, by the National Science Foundation under Grant PHY-1066293) for their hospitality when this work was initiated. O. C. would further like to thank the Simons Foundation for partial support in Aspen. The work of J. D. and A. T. was supported in part by the National Science Foundation under Grant PHY-1316033. The work of O. C. was supported in part by the INCT-Matem´atica and the ICTP-SAIFR in Brazil through a Capes postdoctoral fellowship. Bala-Carter labels In the twisted and untwisted sectors of the A and D series, punctures were in one-toone correspondence with certain classes of partitions [1, 9, 10, 40]. The partition denotes how the fundamental representation (vector representation, in the case of so(N )) of g6 read off the centralizer, f, of su(2) inside g, as well as the decomposition of the fundamental representation of g under su(2) × f, from the partition (see (2.7) in [9]). The decomposition under su(2) × f for each puncture is precisely the information needed to compute the flavour group levels in section 2.4.1, as well as the expansion of the superconformal index in section 2.5. In what follows, we will explain how these decompositions are obtained for the punctures in the e6 theory. In contrast to classical g, nilpotent orbits in the exceptional Lie algebras, which label our punctures, are not naturally classified by partitions. Here, we recall the classification of Bala and Carter [41, 42], following the exposition in [43]. Their theorem states that there is a one-to-one correspondence between nilpotent orbits in g and (conjugacy classes of) pairs (l, Ol) where l is a Levi subalgebra7 of g and Ol is a distinguished 8 nilpotent orbit in l. By the Jacobson-Morozov theorem, any representative X of Ol embeds in a standard l = k∈Z parabolic subalgebra of l, with explicit Levi decomposition into a Levi subalgebra l0 and A nilpotent orbit in g is then given the label XN (ai), called the Bala-Carter label, where XN is the Cartan type of the semisimple part of l, and i is the number of simple in l, which is always distinguished. There are 16 conjugacy classes of Levi subalgebras of E6. These are specified by their semisimple parts: 0, A1, 2A1, 3A1, A2, A2 + A1, 2A2, A2, A2 + 2A1, A3 + A1, D4, A4, A4 + A1, A5, D5, and E6. Here, kAN denotes the direct sum of k copies of AN . The label 0 denotes the Cartan subalgebra, for which the only distinguished orbit is the zero orbit. For l of classical type, distinguished orbits in l are easily specified in terms of their partition: for l of type A, the only distinguished orbit is the principal orbit (which, for which the partition has no repeated parts. It was found by Bala and Carter that, for l of type G2, F4, E6, E7, and E8, there are 2, 4, 3, 6, and 11 distinguished orbits, respectively. The distinguished orbits in the Levi subalgebras listed above give rise to 21 nilpotent orbits in e6. We list these in the table below, along with the centralizer, f, and the de6For untwisted (twisted) punctures in the A and D series, g is of type A (B) and D (C), respectively. section 3.8 of [43] for an introduction. 8A nilpotent orbit, O, in g is distinguished if and only if the only Levi subalgebra of g, containing O, is 9Any su(2) subalgebra of g is spanned by a standard triple {H, X, Y } of nonzero elements of g satisfying composition of the 27 and 78 of e6 under su(2) × f.10 But, before that, let us give a few examples of how to obtain the decomposition of the 27 for various embeddings. [7,1] and [5,3], corresponding to nilpotent orbits D4 and D4(a1), respectively, in e6. The first has centralizer su(3) and the second, u(1)2. We can obtain the decomposition of the 27 The 27 of e6 decomposes under so(10) × u(1) as The 10 and 16 of so(10) decompose under so(8) × u(1) as e6 ⊃ so(10) × u(1) 27 = 1−4 + 102 + 16−1 so(10) ⊃ so(8) × u(1) 10 = 12 + 1−2 + (8v)0 16 = (8s)1 + (8c)−1 e6 ⊃ so(8) × u(1) × u(1) For D4(a1), we embed su(2) in so(8) by taking so we have so(8) ⊃ su(2) 8v,s,c = 5 + 3 so(8) ⊃ su(2) 8v,s,c = 7 + 1 e6 ⊃ su(2) × u(1) × u(1) For D4, we embed su(2) in so(8) by taking e6 ⊃ su(2) × u(1) × u(1) 10The decomposition of the 27 determines a projection matrix, which can be used to obtain the decompoThe decomposition of the 78 determines the levels of the flavor groups, as described in section 2.4.1. For this embedding, the u(1)2 centralizer enhances to su(3). To see this, we can make a change of basis so that the two u(1) charges are given in terms of the old ones by q10 = q20 = (q1 + q2) (q1 − q2) Then the decomposition becomes e6 ⊃ su(2) × u(1) × u(1) where we recognize these u(1)2 charges as the weights (in the Dynkin basis) of the 6 and 3 of su(3). Thus, the decomposition of the 27 is given by e6 ⊃ su(2) × su(3) 27 = (1, 6) + (7, 3) orbits E6, E6(a1), and E6(a3). The decomposition of the 27 for each of these can be obtained by taking the inner product of the Cartan element H of the embedded su(2) (which can be read off from the weighted Dynkin diagram) into the weight vectors of the 27. We conclude this appendix with a summary of the nilpotent orbits in e6 and the corresponding decompositions of the 27 and the 78 under su(2) × f. (1; 1) + (3; 7) + (5; 1) A2 + 2A1 2A2 + A1 so(7) × u(1) su(3) × su(2) su(3) × su(3) su(3) × u(1) su(2) × u(1) sp(2) × u(1) (1; 15) + (2; 6) (1;72 + 1−4) + (2; 8−1) + (3; 12) (1;6, 1) + (2; 3, 2) + (3; 3, 1) (1;3, 3) + (3; 1, 3) + (3; 3, 1) (1; 32) + (2; 3−1 + 11) + (3; 30 + 1−2) + (4; 11) (1;12 + 1−4) + (2; 4−1) + (3; 32) + (4; 2−1) (1;5−2 + 14) + (4; 41) + (5; 1−2) (1; 1) + (2; 2) + (3; 3) + (4; 2) + (5; 1) (1; 35) + (2; 20) + (3; 1) (1;10 + 210) + (2; 83 + 8−3) + (3; 70 + 10) (1;8, 1) + (1; 1, 3) + (2; 8, 2) + (3; 1, 1) + (3; 8, 1) + (4; 1, 2) (1;8, 1) + (1; 1, 8) + (3; 1, 1) + (3; 3, 3) + (3; 3, 3) + (5; 1, 1) (1; 80 + 10) + (2; 31 + 3−1 + 1−3 + 13) + (3; 3−2 + 32 + 10 + 10) + (4; 31 + 3−1) + (5; 10) (1; 14) + (3; 7 + 1) + (5; 7 + 1) (1;10 + 30) + (2; 43 + 4−3) + (3; 10 + 30 + 50) + (4; 23 + 2−3) (1;100 + 10) + (3; 10) + (4; 43 + 4−3) + (5; 50) + (7; 10) (1;3) + (2; 4 + 2) + (3; 3 + 1 + 1) + (4; 2 + 2) + (5; 3 + 1) + (6; 2) su(2) × u(1) u(1) × u(1) su(2) × u(1) (1;14 + 1−2) + (2; 2−2) + (3; 11) + (4; 21) + (5; 11 + 1−2) 12,2 + 10,−4 + 1−2,2 + 31,−1 + 30,2 + 3−1,−1 + 51,−1 + 50,2 + 5−1,−1 (1;2−5) + (3; 1−2) + (5; 21 + 14) + (7; 1−2) (1; 6) + (7; 3) 2−5 + 3−2 + 41 + 54 + 61 + 7−2 1−4 + 2−1 + 32 + 6−1 + 72 + 8−1 (1; 1) + (5; 1) + (6; 2) + (9; 1) 1 + 5 + 5 + 7 + 9 12 + 1−4 + 5−1 + 92 + 11−1 5 + 9 + 13 1 + 9 + 17 (1;10 + 30) + (2; 2)0 + (3; 13 + 1−3 + 10 + 10) + (4; 23 + 2−3 + 20) + (5; 13 + 10 + 1−3) + (6; 2)0 + (7; 10) 10,0 + 10,0 + 30,0 + 32,0 + 31,3 + 31,−3 + 30,0 + 3−2,0 + 3−1,−3 + 3−1,3 + 30,0 + 52,0 + 51,3 + 51,−3 + 50,0 + 5−2,0 + 5−1,−3 + 5−1,3 + 70,0 + 70,0 (1;30 + 10) + (3; 23 + 2−3 + 10) + (5; 16 + 10 + 1−6) + (7; 23 + 2−3 + 10) + (9; 10) (1; 8) + (3; 1) + (7; 8) + (11; 1) 10 + 23 + 2−3 + 30 + 30 + 43 + 4−3 + 56 + 50 + 5−6 + 6−3 + 63 + 70 + 8−3 + 83 + 90 10 + 23 + 2−3 + 30 + 30 + 50 + 63 + 6−3 + 70 + 70 + 83 + 8−3 + 90 + 110 (1; 3) + (3; 1) + (4; 2) + (5; 1) + (6; 2) + (7; 1) + (9; 1) + (10; 2) + (11; 1) 3 + 3 + 3 + 5 + 5 + 5 + 7 + 7 + 9 + 9 + 11 + 11 10 + 30 + 53 + 5−3 + 70 + 90 + 113 + 110 + 11−3 + 150 3 + 5 + 7 + 9 + 11 + 11 + 15 + 17 3 + 9 + 11 + 15 + 17 + 23 Projection matrices Our classification of interacting and mixed fixtures using the superconformal index, carried out in section 2.5, required that we know the decomposition of a number of higherdimensional e6 representations (and not just the 27 and the 78) under su(2) × f. These are trivial to obtain using LieART [44], provided we know a projection matrix for each embedding [44, 45]. From the decomposition of the 27, listed in the table above, one obtains a projection matrix simply by defining a 6 × rk (su(2) × f) matrix, M , such that the LieART command gives the corresponding su(2) × f weights. This projection matrix can then be used to obtain the decomposition of any e6 irrep under su(2) × f. Below, we list a projection matrix for each embedding, following the conventions of su(3)×su(3)  0 0 0 0 1 0  −1 −2 −3 −2 −1 −2 A2+A1 su(3)×u(1)  1 1 −1 −1 −1 −1 2A2 + A1 −2 −1 0 1 2 0 A3 + A1 A4 + A1 −2 −4 −6 −2 2 −3 −1 −2 0 2 1 0 8 14 18 14 8 8 −1 −2 0 2 1 0 12 22 30 22 12 16 16 30 42 30 16 22 As an example, let’s work out the decomposition of the 51975 for the orbit 2A2. Running LieART, we obtain the decomposition with the following two lines of code: + 4(11, 1) + 25(7, 7) + 9(1, 14) + 17(9, 7) + 16(3, 14) + 6(11, 7) + 22(5, 14) + 2(13, 7) + 15(7, 14) + 10(9, 14) + 3(11, 14) + (13, 14) + 6(1, 27) + 25(3, 27) + 23(5, 27) + 21(7, 27) + 9(9, 27) + 4(11, 27) + 5(1, 64) + 12(3, 64) + 13(5, 64) + 9(7, 64) + 4(9, 64) + (11, 64) + 4(1, 77) + 6(3, 77) + 2(3, 770) + 8(5, 77) + (5, 770) + 4(7, 77) + (7, 770) + 2(9, 77) + (3, 182) + (1, 189) + 2(3, 189) + 2(5, 189) + (7, 189) This works for all of the orbits above, except for D4(a1), as the LieART command “DecomposeIrrep” does not seem to work when the target subalgebra has more than one u(1) factor. In this case, getting the decomposition is only slightly more complicated. 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Oscar Chacaltana, Jacques Distler, Anderson Trimm. Tinkertoys for the E 6 theory, Journal of High Energy Physics, 2015, 7, DOI: 10.1007/JHEP09(2015)007