Abstract:We study 4D N = 2 superconformal field theories that arise from the compactification of 6D N = (2, 0) theories of type D N on a Riemann surface, in the presence of punctures twisted by a Z 2 outer automorphism. Unlike the untwisted case, the family of SCFTs is in general parametrized, not by M g,n , but by a branched cover thereof. The classification of these SCFTs is carried out explicitly in the case of the D 4 theory, in terms of three-punctured spheres and cylinders, and we provide tables of properties of twisted punctures for the D 5 and D 6 theories. We find realizations of Spin (8) and Spin(7) gauge theories with matter in all combinations of vector and spinor representations with vanishing β-function, as well as Sp(3) gauge theories with matter in the 3-index traceless antisymmetric representation.
We classify the class S theories of type E 7. These are four-dimensional N = 2 superconformal field theories arising from the compactification of the E 7 (2, 0) theory on a punctured Riemann surface, C. The classification is given by listing all 3-punctured spheres ("fixtures"), and connecting cylinders, which can arise in a pants-decomposition of C. We find exactly 11,000 fixtures with three regular punctures, and an additional 48 with one "irregular puncture" (in the sense used in our previous works). To organize this large number of theories, we have created a web application at https://golem.ph.utexas.edu/class-S/E7/. Among these theories, we find 10 new ones with a simple exceptional global symmetry group, as well as a new rank-2 SCFT and several new rank-3 SCFTs. As an application, we study the strong-coupling limit of the E 7 gauge theory with 3 hypermultiplets in the 56. Using our results, we also verify recent conjectures that the T 2 compactification of certain 6d (1, 0) theories can alternatively be realized in class S as fixtures in the E 7 or E 8 theories.
We study 4D N = 2 superconformal field theories that arise as the compactification of the six-dimensional (2, 0) theory of type E 6 on a punctured Riemann surface in the presence of Z 2 outer-automorphism twists. We explicitly carry out the classification of these theories in terms of three-punctured spheres and cylinders, and provide tables of properties of the Z 2 -twisted punctures. An expression is given for the superconformal index of a fixture with twisted punctures of type E 6 , which we use to check our identifications. Several of our fixtures have Higgs branches which are isomorphic to instanton moduli spaces, and we find that S-dualities involving these fixtures imply interesting isomorphisms between hyperKähler quotients of these spaces. Additionally, we find families of fixtures for which the Sommers-Achar group, which was previously a Coulomb branch concept, acts non-trivially on the Higgs branch operators.
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 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.
N = 2 supersymmetric Spin(n) gauge theory admits hypermultiplets in spinor representations of the gauge group, compatible with β ≤ 0, for n ≤ 14. The theories with β < 0 can be obtained as mass-deformations of the β = 0 theories, so it is of greatest interest to construct the β = 0 theories. In previous works, we discussed the n ≤ 8 theories. Here, we turn to the 9 ≤ n ≤ 14 cases. By compactifying the D N (2,0) theory on a 4-punctured sphere, we find Seiberg-Witten solutions to almost all of the remaining cases. There are five theories, however, which do not seem to admit a realization from six dimensions.
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