Abstract
We classify the class S theories of type E7. These are four-dimensional mathcal{N}=2 superconformal field theories arising from the compactification of the E7 (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 E7 gauge theory with 3 hypermultiplets in the 56. Using our results, we also verify recent conjectures that the T2 compactification of certain 6d (1, 0) theories can alternatively be realized in class S as fixtures in the E7 or E8 theories.
Highlights
As the 6d (2, 0) theories have an ADE classification, the corresponding four-dimensional theories resulting from their compactification come in ADE type
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
For all but one of the free-field fixtures, one of the punctures is an irregular puncture, which we denote by the pair (O, G), where O is the regular puncture obtained as the OPE of the two regular punctures which collide
Summary
The Coulomb branch geometry for our theories can be realized either by studying parabolic Hitchin systems on the punctured Riemann surface, C, or by studying the Calabi-Yau integrable system for a certain family of non-compact Calabi-Yaus fibered over C. where (for definiteness) the determinant is taken in the adjoint representation and λ is the Seiberg-Witten differential. Xu = 0 = −w2 − x3 + 16xy3 + φ2(z)y4 + φ6(z)y3 + φ8(z)xy + φ10(z)y2 + φ12(z)x + φ14(z)y + φ18(z) ⊂ Tot(KC9 ⊕ KC6 ⊕ KC4 ) In both cases, the Seiberg-Witten geometry is expressed in terms of meromorphic kdifferentials, φk(z), on C, which have poles of various orders at the punctures [24]. One finds an elaborate set of constraints on the coefficients of the polar parts of the φk(z) =. When the special piece of ON has more than one element, we have an additional quotient by a finite group (the “Sommers-Achar group”) acting on the coefficients [26]
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