[Paper Review] Product SCFTs for the $E_7$ Theory

Jacques Distler, Behzat Ergun|arXiv (Cornell University)|Mar 6, 2018

Homotopy and Cohomology in Algebraic Topology16 references3 citations

TL;DR

This paper introduces a unitarity-based criterion to identify product superconformal field theories (SCFTs) in class-S constructions of $E_7$ theories. By checking violations of the unitarity bound on levels and central charges, the authors identify 29 out of 11,000 3-punctured sphere fixtures as product SCFTs, providing a computationally efficient and conceptually insightful method that recovers known cases and reveals new ones, particularly involving Minahan-Nemeschansky theories as factors.

ABSTRACT

We present a simple criterion for when an N=2 SCFT must be a product SCFT. Applied to the class-S theories of type $E_7$, we find 29 (out of 11,000) 3-punctured spheres which are product SCFTs.

Motivation & Objective

• To develop a simple, computationally efficient criterion to determine when an $ N=2$ SCFT is a product theory.
• To classify isolated SCFTs arising from 3-punctured spheres in $E_7$ class-S theories, particularly identifying those that are product SCFTs.
• To provide a conceptual explanation for the frequent appearance of Minahan-Nemeschansky theories as factors in product SCFTs via unitarity bounds.
• To extend prior work based on Hall-Littlewood indices by replacing computationally intensive index calculations with a more fundamental criterion rooted in unitarity.

Proposed method

• The authors derive a sufficient condition for a theory to be a product SCFT based on the unitarity bound: $ k_i \geq \frac{24\kappa_{F_i}c}{\dim(F_i)+12c} $, where $k_i$ is the level and $c$ the central charge of a global symmetry factor $F_i$.
• Violation of this bound implies the theory cannot be an irreducible SCFT and must instead be a product of lower-rank SCFTs with reduced $c$ and properly bounded levels.
• The method focuses on fixtures with enhanced global symmetries, as these are most likely to violate the unitarity bound if they are product theories.
• The authors apply the criterion to all 11,000 good 3-punctured sphere fixtures of the $E_7$ theory, systematically identifying 29 that are product SCFTs.
• The analysis explains why Minahan-Nemeschansky theories — which saturate the unitarity bound — frequently appear as factors in such products.
• The method is validated by recovering all previously known product SCFTs and identifying new ones, including cases with multiple $E_7$ and $E_8$ Minahan-Nemeschansky factors.

Experimental results

Research questions

• RQ1 Which 3-punctured sphere fixtures in the $E_7$ class-S theory are product SCFTs, and what criteria can reliably identify them?
• RQ2 Why do Minahan-Nemeschansky theories of rank 1 consistently appear as factors in product SCFTs?
• RQ3 Can a unitarity-based criterion replace computationally expensive Hall-Littlewood index calculations for detecting product structure in SCFTs?
• RQ4 What is the role of the Sommers-Achar group action in transforming product SCFTs with multiple rank-1 factors into higher-rank ones?
• RQ5 How many of the 11,000 $E_7$ fixtures are actually product SCFTs, and what is their global symmetry and central charge structure?

Key findings

• The authors identify exactly 29 product SCFTs among the 11,000 3-punctured sphere fixtures of the $E_7$ theory.
• All previously known product SCFTs are recovered by the unitarity criterion, confirming its validity as a sufficient condition.
• The criterion successfully identifies new product SCFTs, including those with multiple $E_7$ and $E_8$ Minahan-Nemeschansky factors.
• The rank-3 $ (E_7)_{18} \times U(1) $ SCFT is confirmed as a product theory, consistent with prior classifications.
• Fixtures containing $ (E_7)_8 $, $ (E_8)_{12} $, or $ (E_7)_{24} \times SU(2)_{26} $ factors are shown to be product SCFTs due to unitarity constraints.
• The presence of 28 free hypermultiplets transforming as $ \frac{1}{2}(56) $ of $ E_7 $ in fixtures #19–21 is confirmed, with the interacting part being a product of $ E_8 $ and $ E_7 $ Minahan-Nemeschansky theories.