A "Lagrangian" for the E7 Superconformal Theory

TL;DR

The paper constructs an ${ m N}=1$ gauge-theory Lagrangian that flows to the rank-one ${E_7}$ ${ m N}=2$ SCFT, by first realizing the ${R}_{0,N}$ theory and then performing a partial Higgsing to obtain ${E_7}$. It provides a detailed computation of the full superconformal index for ${R}_{0,N}$ and ${E_7}$ via Spiridonov-Warnaar inversion and Higgsing, and confirms consistency with known chiral ring data and representation content, including Hall-Littlewood and Schur limits. Additionally, it analyzes twisted dimensional reduction to ${ m N}=(0,4)$ in two dimensions to study the ${E_7}$ one-instanton string, deriving its elliptic genus through localization and 2d dualities, and verifying the 1-instanton Hilbert series in the ${E_7}$ case. The results provide concrete, computable handles on non-Lagrangian ${E_7}$ theories, enabling quantitative checks of their protected spectra and offering a pathway to study higher-rank instanton strings and potential generalizations to ${E_8}$ and beyond. The approach mirrors prior ${E_6}$ work while highlighting both its power and its limitations, notably the absence of a manifest ${SU(2)}_s$ gauge in the electric frame yet achieving robust, cross-checked invariants.

Abstract

We find an N=1 gauge theory that flows to the rank-one N=2 superconformal field theory with $E_7$ flavor symmetry. We first obtain a Lagrangian description for the $R_{0, N}$ theory, which appears in the S-dual description of the SU(N) gauge theory with 2N fundamental hypermultiplets. This is a straightforward generalization of the proposed Lagrangian description for the $E_6$ theory. The $E_7$ theory is then obtained via partial Higgsing of the $R_{0, 4}$ theory. From this Lagrangian description, we compute the full superconformal index. We also consider twisted dimensional reduction on $S^2$ to obtain N=(0, 4) theory for the $E_7$ one instanton string and compute its elliptic genus.