A “Lagrangian” for the E 7 superconformal theory

We find an $ \mathcal{N}=1 $ gauge theory that flows to the rank-one $ \mathcal{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 $ \mathcal{N}=\left(0,4\right) $ theory for the E$_{7}$ one instanton string and compute its elliptic genus.