Spin chains for ADE quiver theories

The spectral problem of four-dimensional superconformal quiver gauge theories can be mapped to one-dimensional spin chains with restricted Hilbert spaces, where the composition of neighbouring spins follows the path algebra of the quiver. To better understand such spin chains, we compute the one-loop planar dilatation operator for the 4d \mathcal{N}=2 𝒩 = 2 ADE quiver gauge theories obtained by orbifolding the \mathcal{N}=4 𝒩 = 4 Super-Yang-Mills theory and marginally deforming by independently varying the gauge couplings. This extends previous work which was mainly focused on the \mathbb{Z}_2 ℤ 2 quiver. We characterise the general features of the resulting ADE spin-chain models and construct the 2-magnon Bethe Ansatz for holomorphic states. We also evaluate, at large N N , the \mathcal{N}=2 𝒩 = 2 superconformal index of these gauge theories and use it to study their protected spectrum in specific sectors.