Four-dimensional superconformal index reloaded

Abstract

We consider the four-dimensional N≥1 superconformal index and its generalization to the lens space. We discuss reductions of the latter to the three-dimensional N≥2 sphere partition function, the threedimensional N≥2 superconformal index, and the two-dimensional N≥(2, 2) sphere partition function. We apply these reductions to a class of four-dimensional N=1 superconformal field theories dual to toric Calabi-Yau manifolds, and we find surprising connections with integrable spin chains and hyperbolic geometry. We comment on the problem of classifying infrared fixed points of four-dimensional and threedimensional supersymmetric gauge theories.