Four-dimensional superconformal index reloaded

We consider the four-dimensional $\mathcal N\ge 1$ superconformal index and its generalization to the lens space. We discuss reductions of the latter to the three-dimensional $\mathcal N\ge 2$ sphere partition function, the three-dimensional $\mathcal N\ge 2$ superconformal index, and the two-dimensional $\mathcal{N}\ge(2,2)$ sphere partition function. We apply these reductions to a class of four-dimensional $\mathcal N=1$ 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 three-dimensional supersymmetric gauge theories.