Emergent 3-manifolds from four dimensional superconformal indices

We show that the smooth geometry of a hyperbolic 3-manifold emerges from a classical spin system defined on a 2D discrete lattice, and moreover, show that the process of this "dimensional oxidation" is equivalent with the dimensional reduction of a supersymmetric gauge theory from 4D to 3D. More concretely, we propose an equality between (1) the 4D superconformal index of a 4D N=1 superconformal quiver gauge theory described by a bipartite graph on T2 and (2) the partition function of a classical integrable spin chain on T2. The 2D spin system is lifted to a hyperbolic 3-manifold after the dimensional reduction and using the Higgs mechanism in the 4D gauge theory.