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 $\mathcal{N}=1$ superconformal quiver gauge theory described by a bipartite graph on $T^2$ and (2) the partition function of a classical integrable spin chain on $T^2$. 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.

• Received 30 March 2012

DOI:https://doi.org/10.1103/PhysRevLett.109.091602