Emergent 3-manifolds from 4d Superconformal Indices

TL;DR

This paper reveals how hyperbolic 3-manifolds can be derived from 4d superconformal indices via a process called dimensional oxidation, linking supersymmetric gauge theories to geometric structures.

Contribution

It establishes a novel connection between 4d superconformal indices and hyperbolic 3-manifolds through a dimensional reduction and Higgsing process, bridging gauge theories and geometric topology.

Findings

Hyperbolic 3-manifolds emerge from 4d superconformal indices.

Dimensional oxidation corresponds to gauge theory reduction from 4d to 3d.

Equivalence between 4d superconformal index and classical spin chain partition function.

Abstract

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 T^2 and 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 the Higgsing of the 4d gauge theory.