Quivers, YBE and 3-manifolds

TL;DR

This paper explores the deep connections between 4d superconformal indices of quiver gauge theories, integrable systems, and hyperbolic 3-manifolds, revealing new dualities and geometric interpretations.

Contribution

It introduces the double Yang-Baxter move as a gauge theory duality, linking it to Yang-Baxter equations and hyperbolic geometry, and relates indices to 3-manifold volumes and string partition functions.

Findings

Invariance of superconformal index under Seiberg duality expressed as Yang-Baxter equation.

Saddle point of the index integral reproduces hyperbolic volume of 3-manifolds.

Connection between gauge theory indices, hyperbolic geometry, and topological string theory.

Abstract

We study 4d superconformal indices for a large class of N=1 superconformal quiver gauge theories realized combinatorially as a bipartite graph or a set of "zig-zag paths" on a two-dimensional torus T^2. An exchange of loops, which we call a "double Yang-Baxter move", gives the Seiberg duality of the gauge theory, and the invariance of the index under the duality is translated into the Yang-Baxter-type equation of a spin system defined on a "Z-invariant" lattice on T^2. When we compactify the gauge theory to 3d, Higgs the theory and then compactify further to 2d, the superconformal index reduces to an integral of quantum/classical dilogarithm functions. The saddle point of this integral unexpectedly reproduces the hyperbolic volume of a hyperbolic 3-manifold. The 3-manifold is obtained by gluing hyperbolic ideal polyhedra in H^3, each of which could be thought of as a 3d lift of the faces of the 2d bipartite graph.The same quantity is also related with the thermodynamic limit of the BPS partition function, or equivalently the genus 0 topological string partition function, on a toric Calabi-Yau manifold dual to quiver gauge theories. We also comment on brane realization of our theories. This paper is a companion to another paper summarizing the results.