Quivers, YBE and 3-manifolds

TL;DR

The paper develops a bridge between 4d $\mathcal{N}=1$ quiver gauge theories, integrable spin systems, and hyperbolic geometry by encoding quivers via zig-zag paths on $T^2$ and showing that Seiberg dualities materialize as double Yang–Baxter moves. It shows that the 4d superconformal index equals a $T^2$ spin-system partition function whose invariance under dualities mirrors YB equations, and, after dimensional reduction, identifies the saddle point with the hyperbolic volume of a 3-manifold assembled from ideal polyhedra. The authors then relate this volume to the twisted superpotential in 2d and to the topological string prepotential on a dual toric Calabi–Yau, via circle patterns and dimer (BPS) counting, establishing a deep correspondence among gauge theory dynamics, integrable models, 3-manifold topology, and topological strings. This framework provides a new lens on dualities, holography, and BPS counting, with potential extensions to elliptic structures and categorified theories. The work highlights an intricate web connecting Seiberg duality, YB integrable systems on $T^2$, hyperbolic geometry, and topological strings in toric geometries, suggesting rich avenues for further exploration in mathematical physics.

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.

Figures (16)

• Figure 1: Logical structure of this paper. Clearly it is impossible to list all the connections between all the ingredients mentioned here. The main claims of this paper are the two equalities represented in the center of this figure.
• Figure 2: Genericity condition, stating that no three zig-zag paths intersects at a single point. The left figure is allowed, whereas the right is not.
• Figure 3: Admissible (left) and non-admissible (right) configuration of zig-zag paths. Rather than coloring the faces by black and white, we have represented the coloring by placing black and white dots inside (it is hard to represent the white color on a white paper!). We see from this example that moving a zig-zag path across an intersection point breaks the admissibility condition.
• Figure 4: Minimality condition forbids two types of intersections of zig-zag paths. The graph superimposed on it is the bipartite graph $\mathcal{G}^*$.
• Figure 5: A zig-zag path on the bipartite graph $\mathcal{G}^*$ (dotted path) is identified with the zig-zag path defined previously (undotted arrow).