Abstract
We study 4d superconformal indices for a large class of \( \mathcal{N} = 1 \) 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 \( {\mathbb{H}^3} \), 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 [1].
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Yamazaki, M. Quivers, YBE and 3-manifolds. J. High Energ. Phys. 2012, 147 (2012). https://doi.org/10.1007/JHEP05(2012)147
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DOI: https://doi.org/10.1007/JHEP05(2012)147