Quivers, YBE and 3-manifolds

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.

61 pages, 16 figures; v2: typos corrected

References in corpus (50)

• Electric-Magnetic Duality in Supersymmetric Non-Abelian Gauge Theories
• Liouville Correlation Functions from Four-dimensional Gauge Theories
• Superconformal Field Theory on Threebranes at a Calabi-Yau Singularity
• An Index for 4 dimensional Super Conformal Theories
• Exact Results for Wilson Loops in Superconformal Chern-Simons Theories with Matter
• The Exact Superconformal R-Symmetry Maximizes a
• Counting chiral primaries in N=1 d=4 superconformal field theories
• Rigid Supersymmetric Theories in Curved Superspace
• SUSY Gauge Theories on Squashed Three-Spheres
• Gauge Theories from Toric Geometry and Brane Tilings
• Notes on SUSY Gauge Theories on Three-Sphere
• Supersymmetric vacua and Bethe ansatz
• Discrete Heisenberg-Weyl Group and Modular Group
• Quantum Dilogarithm
• A variational principle for domino tilings
• Applications of the Superconformal Index for Protected Operators and q-Hypergeometric Identities to N=1 Dual Theories
• Brane Dimers and Quiver Gauge Theories
• Vortex Counting and Lagrangian 3-manifolds
• Quantization of Integrable Systems and Four Dimensional Gauge Theories
• The 4d Superconformal Index from q-deformed 2d Yang-Mills
• Dilogarithm identities
• S-duality and 2d Topological QFT
• Strongly coupled quantum discrete Liouville theory. I: Algebraic approach and duality
• Brane Tilings
• SL(2,R) Chern-Simons, Liouville, and Gauge Theory on Duality Walls
• From 4d superconformal indices to 3d partition functions
• Quivers, Tilings, Branes and Rhombi
• Abelian Current Algebra and the Virasoro Algebra on the Lattice
• Elliptic hypergeometry of supersymmetric dualities
• Non-commutative Donaldson-Thomas theory and the conifold
• Brane Tilings and Their Applications
• The Laplacian and $\bar{\partial }$ operators on critical planar graphs
• Relation between the 4d superconformal index and the S^3 partition function
• Crystal Melting and Toric Calabi-Yau Manifolds
• Faddeev-Volkov solution of the Yang-Baxter Equation and Discrete Conformal Symmetry
• Properly ordered dimers, $R$-charges, and an efficient inverse algorithm
• On the Superconformal Index of N=1 IR Fixed Points: A Holographic Check
• 3d Partition Function as Overlap of Wavefunctions
• Wall Crossing of BPS States on the Conifold from Seiberg Duality and Pyramid Partitions
• Wall Crossing and M-theory
• Elliptic gamma-function and multi-spin solutions of the Yang-Baxter equation
• Emergent Calabi-Yau Geometry
• Deformations of Toric Singularities and Fractional Branes
• Crystal Melting and Wall Crossing Phenomena
• Exactly marginal deformations of quiver gauge theories as seen from brane tilings
• Wall Crossing As Seen By Matrix Models
• Global symmetries and 't Hooft anomalies in brane tilings
• Dimer Models, Integrable Systems and Quantum Teichmuller Space
• A note on dimer models and McKay quivers
• BPS states, crystals and matrices

Cited by in corpus (42)

• Exact Results in D=2 Supersymmetric Gauge Theories
• 3d Chern-Simons Theory from M5-branes
• Brane Brick Models and 2d (0,2) Triality
• High-Temperature Expansion of Supersymmetric Partition Functions
• Aspects of Defects in 3d-3d Correspondence
• Emergent 3-manifolds from 4d Superconformal Indices
• Network and Seiberg Duality
• A slow review of the AGT correspondence
• New solutions of the star-triangle relation with discrete and continuous spin variables
• The star-triangle relation, lens partition function, and hypergeometric sum/integrals
• Surface defects and elliptic quantum groups
• Integrable Amplitude Deformations for N=4 Super Yang-Mills and ABJM Theory
• New N=1 dualities from orientifold transitions - Part I: Field Theory
• Elliptic hypergeometric sum/integral transformations and supersymmetric lens index
• Quiver gauge theories and integrable lattice models
• A new solution of the star-triangle relation
• Comment on star-star relations in statistical mechanics and elliptic gamma-function identities
• Integrability from 2d N=(2,2) Dualities
• Quasi-classical expansion of the star-triangle relation and integrable systems on quad-graphs
• String Theory Origin of Bipartite SCFTs
• Surface defects as transfer matrices
• Integrable quad equations derived from the quantum Yang-Baxter equation
• Integrable lattice spin models from supersymmetric dualities
• T-Reflection
• Elliptic modular double and 4d partition functions
• Lens elliptic gamma function solution of the Yang-Baxter equation at roots of unity
• Branes and integrable lattice models
• Integrable 3D lattice model in M-theory
• Gauging and Decoupling in 3d $\mathcal{N}=2$ dualities
• Extended Z-invariance for integrable vector and face models and multi-component integrable quad equations
• Quantum Dilogarithm Identities at Root of Unity
• Entanglement in Theory Space
• Hyperbolic and trigonometric hypergeometric solutions to the star-star equation
• Exactly solved models on planar graphs with vertices in ${\mathbb{Z}}^3$
• Mirror Symmetry and Mixed Chern-Simons Levels for Abelian 3d $\mathcal{N}=2$ theories
• Mahler Measure for a Quiver Symphony
• On Bailey pairs for $\mathcal{N}=2$ supersymmetric gauge theories on $S_b^3/{\mathbb{Z}}_r$
• Cluster-Enriched Yang-Baxter Equation from SUSY Gauge Theories
• Thermodynamics of Isoradial Quivers and Hyperbolic 3-Manifolds
• Disk, interval, point: on constructions of quantum field theories with holomorphic action functionals
• Quantum Trilogy: Discrete Toda, Y-System and Chaos
• Tetrahedron duality