On Three Dimensional Quiver Gauge Theories and Integrability

Abstract

In this work we compare different descriptions of the space of vacua of certain three dimensional N=4 superconformal field theories, compactified on a circle and mass-deformed to N=2 in a canonical way. The original N=4 theories are known to admit two distinct mirror descriptions as linear quiver gauge theories, and many more descriptions which involve the compactification on a segment of four-dimensional N=4 super Yang-Mills theory. Each description gives a distinct presentation of the moduli space of vacua. Our main result is to establish the precise dictionary between these presentations. We also study the relationship between this gauge theory problem and integrable systems. The space of vacua in the linear quiver gauge theory description is related by Nekrasov-Shatashvili duality to the eigenvalues of quantum integrable spin chain Hamiltonians. The space of vacua in the four-dimensional gauge theory description is related to the solution of certain integrable classical many-body problems. Thus we obtain numerous dualities between these integrable models.

58 pages, 13 figures, references added

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• Surface Defects and Resolvents
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• Spectrum of Quantum Transfer Matrices via Classical Many-Body Systems
• Gauge/Liouville Triality
• The Coulomb Branch of 3d $\mathcal{N}=4$ Theories
• The Infrared Physics of Bad Theories
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• A slow review of the AGT correspondence
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• Three-Dimensional Mirror Self-Symmetry of the Cotangent Bundle of the Full Flag Variety
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• 3d Mirror Symmetry and Elliptic Stable Envelopes
• Twisted M2 brane holography and sphere correlation functions
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• Quantum K theory of symplectic Grassmannians
• Higgsed network calculus
• The Quantum DELL System
• $q$-Virasoro modular double and 3d partition functions
• Trigonometric version of quantum-classical duality in integrable systems
• The Hasse Diagram of the Moduli Space of Instantons
• Quantum K-theory of Quiver Varieties and Many-Body Systems
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• Equivariant Verlinde formula from fivebranes and vortices
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• Good IR Duals of Bad Quiver Theories
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• New compact forms of the trigonometric Ruijsenaars-Schneider system
• Mirror Symmetry in Three Dimensions via Gauged Linear Quivers
• Refined 3d-3d Correspondence
• Bethe/Gauge correspondence in odd dimension: modular double, non-perturbative corrections and open topological strings
• Supersymmetric quantum spin chains and classical integrable systems
• 3d Mirror Symmetry and Quantum $K$-theory of Hypertoric Varieties
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• Connecting Mirror Symmetry in 3d and 2d via Localization
• Quantum Hydrodynamics from Large-n Supersymmetric Gauge Theories
• 3d Mirror Symmetry for Instanton Moduli Spaces
• Dualities in quantum integrable many-body systems and integrable probabilities -- I
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• Ruijsenaars-Schneider three-body models with N=2 supersymmetry
• The Master T-Operator for Inhomogeneous XXX Spin Chain and mKP Hierarchy
• On the Integrability of Four Dimensional N=2 Gauge Theories in the Omega Background
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• Quiver ${\text{W}}_{{\epsilon }_1,{\epsilon }_2}$ algebras of 4d $\mathcal{N}=2$ gauge theories
• Supersymmetric extension of qKZ-Ruijsenaars correspondence
• Toroidal q-Opers
• Bethe-State Counting and the Witten Index
• Nested coordinate Bethe wavefunctions from the Bethe/gauge correspondence
• BPS states in the Omega-background and torus knots
• Condensates and instanton - torus knot duality. Hidden Physics at UV scale
• On Supersymmetric Interface Defects, Brane Parallel Transport, Order-Disorder Transition and Homological Mirror Symmetry
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• Generalized Devil's staircase and RG flows
• di-Langlands correspondence and extended observables
• Parallel surface defects, Hecke operators, and quantum Hitchin system
• RG limit cycles
• Bispectral duality and separation of variables from surface defect transition
• Quantum-classical duality for Gaudin magnets with boundary
• The Zoo of Opers and Dualities
• Quantum spin chains and integrable many-body systems of classical mechanics
• Solution of quantum integrable systems from quiver gauge theories
• Holographic Duality for three-dimensional Super-conformal Field Theories
• Non-Perturbative Schwinger-Dyson Equations for 3d $\mathcal{N}=4$ Gauge Theories
• Quantum-classical correspondence for gl(1|1) supersymmetric Gaudin magnet with boundary
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• Vacua, Symmetries, and Higgsing of Chern-Simons Matter Theories
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