Crystal bases and three-dimensional $\mathcal{N}=4$ Coulomb branches

Abstract

We establish and develop a correspondence between certain crystal bases (Kashiwara crystals) and the Coulomb branch of three-dimensional $\mathcal{N}=4$ gauge theories. The result holds for simply-laced, non-simply laced and affine quivers. Two equivalent derivations are given in the non-simply laced case, either by application of the axiomatic rules or by folding a simply-laced quiver. We also study the effect of turning on real masses and the ensuing simplification of the crystal. We present a multitude of explicit examples of the equivalence. Finally, we put forward a correspondence between infinite crystals and Hilbert spaces of theories with isolated vacua.

arXiv commentii+60 pages, many figures. v2: minor improvements. v3: clarifications and references added, published version

References in corpus (38)

• S-Duality of Boundary Conditions In N=4 Super Yang-Mills Theory
• Mirror Symmetry in Three Dimensional Gauge Theories
• Symplectic singularities
• Monopole operators and Hilbert series of Coulomb branches of 3d N = 4 gauge theories
• Nonperturbative Tests of Three-Dimensional Dualities
• Towards a mathematical definition of Coulomb branches of $3$-dimensional $\mathcal{N}=4$ gauge theories, II
• The Hilbert Series of the One Instanton Moduli Space
• Coulomb and Higgs Branches from Canonical Singularities: Part 0
• The Higgs Mechanism -- Hasse Diagrams for Symplectic Singularities
• Deformation quantization and superconformal symmetry in three dimensions
• Tropical Geometry and Five Dimensional Higgs Branches at Infinite Coupling
• 3d Coulomb branch and 5d Higgs branch at infinite coupling
• Quiver Subtractions
• Coulomb Branch Operators and Mirror Symmetry in Three Dimensions
• 3d Partition Function as Overlap of Wavefunctions
• 3D-partition functions on the sphere: exact evaluation and mirror symmetry
• Branes and the Kraft-Procesi Transition
• Quiver Theories for Moduli Spaces of Classical Group Nilpotent Orbits
• (Symplectic) Leaves and (5d Higgs) Branches in the Poly(go)nesian Tropical Rain Forest
• Brownian Motion, Chern-Simons Theory, and 2d Yang-Mills
• Mirkovic-Vilonen polytopes and Khovanov-Lauda-Rouquier algebras
• Complete Intersection Moduli Spaces in N=4 Gauge Theories in Three Dimensions
• Coulomb Branch Quantization and Abelianized Monopole Bubbling
• Boundaries, Vermas, and Factorisation
• Hasse Diagrams for $3d$ $\mathcal{N}=4$ Quiver Gauge Theories -- Inversion and the full Moduli Space
• Quiver origami: discrete gauging and folding
• Sphere correlation functions and Verma modules
• Quiver Theories and Hilbert Series of Classical Slodowy Intersections
• On Three-Dimensional Quiver Gauge Theories of Type B
• The Hilbert series of U/SU SQCD and Toeplitz Determinants
• Nilpotent orbits and the Coulomb branch of $T^{\sigma }(G)$ theories: special orthogonal vs orthogonal gauge group factors
• Quantum phase transition in many-flavor supersymmetric QED${}_3$
• Exact equivalences and phase discrepancies between random matrix ensembles
• Resolutions of nilpotent orbit closures via Coulomb branches of 3-dimensional N=4 theories
• A characterization of nilpotent orbit closures among symplectic singularities
• Three-Dimensional Gauge Theories and ADE Monopoles
• Exact results and Schur expansions in quiver Chern-Simons-matter theories
• On ${\mathrm{SQED}}_3$ and ${\mathrm{SQCD}}_3$: phase transitions and integrability

Cited by in corpus (2)

• Fibrations and Hasse diagrams for 6d SCFTs
• Higgs branch RG-flows via Decay and Fission