Towards a mathematical definition of Coulomb branches of $3$-dimensional $\mathcal{N}=4$ gauge theories, II

Abstract

Consider the $3$-dimensional $\mathcal{N}=4$ supersymmetric gauge theory associated with a compact Lie group $G_c$ and its quaternionic representation $\mathbf{M}$. Physicists study its Coulomb branch, which is a noncompact hyper-Kähler manifold with an $\mathrm{SU}(2)$-action, possibly with singularities. We give a mathematical definition of the Coulomb branch as an affine algebraic variety with ${\mathbb{C}}^{\times }$-action when $\mathbf{M}$ is of a form $\mathbf{N}\oplus {\mathbf{N}}^{\ast }$, as the second step of the proposal given in arXiv:1503.03676.

54 pages, 1 figure; v2. 55 pages, Proof of Theorem 4.1 is corrected. Determine Coulomb branches for SL(2) completely. Poisson structure is symplectic on nonsingular locus; v3. Two different usages of the same symbol `z' are corrected. Hyperlinks to a companion paper are added; v4. 57 pages. A proof of independence from a choice of the decomposition N+N^* is given. Several minor changes; v5. typo;

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Cited by in corpus (82)

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• Boundaries, Mirror Symmetry, and Symplectic Duality in 3d $\mathcal{N}=4$ Gauge Theory
• The Higgs Mechanism -- Hasse Diagrams for Symplectic Singularities
• Deformation quantization and superconformal symmetry in three dimensions
• Coulomb Branch Operators and Mirror Symmetry in Three Dimensions
• Coulomb branches of $3d$ $\mathcal{N}=4$ quiver gauge theories and slices in the affine Grassmannian (with appendices by Alexander Braverman, Michael Finkelberg, Joel Kamnitzer, Ryosuke Kodera, Hiraku Nakajima, Ben Webster, and Alex Weekes)
• Magnetic Quivers from Brane Webs with O5 Planes
• Quiver Yangian and Supersymmetric Quantum Mechanics
• Rozansky-Witten geometry of Coulomb branches and logarithmic knot invariants
• Brane Webs and Magnetic Quivers for SQCD
• Shifted Quiver Yangians and Representations from BPS Crystals
• Mirror symmetry and line operators
• Toroidal and Elliptic Quiver BPS Algebras and Beyond
• Coulomb Branch Quantization and Abelianized Monopole Bubbling
• Quantized Coulomb branches of Jordan quiver gauge theories and cyclotomic rational Cherednik algebras
• Three-Dimensional Mirror Self-Symmetry of the Cotangent Bundle of the Full Flag Variety
• Hasse Diagrams for $3d$ $\mathcal{N}=4$ Quiver Gauge Theories -- Inversion and the full Moduli Space
• Fibrations and Hasse diagrams for 6d SCFTs
• Folding Orthosymplectic Quivers
• Balanced B and D-type orthosymplectic quivers -- Magnetic quivers for product theories
• Quiver origami: discrete gauging and folding
• Sphere correlation functions and Verma modules
• Shifted quantum affine algebras: integral forms in type $A$ (with appendices by Alexander Tsymbaliuk and Alex Weekes)
• Feynman diagrams and $\Omega$-deformed M-theory
• Ring objects in the equivariant derived Satake category arising from Coulomb branches (with an appendix by Gus Lonergan)
• The Hasse Diagram of the Moduli Space of Instantons
• Koszul duality between Higgs and Coulomb categories $\mathcal{O}$
• Coulomb Branch Global Symmetry and Quiver Addition
• Mirror symmetry of 3d $\mathcal{N}=4$ gauge theories and supersymmetric indices
• Ungauging Schemes and Coulomb Branches of Non-simply Laced Quiver Theories
• Generators for Coulomb branches of quiver gauge theories
• Blocks and Vortices in the 3d ADHM Quiver Gauge Theory
• The Higgs branch of Heterotic ALE instantons
• Algebraic properties of the monopole formula
• Factorisation of 3d $\mathcal{N}=4$ Twisted Indices and the Geometry of Vortex Moduli Space
• Representation theory of the cyclotomic Cherednik algebra via the Dunkl-Opdam subalgebra
• Boundary vertex algebras for 3d $\mathcal{N}=4$ rank-0 SCFTs
• BFN Springer Theory
• A quantum Mirković-Vybornov isomorphism
• Supersymmetric Ground States of 3d $\mathcal{N}=4$ Gauge Theories on a Riemann Surface
• Monopole Operators and Bulk-Boundary Relation in Holomorphic Topological Theories
• Line bundles over Coulomb branches
• Rational Lax matrices from antidominantly shifted extended Yangians: BCD types
• 3d Mirror Symmetry for Instanton Moduli Spaces
• Towards geometric Satake correspondence for Kac-Moody algebras -- Cherkis bow varieties and affine Lie algebras of type $A$
• BPS States Meet Generalized Cohomology
• A Family of ${\mathrm{GL}}_r$ Multiplicative Higgs Bundles on Rational Base
• Free field realisation of boundary vertex algebras for Abelian gauge theories in three dimensions
• Masses, Sheets and Rigid SCFTs
• Knot Categorification from Mirror Symmetry, Part I: Coherent Sheaves
• Engineering 3D $\mathcal{N}=2$ theories using the quantum affine $\mathfrak{sl}(2)$ algebra
• Complex Symplectic Contractions and 3d Mirrors
• Atomic Higgsings of 6D SCFTs
• Cherkis bow varieties and Coulomb branches of quiver gauge theories of affine type $A$
• Seiberg-Witten monopoles with multiple spinors on a surface times a circle
• Some Open Questions in Quiver Gauge Theory
• Supersymmetric ground states of 3d $\mathcal{N}=4$ SUSY gauge theories and Heisenberg Algebras
• Free field realisation and the chiral universal centraliser
• Quantized Coulomb Branches, Monopole Bubbling and Wall-Crossing Phenomena in 3d $\mathcal{N}=4$ Theories
• Atomic Higgsings of 6D SCFTs II: Induced Flows
• Algebras, traces, and boundary correlators in $\mathcal{N}=4$ SYM
• Crystal bases and three-dimensional $\mathcal{N}=4$ Coulomb branches
• Branes and DAHA Representations
• Instantons on ALE spaces for classical groups
• The Hilbert series of 3d N=2 Yang-Mills theories with vectorlike matter
• Quiver varieties and symmetric pairs
• From representations of the rational Cherednik algebra to parabolic Hilbert schemes via the Dunkl-Opdam subalgebra
• Gaiotto's Lagrangian subvarieties via derived symplectic geometry
• Line Operators in 4d Chern-Simons Theory and Cherkis Bows
• Knot Categorification from Mirror Symmetry, Part II: Lagrangians
• On category $\mathcal{O}$ for affine Grassmannian slices and categorified tensor products
• Tunnels Under Geometries (or Instantons Know Their Algebras)
• Homological mirror symmetry for hypertoric varieties II (with an Appendix written jointly with Laurent Côté and Justin Hilburn)
• Difference operators via GKLO-type homomorphisms: shuffle approach and application to quantum Q-systems
• Homological Mirror Symmetry for Hypertoric Varieties I
• Twisted chiral algebras of class $\mathcal{S}$ and mixed Feigin-Frenkel gluing
• Knizhnik--Zamolodchikov functor for degenerate double affine Hecke algebras : algebraic theory
• Elliptic zastava
• Fermi-gas correlators of ADHM theory and triality symmetry
• Derived equivalences for trigonometric double affine Hecke algebras
• Quasi-lisse vertex (super)algebras
• Elliptic log symplectic brackets on projective bundles