Toroidal and Elliptic Quiver BPS Algebras and Beyond

Abstract

The quiver Yangian, an infinite-dimensional algebra introduced recently in arXiv:2003.08909, is the algebra underlying BPS state counting problems for toric Calabi-Yau three-folds. We introduce trigonometric and elliptic analogues of quiver Yangians, which we call toroidal quiver algebras and elliptic quiver algebras, respectively. We construct the representations of the shifted toroidal and elliptic algebras in terms of the statistical model of crystal melting. We also derive the algebras and their representations from equivariant localization of three-dimensional $\mathcal{N}=2$ supersymmetric quiver gauge theories, and their dimensionally-reduced counterparts. The analysis of supersymmetric gauge theories suggests that there exist even richer classes of algebras associated with higher-genus Riemann surfaces and generalized cohomology theories.

arXiv comment73 pages, 1 figure

References in corpus (52)

• Vortices, Instantons and Branes
• K-theory and Ramond-Ramond charge
• Topological Disorder Operators in Three-Dimensional Conformal Field Theory
• Quantum Foam and Topological Strings
• Lectures on 2D Yang-Mills Theory, Equivariant Cohomology and Topological Field Theories
• Supersymmetric Field Theories on Three-Manifolds
• Moduli Space of Non-Abelian Vortices
• Monopole Operators and Mirror Symmetry in Three Dimensions
• Towards a mathematical definition of Coulomb branches of $3$-dimensional $\mathcal{N}=4$ gauge theories, II
• Comments on twisted indices in 3d supersymmetric gauge theories
• Brane Tilings
• Aspects of 3d N=2 Chern-Simons-Matter Theories
• Phases Of N=2 Theories In 1+1 Dimensions With Boundary
• Non-commutative Donaldson-Thomas theory and the conifold
• The affine Yangian of ${\mathfrak{gl}}_1$ revisited
• A commutative algebra on degenerate CP^1 and Macdonald polynomials
• Brane Tilings and Their Applications
• W-symmetry, topological vertex and affine Yangian
• Crystal Melting and Toric Calabi-Yau Manifolds
• Quantum continuous ${\mathfrak{gl}}_{\infty }$: Semi-infinite construction of representations
• Lectures on localization and matrix models in supersymmetric Chern-Simons-matter theories
• Cohomological Hall algebras, vertex algebras and instantons
• Overview Of K-Theory Applied To Strings
• The cohomological Hall algebra of a preprojective algebra
• Quiver Yangian from Crystal Melting
• $(q,t)$-KZ equations for quantum toroidal algebra and Nekrasov partition functions on ALE spaces
• Higher Spins and Yangian Symmetries
• Brane Brick Models, Toric Calabi-Yau 4-Folds and 2d (0,2) Quivers
• Magnificent Four with Colors
• Quiver Yangian and Supersymmetric Quantum Mechanics
• Zero-dimensional Donaldson-Thomas invariants of Calabi-Yau 4-folds
• Elliptic algebra U_{q,p}(^sl_2): Drinfeld currents and vertex operators
• Shifted Quiver Yangians and Representations from BPS Crystals
• Vector bundles on elliptic curve and Sklyanin algebras
• New solutions of the star-triangle relation with discrete and continuous spin variables
• Analytic vortex solutions on compact hyperbolic surfaces
• Crystal Melting and Wall Crossing Phenomena
• A Note on Quiver Quantum Toroidal Algebra
• Elliptic hypergeometric sum/integral transformations and supersymmetric lens index
• Twisted Indices of 3d $\mathcal{N}=4$ Gauge Theories and Enumerative Geometry of Quasi-Maps
• Quantum toroidal algebra associated with ${\mathfrak{gl}}_{m|n}$
• The MacMahon R-matrix
• Duality in elliptic Ruijsenaars system and elliptic symmetric functions
• Braid actions on quantum toroidal superalgebras
• Volume of Moduli Space of Vortex Equations and Localization
• BPS Hall Algebra of Scattering Hall States
• Supersymmetric Ground States of 3d $\mathcal{N}=4$ Gauge Theories on a Riemann Surface
• Geometry and Energy of Non-abelian Vortices
• Graded Quivers, Generalized Dimer Models and Toric Geometry
• Multi-centered black holes, scaling solutions and pure-Higgs indices from localization
• Lens Generalisation of $\tau$-functions for the Elliptic Discrete Painlevé Equation
• Elliptic Quantum Groups U_{q,p}(gl_N) and E_{q,p}(gl_N)

Cited by in corpus (9)

• Quiver Yangian from Crystal Melting
• Shifted Quiver Quantum Toroidal Algebra and Subcrystal Representations
• A Note on Quiver Quantum Toroidal Algebra
• Quiver Yangians and Crystal Melting: A Concise Summary
• 5d AGT correspondence of supergroup gauge theories from quantum toroidal ${\mathfrak{gl}}_1$
• Crystal Melting, BPS Quivers and Plethystics
• Remarks on Berry Connection in QFT, Anomalies, and Applications
• Quiver Yangians and $\mathcal{W}$-Algebras for Generalized Conifolds
• Supersymmetric ground states of 3d $\mathcal{N}=4$ SUSY gauge theories and Heisenberg Algebras