Crystals and Double Quiver Algebras from Jeffrey-Kirwan Residues

Abstract

We construct statistical mechanical models of crystal melting describing the flavoured Witten indices of $\mathcal{N}\ge 2$ supersymmetric quiver gauge theories. Our results can be derived from the Jeffrey-Kirwan (JK) residue formulas, and generalize the previous results for quivers corresponding to toric Calabi-Yau threefolds and fourfolds to a large class of quivers satisfying the no-overlap condition, including those corresponding to some non-toric Calabi-Yau manifolds. We construct new quiver algebras which we call the double quiver Yangians/algebras, as well as their representations in terms of the aforementioned crystals. For theories with four supercharges, we compare the double quiver algebras with the existing quiver Yangians/BPS algebras, which we show can also be constructed from the JK residues. For theories with two supercharges, the double quiver algebras provide an algebraic description of the BPS states, including the information of the fixed points and their relative coefficients in the full partition functions.

75 pages

References in corpus (37)

• Stability structures, motivic Donaldson-Thomas invariants and cluster transformations
• Dimer models and toric diagrams
• Witten Index and Wall Crossing
• Brane Tilings
• Non-commutative Donaldson-Thomas theory and the conifold
• Brane Tilings and Their Applications
• Crystal Melting and Toric Calabi-Yau Manifolds
• Wall Crossing of BPS States on the Conifold from Seiberg Duality and Pyramid Partitions
• Brane Brick Models in the Mirror
• Shifted Quiver Yangians and Representations from BPS Crystals
• Emergent Calabi-Yau Geometry
• Motivic Donaldson-Thomas invariants and McKay correspondence
• Gauge/Bethe correspondence from quiver BPS algebras
• Crystal Melting and Wall Crossing Phenomena
• A Note on Quiver Quantum Toroidal Algebra
• Quantum toroidal algebras and solvable structures in gauge/string theory
• Affine Yangian of $\mathfrak{gl}(2)$ and integrable structures of superconformal field theory
• R-matrix formulation of affine Yangian of $\widehat{\mathfrak{gl}}(1|1)$
• Quiver Yangians and Crystal Melting: A Concise Summary
• Super-Schur Polynomials for Affine Super Yangian $\mathsf{Y}({\widehat{\mathfrak{gl}}}_{1|1})$
• Counting invariants for the ADE McKay quivers
• A Note on Quiver Yangians and $\mathcal{R}$-Matrices
• The Origin of Calabi-Yau Crystals in BPS States Counting
• Coproduct for affine Yangians and parabolic induction for rectangular $W$-algebras
• Gauge origami and quiver W-algebras
• Integrable structure of $\text{BCD}$ conformal field theory and boundary Bethe ansatz for affine Yangian
• Quiver Yangians and $\mathcal{W}$-Algebras for Generalized Conifolds
• BPS States Meet Generalized Cohomology
• More on Affine Dynkin Quiver Yangians
• Wall-Crossing Effects on Quiver BPS Algebras
• Charging solid partitions
• Gauge origami and quiver W-algebras III: Donaldson--Thomas $qq$-characters
• Deformed Double Current Algebras, Matrix Extended ${\mathcal{W}}_{\infty }$ Algebras, Coproducts, and Intertwiners from the M2-M5 Intersection
• Global magni$4$icence, or: 4G Networks
• Okounkov's conjecture via BPS Lie algebras
• Gauge origami and quiver W-algebras II: Vertex function and beyond quantum $q$-Langlands correspondence
• Virtual invariants of critical loci in GIT quotients of linear spaces

Cited by in corpus (4)

• Gauge origami and quiver W-algebras IV: Pandharipande--Thomas $qq$-characters
• Charge functions for odd dimensional partitions
• Weyl Mutations in Quiver Yangians
• Quiver Yangian algebras associated to Dynkin diagrams of A-type and their rectangular representations