An Overview of Crystals and Double Quiver Yangians

Abstract

In this review, we summarize the recent progress on the crystal melting models and the quiver algebras regarding the BPS counting. We shall consider the constructions of crystals for generic quivers and discuss the so-called double quiver Yangians/algebras. This is an invited review for International Journal of Modern Physics A.

22 pages

References in corpus (39)

• Witten Index and Wall Crossing
• 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
• Shifted Quiver Quantum Toroidal Algebra and Subcrystal Representations
• 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
• Canonical vertex formalism in DT theory of toric Calabi-Yau 4-folds
• R-matrix formulation of affine Yangian of $\widehat{\mathfrak{gl}}(1|1)$
• Fano 3-Folds, Reflexive Polytopes and Brane Brick Models
• Quiver Yangians and Crystal Melting: A Concise Summary
• 5d AGT correspondence of supergroup gauge theories from quantum toroidal ${\mathfrak{gl}}_1$
• Super-Schur Polynomials for Affine Super Yangian $\mathsf{Y}({\widehat{\mathfrak{gl}}}_{1|1})$
• A Note on Quiver Yangians and $\mathcal{R}$-Matrices
• Coproduct for affine Yangians and parabolic induction for rectangular $W$-algebras
• The Origin of Calabi-Yau Crystals in BPS States Counting
• Simple Representations of BPS Algebras: the case of $Y({\widehat{\mathfrak{gl}}}_2)$
• Gauge origami and quiver W-algebras
• Brane Brick Models for the Sasaki-Einstein 7-Manifolds Y^{p,k}(CP^1 x CP^1) and Y^{p,k}(CP^2)
• Mass Deformations of Brane Brick Models
• Integrable structure of $\text{BCD}$ conformal field theory and boundary Bethe ansatz for affine Yangian
• BPS States Meet Generalized Cohomology
• More on Affine Dynkin Quiver Yangians
• Tetrahedron instantons in Donaldson-Thomas theory
• Quiver Yangians and $\mathcal{W}$-Algebras for Generalized Conifolds
• Wall-Crossing Effects on Quiver BPS Algebras
• On the Master Space for Brane Brick Models
• Charging solid partitions
• Algorithms for representations of quiver Yangian algebras
• Gauge origami and quiver W-algebras III: Donaldson--Thomas $qq$-characters
• Towards the theory of Yangians
• Elliptic Quantum Toroidal Algebras, Z-algebra Structure and Representations

Cited by in corpus (2)

• Charge functions for odd dimensional partitions
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