Shifted Quiver Quantum Toroidal Algebra and Subcrystal Representations

Abstract

Recently, new classes of infinite-dimensional algebras, quiver Yangian (QY) and shifted QY, were introduced, and they act on BPS states for non-compact toric Calabi-Yau threefolds. In particular, shifted QY acts on general subcrystals of the original BPS crystal. A trigonometric deformation called quiver quantum toroidal algebra (QQTA) was also proposed and shown to act on the same BPS crystal. Unlike QY, QQTA has a formal Hopf superalgebra structure which is useful in deriving representations. In this paper, we define the shifted QQTA and study a class of their representations. We define 1d and 2d subcrystals of the original 3d crystal by removing a few arrows from the original quiver diagram and show how the shifted QQTA acts on them. We construct the 2d crystal representations from the 1d crystal representations by utilizing a generalized coproduct acting on different shifted QQTAs. We provide a detailed derivation of subcrystal representations of ${\mathbb{C}}^3$, ${\mathbb{C}}^3/{\mathbb{Z}}_n(n\ge 2)$, conifold, suspended pinch point, and ${\mathbb{C}}^3/({\mathbb{Z}}_2\times {\mathbb{Z}}_2)$.

59+23 pages, many figures. v2:fixed typos and added references, v3: added discussions, v4: published version

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• Quiver Yangians and Crystal Melting: A Concise Summary
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• Super-Schur Polynomials for Affine Super Yangian $\mathsf{Y}({\widehat{\mathfrak{gl}}}_{1|1})$
• Gauge origami and quiver W-algebras
• 3-Schurs from explicit representation of Yangian $Y({\widehat{\mathfrak{gl}}}_1)$. Levels 1-5
• Simple Representations of BPS Algebras: the case of $Y({\widehat{\mathfrak{gl}}}_2)$
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• BPS States Meet Generalized Cohomology
• Wall-Crossing Effects on Quiver BPS Algebras
• Algorithms for representations of quiver Yangian algebras
• Macdonald polynomials for super-partitions
• Gauge origami and quiver W-algebras III: Donaldson--Thomas $qq$-characters
• Towards the theory of Yangians
• Supersymmetric polynomials and algebro-combinatorial duality
• Tunnels Under Geometries (or Instantons Know Their Algebras)
• Gauge origami and quiver W-algebras IV: Pandharipande--Thomas $qq$-characters
• An Overview of Crystals and Double Quiver Yangians
• Analytic structure of the high-energy gravitational amplitude: multi-H diagrams and classical 5PM logarithms
• Super-Hamiltonians for super-Macdonald polynomials
• Overcharging Extremal Rotating Black Holes
• Weyl Mutations in Quiver Yangians
• Quiver Yangian algebras associated to Dynkin diagrams of A-type and their rectangular representations
• Reduced quiver quantum toroidal algebras