A Note on Quiver Quantum Toroidal Algebra

Abstract

Recently, Li and Yamazaki proposed a new class of infinite-dimensional algebras, quiver Yangian, which generalizes the affine Yangian ${\mathfrak{gl}}_1$. The characteristic feature of the algebra is the action on BPS states for non-compact toric Calabi-Yau threefolds, which are in one-to-one correspondence with the crystal melting models. These algebras can be bootstrapped from the action on the crystals and have various truncations. In this paper, we propose a $q$-deformed version of the quiver Yangian, referred to as the quiver quantum toroidal algebra (QQTA). We examine some of the consistency conditions of the algebra. In particular, we show that QQTA is a Hopf superalgebra with a formal super coproduct, like known quantum toroidal algebras. QQTA contains an extra central charge $C$. When it is trivial ($C=1$), QQTA has a representation acting on the three-dimensional crystals, like Li-Yamazaki's quiver Yangian. While we focus on the toric Calabi-Yau threefolds without compact 4-cycles, our analysis can likely be generalized to all toric Calabi-Yau threefolds.

35+8 pages, 14 figures. v2:typos are fixed and refs added v3:published version

References in corpus (12)

• Crystal Melting and Toric Calabi-Yau Manifolds
• Notes on Ding-Iohara algebra and AGT conjecture
• Reflection states in Ding-Iohara-Miki algebra and brane-web for D-type quiver
• On W algebras commuting with a set of screenings
• Toroidal and Elliptic Quiver BPS Algebras and Beyond
• Kernel function and quantum algebras
• Shifted Quiver Quantum Toroidal Algebra and Subcrystal Representations
• Drinfeld second realization of the quantum affine superalgebras of $D^{(1)}(2,1;x)$ via the Weyl groupoid
• Five-dimensional SU(2) AGT conjecture and recursive formula of deformed Gaiotto state
• Representations of quantum toroidal $g{l}_n$
• Quantum deformation of Feigin-Semikhatov's W-algebras and 5d AGT correspondence with a simple surface operator
• Combinatorics of vertex operators and deformed $W$-algebra of type D$(2,1;\alpha )$

Cited by in corpus (20)

• Toroidal and Elliptic Quiver BPS Algebras and Beyond
• Shifted Quiver Quantum Toroidal Algebra and Subcrystal Representations
• Gauge/Bethe correspondence from quiver BPS algebras
• Quantum toroidal algebras and solvable structures in gauge/string theory
• Quiver Yangians and Crystal Melting: A Concise Summary
• 5d AGT correspondence of supergroup gauge theories from quantum toroidal ${\mathfrak{gl}}_1$
• The Origin of Calabi-Yau Crystals in BPS States Counting
• Gauge origami and quiver W-algebras
• Simple Representations of BPS Algebras: the case of $Y({\widehat{\mathfrak{gl}}}_2)$
• BPS States Meet Generalized Cohomology
• Quiver Yangians and $\mathcal{W}$-Algebras for Generalized Conifolds
• More on Affine Dynkin Quiver Yangians
• Algorithms for representations of quiver Yangian algebras
• Gauge origami and quiver W-algebras III: Donaldson--Thomas $qq$-characters
• Crystals and Double Quiver Algebras from Jeffrey-Kirwan Residues
• Supersymmetric polynomials and algebro-combinatorial duality
• An Overview of Crystals and Double Quiver Yangians
• Reduced quiver quantum toroidal algebras
• Free field realization of the Ding-Iohara algebra at general levels
• Quiver Yangian algebras associated to Dynkin diagrams of A-type and their rectangular representations