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.

73 pages, 1 figure

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Cited by in corpus (8)

• 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