Toroidal and Elliptic Quiver BPS Algebras and Beyond

TL;DR

This paper introduces new classes of quiver algebras, namely toroidal and elliptic quiver algebras, extending the quiver Yangian framework to encompass more complex algebraic structures related to BPS state counting.

Contribution

It constructs and analyzes trigonometric and elliptic analogues of quiver Yangians, providing their representations via crystal melting models and supersymmetric gauge theories.

Findings

Defined toroidal and elliptic quiver algebras.

Constructed their representations using crystal melting.

Connected algebraic structures to supersymmetric gauge theories.

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.