Crystals and Double Quiver Algebras from Jeffrey-Kirwan Residues

TL;DR

This paper develops a framework connecting Jeffrey-Kirwan residues, crystal melting models, and new double quiver algebras to analyze BPS states in supersymmetric quiver gauge theories, extending previous toric results to more general cases.

Contribution

It introduces double quiver Yangians/algebras and their crystal representations, generalizing known structures to a broader class of quivers including non-toric Calabi-Yau manifolds.

Findings

Constructed statistical models for crystal melting related to supersymmetric quiver theories.

Defined new double quiver Yangians/algebras and linked them to crystal structures.

Compared double quiver algebras with existing BPS algebras for theories with four supercharges.

Abstract

We construct statistical mechanical models of crystal melting describing the flavoured Witten indices of $\mathcal{N}\ge 2$ supersymmetric quiver gauge theories. Our results can be derived from the Jeffrey-Kirwan (JK) residue formulas, and generalize the previous results for quivers corresponding to toric Calabi-Yau threefolds and fourfolds to a large class of quivers satisfying the no-overlap condition, including those corresponding to some non-toric Calabi-Yau manifolds. We construct new quiver algebras which we call the double quiver Yangians/algebras, as well as their representations in terms of the aforementioned crystals. For theories with four supercharges, we compare the double quiver algebras with the existing quiver Yangians/BPS algebras, which we show can also be constructed from the JK residues. For theories with two supercharges, the double quiver algebras provide an algebraic description of the BPS states, including the information of the fixed points and their relative coefficients in the full partition functions.