Crystals and Double Quiver Algebras from Jeffrey-Kirwan Residues

TL;DR

This work develops a comprehensive bridge between JK residue techniques and crystal-melting models for flavoured Witten indices in $\mathcal{N}\ge 2$ quiver gauge theories. It introduces the double quiver Yangians/algebras $\widetilde{\mathsf{Y}}$ and their crystal representations, extending known toric CY3/4 results to a broad class of quivers satisfying the no-overlap condition, including some non-toric Calabi–Yau manifolds. For theories with four supercharges, the authors show JK residues can reconstruct both the quiver Yangians and the enhanced double quiver algebras, while for theories with two supercharges the double algebras encode full BPS-state data, including fixed points and partition-function coefficients, via crystal actions. The framework connects BPS counting with representation theory, DT/invariants, and potential VOAs, and it opens paths to refine counting, study chamber dependence, and explore dualities and higher-structure algebras in non-toric settings.

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.