Shifted Quiver Yangians and Representations from BPS Crystals

TL;DR

This paper introduces shifted quiver Yangians as new algebraic structures capturing BPS states in type IIA string theory on toric Calabi-Yau three-folds, unifying various BPS counting problems through novel representations.

Contribution

The paper defines shifted quiver Yangians and constructs their representations from subcrystals, providing a unified algebraic framework for diverse BPS state counting problems.

Findings

Unified BPS counting via shifted quiver Yangians

Constructed new representations from subcrystals

Revealed a large class of potentially new BPS problems

Abstract

We introduce a class of new algebras, the shifted quiver Yangians, as the BPS algebras for type IIA string theory on general toric Calabi-Yau three-folds. We construct representations of the shifted quiver Yangian from general subcrystals of the canonical crystal. We derive our results via equivariant localization for supersymmetric quiver quantum mechanics for various framed quivers, where the framings are determined by the shape of the subcrystals. Our results unify many known BPS state counting problems, including open BPS counting, non-compact D4-branes, and wall crossing phenomena, simply as different representations of the shifted quiver Yangians. Furthermore, most of our representations seem to be new, and this suggests the existence of a zoo of BPS state counting problems yet to be studied in detail.