  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.
