  We consider a large class of branes in toric strip geometries, both
non-periodic and periodic ones. For a fixed background geometry we show that
partition functions for such branes can be reinterpreted, on one hand, as
quiver generating series, and on the other hand as wave-functions in various
polarizations. We determine operations on quivers, as well as
$SL(2,\mathbb{Z})$ transformations, which correspond to changing positions of
these branes. Our results prove integrality of BPS multiplicities associated to
this class of branes, reveal how they transform under changes of polarization,
and imply all other properties of brane amplitudes that follow from the
relation to quivers.
