  For D=4 BPS state construction, counting, and wall-crossing thereof, quiver
quantum mechanics offers two alternative approaches, the Coulomb phase and the
Higgs phase, which sometimes produce inequivalent counting. The authors have
proposed, in arXiv:1205.6511, two conjectures on the precise relationship
between the two, with some supporting evidences. Higgs phase ground states are
naturally divided into the Intrinsic Higgs sector, which is insensitive to
wall-crossings and thus an invariant of quiver, plus a pulled-back ambient
cohomology, conjectured to be an one-to-one image of Coulomb phase ground
states. In this note, we show that these conjectures hold for all cyclic
quivers with Abelian nodes, and further explore angular momentum and R-charge
content of individual states. Along the way, we clarify how the protected spin
character of BPS states should be computed in the Higgs phase, and further
determine the entire Hodge structure of the Higgs phase cohomology. This shows
that, while the Coulomb phase states are classified by angular momentum, the
Intrinsic Higgs states are classified by R-symmetry.
