BPS States, Refined Indices, and Quiver Invariants

TL;DR

This paper investigates the relationship between Coulomb and Higgs phase counting of BPS states in quiver quantum mechanics, confirming conjectures for cyclic Abelian quivers and analyzing the structure and symmetries of the states.

Contribution

It proves the conjectured relationship between Coulomb and Higgs phases for cyclic Abelian quivers and clarifies the classification of BPS states by angular momentum and R-symmetry.

Findings

Conjectures hold for all cyclic Abelian quivers.

Intrinsic Higgs states are R-symmetry classified.

Hodge structure of Higgs phase cohomology determined.

Abstract

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.