Quiver Invariants from Intrinsic Higgs States

TL;DR

This paper explores the geometric and topological structure of BPS states in quiver quantum mechanics, proposing that intrinsic Higgs states form a fundamental, branch-independent invariant linked to single-center black holes.

Contribution

It introduces a conjecture that intrinsic Higgs states constitute a fundamental quiver invariant, independent of the Coulomb branch and wall-crossing phenomena.

Findings

Higgs phase states can be decomposed into pulled-back and intrinsic parts.

Intrinsic Higgs states are conjectured to be a fundamental, branch-independent invariant.

Intrinsic Higgs states are middle cohomology classes, all angular momentum singlets.

Abstract

In study of four-dimensional BPS states, quiver quantum mechanics plays a central role. The Coulomb phases capture the multi-centered nature of such states, and are well understood in the context of wall-crossing. The Higgs phases are given typically by F-term-induced complete intersections in the ambient D-term-induced toric varieties, and the ground states can be far more numerous than the Coulomb phase counterparts. We observe that the Higgs phase BPS states are naturally and geometrically grouped into two parts, with one part given by the pulled-back cohomology from the D-term-induced ambient space. We propose that these pulled-back states are in one-to-one correspondence with the Coulomb phase states. This also leads us to conjecture that the index associated with the rest, intrinsic to the Higgs phase,is a fundamental invariant of quivers, independent of branches. For simple circular quivers, these intrinsic Higgs states belong to the middle cohomology and thus are all angular momentum singlets, supporting the single-center black hole interpretation.