Pure-Higgs states from the Lefschetz-Sommese theorem

TL;DR

This paper studies a special class of N=4 quiver quantum mechanics relevant for BPS states and black holes, using the Lefschetz-Sommese theorem to identify pure-Higgs states and providing explicit counting formulas.

Contribution

It introduces a novel application of the Lefschetz-Sommese theorem to distinguish pure-Higgs states in a specific quiver class, confirming prior conjectures.

Findings

Derived explicit formulas for counting pure-Higgs states.

Separated induced from singular cohomology using Lefschetz-Sommese theorem.

Validated conjectures about pure-Higgs states in the studied quivers.

Abstract

We consider a special class of N=4 quiver quantum mechanics relevant in the description of BPS states of D4D0 branes in type II Calabi-Yau compactifications and the corresponding 4-dimensional black holes. These quivers have two abelian nodes in addition to an arbitrary number of non-abelian nodes and satisfy some simple but stringent conditions on the set of arrows, in particular closed oriented loops are always present. The Higgs branch can be described as the vanishing locus of a section of a vector bundle over a product of a projective space with a number of Grassmannians. The Lefschetz-Sommese theorem then allows to separate induced from singular cohomology which leads to the notion of pure-Higgs states. We compute explicit formulae for an index counting these pure-Higgs states and prove -- for this special class of quivers -- some previously stated conjectures about them.