BPS invariants for resolutions of polyhedral singularities

TL;DR

This paper investigates the BPS invariants of specific Calabi-Yau resolutions of ADE polyhedral singularities, establishing their relation to ADE root systems and confirming predictions from Gromov-Witten theory.

Contribution

It provides a new computation of genus 0 BPS invariants for these resolutions, linking them explicitly to ADE root systems and confirming theoretical predictions.

Findings

BPS invariants equal half the number of certain positive roots

Results align with Gromov-Witten theory predictions

Establishes a connection between geometric invariants and root systems

Abstract

We study the BPS invariants of the preferred Calabi-Yau resolution of ADE polyhedral singularities C^3/G given by Nakamura's G-Hilbert schemes. Genus 0 BPS invariants are defined by means of the moduli space of torsion sheaves as proposed by Sheldon Katz. We show that these invariants are equal to half the number of certain positive roots of an ADE root system associated to G. This is in agreement with the prediction via Gromov-Witten theory.