Exponential BPS graphs and D brane counting on toric Calabi-Yau threefolds: Part I

TL;DR

This paper introduces exponential BPS graphs for toric Calabi-Yau threefolds, applying nonabelianization to compute BPS spectra and invariants, and demonstrates their ability to encode quivers and potentials through wall-crossing analysis.

Contribution

It develops exponential BPS graphs for toric Calabi-Yau threefolds and uses nonabelianization to compute BPS spectra and invariants, linking them to quiver data.

Findings

Successfully reproduces known BPS spectra for specific geometries.

Introduces exponential BPS graphs encoding quivers and potentials.

Shows wall-crossing behavior of 3d-5d BPS states.

Abstract

We study BPS spectra of D-branes on local Calabi-Yau threefolds $\mathcal{O}(-p)\oplus\mathcal{O}(p-2)\to \mathbb{P}^1$ with $p=0,1$, corresponding to $\mathbb{C}^3/\mathbb{Z}_{2}$ and the resolved conifold. Nonabelianization for exponential networks is applied to compute directly unframed BPS indices counting states with D2 and D0 brane charges. Known results on these BPS spectra are correctly reproduced by computing new types of BPS invariants of 3d-5d BPS states, encoded by nonabelianization, through their wall-crossing. We also develop the notion of exponential BPS graphs for the simplest toric examples, and show that they encode both the quiver and the potential associated to the Calabi-Yau via geometric engineering.