Exponential Networks and Representations of Quivers

TL;DR

This paper extends the geometric framework of BPS states using exponential networks on spectral curves, incorporating new singularities and rules to analyze quivers and wall-crossing phenomena in supersymmetric theories.

Contribution

It introduces a novel geometric approach with logarithmic singularities for BPS states, generalizing existing constructions to local Calabi-Yau threefolds and related models.

Findings

Reproduces BPS quivers for local geometries

Describes wall-crossing of finite-mass bound states

Proposes initial steps for understanding framed BPS spectra

Abstract

We study the geometric description of BPS states in supersymmetric theories with eight supercharges in terms of geodesic networks on suitable spectral curves. We lift and extend several constructions of Gaiotto-Moore-Neitzke from gauge theory to local Calabi-Yau threefolds and related models. The differential is multi-valued on the covering curve and features a new type of logarithmic singularity in order to account for D0-branes and non-compact D4-branes, respectively. We describe local rules for the three-way junctions of BPS trajectories relative to a particular framing of the curve. We reproduce BPS quivers of local geometries and illustrate the wall-crossing of finite-mass bound states in several new examples. We describe first steps toward understanding the spectrum of framed BPS states in terms of such "exponential networks."