Quantum Holonomies from Spectral Networks and Framed BPS States

A note on discrete dynamical systems in theories of class $S$

“…These theories admit supersymmetric line defects which are associated to the spectral problem of computing framed BPS states [26]. In many cases such a problem can be approached by localization on quiver moduli spaces [9,12,13,14,16], or on the moduli spaces of semiclassical configurations [5,33], by using cluster algebras [3,10,11,41], or via BPS graphs or spectral networks [22,23,28,35].…”

Quantum line defects and refined BPS spectra

“…From the spectral network one can determine the BPS quiver in some region of the moduli space [24,27]. The core charge can in principle by determined by a careful study of the abelianiziation map for spectral networks [23,25] which relates flat connections on C and on Σ u . There is however no general algorithm to determine the superpotential W L .…”

BPS spectra from BPS graphs

“…with the phase ϑ ∈ [0, π) increasing from right to left in the product (making sense of this ordering requires moving slightly away from the walls of marginal stability into a well-defined BPS chamber, which as we will see corresponds to choosing a sequence of edges in the BPS graph). Now since the generating functions F ± (℘) can be computed as parallel transports with spin [13], or a quantum holonomies [14], the idea is to use the framed wall-crossing formula (4.1) to determine the quantum spectrum generator S.…”

“…Given that F (℘, ϑ) can be computed as a quantum holonomy along the path ℘ on C [13,14], the framed wall-crossing formula can be turned around and used to determine the quantum spectrum generator S. The method consists in choosing a path ℘ that crosses an edge γ α of the BPS graph and computing F (℘, ϑ), or rather some building block Q α , expressed as a product of quantum dilogarithms. This only provides some partial information about S, but the procedure can be repeated for another edge γ β after having deleted the edge γ α from the BPS graph.…”