Quivers, Line Defects and Framed BPS Invariants

A large class of N=2 quantum field theories admits a BPS quiver description, and the study of their BPS spectra is then reduced to a representation theory problem. In such theories the coupling to a line defect can be modeled by framed quivers. The associated spectral problem characterizes the line defect completely. Framed BPS states can be thought of as BPS particles bound to the defect. We identify the framed BPS degeneracies with certain enumerative invariants associated with the moduli spaces of stable quiver representations. We develop a formalism based on equivariant localization to compute explicitly such BPS invariants, for a particular choice of stability condition. Our framework gives a purely combinatorial solution to this problem. We detail our formalism with several explicit examples.