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 modelled by
framed quivers. The associated spectral problem characterises 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 of this problem. We detail our
formalism with several explicit examples.