We consider a class of line operators in d=4, N=2 supersymmetric field
theories which leave four supersymmetries unbroken. Such line operators support
a new class of BPS states which we call "framed BPS states." These include halo
bound states similar to those of d=4, N=2 supergravity, where (ordinary) BPS
particles are loosely bound to the line operator. Using this construction, we
give a new proof of the Kontsevich-Soibelman wall-crossing formula for the
ordinary BPS particles, by reducing it to the semiprimitive wall-crossing
formula. After reducing on S1, the expansion of the vevs of the line operators
in the IR provides a new physical interpretation of the "Darboux coordinates"
on the moduli space M of the theory. Moreover, we introduce a "protected spin
character" which keeps track of the spin degrees of freedom of the framed BPS
states. We show that the generating functions of protected spin characters
admit a multiplication which defines a deformation of the algebra of functions
on M. As an illustration of these ideas, we consider the six-dimensional (2,0)
field theory of A1 type compactified on a Riemann surface C. Here we show
(extending previous results) that line operators are classified by certain
laminations on a suitably decorated version of C, and we compute the spectrum
of framed BPS states in several explicit examples. Finally we indicate some
interesting connections to the theory of cluster algebras.
