Framed BPS states

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 (WCF) for the ordinary BPS particles, by reducing it to the semiprimitive WCF. After reducing on S-1, 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 (PSC) that keeps track of the spin degrees of freedom of the framed BPS states. We show that the generating functions of PSCs admit a multiplication, which defines a deformation of the algebra of holomorphic functions on M. As an illustration of these ideas, we consider the six-dimensional (2,0) field theory of A(1) 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.