Framed BPS States

TL;DR

This paper introduces framed BPS states in 4d N=2 theories, providing new insights into wall-crossing, moduli space coordinates, and connections to cluster algebras, with explicit examples and a new proof of known formulas.

Contribution

It defines framed BPS states supported by line operators, offers a new proof of the Kontsevich-Soibelman wall-crossing formula, and explores their algebraic and geometric structures.

Findings

Proof of wall-crossing formula via framed BPS states

Classification of line operators by laminations on Riemann surfaces

Explicit computation of framed BPS spectra in examples

Abstract

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.