Framed BPS States

TL;DR

The paper develops a comprehensive framework where framed BPS states arise from line operators in 4d N=2 theories, with their protected spin characters organizing into a noncommutative deformation of the moduli-space functions. It proves a framed version of the Kontsevich–Soibelman wall-crossing formula via halo configurations and quantum dilogarithms, and shows how Darboux coordinates on the Coulomb-branch moduli space are physically realized as line-operator vevs. The authors extend the analysis to class S theories, unveiling a rich interplay between laminations on Riemann surfaces, cluster algebras, and Hitchin moduli spaces, and they provide explicit computations in A1 theories including laminations, N=3/4 Argyres–Douglas, and ${ m SU}(2)$ with various flavors. They also connect the 4d construction to a six-dimensional perspective, offering a unified picture in terms of twisted local systems and twisted moduli spaces, with broader implications for quantum holonomies and tropical labels. The work thus links BPS spectra, algebraic structures, and geometric moduli spaces, suggesting deep interactions with cluster algebras, quantum Teichmüller theory, and open problems in higher-rank theories.

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.

Figures (52)

• Figure 1: A schematic picture of the spectrum of energies of states with charge $\gamma$, in the theory with the line operator $L_\zeta$ inserted. At the bottom we see the framed BPS states. Next there are discrete excited states bound to the line operator. Finally there are various continua of unbound states, corresponding to different possible decompositions of $\gamma$ as $\gamma_c + \gamma_h$. The framed BPS states are safely separated from the continua except when there is a $\gamma_h$ with $-{\rm Re } \, (Z_{\gamma_h}(u) / \zeta)= \vert Z_{\gamma_h}(u)\vert$.
• Figure 2: Two paths from $(\zeta_1, u_1)$ to $(\zeta_2, u_2)$. The BPS rays appear as codimension-1 loci in the joint $(\zeta, u)$ space. Each BPS wall $\widehat{W}(\gamma)$ has a coordinate transformation attached. The walls can coalesce and split as shown, and hence the two paths in general cross different sets of BPS walls. Nevertheless, the composition of the corresponding coordinate transformations must be independent of the path.
• Figure 3: Chambers and strongly positive formal line operators in the very simplest case, a $U(1)$ gauge theory with a single BPS hypermultiplet. Here $n=0 \mod 2$ and $p>0$. Regions which are stable to halo formation are shaded.
• Figure 4: Chambers and strongly positive formal line operators in the very simplest case, a $U(1)$ gauge theory with a single BPS hypermultiplet. Here $n=1 \mod 2$ and $p<0$. Regions which are stable to halo formation are shaded.
• Figure 5: BPS walls for the $N=3$ Argyres-Douglas theory at $u=0$ with $\Lambda>0$ projected to the $\zeta$-plane.