Line Defects, Tropicalization, and Multi-Centered Quiver Quantum Mechanics

TL;DR

This work develops a comprehensive framework for BPS line defects in 4D ${\mathcal N}=2$ theories, focusing on quiver-type models where BPS spectra are captured by quiver quantum mechanics. It establishes a bijective RG flow between UV defects and IR abelian-defect data for broad classes, enabling reconstruction of UV framed BPS spectra from IR information, and introduces a precise framed-quiver algorithm with a Coulomb-branch basis. The authors uncover universal OPE structures and tropical/discontinuity phenomena tied to quiver mutation, dual cones, and defect mutations, providing both general theory and explicit checks in Argyres–Douglas and ${SU(2)}$ Yang–Mills cases. The results bridge UV defect data with IR cluster/tropical geometry, offering practical tools for computing framed spectra and advancing understanding of line-defect algebras in non-Lagrangian theories.

Abstract

We study BPS line defects in N=2 supersymmetric four-dimensional field theories. We focus on theories of "quiver type," those for which the BPS particle spectrum can be computed using quiver quantum mechanics. For a wide class of models, the renormalization group flow between defects defined in the ultraviolet and in the infrared is bijective. Using this fact, we propose a way to compute the BPS Hilbert space of a defect defined in the ultraviolet, using only infrared data. In some cases our proposal reduces to studying representations of a "framed" quiver, with one extra node representing the defect. In general, though, it is different. As applications, we derive a formula for the discontinuities in the defect renormalization group map under variations of moduli, and show that the operator product algebra of line defects contains distinguished subalgebras with universal multiplication rules. We illustrate our results in several explicit examples.

Figures (10)

• Figure 1: A semiclassical cartoon of a framed BPS state. The blue object indicates the infinitely massive core charge. It is modeled in the quiver description by a new node with charge determined by the $\mathbf{RG}$ map. The red objects are ordinary BPS states of finite mass. They comprise the halo charge which is bound to the defect, and orbit the core in a multi-centered Ptolemaean system. The core charge is spatially separated from any constituent of the halo.
• Figure 2: A jump in the cone of particles induced by a change in the particle half-space. Red lines indicate the central charges of particles, while blue lines denote central charges of antiparticles. The gray shaded region indicates the projection of the cone of particles, and the boundary of the particle half-space is indicated with a dashed line. In passing from (a) to (b) the particle with central charge $Z_{\gamma}$ changes its identity to an antiparticle and the cone jumps. At the critical value $\vartheta_{c},$ the dashed line coincides with the ray $Z_{\gamma}$ and the quiver does not exist.
• Figure 3: The particle half-space for the framed quiver implied by the RG map. The red ray indicates the central charge of the defect $\gamma_{c}+\gamma_{F}$, while black rays are the central charges of ordinary BPS states. The gray shaded region indicates the cone containing the framed BPS states. In the framing limit, the length of the red ray tends to infinity and the width of the cone is infinitesimal. The boundary of the half-space is indicated by the dashed line. As moduli are varied and a central charge of an ordinary BPS state exits the particle half-space, the core charge changes by a mutation.
• Figure 4: Topology of a framed quiver $Q[\gamma_{c}]$ when the core charge occupies $\check{\mathcal{C}}$. The framing node is indicated by a square, while the ordinary nodes of $Q$ are circles. Arrows to between the nodes of $Q$ are suppressed.
• Figure 5: Adapted stability conditions plotted in the particle half-space. The framing ray is indicated in red and possible halo rays are indicated in black. The gray shaded region is the infinitesimal cone containing framed BPS states. In (a) the stability condition is left adapted. In (b) the stability condition is right adapted.

Theorems & Definitions (16)

• Proposition 1
• proof
• Proposition 2
• proof
• Proposition 3
• Proposition 4
• proof
• Proposition 5
• proof
• Proposition 6