N=2 Quantum Field Theories and Their BPS Quivers

TL;DR

This work develops a comprehensive framework connecting 4d N=2 QFTs to BPS quivers and introduces a mutation-based method to compute BPS spectra across moduli spaces. By formulating a quiver quantum mechanics with stability criteria and superpotentials, the authors derive a robust mutation algorithm that generates the full BPS spectrum in finite chambers and connects different dual descriptions via wall-crossing. They apply the method to wide classes of theories, including SU(2) and SU(N) gauge theories with matter, ADE-type groups, and Gaiotto-type constructions from M5-branes, obtaining strong-coupling spectra and finite chambers, and clarifying the roles of flavor symmetries and gauging in the quiver language. The approach yields new insights into dualities (e.g., Argyres-Seiberg), reproduces known spectra from Seiberg–Witten data, and provides a unified path to construct quivers for Gaiotto theories, highlighting the deep link between geometry, dualities, and BPS state counting.

Abstract

We explore the relationship between four-dimensional N=2 quantum field theories and their associated BPS quivers. For a wide class of theories including super-Yang-Mills theories, Argyres-Douglas models, and theories defined by M5-branes on punctured Riemann surfaces, there exists a quiver which implicitly characterizes the field theory. We study various aspects of this correspondence including the quiver interpretation of flavor symmetries, gauging, decoupling limits, and field theory dualities. In general a given quiver describes only a patch of the moduli space of the field theory, and a key role is played by quantum mechanical dualities, encoded by quiver mutations, which relate distinct quivers valid in different patches. Analyzing the consistency conditions imposed on the spectrum by these dualities results in a powerful and novel mutation method for determining the BPS states. We apply our method to determine the BPS spectrum in a wide class of examples, including the strong coupling spectrum of super-Yang-Mills with an ADE gauge group and fundamental matter, and trinion theories defined by M5-branes on spheres with three punctures.

Figures (10)

• Figure 1: The spectrum and BPS quiver of $SU(2)$ Yang-Mills. In (a) the weak-coupling BPS spectrum, both particles and antiparticles, is plotted in the $(e,m)$ plane. Red dots denote the lattice sites occupied by BPS states. The green arrows show the basis of particles given by the monopole and dyon. We have represented our choice of particle central charge half-plane by the grey region. In (b) the BPS quiver is extracted from this data. It has one node for each basis vector, and the double arrow encodes the symplectic product.
• Figure 2: The chambers of the $A_{2}$ Argyres-Douglas theory. The BPS spectrum is plotted in the central charge plane. Particles are shown in red, antiparticles in blue. The cone of particles is the shaded grey region. In (a) the particles form a bound state. In (b) the bound state is unstable and decays.
• Figure 3: A discontinuity in the quiver description results in a quantum mechanical duality described by quiver mutation. In both diagrams the BPS spectrum is plotted in the central charge plane. Red lines denote particles while blue lines denote antiparticles. The gray shaded region indicates the cone of particles. In passing from (a) to (b) the particle with central charge $\mathcal{Z}_{1}$ changes its identity to an antiparticle. The cone of particles jumps discontinuously and a new quiver description is required.
• Figure 4: A cartoon of the moduli space and its relation to various BPS quiver descriptions. The red lines denote walls of marginal stability where the BPS spectrum jumps. The gray shaded region is the domain in moduli space where $Q$ describes the BPS spectrum. The gray checkered region is the domain where $\widetilde{Q}$ describes the spectrum. The two descriptions are glued together smoothly away from the walls of marginal stability. Their interface is a wall of the second kind.
• Figure 5: The BPS spectrum of pure $SU(2)$ gauge theory, plotted in the central charge $\mathcal{Z}$-plane. The spectrum contains a vector state with charge $\mathcal{Z}_1+\mathcal{Z}_2$ (plotted in green), which is forced to occur in the $\mathcal{Z}$-plane at an accumulation ray of hypermultiplet states. On either side of the vector state, there is an infinite sequence of dyons whose central charges asymptotically approach the ray on which the vector lies. The mutation method is able to capture the full spectrum of the theory by rotating the half-plane to the left (yielding particles on the left of the vector particle) and the to right (yielding particles on the right of the vector particle).