BPS Quivers and Spectra of Complete N=2 Quantum Field Theories

TL;DR

The paper develops a full framework linking BPS spectra of 4d N=2 complete theories to quivers and superpotentials derived from triangulations of punctured Riemann surfaces (Gaiotto curves). It introduces an algorithmic mutation-based method to compute Π-stable BPS states and proves the existence of finite chambers for broad classes, including asymptotically free, Argyres–Douglas, and conformal theories on spheres and tori. By geometrically engineering these theories and exploiting special-Lagrangian flows, the authors provide explicit quivers and potentials for SU(2) gauge theories and extend the analysis to exceptional complete theories, with glueing rules that preserve finite chambers. The results offer a practical pathway to compute BPS spectra across large families of complete theories and motivate extensions to higher-rank Gaiotto theories and a complete mutation-class classification. Overall, the work demonstrates a deep and computationally tractable bridge between geometry, quiver representations, and BPS state counting in N=2 QFTs.

Abstract

We study the BPS spectra of N=2 complete quantum field theories in four dimensions. For examples that can be described by a pair of M5 branes on a punctured Riemann surface we explain how triangulations of the surface fix a BPS quiver and superpotential for the theory. The BPS spectrum can then be determined by solving the quantum mechanics problem encoded by the quiver. By analyzing the structure of this quantum mechanics we show that all asymptotically free examples, Argyres-Douglas models, and theories defined by punctured spheres and tori have a chamber with finitely many BPS states. In all such cases we determine the spectrum.

Figures (15)

• Figure 1: Special-Lagrangian geometry in the Calabi-Yau. The blue denotes a patch of the surface $\mathcal{C}$. The red trajectory denotes the cycle $\eta$ and the $S^{2}$ fibers are indicated schematically above $\mathcal{C}$. In (a) the topology of the cycle $\eta$ is an interval which terminates at two zeros of $\phi$. The $S^{2}$ fibers shrink at these end points yielding a total space of an $S^{3}$. In (b), the cycle $\eta$ has the topology of a circle, and the total space is $S^{1}\times S^{2}$. Such special-lagrangians always come in one parameter families indicated in orange.
• Figure 2: The local structure of the flow near a zero of $\phi$ shown as a black dot at the center of the diagram. The red trajectories are the three flow lines which pass through the zero. The black trajectories denote other generic flow lines.
• Figure 5: An example flow diagram and its associated triangulation. In (a) we have a global flow diagram on a disc with four marked points on the boundary. The red dots are the zeros of $\phi$ and the associated separating trajectories are the red lines. The gray cells denote one parameter families of generic flows. All flow lines end on the four marked blue dots on the boundary. In (b) we have extracted the associated triangulation. Each black line is a generic flow line selected from each one parameter family. The resulting triangles each contain one zero of $\phi$ by construction.
• Figure 6: Evolution of the special lagrangian flows with the BPS angle $\theta$. In each picture the black dots indicate the branch points of the cover where flows emerge. Red trajectories are flows that emerge from the branch points and terminate on the boundary at $|x|=\infty$, while gray trajectories indicate generic flow lines. The green trajectory denotes a representative of a generic flow line which can serve as an edge in the triangulation. In (b) the BPS angle of the flow aligns with the phase of the central charge and a new kind of trajectory, shown in blue, traverses between branch points. Afterwards in (c) the green line has flipped.
• Figure 7: A bivalent puncture in the triangulation gives rise to a two-cycle in $Q$. The blue denotes a patch of $\mathcal{C}$. Red lines indicate diagonals and marked points are punctures. The nodes of the quiver for the two indicated diagonals are drawn. The bivalent puncture implies that there is a two cycle in the quiver indicated by the black arrows.