On 5d SCFTs and their BPS quivers. Part I: B-branes and brane tilings

TL;DR

This work develops a comprehensive BPS-quiver framework for 5d SCFTs compactified on a circle, identifying the BPS category with the derived category D^b( hat{X}) of D-branes on a resolved CY_3 singularity and deriving 5d BPS quivers as fractional-brane quivers. For toric X, brane tilings provide explicit constructions of the 5d BPS quiver and Beilinson quivers, enabling detailed analysis of rank-one and rank-two theories through geometric engineering and Higgsing, and revealing how UV dualities correspond to mutations and autoequivalences of the BPS category. The paper tests conjectures on BPS spectra in tame chambers, elucidates the structure of electric subcategories, and demonstrates that 4d KK theories inherit well-controlled BPS substructures from their 5d origins. Together, these results link M-theory/M-theory uplifts, geometric engineering, and category-theoretic methods to systematically study the BPS spectrum and dualities of 5d SCFTs and their 4d reductions. The framework offers a tractable path to compute BPS spectra, study dualities, and explore higher-rank and non-Lagrangian theories via quivers, mutations, and tilting techniques, with potential applications to partition functions and wall-crossing invariants.

Abstract

We study the spectrum of BPS particles on the Coulomb branch of five-dimensional superconformal field theories (5d SCFTs) compactified on a circle. By engineering these theories in M-theory on ${\mathbf X} \times S^1 $, for ${\mathbf X}$ an isolated Calabi-Yau threefold singularity, we naturally identify the BPS category of the 5d theory on a circle with the derived category of coherent sheaves on a resolution of ${\mathbf X}$. It follows that the BPS spectrum can be studied in terms of 5d BPS quivers, which are the fractional-brane quivers for the singularity ${\mathbf X}$. 5d BPS quivers generalize the well-studied 4d BPS quivers for 4d $\mathcal{N}{=}2$ gauge theories that can be obtained from ${\mathbf X}$ in so-called geometric engineering limits. We study the interplay between 4d and 5d BPS quivers in detail. We particularly focus on examples when ${\mathbf X}$ is a toric singularity, in which case the 5d BPS quiver is given in terms of a brane tiling. For instance, the well-studied $Y^{p,q}$ brane tiling gives a 5d BPS quiver for the $SU(p)_q$ 5d gauge theory. We present a conjecture about the structure of the BPS spectra of a wide class of models, which we test in the simple case of the 5d $SU(2)_0$ theory (more precisely, the $E_1$ SCFT). We also argue that 5d UV dualities can be realized in terms of mutation sequences on the BPS quivers, which are in turn interpreted as autoequivalences of the BPS category.

Figures (28)

• Figure 1: Schematic representation of the quantum Schottcky problem: not all the possible assignments of formal stability conditions are physically realized. Since the space of stability conditions is divided into BPS chambers separated by walls of marginal stability, and since the physical central charges are on a locus of positive codimension, assigning an arbitrary central charge might lead to a formal BPS chamber which is not physically realized. Determining such a space from first principles is an interesting open problem.
• Figure 2: Some examples of toric diagrams.
• Figure 3: Conifold singularity, associated brane tiling and quiver. The brane tiling is a double-periodic tiling on the plane or, equivalently, a tiling of the torus. The torus fundamental domain is shown is red.
• Figure 4: Toric diagrams for the toric del Pezzo singularities.
• Figure 5: Toric diagram and brane tiling for the $dP_3$ geometry. The perfect matchings are associated to toric divisors as indicated on the toric diagram. The brane tiling consists of 6 faces, 14 edges and 8 vertices (4 white and 4 black); the torus fundamental domain is shown in red. The perfect matching $p_{13}$ is shown in light blue.