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

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 X × S¹, for 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 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 X. 5d BPS quivers generalize the well-studied 4d BPS quivers for 4d 𝒩 = 2 gauge theories that can be obtained from in so-called geometric engineering limits. We study the interplay between 4d and 5d BPS quivers in detail. We particularly focus on examples when 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,d brane tiling gives a 5d BPS quiver for SU (p)_q the 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 E₁ 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.