Higgs branches of 5d rank-zero theories from geometry

TL;DR

We address the problem of determining Higgs branches for 5d $\mathcal{N}=1$ rank-zero theories arising from M-theory on non-toric Calabi–Yau threefolds (Reid's pagodas and Laufer-type geometries). The main approach reduces the problem to IIA with D6/O6 setups, analyzes open-string spectra, and employs T-brane backgrounds to access bound states, revealing discrete gauging and HB geometries as orbifolds of quaternionic spaces. Key contributions include explicit Higgs-branch formulas for Reid's pagodas $\mathcal{H}(\mathrm{RP}_k)=\mathbb{H}^k/\mathbb{Z}_k$, Brown–Wemyss/Laufer cases $\mathcal{H}=\mathbb{H}^k\times\mathbb{H}^{2k+2}/\mathbb{Z}_2\times\mathbb{H}/\mathbb{Z}_2$ (with flavor group $U(1)$), and the non-simple flop example giving $\mathcal{H}=\mathbb{H}^3$, plus discussions of T-brane data yielding multiple HB possibilities. The work broadens Higgs-branch geometry beyond toric examples, connecting M-/IIA data to geometric moduli and discrete gaugings, and points to future links with monodromy and D6-brane worldvolume physics on Riemann surfaces.

Abstract

We study the Higgs branches of five-dimensional N=1 rank-zero theories obtained from M-theory on two classes non-toric non-compact Calabi-Yau threefolds: Reid's pagodas, and Laufer's examples. Our approach consists in reducing to IIA with D6-branes and O6-planes, and computing the open-string spectra giving rise to hypermultiplets. Starting with the seven-dimensional worldvolume theories, we switch on T-brane backgrounds to give rise to bound states with angles. We observe that the resulting partially Higgsed 5d theories have discrete gauge groups, from which we readily deduce the geometry of the Higgs branches as orbifolds of quaternionic varieties.