On brane systems with O${}^+$ planes -- 5d and 6d SCFTs

TL;DR

The paper develops a framework to study Higgs branches of 5d and 6d 8-supercharge theories realized on brane webs in the presence of orientifold planes, focusing on O7^+ and O8^+. By deriving magnetic quivers and formalizing how the O7^+ modifies stable intersections, it identifies non-simply-laced MQs and relates FI deformations to mass deformations, enabling phase diagrams and HWGs to reveal global flavor symmetries. The construction connects 6d (1,0) SCFTs to 5d fixed points via twisted compactification and folding, and demonstrates consistency across brane-web moves and MQ manipulations. A notable result is the invariant-geometry rule SI_{O7^+[u,v]} that captures how O7^+ monodromies affect intersections, and the appearance of a D_4 slice in the 6d tensionless limit, suggesting rich structure for Higgs branches in 5d/6d theories with orientifolds.

Abstract

We study Higgs branches of field theories with 8 supercharges in 5 and 6 dimensions, focusing on theories realised on 5-brane webs in Type IIB with an O$7^+$ plane, or a D6-D8-NS5 brane system in Type IIA in the presence of an O$8^+$ plane. We find magnetic quivers for the Higgs branches of these theories. The main consequence of the presence of the orientifold is that it renders the magnetic quiver to be non-simply-laced. We propose a contribution of the O$7^+$ to the usual stable intersection number of 5-branes from tropical geometry, and show that it is consistent with Fayet-Iliopoulos deformations of magnetic quivers which represent mass deformations of 5d SQFTs. From the magnetic quivers, we compute phase diagrams and highest weight generating functions for the Higgs branches, enabling us to identify the global form of the flavour symmetry for several families of 5d SQFTs; among them Bhardwaj's rank-1 theory. For 6d theories realised on a $-4$ curve, we observe the appearance of an additional $D_4$ slice on top of the phase diagram as one goes to the tensionless limit.