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Two highly successful approaches to constructing 5d SCFTs are geometric engineering using M-theory on a Calabi-Yau 3-fold and the use of 5-brane webs suspended from 7-branes in Type IIB string theory. In the brane web realization, the extended Coulomb branch of the 5d SCFT can be studied by opening the web using rigid triple intersections of branes — i.e. configurations with no deformations. In this paper, we argue that the geometric engineering counterpart of these rigid triple intersections are the T-cones introduced in the mathematical literature. We extend the class of rigid brane webs to include locked superpositions of the minimal ones. These rigid brane webs serve as fundamental building blocks for supersymmetrically tessellating Generalized Toric Polygons (GTPs) from first principles. Interestingly, we find that the extended Coulomb branch generally exhibits a structure consisting of multiple cones intersecting at a single point. Hanany-Witten (HW) transitions in the web have been conjectured to correspond geometrically to flat fibrations over a line, where the central and generic fibers represent the geometries dual to the webs before and after the transition. We demonstrate this explicitly in an example, showing that for GTPs reducing to standard toric diagrams, the HW transition corresponds to a deformation of the BPS quiver that we map to the geometric deformation. 

A new type of quiver theories, denoted twin quivers, was recently introduced for studying 5d SCFTs engineered by webs of 5-branes ending on 7-branes. Twin quivers provide an alternative perspective on various aspects of such webs, including Hanany-Witten moves and the s-rule. More ambitiously, they can be regarded as a first step towards the construction of combinatorial objects, generalizing brane tilings, encoding the corresponding BPS quivers. This paper continues the investigation of twin quivers, focusing on their non-uniqueness, which stems from the multiplicity of toric phases for a given toric Calabi-Yau 3-fold. We find that the different twin quivers are necessary for describing what we call quiver tails, which in turn correspond to certain sub-configurations in the webs. More generally, the multiplicity of twin quivers captures the roots of the Higgs branch in the extended Coulomb branch of 5d theories. 

A<sc>bstract</sc> We make progress in understanding the geometry associated to the Generalized Toric Polygons (GTPs) encoding the Physics of 5d Superconformal Field Theories (SCFTs), by exploiting the connection between Hanany-Witten transitions and the mathematical notion of polytope mutations. From this correspondence, it follows that the singular geometry associated to a GTP is identical to that obtained by regarding it as a standard toric diagram, but with some of its resolutions frozen in way that can be determined from the invariance of the so-called period under mutations. We propose the invariance of the period as a new criterion for distinguishing inequivalent brane webs, which allows us to resolve a puzzle posed in the literature. A second mutation invariant is the Hilbert Series of the geometry. We employ this invariant to perform quantitative checks of our ideas by computing the Hilbert Series of the BPS quivers associated to theories related by mutation. Lastly, we discuss the physical interpretation of a mathematical result ensuring the existence of a flat fibration over ℙ¹interpolating between geometries connected by mutation, which we identify with recently introduced deformations of the corresponding BPS quivers.