The 5d Tangram: Brane Webs, 7-Branes and Primitive T-cones

TL;DR

This paper explores the geometric and brane web constructions of 5d SCFTs, introducing T-cones as the geometric counterparts of rigid brane intersections and analyzing the structure of the Coulomb branch.

Contribution

It extends the class of rigid brane webs, introduces T-cones as their geometric counterparts, and connects Hanany-Witten transitions to geometric deformations in 5d SCFTs.

Findings

Rigid brane webs serve as fundamental building blocks for GTP tessellations.

The Coulomb branch often consists of multiple intersecting cones.

HW transitions correspond to geometric deformations and quiver changes.

Abstract

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.