Non-toric brane webs, Calabi-Yau 3-folds, and 5d SCFTs

TL;DR

This paper explores the geometric structure of 5-brane webs with 7-branes in Type IIB string theory, linking them to Calabi-Yau surfaces and 5d superconformal field theories, and provides algorithms and constructions for their analysis.

Contribution

It introduces a geometric framework connecting brane webs, tropical curves, and Calabi-Yau surfaces, and develops methods to test web consistency and construct mirror geometries for 5d SCFTs.

Findings

Webs correspond to tropical curves with focus-focus singularities.

An algorithm to test web consistency via dual polygons.

Construction of mirror non-toric Calabi-Yau 3-folds for 5d SCFTs.

Abstract

We study webs of 5-branes with 7-branes in Type IIB string theory from a geometric perspective. Mathematically, a web of 5-branes with 7-branes is a tropical curve in $\mathbb{R}^2$ with focus-focus singularities introduced. To any such a web $W$, we attach a log Calabi-Yau surface $(Y,D)$ with a line bundle $L$. We then describe supersymmetric webs, which are webs defining 5d superconformal field theories (SCFTs), in terms of the geometry of $(Y,D,L)$. We also introduce particular supersymmetric webs called ``consistent webs", and show that any 5d SCFT defined by a supersymmetric web can be obtained from a consistent web by adding free hypermultiplets. Using birational geometry of degenerations of log Calabi-Yau surfaces, we provide an algorithm to test the consistency of a web in terms of its dual polygon. Moreover, for a consistent web $W$, we provide an algebro-geometric construction of the mirror $\mathcal{X}^{\mathrm{can}}$ to $(Y,D,L)$, as a non-toric canonical 3-fold singularity, and show that M-theory on $\mathcal{X}^{\mathrm{can}}$ engineers the same 5d SCFT as $W$. We also explain how to derive explicit equations for $\mathcal{X}^{\mathrm{can}}$ using scattering diagrams, encoding disk worldsheet instantons in the A-model, or equivalently the BPS states of an auxiliary rank one 4d $\mathcal{N}=2$ theory.