(Symplectic) Leaves and (5d Higgs) Branches in the Poly(go)nesian Tropical Rain Forest

TL;DR

This paper develops a geometric approach using generalized toric polygons and tropical curves to analyze the Higgs branch structure of 5d superconformal theories, revealing symplectic leaves and foliation structures.

Contribution

It introduces a novel method linking tropical curve decomposition and Minkowski sums to compute magnetic quivers and Higgs branch structures in 5d theories.

Findings

Decomposition of Higgs branches into symplectic leaves.

Identification of Coulomb and Higgs branches via magnetic quivers.

Reduction to Calabi-Yau deformations for toric polygons.

Abstract

We derive the structure of the Higgs branch of 5d superconformal field theories or gauge theories from their realization as a generalized toric polygon (or dot diagram). This approach is motivated by a dual, tropical curve decomposition of the $(p,q)$ 5-brane-web system. We define an edge coloring, which provides a decomposition of the generalized toric polygon into a refined Minkowski sum of sub-polygons, from which we compute the magnetic quiver. The Coulomb branch of the magnetic quiver is then conjecturally identified with the 5d Higgs branch. Furthermore, from partial resolutions, we identify the symplectic leaves of the Higgs branch and thereby the entire foliation structure. In the case of strictly toric polygons, this approach reduces to the description of deformations of the Calabi-Yau singularities in terms of Minkowski sums.