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

TL;DR

The paper develops a geometric program that derives the Higgs branch HB of 5d SCFTs/gauge theories from generalized toric polygons (GTPs) via tropical decompositions of (p,q) 5-brane webs. By coloring edges and performing refined Minkowski-sum decompositions, they construct tropical quivers whose Coulomb branches reproduce HB, with the union over all colorings giving the full HB; partial resolutions yield the Hasse diagram of symplectic leaves. In the strictly toric limit this reduces to Altmann’s Minkowski-sum deformations of Calabi-Yau singularities. A broad suite of examples (including SQCD-like theories, T4, and certain non-Lagrangian models) demonstrates the method, and the authors derive the construction from the brane-web/tropical geometry perspective, also providing a Mathematica tool for automatic MQ computation.

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.

Figures (1)

• Figure 1: Screenshot of the execution of the polyFullAnalysis function on the polygon (\ref{['polyMathematica']}).