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.
Highlights
Calabi-Yau three-fold singularity gives rise to a 5d SCFT, whereas resolving the singularity, i.e. Kähler deformations, correspond to the Coulomb branch, of vacuum expectation values of adjoint scalars in the vector multiplet
We propose in this paper a reverse approach: utilizing the map of 5-brane-webs to generalized toric polygons (GTP), or dot-diagrams [25, 26, 41, 42], we identify the rules for determining the magnetic quiver and Hasse diagrams in terms of the polygons
More importantly, we hope this provides the first step to understanding the Higgs branch from a geometric point of view: when the GTP is a conventional toric diagram, there is a known map to an actual Calabi-Yau geometry
Summary
In this paper we consider 5d SCFTs defined by so-called generalized toric polygons (GTP). The GTPs are lattice polygons, which generalize the concept of a toric fan for a Calabi-Yau three-fold They map one-to-one to a 5-brane-web WP (which in the toric case corresponds equivalently to a tropical geometry). The conjecture in [31] is that the Coulomb branch of the magnetic quiver MQP associated to a GTP P can be identified with the Higgs branch of the 5d SCFT. The elementary transverse slices of the Hasse diagram of the 5d Higgs branches for the theories we consider here can be closures of minimal nilpotent orbits g or Kleinian singularities — see table 1, which includes their magnetic quivers [51, 52]. Rules — which simplify the brane-web based analysis, and connect to the known geometric constructions in toric geometry
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