Webs of (p,q) 5-branes, Five Dimensional Field Theories and Grid Diagrams

TL;DR

This paper introduces (p,q) Webs, a geometric model for 5d N=1 supersymmetric field theories, enabling simplified calculations and insights into their symmetries, spectra, and vacuum structures.

Contribution

The authors develop (p,q) Webs and Grid Diagrams to geometrically analyze 5d supersymmetric theories, reproducing known classifications and providing new methods to study BPS spectra.

Findings

(p,q) Webs reproduce known 5d field theory classifications.

Grid Diagrams determine Coulomb branch dimensions.

New methods to analyze BPS spectra and states.

Abstract

We continue to study 5d N=1 supersymmetric field theories and their compactifications on a circle through brane configurations. We develop a model, which we call (p,q) Webs, which enables simple geometrical computations to reproduce the known results, and facilitates further study. The physical concepts of field theory are transparent in this picture, offering an interpretation for global symmetries, local symmetries, the effective (running) coupling, the Coulomb and Higgs branches, the monopole tensions, and the mass of BPS particles. A rule for the dimension of the Coulomb branch is found by introducing Grid Diagrams. Some known classifications of field theories are reproduced. In addition to the study of the vacuum manifold we develop methods to determine the BPS spectrum. Some states, such as quarks, correspond to instantons inside the 5-brane which we call strips. In general, these may not be identified with (p,q) strings. We describe how a strip can bend out of a 5-brane, becoming a string. A general BPS state corresponds to a Web of strings and strips. For special values of the string coupling a few strips can combine and leave the 5-brane as a string.