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

TL;DR

The paper develops a cohesive geometric framework for 5d N=1 theories via $(p,q)$ Webs, Grid diagrams, and Seiberg-Witten–like curves, enabling direct geometric readouts of vacuum structure, Coulomb and Higgs branches, and the BPS spectrum. It shows how local and global deformations encode gauge couplings and global symmetries, and how monopole tensions and BPS masses emerge from Web areas and string/strip configurations, respectively. The Grid diagrams classify 5d theories, reproduce known pure $SU(2)$ and $SU(N_c)$ results, and expose non-Lagrangian fixed points (E0, etc.) and their enhanced global symmetries. The work also explores flows beyond infinite coupling, including continuations to dual gauge theories and transitions to theories without a gauge description, and connects the brane picture to M-theory/CY geometries while aligning with existing results like Nekrasov’s analyses. Collectively, the framework offers a robust, diagrammatic toolkit for analyzing 5d gauge theories, their circle compactifications, and non-perturbative spectra and dualities.

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.

Figures (25)

• Figure 1: Three basic brane configurations : (a) a vertex, (b) pure $SU(2)$, (c) $N_c=3$, $N_f=2$ SQCD. Henceforth, all figures will be drawn for $\tau=i$.
• Figure 2: The basic BPS states in the pure $SU(2)$ gauge theory -- the W boson and the instanton. The solid lines correspond to a generic brane configuration on the Coulomb branch of this theory, while the dotted lines correspond to the origin of the Coulomb branch.
• Figure 3: Deformations of a $(p,q)$ Web. Dashed lines denote the old configuration.
• Figure 4: A "hidden" face in the $E_0$ Web, realizing the 1d Coulomb branch.
• Figure 5: The Grid diagram for the simple vertex of figure \ref{['basic']}a. Vertices and corresponding polygons are marked a,b,c,..., edges and corresponding lines are marked A,B,C,..., and faces and corresponding points are marked $1, 2, 3, \dots$