Notes on Supersymmetric Gauge Theories in Five and Six Dimensions

TL;DR

The paper analyzes consistency conditions for supersymmetric gauge theories in five and six dimensions, reviewing Seiberg's criteria and exploring how higher-dimensional theories relate to lower-dimensional physics through dimensional reduction. It details 5D anomaly-free matter content via the Coulomb-branch prepotential convexity condition and enumerates allowed representations for several gauge groups, while 6D analysis distinguishes groups by quartic Casimir structure and provides explicit anomaly-cancellation solutions. The work then investigates dimensional crossovers, showing that an extra dimension can avert Landau poles for certain 4D theories but not universally, and argues that tensor multiplet mixing does not generally produce a consistent 5D uplift. A brief note references global anomaly constraints refined in later work, illustrating how such refinements further shape the allowed spectrum.

Abstract

We investigate consistency conditions for supersymmetric gauge theories in higher dimensions. First, we give a survey of Seiberg's necessary conditions for the existence of such theories with simple groups in five and six dimensions. We then make some comments on how theories in different dimensions are related. In particular, we discuss how the Landau pole can be avoided in theories that are not asymptotically free in four dimensions, and the mixing of tensor and vector multiplets in dimensional reduction from six dimensions.

Figures (1)

• Figure 1: The effective coupling for $SU(2)$ with ${n_\mathbf{f}}$ fundamental hypermultiplets on ${\bf R}^4 \times S^1$.