Six-dimensional superconformal field theories from principal 3-bundles over twistor space

We construct manifestly superconformal field theories in six dimensions which contain a non-Abelian tensor multiplet. In particular, we show how principal 3-bundles over a suitable twistor space encode solutions to these self-dual tensor field theories via a Penrose-Ward transform. The resulting higher or categorified gauge theories significantly generalise those obtained previously from principal 2-bundles in that the so-called Peiffer identity is relaxed in a systematic fashion. This transform also exposes various unexplored structures of higher gauge theories modelled on principal 3-bundles such as the relevant gauge transformations. We thus arrive at the non-Abelian differential cohomology that describes principal 3-bundles with connective structure.