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We construct manifestly superconformal field theories in six dimensions whichcontain a non-Abelian tensor multiplet. In particular, we show how principal3-bundles over a suitable twistor space encode solutions to these self-dualtensor field theories via a Penrose-Ward transform. The resulting higher orcategorified gauge theories significantly generalise those obtained previouslyfrom principal 2-bundles in that the so-called Peiffer identity is relaxed in asystematic fashion. This transform also exposes various unexplored structuresof higher gauge theories modelled on principal 3-bundles such as the relevantgauge transformations. This enables us to formulate the non-Abeliandifferential cohomology that describes principal 3-bundles with connectivestructures.

Six-Dimensional Superconformal Field Theories from Principal 3-Bundles over Twistor Space