Higher Poincare lemma and integrability

We prove the non-abelian Poincare lemma in higher gauge theory in two different ways. That is, we show that every flat local connective structure is gauge trivial. The first method uses a result by Jacobowitz [J. Differ. Geom. 13, 361 (1978)] which states solvability conditions for differential equations of a certain type. The second method extends a proof by Voronov [Proc. Am. Math. Soc. 140, 2855 (2012)] and yields the explicit gauge parameters connecting a flat local connective structure to the trivial one. Finally, we show how higher flatness appears as a necessary integrability condition of a linear system which featured in recently developed twistor descriptions of higher gauge theories. (C) 2015 AIP Publishing LLC.