Higher Poincare Lemma and Integrability

TL;DR

This paper proves a higher-dimensional non-abelian Poincare lemma in gauge theory using two methods, linking flatness conditions to integrability and twistor theory applications.

Contribution

It introduces two novel proofs of the higher Poincare lemma and connects higher flatness to integrability in advanced gauge theories.

Findings

Established the non-abelian Poincare lemma in higher gauge theory.

Connected higher flatness to integrability conditions in linear systems.

Extended twistor descriptions to include higher gauge structures.

Abstract

We prove the non-abelian Poincare lemma in higher gauge theory in two different ways. The first method uses a result by Jacobowitz which states solvability conditions for differential equations of a certain type. The second method extends a proof by Voronov 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.