High gauge theory with string 2-groups and higher Poincare lemma

Abstract

This thesis is concerned with the mathematical formulations of higher gauge theory. Firstly, we develop a complete description of principal 2-bundles with string 2-group model of Schommer-Pries, which is obtained by defining principal smooth 2-group bundles as internal functors in the weak 2-category Bibun of Lie groupoids, right principal smooth bibundles and bibundle maps. Furthermore, this formalism allows us to construct the known string Lie 2-algebra by differentiating this model of the string 2-group. Generalizing the differentiation process, we provide Maurer- Cartan forms leading us to higher non-abelian Deligne cohomology, encoding the kinematical data of higher gauge theory together with their (finite) gauge symmetries. Secondly, 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 in strict principal 2-bundles is gauge trivial. The first proof is based on the result by Jacobowitz, which explains solvability conditions for equations of differential forms. The second is an extension of a proof by T. Voronov and yields the explicit gauge parameters connecting a flat local connective structure to the trivial one. Finally, we develop a method that shows how higher flatness appears as a necessary integrability condition of a linear system by translating the usual matrix product into categorised settings. Moreover, we comment how this notion can be also generalized to the case of higher principal bundles with connective structures.