Higher Structures, Self-Dual Strings and 6d Superconformal Field Theories

I summarize and discuss some recent results on formulating actions of six-dimensional superconformal field theories using the language of higher gauge theory. The latter guarantees mathematical consistency of our constructions and we review crucial aspects of this framework, such as $L_\infty$-algebras and corresponding kinematical data given by higher connections. We then show that there is a mathematically consistent non-Abelian extension of the self-dual string equation which satisfies many physical expectations. Our construction favors a particular higher gauge group leading us to higher principal bundles known as string structures. Using these, we manage to formulate a six-dimensional action which shares many properties with the famous $(2,0)$-theory but also still differs from it in some key points.