Super Cartan geometry and the super Ashtekar connection

This work is devoted to the geometric approach to supergravity. More precisely, we interpret $\mathcal{N}=1$, $D=4$ supergravity as a super Cartan geometry which provides a link between supergravity and Yang-Mills theory. To this end, we first review important aspects of the theory of supermanifolds and we establish a link between various different approaches. We then introduce super Cartan geometries using the concept of so-called enriched categories. This, among other things, will enable us to implement anticommutative fermionic fields. We will then also show that non-extended $D=4$ supergravity naturally arises in this framework. Finally, using this gauge-theoretic interpretation as well as the chiral structure of the underlying supersymmetry algebra, we will derive graded analoga of Ashtekar's self-dual variables and interpret them in terms of generalized super Cartan connections. This gives canonical chiral supergravity the structure of a Yang-Mills theory with gauge supergroup similar to the self-dual variables in ordinary first-order Einstein gravity which was first observed by F\"ul\"op \cite{Fulop:1993wi}. We then construct the parallel transport map corresponding to the super connection in mathematical rigorous way using again enriched categories. This provides the possibility of quantizing gravity and matter degrees of freedom in loop quantum gravity in a unified way.