论文标题
超级纸箱几何形状和超级Ashtekar连接
Super Cartan geometry and the super Ashtekar connection
论文作者
论文摘要
这项工作致力于几何方法。更确切地说，我们将$ \ Mathcal {n} = 1 $，$ d = 4 $ Supergravity作为超级纸箱几何形状，提供了超级格雷德和扬米尔斯理论之间的联系。为此，我们首先回顾了超曼菲尔德理论的重要方面，并在各种不同的方法之间建立了联系。然后，我们使用所谓的丰富类别的概念介绍了超级卡坦几何形状。除其他外，这将使我们能够实施抗官方的费米文领域。然后，我们还将证明非扩展的$ d = 4 $超级自然在此框架中出现。最后，使用这种量规理论解释以及基础超对称代数的手性结构，我们将得出Ashtekar的自动偶联变量的分级类似物，并根据广义的超级盒子连接来解释它们。这给出了规范性手性超级重力的结构，Yang-Mills理论的结构具有仪表超组类似于普通一阶爱因斯坦重力中的自偶变量，而Fülöp\ cite {fulop {fulop：1993wi}首先观察到。然后，我们使用再次富集的类别构建了以数学严格的方式构建与超级连接相对应的并对应于超级连接的平行传输图。这提供了以统一的方式量化引力和物质自由度的可能性。
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 Fülöp \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.
