非交换高规范理论与范畴丛
Non-abelian higher gauge theory and categorical bundle
摘要：
一个规范理论与一个主丛相联系，主丛具有一个允许定义路径水平提升的联系。表面的水平升力不能被定义为主束结构。高规范理论试图推广丛结构，以描述表面的水平升力。这样的尝试对于非阿贝尔的情况是特别困难的。已经提出了一些结构来实现这一目标（扭曲束，gerbes与连接，束gerbe，2束）。它们中的每一个都用一个范畴来代替通常的主丛结构的全空间流形。他们中的一些人也取代结构群的范畴（更确切地说，一个李交叉模视为一个范畴）。但是基空间仍然是一个简单的流形（可能被看作是一个只有单位元箭头的平凡范畴）。我们提出了一个新的主范畴丛结构，一个李交叉模作为结构广群，但与一个基空间属于一个更大的类别（其中包括非平凡范畴），我们称之为仿射2-空间。我们研究了建立在这些范畴上的范畴丛的几何结构（这是一种比2-丛更复杂的结构）以及这些范畴丛上的联结结构。最后，我们处理一个有趣的例子，量子动力学是与布洛赫波算子理论。
英文摘要：
A gauge theory is associated with a principal bundle endowed with a connection permitting to define horizontal lifts of paths. The horizontal lifts of surfaces cannot be defined into a principal bundle structure. An higher gauge theory is an attempt to generalize the bundle structure in order to describe horizontal lifts of surfaces. A such attempt is particularly difficult for the non-abelian case. Some structures have been proposed to realize this goal (twisted bundle, gerbes with connection, bundle gerbe, 2-bundle). Each of them uses a category in place of the total space manifold of the usual principal bundle structure. Some of them replace also the structure group by a category (more precisely a Lie crossed module viewed as a category). But the base space remains still a simple manifold (possibly viewed as a trivial category with only identity arrows). We propose a new principal categorical bundle structure, with a Lie crossed module as structure groupoid, but with a base space belonging to a bigger class of categories (which includes non-trivial categories), that we call affine 2-spaces. We study the geometric structure of the categorical bundles built on these categories (which is a more complicated structure than the 2-bundles) and the connective structures on these bundles. Finally we treat an example interesting for quantum dynamics which is associated with the Bloch wave operator theory.
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