回路空间和路空间与四维BF理论：联系、完整性和可观测性
Loop and Path Spaces and Four-Dimensional BF Theories: Connections, Holonomies and Observables
摘要：
研究了流形M上自由路（圈）空间为底空间的主G-丛的微分几何。特别地，我们考虑用对（A，B）定义的联络，其中A是固定主丛P（M，G）的联络，B是M上的2-形式。计算了相应的曲率、平行输运和完整性，并给出了它们在局部坐标下的表达式。当2-形式B由A的曲率给出时，则遵循所谓的非阿贝尔斯托克斯公式。对于一般的2-形式B，我们区分了平行运输依赖于路径的整个路径和仅依赖于张成曲面的情况。特别是，我们讨论了非阿贝尔斯托克斯公式的推广。我们还研究了在适当的变换群作用于对（A，B）下完整性（迹）的不变性。用这种方法，我们能够为BF型的拓扑和非拓扑量子场论定义观测量。在非拓扑情况下，表面项可能与理解夸克禁闭问题有关。在拓扑情况下，（微扰）四维量子BF理论预计将产生嵌入（或浸入）表面在一个4-流形M的不变量。
英文摘要：
We study the differential geometry of principal G-bundles whose base space is the space of free paths (loops) on a manifold M. In particular we consider connections defined in terms of pairs (A,B), where A is a connection for a fixed principal bundle P(M,G) and B is a 2-form on M. The relevant curvatures, parallel transports and holonomies are computed and their expressions in local coordinates are exhibited. When the 2-form B is given by the curvature of A, then the so-called non-abelian Stokes formula follows. For a generic 2-form B, we distinguish the cases when the parallel transport depends on the whole path of paths and when it depends only on the spanned surface. In particular we discuss generalizations of the non-abelian Stokes formula. We study also the invariance properties of the (trace of the) holonomy under suitable transformation groups acting on the pairs (A,B). In this way we are able to define observables for both topological and non-topological quantum field theories of the BF type. In the non topological case, the surface terms may be relevant for the understanding of the quark-confinement problem. In the topological case the (perturbative) four-dimensional quantum BF-theory is expected to yield invariants of imbedded (or immersed) surfaces in a 4-manifold M.
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