论文标题
BF理论的3个关于同源性的定义3秒
3 Definitions of BF Theory on Homology 3-Spheres
论文作者
论文摘要
具有量规组的3维BF理论$ g $（= chern-simons理论，具有非紧凑型量规$ tg $）是一种看似简单而微妙的拓扑仪表理论。正式地，其分区函数是模量空间$ \ MATHCAL {m} $的平面连接的总和/积分，并由射线手扭转加权。但是，实际上，这种形式的表达几乎总是奇异的且定义不明的。 为了改善这一点，我们直接评估了某些三个序列类别的路径积分（即积分和理性的Seifert同源性领域）。通过适当的量规选择，我们避开了必须集成在$ \ Mathcal {M} $上并将分区函数减少到有限维的Abelian Matrix积分的问题，但是，该功能本身需要定义。我们提供了3个整体定义，首先是通过残基，然后通过相应的$ g \ times g $或$ g_c $ g_c $ chern-simons矩阵积分（以前获得）的$ k $限制。然后，我们检查并讨论结果在何种程度上捕获所有平坦连接上的预期总和/积分。
3-dimensional BF theory with gauge group $G$ (= Chern-Simons theory with non-compact gauge group $TG$) is a deceptively simple yet subtle topological gauge theory. Formally, its partition function is a sum/integral over the moduli space $\mathcal{M}$ of flat connections, weighted by the Ray-Singer torsion. In practice, however, this formal expression is almost invariably singular and ill-defined. In order to improve upon this, we perform a direct evaluation of the path integral for certain classes of 3-manifolds (namely integral and rational Seifert homology spheres). By a suitable choice of gauge, we sidestep the issue of having to integrate over $\mathcal{M}$ and reduce the partition function to a finite-dimensional Abelian matrix integral which, however, itself requires a definition. We offer 3 definitions of this integral, firstly via residues, and then via a large $k$ limit of the corresponding $G\times G$ or $G_C$ Chern-Simons matrix integrals (obtained previously). We then check and discuss to which extent the results capture the expected sum/integral over all flat connections.
