四维流形上的Chern-Simons预量子化
Chern-Simons pre-quantization over four-manifolds
摘要：
本文在四维流形上G-主丛的联络空间上引入了一个预辛结构，并在其上引入了一个在边界上平凡的规范变换群的Hamilton作用.矩映射由曲率的平方给出，使得0-水平集是平坦连通的空间。因此，平坦联络的模空间被赋予了一种预辛结构。在四流形为零协边的情况下，我们将在联络的模空间上以及在平坦联络的模空间上构造一个厄米特线丛，其曲率由前辛形式给出。这是陈-西蒙斯前量子线丛。基流形边界上的规范变换群以无穷小辛方式作用于平坦联络的模空间。当基流形是一个4维圆盘时，我们证明了这个作用量通过它的阿贝尔扩张提升到前量子线丛。后者的几何描述与四维Wess-Zumino-Witten模型有关。这个arxiv文本的前一个版本与微分几何及其应用vol.29中发表的文章有几处不一致，因此作者更正了它们。
英文摘要：
We introduce a pre-symplectic structure on the space of connections in a G-principal bundle over a four-manifold and a Hamiltonian action on it of the group of gauge transformations that are trivial on the boundary. The moment map is given by the square of curvature so that the 0-level set is the space of flat connections. Thus the moduli space of flat connections is endowed with a pre-symplectic structure. In case when the four-manifold is null-cobordant we shall construct, on the moduli space of connections, as well as on that of flat connections, a hermitian line bundle with connection whose curvature is given by the pre-symplectic form. This is the Chern-Simons pre-quantum line bundle. The group of gauge transformations on the boundary of the base manifold acts on the moduli space of flat connections by an infinitesimally symplectic way. When the base manifold is a 4-dimensional disc we show that this action is lifted to the pre-quantum line bundle by its abelian extension. The geometric description of the latter is related to the 4-dimensional Wess-Zumino-Witten model. The previous version of this arxiv text had several incoincidence with the published article in the Differential Geometry and its Applications vol.29, so the author corrected them.
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