Quantization of the Chem-Simons Gauge Theory on Four-manifolds

本文作者从2004年到2006年研究了四维流形上的Chern-Simons规范理论的量子化，给出了一般带边界的四维流形上联络的模空间的几何量子化。他给出了连通模空间上的预辛结构。他在模空间上构造了一个具有连通性的厄米线丛，其曲率由预辛形式给出。线束的跃迁函数用5维的Chern-Simons泛函描述。在联络空间上，存在边界上一致的规范变换群的哈密顿作用量，其辛约化为平坦联络空间，这就是平坦联络空间的几何量子化。从边界三维流形到结构李群的映射群在这个平坦联络的模空间上以无限辛方式作用，并且证明了映射群的阿贝尔扩张将作用提升到量子线丛。最后一个扩展是由Mickelsson构造的，并通过作者先前对四维Wess-Zumino-Witten理论的研究而扩展。研究结果发表在《微分几何》杂志上。在这一研究之后，记者还研究了三维流形上平坦联络空间上的拟辛结构，以及联络扩张到四维流形的条件，并确定了这类联络的类别。此外，作者还研究了哈密顿-杨-米尔斯方程的涡旋表示及其与哈密顿流螺旋度的关系。

The reporter of this note studied from the year 2004 to 2006 the quantization of the Chern-Simons gauge theory on four-manifolds, he gave the geometric quantization of the moduli space of connections on a four-manifolds generally with boundary. He gave a pre- symplectic structure on the moduli-space of connections. He constructed an hermitian line bundle with connection on the moduli space whose curvature is given by the pre-symplectic form. The transition function of the line bundle is described by the 5- dimensional Chern-Simons functional. On the space of connections there is a Hamiltonian action of the group of gauge transformations that are identity on the boundary, whose symplectic reduction becomes the space of flat connections, this is the geometric quantization of the space of flat connections. The mapping group from the boundary three-manifold to the structure Lie group acts infinitesimally symplectic way on this moduli space of flat connections, and the author showed that the abelian extension of the mapping group lifts the action to the quantium line bundle. The last extension was constructed by Mickelsson and extended by the author's previous research on four-dimensional Wess-Zumino-Witten theory. The results were submitted to a journal of differential geometry. After this research the reporter investigated also the quasi-symplectic structure on the space of flat connections on three- manifolds and the condition for a connection to be extended to the four-manifold that cobord the first one and decided the class of such connections. Other than these research the author investigated the vortex representation of the Hamilton-Yang-Mills equation and the relation of it to the helicity of Hamiltonian flows.