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

四流形上 Chem-Simons 规范理论的量化

基本信息

批准号：
16540084
负责人：
KORI Toshiaki
金额：
$ 1.98万
依托单位：
Waseda University
依托单位国家：
日本
项目类别：
Grant-in-Aid for Scientific Research (C)
财政年份：
2004
资助国家：
日本
起止时间：
2004 至 2006
项目状态：
已结题

项目摘要

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.

本文作者从2004年到2006年研究了四流形上的chen - simons规范理论的量子化，给出了一般有边界的四流形上的连接模空间的几何量子化。他在连接的模空间上给出了一个预辛结构。他在模空间上构造了一个带连接的厄米线束，其曲率由前辛形式给出。线束的跃迁函数用5维陈-西蒙斯泛函来描述。在连接空间上存在边界上恒等的规范变换群的哈密顿作用，其辛约化成为平坦连接空间，这就是平坦连接空间的几何量化。从边界三流形到结构李群的映射群在平面连接模空间上以无限辛的方式作用，并证明了映射群的阿贝尔扩展将这种作用提升到量子线束。最后一个扩展是由Mickelsson构建的，并由作者之前对四维Wess-Zumino-Witten理论的研究进行了扩展。研究结果发表在一本微分几何杂志上。在此基础上，还研究了三流形平面连接空间上的拟辛结构，以及一种连接扩展到与第一种连接相连的四流形的条件，并确定了这种连接的类别。除此之外，作者还研究了哈密顿-杨-米尔斯方程的涡表示及其与哈密顿流螺旋度的关系。

项目成果

期刊论文数量（7）

专著数量（0）

科研奖励数量（0）

会议论文数量（0）

专利数量（0）

Yang-Mills方程式のハミルトン形式とHelicity

Yang-Mills 方程的哈密顿形式和螺旋度

DOI：
发表时间：
2005
期刊：
京都大学数理解析研究所講究録[力学系と微分幾何] 1408
影响因子：
0
作者：
K.Kiyohara;J.Itoh;郡敏昭
通讯作者：
郡敏昭

Yang-Mills 方程式のハミルトン形式

Yang-Mills 方程的哈密顿形式

DOI：
发表时间：
2004
期刊：
数理解析研究所講究録 1408
影响因子：
0
作者：
X.Chen;M.Fukushima;Yoshiyuki Ohyama;Masayo Fujimura;M. Yamasaki;Masayuki Yamasaki;山崎正之;山崎正之;Masayuki Yamasaki;山崎正之;Masayuki Yamasaki;Kiyoko Nishizawa;T.Kori;郡敏昭
通讯作者：
郡敏昭

Hamiltonian formalism of Yang-Mills equation : vortex formula.

Yang-Mills 方程的哈密顿形式主义：涡旋公式。