Mathematical Sciences: The Topology of Lattice Gauge Fields

数学科学：格子规范场的拓扑

• 批准号: 8907753
• 负责人: Anthony Phillips
• 金额: $ 9.98万
• 依托单位: SUNY at Stony Brook
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 1989
• 资助国家: 美国
• 起止时间: 1989-07-01 至 1993-06-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=8907753&HistoricalAwards=false
• 关键词: Mathematical; Sciences; Topology; Lattice; Gauge

Lattice gauge fields were invented by physicists as discrete models of connections in principal fibre bundles. Anthony Phillips' previous work (with David Stone) on calculations of topological charge led to a general scheme for a local calculation of cocycles representing real characteristic classes. Part of the planned research involves extending this scheme to a wider class of structure groups. Another part involves modification of this scheme to detect U(1)-monopoles in SU(n)- theories in a way that is both reliable and efficient for lattice computations. Phillips also plans to investigate the topology and geometry of H-monopoles in G-bundles for general G including H. Finally, he intends to attack a combinatorial problem, the enumeration of a certain class of mazes which arose in his prior work on lattice gauge fields. Overall, this project employs some of the most modern mathematical tools of differential geometry and topology to analyze a type of structure which arises in one formulation of quantum mechanics.

晶格规范场是物理学家作为主要纤维束连接的离散模型而发明的。安东尼·菲利普斯（Anthony Phillips）之前与大卫·斯通（David Stone）在拓扑电荷计算方面的工作，为表示真实特征类的环的局部计算提供了一个通用方案。计划研究的一部分涉及将该方案扩展到更广泛的结构组类别。另一部分涉及对该方案的修改，以一种既可靠又有效地用于晶格计算的方式检测SU(n)-理论中的U(1)-单极子。Phillips还计划研究G束中包括h的一般G的h单极子的拓扑结构和几何结构。最后，他打算研究一个组合问题，即在他之前关于晶格规范场的工作中出现的某类迷宫的枚举。总的来说，这个项目采用了一些最现代的微分几何和拓扑学数学工具来分析一种出现在量子力学公式中的结构。