Mathematical Sciences: The Topology of Lattice Gauge Fields

点击翻译按钮获取中文摘要

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.