Connections on discrete fibre bundles
Considering the unit 3-sphere S 3 as the set of ordered pairs of complex numbers \((z_ 1,z_ 2)\) with \(| z_ 1|\) \(2+| z_ 2|\) \(2=1\), the multiplication by complex numbers of unit magnitude: \((z_ 1,z_ 2)\to (z_ 1e^{i\theta},z_ 2e^{i\theta})\) (\(\theta\in R)\), defines an action of U(1) on S 3, which gives us the U(1) Hopf bundle S \(3\to S\) 2. Replacing the complex numbers by quaternions, the similar consideration gives us the SU(2) Hopf bundle S \(7\to S\) 3. The author tries to define the discretization D of the U(1) Hopf bundle by taking 24 points in S 3, having the coordinates \((\pm 1,0)\), \((0,\pm 1)\), \((0,\pm i)\), \((\pm i,0)\), \((\pm 1\pm i,\pm 1\pm i)/2\), which are the vertices of a regular self-dual polytope in R 4. The gauge group is the \(Z_ 4\) subgroup of U(1) and the base space B therefore has six points. The author next defines neighbors of each point in D, the section of the bundle \(D\to B\), a connection of this bundle and the holonomy of a connection. The author asserts that the description of a connection can be related to Wilson's description of a lattice gauge field, where the element of the gauge group is associated to each link in the base space. Similar considerations can be applied for a discrete SU(2) Hopf bundle.
scientific article; zbMATH DE number 4040753