Classification of singular Sobolev connections by their holonomy. (English) Zbl 0747.53024
In topological quantum dynamics several authors have encountered the existence of non-integral charge, which depend critically on holonomy [see, for example, M. F. Atiyah, Vector bundles on algebraic varieties, Pap. Colloq., Bombay 1984, Stud. Math., Tata Inst. Fundam. Res. 11, 1-33 (1987; Zbl 0688.53027), P. Forgacs, Z. Horvath and L. Palla, Z. Phys. C 12, 359-360 (1982) and P. J. Braam, J. Differ. Geom. 30, No. 2, 425-464 (1989; Zbl 0689.53028)]. The present work goes some way in clarifying and classifying the situation. A limit holonomy condition is stated for a connection on a principal \(SU(2)\)- bundle over a base space with a codimension two singular set. It is shown, that the condition is satisfied in dimension four for finite actions and the singularity can be classified in terms of holonomy. A codimension two singularity theorem is shown to hold in the absence of holonomy.
Reviewer: C.S.Sharma (London)
MSC:
| 53C07 | Special connections and metrics on vector bundles (Hermite-Einstein, Yang-Mills) |
| 81T13 | Yang-Mills and other gauge theories in quantum field theory |
| 57N13 | Topology of the Euclidean \(4\)-space, \(4\)-manifolds (MSC2010) |
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