The Chern classes of Sobolev connections. (English) Zbl 0586.53018

Assuming that F is the curvature (field) of a connection (potential) on \(R^ 4\) with finite \(L^ 2\) norm, the author proves that the Chern number \(c_ 2=1/8\pi^ 2\int_{R^ 4}F\wedge F\) (topological quantum number) is an integer. This generalizes previous results which showed that the integrality holds for F satisfying the Yang-Mills equations. Actually, the author proves general even dimensional results. The main idea of the proof is to choose a good gauge near (\(\infty)\). This relies on an earlier theorem of the author on the existence of good (Coulomb) gauges. It should be pointed out that the proof can be shortened considerably for the case of smooth connections.
Reviewer: Y.L.Pan


53C05 Connections (general theory)
57R20 Characteristic classes and numbers in differential topology
81T08 Constructive quantum field theory
Full Text: DOI


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