×

zbMATH — the first resource for mathematics

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

MSC:
53C05 Connections, general theory
57R20 Characteristic classes and numbers in differential topology
81T08 Constructive quantum field theory
PDF BibTeX XML Cite
Full Text: DOI
References:
[1] Uhlenbeck, K.: Removable singularities in Yang-Mills fields. Commun. Math. Phys.83, 11-29 (1982) · Zbl 0491.58032 · doi:10.1007/BF01947068
[2] Parker, T.: Gauge theories on four dimensional Riemannian manifolds. Commun. Math. Phys.85, 563-602 (1982) · Zbl 0502.53022 · doi:10.1007/BF01403505
[3] Witten, E.: Instantons the quark model, and the 1/N expansion. Nucl. Phys. B149, 285-320 (1979) · doi:10.1016/0550-3213(79)90243-8
[4] Schlafly, R.: A Chern number for gauge fields on ?4. J. Math. Phys.23, 1379-1394 (1982) · Zbl 0521.53060 · doi:10.1063/1.525505
[5] Uhlenbeck, K.: Connections withL p bounds on curvature. Commun. Math. Phys.83, 31-42 (1982) · Zbl 0499.58019 · doi:10.1007/BF01947069
[6] Sibner, L. M., Sibner, R. J.: Removeable singularities of coupled Yang-Mills fields in ?3, Commun. Math. Phys.93, 1-17 (1984) · Zbl 0552.35028 · doi:10.1007/BF01218636
[7] Sedlacek, S.: A direct method for minimizing the Yang-Mills functional over 4 manifolds. Commun. Math. Phys.86, 515-528 (1982) · Zbl 0506.53016 · doi:10.1007/BF01214887
[8] Schoen, R., Uhlenbeck, K.: J. Differ. Geom.18, 253-268 (1983)
[9] Schoen, R., Uhlenbeck, K.: J. Differ. Geom.17, 307-335 (1982)
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.