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The Chern classes of Sobolev connections

Authors:	Karen K Uhlenbeck

Institution:	(1) Department of Mathematics, University of Chicago, 60637 Chicago, Illinois, USA

Abstract:	AssumeF is the curvature (field) of a connection (potential) onR ⁴ with finiteL ² norm $\left( {\int\limits_{R^4 } {\left\| F \right\|^2 dx< \infty } } \right)$ . We show the chern number $c_2= {1 \mathord{\left/ {\vphantom {1 8}} \right. \kern-\nulldelimiterspace} 8}\pi ^2 \int\limits_{R^4 } {F \wedge} F$ (topological quantum number) is an integer. This generalizes previous results which showed that the integrality holds forF satisfying the Yang-Mills equations. We actually prove the natural general result in all even dimensions larger than 2.

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