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We consider local Sobolev connections on SU(2) bundles over the complement, in R, of a smoothly embedded compact 2-manifold. Finite action implies that a holonomy condition is satisfied and we obtain an a priori estimate for the connection 1-form in terms of curvature and the flat connection carrying the holonomy. The a priori estimate classifies the possible singularities in these connections by the set of flat connections. In a certain case, this leads to smoothness and extendability results. Let N be a full 4-dimensional neighborhood of the singular set S. The objects of study are connections D = d + A defined on SU(2) bundles over X = N\S. We assume that A € Hloc(X) and that the action is finite, i.e., the curvature F = dA + A A A is in L(N). The following holonomy condition was first stated by Cliff Taubes. Choose coordinates (r, 0, u, v) with (u, v) coordinates on S and (r, 9) coordinates in a plane normal to S. Fixing u and v, and denoting by Ae the 6 component of A, the initial value problem for an SU(2) valued function,

Singular Sobolev connections with holonomy