Abstract
In an earlier paper on the field copy problem, we proved that there exists a generic set of connections (gauge potentials) on a principle bundle with a semi-simple structure group over a four-dimensional base manifold for which the connection is uniquely determined by its curvature (gauge field). We conjectured that there exists a smaller, but still generic, set of connections for which the curvature map sending a connection to its curvature admits a continuous inverse with respect to the appropriate function space topologies. The conjecture says, in other words, that restricting to certain generic curvature 2-forms, one can determine and uniformly estimate the connection and its derivatives from the curvature and uniform estimates of its derivatives. In this Letter we give an affirmative answer to the conjecture and show, moreover, that the set of such connections contains an open dense set in the Whitney C ∞ topology.
Similar content being viewed by others
References
Mostow, Mark and Shnider, Steve, ‘Does a Generic Connection Depend Continuously on its Curvature?’, Commun. Math. Phys. 90, 417–432 (1983).
Mostow, Mark and Shnider, Steve, ‘Joint Continuity of Division of Smooth Functions I. Uniform Lojasiewicz Estimates’, Trans. A.M.S. 292, 573–585 (1985).
Mostow, Mark, ‘Joint Continuity of Division II. The Distance to a Whitney Stratified Set from a Transversal Submanifold’, Trans. A.M.S. 292 585–595 (1985).
Golubitsky, M. and Guillemin, V., Stable Mappings and Their Singularities, Springer-Verlag, New York, 1973, 1980.
Trotman, D. J. A., ‘Stability of Transversality to a Stratification Implies Whitney (a)-regularity’, Invent. Math. 50, 273–277 (1979).
Author information
Authors and Affiliations
Rights and permissions
About this article
Cite this article
Mostow, M., Shnider, S. Determining and uniformly estimating the gauge potential corresponding to a given gauge field on M 4 . Lett Math Phys 12, 157–161 (1986). https://doi.org/10.1007/BF00416466
Received:
Issue Date:
DOI: https://doi.org/10.1007/BF00416466