Abstract
For a principal bundle with semi-simple structure group over a smooth four-dimensional base manifold, the set of connections (gauge potentials)A which are uniquely determined by their curvature (field or field strength)F is generic in the set of all potentials, endowed with the WhitneyC ∞ topology. However, the operator taking each such fieldF to its potentialA is not continuous. Partial negative results are given concerning the existence of a smaller generic set on which this operator is continuous.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Belifante, J.G.F., Kolman, B.: A Survey of Lie groups and Lie algebras. Philadelphia: SIAM 1972
Calvo, M.: Connection between Yang-Mills potentials and their field strengths. Phys. Rev. D15, 1733–1735 (1977)
Dao-xing, X.: On field strengths and gauge potentials of Yang-Mills' fields. Scientia Sinica20, 145–157 (1977)
Deser, S., Drechsler, W.: Generalized gauge field copies. Phys. Lett.86B, 189–192 (1979)
Deser, S., Teitelboim, C.: Duality transforms of abelian and non-abelian gauge fields. Phys. Rev. D13, 1592–1597 (1976)
Deser, S., Wilczek, F.: Non-uniqueness of gauge field potentials. Phys. Lett.65B, 391–393 (1976)
Doria, F.A.: The geometry of gauge field copies. Commun. Math. Phys.79, 435–456 (1981)
Doria, F.A.: Quasi-abelian and fully non-abelian gauge field copies: a classification. J. Math. Phys.22, 2943–2951 (1981)
Flanders, H.: Differential forms with applications to the physical sciences. New York: Academic Press 1963
Golubitsky, M., Guillemin, V.: Stable mappings and their singularities. New York: Springer 1973
Greub, W., Halperin, S., Vanstone, R.: Connections, curvature, and cohomology, Vol. II. New York: Academic Press 1973
Gu, C.-H., Yang, C.-N.: Some problems on the gauge field theories, II. Sci. Sin.20, 47–55 (1977)
Halpern, M.B.: Field strength formulation of quantum chromodynamics. Phys. Rev. D16, 1798–1801 (1977)
Halpern, M.B.: Field strength copies and action copies in quantum chromodynamics. Nucl. Phys. B139, 477–489 (1978)
Halpern, M.B.: Field strength and dual variable formulations of gauge theory. Phys. Rev. D19, 517–530 (1979)
Kobayashi, S., Nomizu, K.: Foundations of differential geometry, Part 1. New York: Interscience 1963
Kugler, M., Castillejo, L.: When does the Yang-Mills field determine the potential uniquely? Unpublished notes
Milnor, J.: Singular points of complex hypersurfaces. Princeton: Princeton Univ. Press 1968
Mostow, M.A.: The field copy problem: to what extent do curvature (gauge field) and it covariant derivatives determine connection (gauge potential)? Commun. Math. Phys.78, 137–150 (1980)
Mostow, M.A., Shnider, S.: Counterexamples to some results on the existence of field copies. Commun. Math. Phys.
Roskies, R.: Uniqueness of Yang-Mills potentials. Phys. Rev. D15, 1731–1732 (1977)
Singer, I.M.: Some remarks on the Gribov ambiguity. Commun. Math. Phys.60, 7–12 (1978)
Solomon, S.: On the field strength-potential connection in non-abelian gauge theory. Nucl. Phys. B147, 174–188 (1979)
Weiss, N.: Determination of Yang-Mills potentials from the field strengths. Phys. Rev. D20, 2606–2609 (1979)
Wu, T.-T., Yang, C.-N.: Some remarks about unquantized non-abelian gauge fields. Phys. Rev. D12, 3843–3844 (1975)
Author information
Authors and Affiliations
Additional information
Communicated by A. Jaffe
Rights and permissions
About this article
Cite this article
Mostow, M.A., Shnider, S. Does a generic connection depend continuously on its curvature?. Commun.Math. Phys. 90, 417–432 (1983). https://doi.org/10.1007/BF01206891
Received:
Revised:
Issue Date:
DOI: https://doi.org/10.1007/BF01206891