Abstract
Several criteria are known for determining which connectionsA are determined uniquely by their curvatureF, or byF and its covariant derivatives. On a principal bundle with semi-simple gauge groupG over a 4-manifoldM, a sufficient condition forF to determineA uniquely is that the linear mapB → [F ∧B] from Lie algebra-valued 1-forms to 3-forms (pulled back toM via a local gauge) be invertible on an open dense set inM. Recently F. A. Doria has claimed that this condition is also necessary. We present counterexamples to this claim, and also to his assertion thatF determinesA uniquely if the restriction of the bundle to every open subset ofM has holonomy group equal toG andF is “not degenerate as a 2-form over spacetime.”
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References
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. Sci. Sin.20, 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)
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, H. B.: Field strength and dual variable formulations of guage 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 (Unpublished notes)
Mostow, M. A.: The field copy problem: to what extent do curvature (gauge field) and its covariant derivatives determine connection (gauge potential)? Commun. Math. Phys.78, 137–150 (1980)
Mostow, M. A., Shnider, S.: Does a generic connection depend continuously on its curvature? (to appear in Commun. Math. Phys.).
Roskies, R.: Uniqueness of Yang-Mills potentials. Phys. Rev.D15, 1731–1732 (1977)
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)
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Communicated by A. Jaffe
This research was supported in part by N. S. F. grant MCS80-03419 (first author) and by an NSERCC operating grant (second author)
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Mostow, M.A., Shnider, S. Counterexamples to some results on the existence of field copies. Commun.Math. Phys. 90, 521–526 (1983). https://doi.org/10.1007/BF01216183
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DOI: https://doi.org/10.1007/BF01216183