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On the Atiyah-Drinfeld-Hitchin-Manin construction for self-dual gauge fields

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Abstract

The vector spacesA, B, C, in terms of which the general construction due to Atiyah, Drinfeld, Hitchin and Manin for self-dual gauge fields defined over some region of Euclidean space is phrased, are shown to be expressible in terms of the spaces spanned by the solutions of certain linear covariant differential equations depending on the gauge field. The corresponding linear maps betweenA andB, B andC are given with the properties required by ADHM and the results then necessary to verify the construction informally proved. The local problems associated with assuming the gauge field to obey the self-duality equations are separated from the global problems of assuring the required boundary conditions for a particular solution. With suitable global conditionsC is shown to be the dual ofA and a natural scalar product defined onB so as to reconstruct the gauge field in the standard form given by the construction. A discussion is given of the requirements entailed by the condition of a symmetry on the gauge field and the relation to the usual cohomological treatment is outlined in an appendix.

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  1. CERN, CH-1211, Geneva 23, Switzerland

    H. Osborn

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  1. H. Osborn
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Communicated by R. Stora

On leave of absence from DAMTP, Silver Street, Cambridge, England

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Osborn, H. On the Atiyah-Drinfeld-Hitchin-Manin construction for self-dual gauge fields. Commun.Math. Phys. 86, 195–219 (1982). https://doi.org/10.1007/BF01206010

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  • DOI: https://doi.org/10.1007/BF01206010

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