Eur. Phys. J. C 24, 165-176 (2002)
https://doi.org/10.1007/s100520100857

Non-commutative Lorentz symmetry and the origin of the Seiberg–Witten map

¹ Institut für Theoretische Physik, Technische Universität Wien, Wiedner Hauptstrasse 8–10, 1040 Wien, Austria
² Institut für Theoretische Physik, Universität Wien, Boltzmanngasse 5, 1090 Wien, Austria
³ Physikalisches Institut der Universität Bonn, Nussallee 12, 53115 Bonn, Germany

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Received: 6 November 2001
Published online: 5 April 2002

Abstract

We show that the non-commutative Yang–Mills field forms an irreducible representation of the (undeformed) Lie algebra of rigid translations, rotations and dilatations. The non-commutative Yang–Mills action is invariant under combined conformal transformations of the Yang–Mills field and of the non-commutativity parameter θ. The Seiberg–Witten differential equation results from a covariant splitting of the combined conformal transformations and can be computed as the missing piece to complete a covariant conformal transformation to an invariance of the action.