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According to the Haag-Lopuszanski-Sohnius theorem the most general symmetry that a consistent 4 dimensional field theory can enjoy is supersymmery, seen as an extension of Poincarè symmetry, in direct product with the internal gauge symmetry.

But we know that conformal theories, having as a symmetry group the conformal group (which is indeed an extension of the Poincarè group) in direct product with the internal gauge group exist. Also there exist superconformal theories, which enjoy both conformal symmetry, supersymmetry and gauge internal symmetry. All this theories are consistent, from a theoretical point of view, and well definite in $d=4$.

Therefore I ask, how does superconformal field theories avoid the Haag-Lopuszanski-Sohnius theorem?

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2 Answers 2

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Conformal field theories do not have a mass-gap, which is one of the assumptions [for the strong conclusions of non-mixing of Poincare spacetime symmetries vs internal symmetries] of the Coleman-Mandula no-go theorem. Similarly, for its superversion: the Haag-Lopuszanski-Sohnius no-go theorem. [In the supercase, the Poincare algebra is replaced with the super-Poincare algebra.]

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The actual paper by Haag, Łopuszański and Sohnius covers Conformal Supersymmetry, and it states explicitly that this extension is achieved by NOT assuming the mass gap.

Historical note added 7 years later: Rudolf Haag was at the time very interested in Conformal Field Theories and was all in favour of exploring their supersymmetric extensions. I remember endless hours sitting at the kitchen table in his flat in Geneva working out the details of the generic N-extended case (it had been done previously for N=1 by Wess and Zumino and for N=2 by Peter Dondi and me). As Julius Wess's graduate student, I was the "supersymmetry guy" in that collaboration.

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