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?
This post imported from StackExchange Physics at 2016-01-11 12:16 (UTC), posted by SE-user Federico Carta