Beyond 11 -dimensional supergravity and Cartan integrable systems

Although one can match in 12 dimensions the numbers of Bose and Fermi states, no corresponding supergravity model exists. In the component approach the gravitino and Maxwell sectors are incompatible already at the linearized level, while in the group-manifold approach an even simpler explanation exists: the Maurer-Cartan equations for the four-index photon are nonintegrable.