We construct the complete Lagrangian of extended supergravity in ten dimensions by dimensional reduction of the eleven-dimensional theory. We find a compactification of the D = 10 model to the D =4 model with internal manifold M4(compact )%S2.
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.The nonexistence of supergravity models beyond d = 11 dimensions was shown by Nahml using group theory. His result confirms another no-go theorem: If d = 12 supergravity would have existed, its dimensional reduction would have produced in d = 4 a theory with particles whose spin exceeds 2. Their coupling to gravity is known to be incon~isbent,~ at least if their energy is positive-definite.However, no-go theorems often make implicit assumptions which can be weakened; for example, the higher-spin fields could be nonpropagating or have nonpositive energies. Thus we find it still worthwhile to investigate the possibility of a viable theory of supergravity in d = 12, even more so because we found, somewhat to our surprise, that one can actually balance the bosonic and fermionic degrees of freedom in 12 dimensions. To show this latter result, consider real antisymmetric tensor gauge fields up to four indices, and note that in d = 12 Majorana spinors exCounting the degrees of freedom in the usual manner,4 one constructs a multiplet of 320 bosons and 320 fermions as follows:V: : 54
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