The very-extended trombone

Starting from the very-extended Kac-Moody algebra $E_{11}$, we consider the algebra $E_{11,D}^{local}$, obtained by adding to the non-negative level $E_{11}$ generators the $D$-dimensional momentum operator and an infinite set of additional generators that promote the global $E_{11}$ symmetries to gauge ones. We determine all the possible trombone deformations of this algebra, that is the deformations that involve the $D$-dimensional scaling operator. The Jacobi identities imply that such deformations are uniquely determined by a single tensor belonging to the same representation of the internal symmetry group as the vector generators and satisfying additional quadratic constraints. The non-linear realisation of the deformed algebra gives the field strengths of the theory which are those of any possible maximal supergravity theory in which the global scaling symmetry is gauged in any dimension. All the possible deformed algebras are in one to one correspondence with all such maximal supergravity theories. The tensor that parametrises the deformation is identified with the embedding tensor that is used to parametrise all maximal supergravity theories with gauged scaling symmetry, and the quadratic constraints that we determine exactly coincide with the field theory results.

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