Local E(11) and the gauging of the trombone symmetry

In any dimension, the positive level generators of the very-extended Kac-Moody algebra $E_{11}$ with completely antisymmetric spacetime indices are associated to the form fields of the corresponding maximal supergravity. We consider the local $E_{11}$ algebra, that is the algebra obtained enlarging these generators of $E_{11}$ in such a way that the global $E_{11}$ symmetries are promoted to gauge symmetries. These are the gauge symmetries of the corresponding massless maximal supergravity. We show the existence of a new type of deformation of the local $E_{11}$ algebra, which corresponds to the gauging of the symmetry under rescaling of the fields. In particular, we show how the gauged IIA theory of Howe, Lambert and West is obtained from an eleven-dimensional group element that only depends on the eleventh coordinate via a linear rescaling. We then show how this results in ten dimensions in a deformed local $E_{11}$ algebra of a new type.