We construct the nonlinear realisation of the semidirect product of the very extended algebra [Formula: see text] and its vector representation. This theory has an infinite number of fields that depend on a space–time with an infinite number of coordinates. Discarding all except the lowest level field and coordinates the dynamics is just Einstein’s equation for the graviton field. We show that the gravity field is related to the dual graviton field by a duality relation and we also derive the equation of motion for the dual gravity field.

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We study in detail the irreducible representation of [Formula: see text] theory that corresponds to massless particles. This has little algebra [Formula: see text] and contains 128 physical states that belong to the spinor representation of [Formula: see text]. These are the degrees of freedom of maximal supergravity in eleven dimensions. This smaller number of the degrees of freedom, compared to what might be expected, is due to an infinite number of duality relations which in turn can be traced to the existence of a subaglebra of [Formula: see text] which forms an ideal and annihilates the representation. We explain how these features are inherited into the covariant theory. We also comment on the remarkable similarity between how the bosons and fermions arise in [Formula: see text] theory.

We consider the string, like point particles and branes, to be an irreducible representation of the semi-direct product of the Cartan involution invariant subalgebra of [Formula: see text] and its vector representation. We show that the subalgebra that preserves the string charges, the string little algebra, is essentially the Borel subalgebra of [Formula: see text]. We also show that the known string physical states carry a representation of parts of this algebra.