Generalised vielbeins and non-linear realisations

Understand

We briefly review why the non-linear realisation of the semi-direct product of a group with one of its representations leads to a field theory defined on a generalised space-time equipped with a generalised vielbein.

• We give formulae, which only involve matrix multiplication, for the generalised vielbein, the Cartan forms and their transformations.
• We consider the generalised space-time introduced in 2003 in the context of the non-linear realisation of the semi-direct product of E(11) and its first fundamental representation.
• For this latter theory we give explicit expressions for the generalised vielbein up to and including the levels associated with the dual graviton in four, five and eleven dimensions and for the IIB theory in ten dimensions.