Generalised vielbeins and non-linear realisations

TL;DR

This paper reviews the construction of generalised vielbeins in non-linear realisations of semi-direct product groups, providing explicit formulae and examples in various dimensions related to E(11) and SL(2).

Contribution

It introduces matrix-based formulae for generalised vielbeins and applies them to complex theories like E(11) and very extended SL(2) in multiple dimensions.

Findings

Explicit expressions for generalised vielbeins in E(11) theories in 4, 5, 11 dimensions.

Generalised vielbein calculations for IIB supergravity in 10 dimensions.

Application of the formalism to gravity-related theories in four dimensions.

Abstract

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. We also compute the generalised vielbein, up to the analogous level, for the non-linear realisation of the semi-direct product of very extended SL(2) with its first fundamental representation, which is a theory associated with gravity in four dimensions.