Adjoint orbits, generalised parallelisable spaces and consistent truncations

TL;DR

This paper constructs explicit examples of generalized parallelisable spaces from adjoint orbits of semi-simple Lie groups, enabling consistent supergravity truncations via generalized Scherk-Schwarz reductions.

Contribution

It introduces new explicit constructions of $O(d,d)$-generalized Leibniz parallelisable spaces from adjoint orbits, with detailed expressions for regular cases and insights into degenerate orbits.

Findings

Explicit generalized frames for regular orbits

Spaces form flat fiber bundles over orbits

Potential for consistent supergravity truncations

Abstract

The aim of this note is to present some new explicit examples of $O(d,d)$-generalised Leibniz parallelisable spaces arising as the normal bundles of adjoint orbits $\mathcal{O}$ of some semi-simple Lie group $G$. Using this construction, an explicit expression for a generalised frame is given in the case when the orbits are regular, but subtleties arise when they become degenerate. In the case of regular orbits, the resulting space is a globally flat fiber bundle over $\mathcal{O}$ which can be made compact, allowing for a generalised Scherk-Schwartz reduction. This means these spaces should admit consistent supergravity truncations. For degenerate orbits, the procedure hinges on the existence of a suitable metric, allowing for a consistent normalisation of the generalised frame.