Spheres, generalised parallelisability and consistent truncations

TL;DR

The paper develops a unified framework in which maximally supersymmetric consistent truncations are understood as generalised Scherk–Schwarz reductions on generalised parallelisable spheres. By constructing global generalised frames with constant frame algebras on $S^d$, it shows that the embedding tensor of the truncated theory is encoded in the frame torsion, yielding an $SO(d+1)$ gauging for all spheres and recovering the known truncations on $S^3$, $S^4$, $S^5$, and $S^7$ within the appropriate $E_{d(d)}\times\mathbb{R}^+$ (or exceptional) geometries. The work provides explicit scalar ansaetze, including the full scalar ansatz for type IIB on $S^5$, and explains how these truncations fit into a generalised geometry analogue of Scherk–Schwarz reductions on local group manifolds. This viewpoint offers a route to classifying potential maximally supersymmetric truncations and connects traditional consistent truncations to the broader framework of exceptional generalised geometry and its Weitzenböck-type connections. In short, sphere truncations are manifestations of generalised parallelisable structures, unifying previously disparate results under a single geometric mechanism.

Abstract

We show that generalised geometry gives a unified description of maximally supersymmetric consistent truncations of ten- and eleven-dimensional supergravity. In all cases the reduction manifold admits a "generalised parallelisation" with a frame algebra with constant coefficients. The consistent truncation then arises as a generalised version of a conventional Scherk-Schwarz reduction with the frame algebra encoding the embedding tensor of the reduced theory. The key new result is that all round-sphere $S^d$ geometries admit such generalised parallelisations with an $SO(d+1)$ frame algebra. Thus we show that the remarkable consistent truncations on $S^3$, $S^4$, $S^5$ and $S^7$ are in fact simply generalised Scherk-Schwarz reductions. This description leads directly to the standard non-linear scalar-field ansatze and as an application we give the full scalar-field ansatz for the type IIB truncation on $S^5$.