Spheres, generalised parallelisability and consistent truncations (1401.3360v1)

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 $S^5$.

• The paper unifies maximally supersymmetric consistent truncations in supergravity using generalised geometry and the concept of generalised parallelisation.
• It shows that round spheres like S³, S⁴, S⁵, and S⁷ admit generalised parallelisations, which encode the embedding tensor for consistent truncations into gauged supergravities.
• This framework predicts consistent truncations and simplifies the derivation of scalar field ansatze, clarifying the structure of resulting gauged supergravity theories.

Generalised Geometry and Consistent Truncations in Supergravity

The paper presents a comprehensive understanding of maximally supersymmetric consistent truncations in the context of generalised geometry, specifically focusing on ten- and eleven-dimensional supergravity. The authors explicate how generalised geometry provides a unified framework for consistent truncations by introducing the concept of "generalised parallelisation." Here, they establish that various round-sphere geometries, such as $S^3$, $S^4$, $S^5$, and $S^7$, which have been pivotal in supergravity reductions, all admit generalised parallelisations.

Key Findings and Framework

The authors demonstrate that the maxmially supersymmetric consistent truncations arise through a generalised version of the Scherk–Schwarz reduction. In this framework, the reduction space — a manifold equipped with a global frame — is characterised by a frame algebra with constant coefficients. This algebra encodes the embedding tensor facilitating the truncation, echoing the structure found in gauged supergravities.

• Generalised Parallelisation: The paper's critical revelation is that round-sphere geometries inherently support generalised parallelisations. For example, they show that every $S^d$ sphere can be equipped with an SO($d+1$) frame algebra, identifying them as a form of generalised Lie algebra. This insight extends the notion of parallelisability, previously understood only in limited cases within classical Riemannian geometry.
• Embedding Tensor and Consistent Truncations: The embedding tensor, a pivotal element in defining the gauged symmetries of the truncated theory, is naturally encoded within the generalised geometric framework. This leads to the assertion that consistent truncations on these spheres result in gauged supergravity theories with the same number of supersymmetries as the original higher-dimensional theories.
• Example Applications: The paper derives non-linear scalar field ansatze for supergravity truncations, such as type IIB on $S^5$, illustrating how standard scalar field reductions are seen as natural outcomes of generalised Scherk–Schwarz reductions. They extensively verify these reductions by computing components of the generalised geometry that align with known embedding tensors for various gauged supergravities, such as SO(8) in four dimensions.

Implications and Future Directions

The results potentially simplify and unify the approach to consistent truncations by abstracting them under a broad generalised geometric framework. This not only clarifies existing theories but opens pathways to exploring new truncations of supergravity theories, leveraging generalised geometry's intrinsic structure to predict where consistent models might emerge. It encourages further exploration into generalised manifolds, extending beyond typical Lie group manifolds, and motivates the development of a complete classification of consistent truncation spaces.

Moreover, this framework's implications for string theory and M-theory reformulations are significant. It could inform the construction of new theoretical models or extend existing models to incorporate a richer symmetry structure afforded by generalised geometrical settings. Given the trend towards understanding dualities and their geometrical representations in string theory, these insights have the potential to make foundational contributions to the field.

In summary, this paper illuminates the path toward a unified understanding of supergravity reductions by employing the sophisticated language of generalised geometry. This approach not only consolidates existing truncation processes but also sets a fertile ground for future theoretical exploration and development in the field of high-dimensional supergravity theories.