Spheres, Generalised Parallelisability and Consistent Truncations

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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 geometries admit such generalised parallelisations with an frame algebra. Thus we show that the remarkable consistent truncations on S3, S4, S5 and S7 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 S5.
Funding Information
  • Ministry of Science ICT and Future Planning
  • Engineering and Physical Sciences Research Council (EP/K034456/1)
  • Science and Technology Facilities Council (ST/J000353/1)
  • National Research Foundation of Korea (2005-0049409)
  • Deutsche Forschungsgemeinschaft (SFB 676)

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