Infinite and finite consistent truncations on deformed generalised parallelisations

Given a manifold $\mathbb{M}$ \( \mathbbm{M} \) admitting a maximally supersymmetric consistent truncation, we show how to formulate new consistent truncations by restricting to a set of Kaluza-Klein modes on $\mathbb{M}$ \( \mathbbm{M} \) invariant under some subgroup of the group of isometries of $\mathbb{M}$ \( \mathbbm{M} \) . These truncations may involve either finite or infinite sets of modes. We provide their global description using exceptional generalised geometry to construct a ‘deformed’ generalised parallelisation starting with that on $\mathbb{M}$ \( \mathbbm{M} \) . This allows us to explicitly embed known consistent truncations directly into exceptional generalised geometry/exceptional field theory, and to obtain the equations governing situations where the consistent truncation retains an infinite tower of modes.