We show how to construct seven-dimensional half-maximally supersymmetric consistent truncations of 11-/10-dimensional SUGRA using $\mathrm{ SL}(5)$ exceptional field theory. Such truncations are defined on generalised $\mathrm{ SU}(2)$-structure manifolds and give rise to seven-dimensional half-maximal gauged supergravities coupled to $n$ vector multiplets and thus with scalar coset space $\mathbb{ R}^+ \times \mathrm{ O}(3,n)/\mathrm{ O}(3)\times\mathrm{ O}(n)$. The consistency conditions for the truncation can be written in terms of the generalised Lie derivative and take a simple geometric form. We show that after imposing certain "doublet" and "closure" conditions, the embedding tensor of the gauged supergravity is given by the intrinsic torsion of generalised $\mathrm{ SU}(2)$-connections and automatically satisfies the linear constraint of seven-dimensional half-maximal gauged supergravities, as well as the quadratic constraint when the section condition is satisfied.
Categories: hep-th gr-qc