7-dimensional $\mathcal{N}=2$ Consistent Truncations using $\mathrm{S}\mathrm{L}(5)$ Exceptional Field Theory

We show how to construct seven-dimensional half-maximally supersymmetric consistent truncations of 11-/10-dimensional SUGRA using $\mathrm{S}\mathrm{L}(5)$ exceptional field theory. Such truncations are defined on generalised $\mathrm{S}\mathrm{U}(2)$-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 $\mathrm{S}\mathrm{U}(2)$-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.