Supersymmetric gauge theory on curved 7-branes

We construct novel $7d$ supersymmetric gauge theories which include a Chern-Simons-like term on curved spaces. In order to so, we examine the supersymmetry constraints for E7-branes in type IIA$^*$ theory, rather than making use of an off-shell supergravity. We find two classes of solutions to the constraints, expressed in terms of a $G_2$-structure with non-vanishing intrinsic torsion. The supersymmetric gauge theories are then obtained by coupling flat gauge theory to such supersymmetric backgrounds. Various examples are given, including round and squashed $S^7$ as well as $S^3 \times M_4$ with $M_4$ hyperk\"{a}hler.