We formulate and study the negative gradient flow of an energy functional of Spin(7)-structures on compact 8-manifolds. The energy functional is the L 2 -norm of the torsion of the Spin(7)-structure. Our main result is the short-time existence and uniqueness of solutions to the flow. We also explain how this negative gradient flow is the most general flow of Spin(7)-structures. We also study solitons of the flow and prove a nonexistence result for compact expanding solitons.

We obtain non-trivial approximate solutions to the heterotic G 2 \mathrm {G}_2 system on the total spaces of non-trivial circle bundles over Calabi–Yau 3 3 -orbifolds, which satisfy the equations up to an arbitrarily small error, by adjusting the size of the S 1 S^1 fibres in proportion to a power of the string constant α ′ \alpha ’ . Each approximate solution provides a cocalibrated G 2 \mathrm {G}_2 -structure, the torsion of which realises a non-trivial scalar field, a constant (trivial) dilaton field and an H H -flux with non-trivial Chern–Simons defect. The approximate solutions also include a connection on the tangent bundle which, together with a non-flat G 2 \mathrm {G}_2 -instanton induced from the horizontal Calabi–Yau metric, satisfy the anomaly-free condition, also known as the heterotic Bianchi identity. The approximate solutions fail to be genuine solutions solely because the connections on the tangent bundle are only G 2 \mathrm {G}_2 -instantons up to higher order corrections in α ′ \alpha ’ .