The heterotic $\rm{G}_2$ system on contact Calabi--Yau $7$-manifolds

We obtain non-trivial solutions to the heterotic $\rm{G}_2$ system, which are defined on the total spaces of non-trivial circle bundles over Calabi--Yau $3$-orbifolds. By adjusting the $S^1$ fibres in proportion to a power of the string constant $\alpha'$, we obtain a cocalibrated $\rm{G}_2$-structure the torsion of which realises an arbitrary constant (trivial) dilaton field and an $H$-flux with nontrivial Chern--Simons defect. We find examples of connections on the tangent bundle and a non-flat $\rm{G}_2$-instanton induced from the horizontal Calabi--Yau metric which satisfy together the anomaly-free condition, also known as the heterotic Bianchi identity. The connections on the tangent bundle are $\rm{G}_2$-instantons up to higher order corrections in $\alpha'$.