The heterotic G₂ system on contact Calabi–Yau 7-manifolds

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 ’



.