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

Abstract

We obtain non-trivial solutions to the heterotic ${\mathrm{G}}_2$ 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 ${\alpha }^{\prime }$, we obtain a cocalibrated ${\mathrm{G}}_2$-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 ${\mathrm{G}}_2$-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 ${\mathrm{G}}_2$-instantons up to higher order corrections in ${\alpha }^{\prime }$.

arXiv commentMinor corrections and comments added. To appear in Transactions of the AMS

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• The heterotic G${}_2$ system with reducible characteristic holonomy
• Some remarks on strong ${\mathrm{G}}_2$-structures with torsion
• Isolated singularities in $G_2$-structures with torsion