Isometric solutions to the heterotic $\mathrm{G}_2$-system

TL;DR

This paper constructs new isometric solutions to the heterotic G2-system on specific manifolds, varying both G2-structures and gauge data, and extends known examples within a unified framework.

Contribution

It introduces an ansatz that varies G2-structures and gauge data simultaneously, producing new isometric solutions with different gauge groups on the same manifold.

Findings

Constructed solutions on nilmanifolds and 3-Sasakian manifolds.

Generated isometric solutions with different gauge groups.

Extended known solutions using an S^1-invariant approach.

Abstract

In this note, we construct new solutions to the heterotic $\mathrm{G}_2$-system with non-abelian gauge group, both compact and non-compact, on certain $2$-step nilmanifolds and $3$-Sasakian manifolds. Our approach is based on an ansatz that allows us to vary both the $\mathrm{G}_2$-structure and the gauge data while keeping the underlying metric and orientation fixed. This leads, in particular, to distinct isometric solutions on the same manifold but with different gauge groups, and in some cases the resulting connection coincides with the characteristic connection of the $\mathrm{G}_2$-structure. We also investigate an $S^1$-invariant construction that yields further isometric solutions and with varying cosmological constant. Our results recover and extend several known examples solving the heterotic $\mathrm{G}_2$-system within a unified framework.