The heterotic G$_2$-system on 2-step nilmanifolds endowed with principal torus bundles

TL;DR

This paper investigates the heterotic G$_2$-system on 7-dimensional 2-step nilmanifolds with principal torus bundles, establishing conditions for solutions and providing explicit examples with various cosmological constants.

Contribution

It proves that invariant G$_2$-structures solving the system are coclosed under certain conditions and explores solution existence across all 7-dimensional 2-step nilpotent Lie algebras.

Findings

Invariant G$_2$-structures are coclosed under additional calibration.

Solutions exist for all 7-dimensional 2-step nilpotent Lie algebras.

Examples with constant dilaton are constructed for zero and nonzero cosmological constants.

Abstract

We study the heterotic G$_2$-system on 7-dimensional 2-step nilmanifolds $M=\Gamma\backslash N$ endowed with principal torus bundles. We first prove that every invariant G$_2$-structure solving the system must be coclosed (under an additional calibration assumption when the dimension of the derived Lie algebra of $N$ is $3$). Then, we discuss the existence of solutions for all possible isomorphism classes of 7-dimensional 2-step nilpotent Lie algebras, and we provide examples with constant dilaton function both when the cosmological constant of the spacetime is zero and when it is nonzero.