Torsion-free $G_{2(2)}^*$-structures with full holonomy on nilmanifolds

We study the existence of invariant metrics with holonomy $G_{2(2)}^* \subset SO(4,3)$ on compact nilmanifolds, i.e. on compact quotients of nilpotent Lie groups by discrete subgroups. We prove that, up to isomorphism, there exists only one indecomposable nilpotent Lie algebra admitting a torsion-free $G_{2(2)}^*$-structure such that the center is definite with respect to the induced inner product. In particular, we show that the associated compact nilmanifold admits a 3-parameter family of invariant metrics with full holonomy $G_{2(2)}^*$.

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