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The moduli space of null Sasaki $η$-Einstein structures for simply connected compact 5-dimensional manifold $M$ is determined by the following quadric

$$\{[\alpha]\in H^2(M,\mathbb C) \; \text{such that} \; |([\alpha],[\alpha])|+|([\alpha],[\bar\alpha])|>0\}/\mathbb C^*$$

Is there any explicit moduli space of null Sasaki $η$-Einstein structures for higher dimensions?

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