Consistent truncation on Calabi-Yau and Nearly-K\"ahler manifolds

We complete and extend the analysis of arXiv:1903.10504 in several directions: we put the 4d theory, arising from the IIA consistent truncation of the universal sector of Calabi-Yau compactification, in a form manifestly consistent with 4d $\mathcal{N}=2$ supergravity. We go beyond the universal sector and construct the 4d effective action of IIA compactified on Calabi-Yau's with $h^{1,1}=1$, $h^{2,1}\geq1$, in the presence of background flux and fermionic condensates. For ALE gravitational instantons, we show that the putative quartic gravitino condensate is non-negative, as required for the existence of (formal) de Sitter solutions of the 4d theory. We discuss some of the issues in promoting these formal solutions to full-fledged string theory de Sitter solutions. We also extend the Nearly-K\"ahler consistent truncation of arXiv:1810.06344 to the complete bosonic sector of one vector multiplet and one hypermultiplet.