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

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