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We derive the Free Differential Algebra for type IIA supergravity in 10dimensions in the string frame. We provide all fermionic terms for allcurvatures. We derive the Green-Schwarz sigma model for type IIA superstringbased on the FDA construction and we check its invariance under kappa-symmetry.Finally, we derive the pure spinor sigma model and we check the BRSTinvariance. The present derivation has the advantage that the resulting sigmamodel is constructed in terms of the superfields appearing in the FDA andtherefore one can directly relate a supergravity background with thecorresponding sigma model. The complete explicit form of the BRSTtransformations is given and some new pure spinor constraints are obtained.Finally, the explicit form of the action is given.

Pure Spinor Superstrings on Generic type IIA Supergravity Backgrounds