IIB Supergravity from worldsheet (super)conformal invariance of Green-Schwarz string After reading this question
How are low energy effective actions derived in string theory?
I began to wonder what is the coupling of the string to the other sugra fields. In almost all textbooks there is information how the string can be described in an arbitrary background with fields $g_{\mu\nu}$, $b_{\mu\nu}$ and $\phi$. Then, conditions for worldsheet conformal invariance give us beta functions for each field and the corresponding equations of motion for the metric, Kalb-Ramond field and dilaton.
So, if we want to get the equations of motion of IIB sugra, I guess we first need to have an action for the Green-Schwarz string in background fields: $g_{\mu\nu}$, $b_{\mu\nu}$ and $\phi$, RR forms $F_1$, $F_3$, $F_5$ and gravitini and dilatini. 
But, as far as I know, the string cannot couple to RR-fields for instance.
The idea is to have that string action (in arbitrary IIB background) and then, form beta functions, obtain the equations of motion of all the background fields (bosonic and fermionic). 
Has this route to the supergravity effective theory been taken in some paper? (if YES, could you sketch the procedure and give some reference)
Is this route imposible? perhaps just because the string doesn't couple to all sugra fields.
 A: First, strings does couple with RR flux. What is correct to say is that strings are not sources for RR flux. Saying differently, strings does not couple directly with the RR gauge fields, only with their field strength. The sources for the RR fields are the D-branes.
For the RNS formalism it is very hard to add RR flux since the picture changing phenomena. The vertex operator for the RR states have half integer picture number. For similar reasons the target space supersymmetry is not manifest in this formalism.
For the GS formalism, the supersymmetry is manifest and it is simple to add RR flux. The supergravity equations follows from 2d Weyl invariance at one loop. Also, it is very hard to quantize strings in this formalism and obtain the spectrum in a covariant manner because of the presence of the kappa symmetry. What is puzzling is that requiring kappa symmetry implies almost the supergravity equations, up to some extra restrictions on the dilaton field. You can see more about it here.
For the Pure spinor formalism, the supersymmetry is manifest, so the RR flux is simple to add. The supergravity equations follows from tree-level BRST holomorphicity+nilpotence. Futhermore the strings are more easy to be covariantly quantized in this formalism than in the GS formalism. It is very interesting to note that the BRST invariance and the Kappa symmetry can be related as comming from different gauge fixings of a twistor like action.
