R-R fields in RNS formalism In string theory I came across the fact that there are difficulties in describing the coupling of R-R fields with world sheets in RNS formalism and it can be done in GS formalism only. Can someone explain the reason(s) behind that?
Also, I think if you have a space time where you can quantize strings and you have non zero R-R fields, then you need to use GS formalism to quantize strings there. It seems to me that these two are related and again my question is- why is that? Do we always need to use GS formalism there?
 A: In the RNS formalism the vertex operators for Ramond fields are not polynomials in the worldsheet fields. If you want to include them as part of the background, you'd have to deal with non-polynomial actions on the worldsheet. This is technically difficult. 
One way to deal with this difficulty is to use a different method to quantize the worldsheet of the string, like the GS method or the Berkovits method. These are all equivalent, but some things that are difficult in one language are easier in another (and vice versa). In the quantization methods where spacetime SUSY is manifest on the worldsheet, RR fields are simpler, but seems to me many other things are more difficult, but maybe I am biased by my own education here. 
For an example how such quantization works, take a look at this paper by Berkovits-Vafa-Witten.
A: The easiest way to write the curved background action is to covariantize the vertex operator. But in the RNS version of the superstring, to write the RR vertex you need to break worldsheet supersymmetry and mix ghost and matter fields. You can look at that famous paper by Friedan, Martinec and Shenker if you don't know how it is done:
Conformal Invariance, Supersymmetry and String Theory
Without supersymmetry as a guiding principle, is very difficult to write the right action and no one really knows how to do it.
In the Green-Schwarz or pure spinor version, there is not much problem to do so. See, for instance:
Ten-Dimensional Supergravity Constraints from the Pure Spinor Formalism for the Superstring
The GS superstring has problems to be quantized and this is usually made in the light-cone gauge, but it is not always that this gauge exists. So, I guess that the the real known way to treat a superstring in a RR background as a quantum CFT is with the pure spinor action.
