How does one get supergravity from string theory? I know one can get the Fermi theory of weak interactions as a low energy effective theory of the electroweak interaction by writing down Feynman amplitudes for weak decay processes, noting that $\tfrac{1}{p^2-M_W^2}\approx-\tfrac{1}{M_W^2}$ for small momenta $p^2\ll M_W^2$, and then identifying the new effective coupling $\tfrac{g^2}{M_W^2}\sim G_F$ (up to some conventional factors like $\sqrt{2}$ that I don't precisely recall).
I know a bit about string theory and supergravity and I've always heard that supergravity is a low energy effective theory for superstring theory, so I was wondering how that comes about? Is it a similar process or something totally different?
 A: I will assume a rudimentary understanding of string theory, such as the key players in the story, like the Nambu-Goto action.
Essentially, you take your 2D action that you start with to describe string theory, and you quantise, finding the bosonic and fermionic content of the theory.
Once you know this information, you write down that 2D action, including the string propagating in the background of fields arising in the bosonic and fermionic content of the theory. For example, for the bosonic string, you would add in a scalar coupling and a coupling to a 2-form potential.
Finally, with this 2D action with the added terms, you compute the beta functions. The low energy effective action (which is supergravity if you work with the superstring) is the action whose equations of motion are those beta functions.
It is thus a bit of an inverse problem, and I will be honest most textbooks will say what the effective action is that gives rise to those beta functions, without systematically working backwards. I suspect this is because it takes some guesswork and experimentation, but I don't know how it was done originally in the literature.
I would recommend section 7 of David Tong's notes on string theory, and you can find the messy one loop computation of the beta functions for the bosonic string here.
