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On page 5 of the notes by Veronika Hubeny on The AdS/CFT correspondence, we find the following:

Given that any string theory is a quantum theory which necessarily includes gravity, it naturally invites the examination of its consequences and implications for quantum gravity. Since spacetime plays a central role in general relativity, the obvious question to ask is how does string theory give rise to the curved dynamical spacetime of general relativity, and what happens to the 'stringy geometry' when the classical description breaks down. In the perturbative formulation valid at small string coupling $g_s$, the spacetime coordinates specifying the position of the string appear as scalar fields in the $2$-dimensional sigma model describing the dynamics of the string worldsheet, and the spacetime metric then enters as a coupling constant. But because the strength of gravitational interactions is governed by the string coupling – the $d$-dimensional Newton’s constant is $G_{N} = g^{2}_{s}\ell^{d−2}_{s}$ in units of the string length $\ell_s$ – it is difficult to directly access the most interesting, strongly gravitational, regime.

To an audience of graduate students without a background in string theory, how would you mathematically formulate the following sentence of the above paragraph:

In the perturbative formulation valid at small string coupling $g_s$, the spacetime coordinates specifying the position of the string appear as scalar fields in the $2$-dimensional sigma model describing the dynamics of the string worldsheet, and the spacetime metric then enters as a coupling constant.

In particular, could you

  1. write down the $2$-dimensional sigma model describing the dynamics of the string worldsheet?
  2. explain why the spacetime coordinates specifying the position of the string appear as scalar fields in the $2$-dimensional sigma model?
  3. explain how the spacetime metric then enters as a coupling constant?

The answers are all available in textbooks, but the details are overwhelming for a graduate student without a background in string theory. The purpose of the question is to provide a watered-down explanation of the emergence of the dynamical spacetime of general relativity from string theory.

Edit:

In lieu of ACuriousMind's comment, I ask a different question.

Why is the string worldsheet taken to be more fundamental than the spacetime coordinates in the Polyakov action of string theory?

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    $\begingroup$ The "2-dimensional sigma model" is just the standard formulation of string theory in terms of the Polyakov action, and it is the topic of basically every introduction to string theory. I'm not quite sure what you want as an answer to this question except a typical introduction to string theory - the lack of "background" is not relevant because this is the very starting point of string theory (in a modern view). $\endgroup$ – ACuriousMind May 30 '17 at 11:33
  • $\begingroup$ Okay, reflecting your comment, let me edit my question. $\endgroup$ – nightmarish May 30 '17 at 19:20
  • $\begingroup$ Isn't it the whole point of string theory — that strings are fundamental? $\endgroup$ – Prof. Legolasov Jun 1 '17 at 10:57

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