String theory and background independence I have read that string theory assumes strings live in spacetime defined by general relativity which make the theory background dependent (although general relativity is a background independent theory). Background independence dictates that spacetime emerge from more fundamental ingredient than spacetime. Quoting Brian Greene, “Then, the theories ingredients - be they strings, branes, loops, or something else discovered in the course of further research - coalesced to produce a familiar, large-scale spacetime” (Greene, The Fabric of the Cosmos, 2004: 491).
My question: why couldn’t spacetime just be spacetime, a fundamental entity? If so, string theory, based on general relativity, would not be a “great unsolved problem” facing string theory. 
 A: I think you might be slightly misinterpreting what string theorists believe their theory says about quantum spacetime.
Start with a classical spacetime with a certain pseudo-Riemannian metric, and consider the theory of strings propagating on this spacetime. There are two very important results:


*

*Unless $R_{\mu \nu} = 0$, the theory acquires a non-zero beta function under conformal transformations. Afaik this is widely believed to be unphysical, because strings theorists don't know how to make sense of the theory when it isn't invariant under conformal transformations. Therefore we conclude that string theories exist only on Ricci-flat spacetimes, that is, on spacetimes solving the vacuum EFE.

*In the spectrum of the string, you will find spin-2 states which have been conjectured to model physical gravitons. They have all properties that one would expect gravitons of General Relativity to have, except that their ultraviolet completion is finite.


The question then becomes – why did we have to choose the classical background in the first place? After all, isn't quantum gravity supposed to model classical spacetime as a certain limit of the full theory?
An interesting observation is that if you try to build a "coherent" state from string theory gravitons, the setup can be re-interpreted exactly as if the string was in the vacuum state, but propagating in a different spacetime.
So it can be conjectured that there's a concealed duality between the excited states of the string/superstring propagating in one classical spacetime, and an unexcited string propagating in another spacetime. Thus, maybe string theory is background-independent after all, even though the perturbative formulation that we've found isn't manifestly background independent?
Though afaik there's no background independent nonperturbative formulations of string theory known (I am not considering any of the AdS/CFT stuff here because it probably isn't directly related to the perturbative string/superstring theories and still remains just a conjecture).
From my personal correspondence with people working in the field, I conclude that there's no consensus on this subject, despite some individuals' belief that there is consensus :) I personally know people working on topics related to string theory, who are absolutely convinced that background independence is a must-have property of the non-perturbative definition of string theory, whatever it is. But I also know at least one senior lecturer who is perfectly satisfied with a special-relativistic flat spacetime entering the definition of the theory.
A: 
If so, string theory, based on general relativity, would not be a “great unsolved problem” facing string theory.

The unsolved problem  comes because the holy grail of physics at present is unification of  all four forces,  in one quantum mechanical theory. The weak the electromagnetic are unified , with the discovery of the Higgs validating the theory, and the strong has all the indications of unification. It is the quantization of gravity  that is still elusive so cannot be harnessed except with speculations in one theory of everything.
At the moment it is only string theories that seem to be able to quantize gravity and also embed the standard model of physics in a unified way. The draw back is that there are a very large number of possible string theories, and research is ongoing for finding a model of our universe.

My question: why couldn’t spacetime just be spacetime, a fundamental entity?

The premise for present research  is that the underlying level of nature is quantum mechanical. The aim is to show that classical ( and general relativity is  a classical theory ) emerges from the  basic underlying quantum  level of reality. This can be mathematically shown for electromagnetism, and the hope is it will be true for gravity too. Note "hope", no definite theory yet.  So spacetime has to be emergent in this theoretical outlook.
A: The idea of presenting gravity as a standard quantum field of interacting gravitons is naive, because it does not explain how spacetime becomes curved. All other fields do not affect the background like pens writing on a sheet of paper. The sheet may be flat or curved, but writing on it doesn't change the curvature. GR defines the curvature and thus cannot be presented as a standard QFT.
Quantum gravity is not just about quantizing gravity. It is about building a theory on a background independent of spacetime. A theory built on this background can define spacetime as flat or curved as appropriate, like a projector projecting different images of reality. Then all other fields must be defined on the same background to be able to interact. So quantum gravity is about redefining both gravity and QFT to unite them on a new background.
A: You're asking a question that deserves an enormous amount and clarifications. I will just want to share some very illuminating references on the idea of "Background independence" in the context of string theory. 
What is background independence and how important is it?
At least two philosophers understood background independence
Is space and time emergent? ER-EPR correspondence adds a voice
