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In a 2007 note on ArXiv, it said:

String theory unifies all interaction but provides a perturbative background dependent formulation which violates general covariance.

However, another 2012 paper on ArXiv said that:

(4.2.1)

Instinctively, one identifies the constraint of the geometry being asymptotically AdS as a background and concludes that AdS/CFT is background dependent by construction. However, it has been argued that the situation is different...

and

...If the AdS/CFT conjecture is true, background independence of string theory could be proven, following this argument.

However, as we know, our universe is not AdS-like, but more similar to dS spacetime. The author also mentioned this in Sec.4.2.2:

The superselection sector of AdS/CFT is not the one we are living in since the cosmological constant was measured to be positive. If there were a duality to string theory in asymptotic de Sitter spacetime, a vacuum solution with positivecosmological constant, this would be a much more realistic model. This is being studied, see for example [7]. Unfortunately, this has not been achieved yet for technical reasons.

Nevertheless, even in an AdS universe, the problem still exists:

The major problem is the lack of an actual proof of the conjecture...Part of the problem is also the AdS part of the duality being ill-defined. String theory can be defined perturbatively, but as argued before this cannot be the fundamental definition, which is still lacking. Attempts to define it with e.g. string field theory have not succeeded yet.

So, does string theory violate general covariance or not? Or we are just still not sure about it?

Thank you.

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    $\begingroup$ Are the preprints published in peer-reviewed journals? $\endgroup$
    – Qmechanic
    Feb 5, 2019 at 12:38
  • $\begingroup$ According to the abstract pages, the first one seems not, and the second one was submitted as essay in philosophy of physics for the Cambridge Mathematical Tripos. $\endgroup$
    – pinchun
    Feb 5, 2019 at 16:14
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    $\begingroup$ a quick answer is that the full general covariance (and corresponding background independence) will become manifest after summing over string loops, which is hard. without summing over loops we see the infinitesimal version which corresponds to brst invariance. so yes, string theory is background independent but not manifestly so. $\endgroup$ Feb 5, 2019 at 22:53
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    $\begingroup$ Sen proved that closed superstring field theory is background independent in arxiv.org/abs/1711.08468 $\endgroup$
    – Mtheorist
    Apr 13, 2019 at 9:01

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Superstring theory has many local symmetries that are not manifest, coming from the BRST charge identifications in the covariant formalism, $|\psi\rangle\cong|\psi\rangle + Q|\chi\rangle$. Those symmetries are spontaneously broken by the solution (i.e. the background). Diffeomorphism (i.e. general covariance) is one of those. This is not different the General Relativity, where all the solutions spontaneously breaks the diffeomorphism by fixing a configuration for the metric tensor, which is not invariant under diffeomorphism. Like any other spontaneously symmetry breaking the symmetry is still there but the solution is not invariant.

The main difference is that for the GR we have a background independent formulation, so we can write an equation that is explicitly invariant under diffeomorphism, without referring to a particular solution. For the superstring theory the situation is different. Usually we start with a background (i.e. a solution) and then compute what quantum gravity is suppose to compute for that solution. In the case of asymptotically flat solutions it computes S-matrix, and for asymptotically AdS it computes correlation functions for the CFT that lives in the boundary. So it is because we do not have a background independent formulation for the superstring that we do not have a manifest covariant formulation, since generally a solution breaks spontaneously the diffeomorphism.

Since string theory is full of dualities relating different backgrounds and objects, preserving only the spectrum (as a duality should do), it is expected that all the local symmetries that comes from BRST charge, even the ones that act on massive states, should enter in the background independent formulation, i.e. the background independent formulation of string theory should be invariant under all these local symmetries. This is why is more hard to obtain a background independent formulation for string theory than usual GR.

For low energy approximation, so restricting to space times that are nice and flat for small distances, these gauge symmetries for the massive states does not enter and the GR is a good approximation. The challenge in formulate the string theory in a background independent way is that we don't understand very well situations in which these others gauge symmetries enters in the game.

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