Space-time in String Theory

I think Anna s comment is correct, in LQG spacetime consists of discrete atoms and in ST it is continuous.

In addition, This article contains an interesting and quite accessible Nima talk related to the topic. Therein Nima explains why the present notions of spacetime are doomed and introduces the recent cutting edge ideas about how spacetime could emerge from a newly discovered and not yet fully explored structure called T-theory.

This is a very good question, because no one knows the answer. In a recent talk I asked the very same question to Dr. Brian Greene. I also asked him why we don’t' see many papers dealing for example with the quantum dynamics of branes in M-Theory, just many low energy solitonic semiclassical physics from some low energy 11-D supergravity lagrangian. Your questions about quantum spacetime are deeply related with the nature of physical quantum branes. His answer was straight foward he said "you aren't missing anything, we just don't know". In string theory, in principle, space time can be a fully quantum membrane in some dimension with open string excitations and a bulk probably another space filling brane with also closed quantum strings degrees of freedom. But brane quantization is still not well known. At the present state of string theory most calculations assume a space time continuum. But its very difficult to reconcile this notion with space time producing close string states, or a string state warping space. Maybe in the future it will be possible to make calculations or formulate string theory in quantized backgrounds that maybe branes themselves

So far only a perturbative formulation of string theory is known, despite the fact that there are some hints for what a non-perturbative formulation should contain. As far as I understand, it is expected that the geometry of the background spacetime in which the string propagates in the perturbative formulation, will ultimately be encoded in some other way in a non-perturbative formulation.

Roughly you can think about it as follows: Instead of quantizing General Relativity directly, which fails in a naive perturbative approach, perturbative string theory contains a field, that would arise in a perturbative quantization of General Relativity, too. In string theory this field is the massless part of a whole tower of massive fields. This together with the fact that a consistency condition gives you the vacuum equations $R_{\mu\nu} = 0$ of General Relativity (this is true at least in the bosonic sigma model, without $B$ field or Dilaton) are two reasons to believe that perturbative string theory contains a perturbative quantization of gravity at least in 26 or 10 dimensions. Contrary to the naive quantization, it yields (some) finite results at loop level (for superstring theory, this is actually only known up to two loops I believe).

In a sense that can be made somewhat precise: Certain 2-dimensional QFTs should be thought of as generalized (semi-)Riemannian manifolds.

Since there is actually no known non perturbative formulation one studies instead low-energy effective theories (supergravity theories), compactifications (here is where the calabi-yau manifolds come in), F-Theory and so on. Always in the hope that they might give a clue to what a non perturbative formulation should contain. In that way the fact that there is a 12-dimensional supergravity theory that dimensionally reduces to 11-dimensional theories leads to the idea that there should be an $M$-Theory or the presence of $p$-Form fields leads to the idea that there should be charged "branes".