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It seems like, at this point, string theory is a reasonably well-developed mathematical framework. Tools from string theory are being used to understand things in QCD and condensed matter with no small success.

So what is the obstacle to actually writing down a string theory (using 'string theory' in the most general sense) in $N+1$ dimensions with $N-3$ dimensions compactified in some way? Presumably there is one, but I can't find it discussed in any of the overview papers I've read (and I'm not really in a position to be able to understand anything deeper than an overview paper).

To clarify, I don't mean obstacle in the sense of 'makes predictions we can't yet test', but either mathematical obstacles, or making predictions we already know to be false. I understand that there's a challenge in finding vacuum that has the right value of $\Lambda$, but is this the only thing?

Edit: Just to doubly clarify: it's well known that string theory has a large parameter space. Is finding the correct set of parameters the only thing stopping us from writing down a string theory that describes our universe?

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    $\begingroup$ There's a lot of 'em $\endgroup$
    – Slereah
    Apr 27 '20 at 11:18
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    $\begingroup$ @Slereah One could even say that there are manifold reasons... $\endgroup$
    – PM 2Ring
    Apr 27 '20 at 11:23
  • $\begingroup$ This post (v3) seems very broad. $\endgroup$
    – Qmechanic
    Apr 27 '20 at 12:14
  • $\begingroup$ @Slereah it's not a reason though. It's just a statement that reasons exist, somewhere in the ether. $\endgroup$
    – gautampk
    Apr 27 '20 at 12:17
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    $\begingroup$ There are infinitely many such manifolds, therefore the probability of finding the correct one by random chance is zero. $\endgroup$
    – Slereah
    Apr 27 '20 at 12:18
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Even though string theorists have constructed huge classes of exciting semi-realistic compactifications (see this for a modern example), the problem is our lack of understanding of how to systematically achieve string theory compactifications that share properties with the universe we observe.

The paper Life at the Interface of Particle Physics and String Theory is an excellent divulgative discussion on the problems and advantages of many specific string constructions.

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Found this quote from a 2011 paper , String theory: a perspective over the last 25 years :

Flux compactifications are the closest we have come to realising the Standard Model of particle physics, as well as a realistic gravitational sector including a cosmological constant, in the context of string theory. Their success has paradoxically raised a problem: it seems likely that string theory admits an immense number of vacua, known as the “landscape”, and many of them lie arbitrarily close to the real world. The problem of finding “the right vacuum” in this situation looks very daunting. We have ended up very far from the original hope that a simple, almost unique compactification of string theory would lead to a viable description of the real world

Italics mine.

The dificulty seems to be finding which vacuum is the one our universe has

This link also describes a multiplicity of ways the standard model could arise in string theories.

My summary is that there exist too many similar string theories that describe universes like ours (i.e. the standard model), and the difficulty is to choose one out of the plethora.

If a string theorist comes up with a good answer, I will delete this.

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