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I've seen one paper that says finding string theory vacua with particular low-energy properties is totally intractable, from a computational complexity standpoint. I've also seen people saying that they have figured out how to compactify the extra dimensions such that at low energies a SUSY standard model emerges. What's the truth about string phenomenology? Are we regularly finding new vacuua that look like our universe? Have we only ever found one? Any? How hard is it?

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    $\begingroup$ That sounds interesting, but why is intractable from a computational complexity standpoint? The algorithm for finding string theory vacua reduces to 3SAT? $\endgroup$ – Hank Igoe Dec 30 '20 at 23:50
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    $\begingroup$ I don't remember if this was the exact paper I saw, but here's one that contains the right information: arxiv.org/abs/1809.08279 $\endgroup$ – Retracted Dec 30 '20 at 23:53
  • $\begingroup$ @HankIgoe It's difficult for myself, a non-expert in complexity, to repeat statements made about it without making an error, but in broad strokes they take a look at the problem of finding stable and metastable vacuua in scalar potentials, and deduce that if you had a black box that could do it, you could use the black box to solve 3SAT. They don't do the black box argument proof themselves, instead they identify sub-problems of the vacuum problem that are already known to have such a property. $\endgroup$ – Retracted Dec 30 '20 at 23:58
  • $\begingroup$ Wow, that is interesting. I didn't know they were applying complexity theory to string theory like that, or even that such a thing was possible. I don't have time to read the paper in full, and I doubt I'd understand most of it if I did, but that's good to know. Thanks for the great question. $\endgroup$ – Hank Igoe Dec 31 '20 at 0:00
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    $\begingroup$ Related: What is the current status of string theory (2013)? and links therein. $\endgroup$ – Qmechanic Dec 31 '20 at 17:26
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First, the entire subject of string phenomenology is not reduced to search by brute force for some internal space that produces a low energy spectrum that reproduces the SM one. See life at the interface between string theory and particle physics for an excellent (and conceptual) string phenomenology overview.

Are we regularly finding new vacuua that look like our universe? Yes. See A Quadrillion Standard Models from F-theory for a quadrillion of examples, an this nice blog post for divulgative information.

How hard is it? Its pretty hard. A navie argument that exhibits why is so hard to find new Calabi-Yau spaces, let alone one with semi-realistic properties, would be to recognize that that the typical number of moduli of a Calabi-Yau space is huge (sometimes of the order of hundreds), and those fields are subject to very few restrictions; then the expectation to write new CY metrics in an explicit way is hopless.

For a clear argument about the computational complexity of finding a particular compactification, with fluxes, and capable to produce a small cosmological constant in the Bousso-Polchinski scenario, see Computational complexity of the landscape.

Observation: The fact that a problem is $NP$-complete does not imply that we can not use other computational techniques, equipped with a set of reasonable assumptions, to give some special answers to this problem in a reasonable time. See Deep learning the landscape for an example of this.

Update: A nice paper on neural networks approximating Calabi-Yau metrics Neural Network Approximations for Calabi-Yau Metrics.

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  • $\begingroup$ Deep learning can't get around NP-completeness, it's not magic. If deep learning works, then the problem must have some "not really NP complete" character, for example it may be possible to have arbitrarily good approximations in polynomial time. $\endgroup$ – Retracted Jan 1 at 4:13
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    $\begingroup$ @Retracted I'm definitely not claiming at all that deep learning can solve arbitrary NP -problems in reasonable (polynomial) time. What I'm showing is a concrete example in which particular deep learning architectures, equipped with a set of reasonable assumptions, are shown to be efficient in predicting the answers for a very specific NP-hard problem. Does that imply that we can use those schemes to predict geometric properties of a given CY with arbitrary precision? No. The observation is just that Deep learning does his work, well enough, to deserve the attention of the experts. $\endgroup$ – Ramiro Hum-Sah Jan 1 at 8:41
  • $\begingroup$ @Retracted Consider the problem of finding hamiltonian paths on a graph (an NP-hard problem). Suppose that we have discovered that a deep learning scheme (plus some assumptions and biases over the inputs) is good enough to construct hamiltonian paths in graphs of size 17x17. Does that imply that you have reduced an NP-Problem to a problem in P? No, and that's because that procedure is not general, and does not provide answers with arbitrary high precision. Something similar happens here. The procedure is not as effective as we would want and is not general; it simply gives interesting results. $\endgroup$ – Ramiro Hum-Sah Jan 1 at 9:03

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