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We know, string theory, from its inception, has a technical problem of the instability of the extra dimensions. I have heard of the partial solution of the problem quite a few years ago. My question is, has it been completely solved now?

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This question has many aspects because there are many groups of stringy vacua, some of them have been showed to have (unstabilized) moduli, the status of others was (or is) unknown. Those that were (or are) almost certainly stabilized may be stabilized at the classical level, or by calculable non-perturbative effects, or by not-quite-understood non-perturbative effects which may required fluxes or antibranes, and so on.

In most classes of stringy vacua, the boundaries have been understood which means that it is known whether the vacua have some unstabilized moduli or not, and if they don't, the relevant potentials became largely understood, at least qualitatively. See papers such as

So if you ask whether the question has been solved for all vacua, the answer is surely No. But for many big classes etc., the answer is Yes, at least morally. And yes, there are fully stabilized vacua.

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