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Of the $10^{500}$ or so vaccua in string theory. Do they all have the same energy? i.e. are the vaccua degenerate? Or do they have different energies? (If this is even possible.)

It is said that the different vaccua correspond to different modes of curling up the extra dimensions. From this description, it is not clear to me why each vaccua would have the same energy. e.g. the extra dimensions might have different modes of vibration and in particular the lowest mode would be different for each 'shape'.

On the other hand vibrations on different shaped drums probably all are quantised in the same way. But is is not clear if holes would make a difference.

I guess if each vacuum had the same ground energy this would mean the cosmological constant could be predicted without knowing which vacuum we were in.

Qmechanic
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A string theory vacuum is indeed just a particular choice of compactification of the additional six dimensions. It does not have an "energy" in the straightforward sense because it's not a quantum state or something like that - it's just a choice of compacification. However, what we can talk about is the effective cosmological constant - vacuum energy - in the 4d theory. See also this answer of mine for more about energy in the string landscape and this question for discussion of why there might be "$10^{500}$" vacua.

All the different vacua produce different effective cosmological constants because the matter/field/brane content in the compact dimensions is different. Notably, a vanishing cosmological constant corresponds to unbroken supersymmetry (i.e. a pure Calabi-Yau compactification without fluxes).

ACuriousMind
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