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Nowadays it's assumed (it can still turn out to be different) that elementary particles are "given" mass by interaction with the Higgs field.

I read in the only answer to this question:

If you know which string theory vacuum you have (which compactification, which fluxes etc) and you know how the symmetries are broken, then you could in principle compute the masses.

Is this a different mechanism or is the Higgs field somehow included in this answer? The Higgs-particle itself is also an elementary particle. That is, an excitation in the Higgs field.

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In string theory particles exist as a tower of states starting with a massless particle and with the higher energy states having masses of the order of the Planck mass. The particles we could observe are the bottom of the tower i.e. at high energy they are all massless.

The observed masses of the Standard Model are produced by a symmetry breaking mechanism analogous to the Higgs mechanism, but the fields involved depend on the way the original symmetry is broken. String theory does not specify exactly how the symmetry is broken, only how to compute the masses given some symmetry breaking mechanism. The Higgs mechanism is probably one of the possibilities that string theory allows. I say probably because as far as I know no-one has succeeded in getting the Standard model from stringy theory (though they have got close). Given the alleged $10^{500}$ possible string vacua this isn't all that surprising.

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