Is there quantum foam between strings?
To the best of my knowledge - no.
Its part of Loop Quantum Gravity.
The true nature of spacetime isn't fully known in String Theory. It's certainly not smooth like in GR. Whether or not it would be correct to describe it as "quantum foam", I wouldn't like to say, but the incompaibility between quantum foam and GR, is one of the motivations for String Theory.
Quantum foam is a poetic name that Wheeler coined to describe the analogy of the QCD vacuum. One is supposed to picture teeming geometries bubbling in and out of existence.
It is a vague concept that has no clear mathematical definition, and depending on how one views it, it could apply to both string theory, lqg or to neither one.
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