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3

If you take a classical string with a constant tension (NB unlike a rubber band the tension doesn't depend on how far the string it stretched) and let it relax then it will shrink to a point. However once you quantise the string you have the Heisenberg uncertainty principle to contend with. That means if you were to shrink the string to a point its ...

0

The quotient of locally free sheaves is not necessarily locally free, i.e. the first two terms being locally free does not force the third to be locally free, cf. this math.SE post. You have no guarantee that, given a map $f: E\to F$ of vector bundles, that the quotient $\mathrm{coker}(E\to F)$ (which we would also like to write as $F/\mathrm{im}(E)$) ...

2

Let me take parts 2. and 3. of the question first: The 10 dimensions of string theory are, a priori, not "coiled up" or anything else. They are derived for a string theory where the classical version of the string propagates in d-1 spatial dimensions and 1 temporal dimension, i.e. Minkowski space $\mathbb{R}^{1,d-1}$. "Dimension" here is dimension of a ...

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