Matrix string theory

http://arxiv.org/abs/hep-th/9701025

http://arxiv.org/abs/hep-th/9702187

http://arxiv.org/abs/hep-th/9703030

is indeed an exact description of fundamental type IIA strings (and similarly $E_8\times E_8$ heterotic strings) at any (e.g. weak) coupling where you can explicitly see the off-diagonal degrees of freedom. You could say that this description is was obtained by dualities from the low-energy dynamics of D1-branes and you would be right. However, when properly interpreted etc., it's a description of fundamental strings, too.

The reason why we normally (outside matrix string theory) don't see the off-diagonal degrees of freedom is that these off-diagonal degrees of freedom sit in their ground state for generic quantum states. For D1-branes, which are heavy, you may imagine a stack of several D1-branes which are located at the same point (along the same curve, to be more precise), which subsequently guarantees that the open strings connecting 2 different D1-branes – the off-diagonal modes – are light.

However, if the objects we want to connect are fundamental strings, which are light, the uncertainty principle guarantees that they will not be sitting in a fixed position determined with the accuracy better than $L_{\rm string}$ which is why the description of the perturbations in terms of off-diagonal open strings is impossible.

The asymmetry is particularly obvious in type IIB string theory. Two different D1-branes may be connected by light F1-strings. By S-duality, F1-strings may also be connected by D1-branes. However, D1-branes are heavy and F1-strings' separation is at least a string length. So the mass of the D1-branes connecting two different F1-strings, or two different points of an F1-string, will be much greater than the string mass. So there's no systematic description of physics that would consistently incorporate such massive degrees of freedom: there are many more additional degrees of freedom that are lighter and that should be incorporated before the D1-branes connecting the F1-strings.

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