# Matrix geometry for F-strings

A stack of N D-branes has the strange property that the traverse D-brane coordinates are matrix-valued. When the matrices commute, they can be interpreted as ordinary coordinates for N indistinguishable objects. But in general they correspond to something classical geometry doesn't describe. In particular this leads to a non-perturbative description of string theory on asymptotically Minkowski spacetime: Matrix Theory.

S-Duality exchanges F-strings and D1-strings. This means this odd "matrix geometry" should happen for F-strings as well. The question is, how can we see it directly, without invoking S-duality?

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I apologize for posting additional questions before carefully reading and replying to the discussion around the answers to my previous questions. This is not out of disrespect to the effort put into writing these answers, for which I am very grateful. This is merely because I suspect the site will be closed in 2 days for which reason I'm shooting all the questions I got. – Squark Dec 11 '11 at 21:44

Matrix string theory

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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