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According to this

the average size of a string should be somewhere near the length scale of quantum gravity, called the Planck length, which is about $10^{-33}$

As far as I understand string theory - which is not very far - D-branes are extended objects on which open strings both begin and end.

As they are extended objects what are the size of their extensions? Are they of the order planck length squared, cubed etc depending on the dimension of the D-brane in question? I haven't come across anything that explicitly says that this is the case.

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The D-branes, in the simplest setup, are infinite in the directions in which they are extended. This is why you might have read that our $1+3$-dimensional universe could in principle be the world-volume of a D3 brane.

It is also possible to have D-branes ending on other D-branes, much like strings end on D-branes. In this case, the "size" of the D-brane is finite, but it can be as big as you want. So in principle, you can perfectly imagine a D1 brane one meter long, ending on other branes.

In fact, the difficulty arises at very short distances, when you want to explore what happens exactly at the junction points (between strings and branes, or between branes and other branes). In some situations, you have to give up your classical vision, and things are much more complicated.

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According to Lectures on D-Branes by Constantin Bachas.

We this dynamical size of D-particles is comparable to the inverse cubic root of the membrane tension, that is the 11d Planck scale of M-theory.

Since this is much smaller than the string scale at weak coupling perturbative string theory does not capture all the degrees of freedom.

The fact that D-branes were much smaller than fundamental strings was conjectured early on by Shenkar in 1995.

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