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I am currently studying Zweibach's "A First Course in String Theory" CH:9 and I have managed to understand mode expansion in the light-cone gauge, at the classical level. Right before deriving the explicit solution of the wave equation for the open string, he remarks "We will assume that we have a space-filling $D$-Brane".

In the classical sense, $D$-Branes are just hyper-surfaces that imbose Dirichlet boundary conditions at the ends of open strings. So in that sense, the existence of a space-filling $D$-Brane is equivalent to having no $D$-Branes at all. But in that case, how can we dynamically reason the existence or the significance of space-filling $D$-Branes. How does the picture change when we quantize the open string? Do we quantize the $D$-Brane, and if yes, how so? In the theory of quantized open strings, what additional features other than Dirichlet boundary conditions do space-filling $D$-Branes have?

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    $\begingroup$ If you would like a historical background, Polchinski's insight in 1995 that D-branes were not just boring boundary-condition enforcers is essentially what triggered the "second superstring revolution" [keyword to search for] $\endgroup$ Apr 1 at 15:49
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A space-filling brane has three effects:

  1. It carries a tension, thus it contributes to the overall energy of your setup, which is especially important for phenomenological applications.

  2. The brane allows open strings to end on it. An open string needs a brane to end on. Thus if you would have no brane you would also have no open strings ( actually they are equivalent, you can use either open strings ending on the brane or the D-brane itself to describe the same degrees of freedom)

  3. It contributes to certain RR-tadpoles. Again mostly important for construction phenomenological setups.

In general D-branes are dynamical objects, which can even be created and annihilated dynamically. The definition in terms of boundary conditions is just to get some intuition, but a brane is much more.

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Initially, D-branes were just the shape of the dimensions in which a string obeys Neumann boundary conditions. Before the series of realizations now called the "second superstring revolution" and in particular Polchinski's "Dirichlet Branes and Ramond-Ramond Charges", "space-filling D-brane" would have been just another way to say "the string obeys Neumann boundary conditions in all spatial dimensions", and in some ways, it still is. But Polchinski realized that under T-duality between type I and type II string theory, the D-branes need to be charged objects under the Ramond-Ramond fields of type II supergravity.

Polchinski even discusses the 9-brane - the space-filling brane - explicitly in his seminal paper:

It [the 10-form of type II string theory] couples to a 9-brane, but what is that? A 9-brane fills space, so the open string endpoints are allowed to go anywhere: This is simply a Neumann boundary condition. If there are $n$ 9-branes (which must of course lie on top of one another), the endpoints have a discrete quantum number: This is the Chan-Paton degree of freedom.

So in the type II view, the D-branes are charged objects under the Ramond-Ramond gauge fields, and their number produces Chan-Paton gauge groups in the effective theory. The existence of a charge is obviously different from the existence of no charge.

Your question of what distinguishes a D-brane that is everywhere from no D-brane at all is philosophically very similar to the question of what distinguishes a single D-brane from a stack of coincident D-branes, to which the answer also is essentially "the Chan-Paton factors", on which I've previously written here.

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