How does the $D$-brane geometry influence the gauge symmetries? I am trying to understand why the $D$-brane structure of a string theory matters for string phenomenology.
From what I understood from Wikipedia, the $D$-brane structure determines the gauge symmetries of the theory, the intuition being as follows: the position of the D-branes determines the minimum length of the open strings, and thus the possible vibrational modes. Since the vibrational modes correspond to the particles (including the gauge particles), we can deduce the gauge symmetries of the system from the particles that we can obtain in the $D$-brane configuration of the system.
Question:
Aren't $D$-branes dynamical objects? Then shouldn't the distance between them vary? (This argument seems to assume that it does not vary).
 A: Moving a D-branes is the same thing as changing the expectation value of the scalar fields that lives on it. The D-branes are static in the stringy description of the theory but this is a perturbative illusion. The tension of the D-brane is proportional to $g_s^{-1}$, where $g_s$ is the string coupling.
$$
T\propto g_s^{-1}
$$
The description of the theory in terms of "fundamental strings" is suppose to compute the various quantities of the theory as an "Taylor expansion" in the string coupling at $g_s=0$. The tension of the D-brane is, at leading order in $g_s$, infinite.
$$
\lim_{g_s\rightarrow 0} g_s^{-1}=\infty
$$
Computing some scattering amplitude or effective action in the presence D-branes, the open strings states which are related to the scalar fields will reproduce a non-relativistic approximation of the D-brane, i.e. an expansion of $v/c$, where $c$ is the speed of light and $v$ is the transverse velocity of the D-brane. The D-branes moves just at first and subsequent order in perturbation theory.
This is how the D-branes moves in this picture of "fundamental strings" and "static D-branes". The open strings will describe the motion of the D-brane, in the same lines as the ones in which closed string describe the fluctuation of the metric and others fields. The metric in string theory is dynamical but in the stringy description is "static". The same is true for all the fields that compose the background in which the strings lives.
