A stack of N coincident D3-branes on its world-volume describe, at the lowest order in $\alpha'$ and in absence of non-trivial background fields, a supersymmetric $U(N)$ gauge theory as explained in Becker-Becker-Schwartz section 12.3.
IN this same book they go on and argue that a $U(1)$ factor of the $U(N)$ gauge group decouples as a free theory.
More precisely, the $U(1)$ lives on the boundary and the $SU(N)$ lives in the bulk, which is why the $U(1)$ is not relevant.
- First: the boundary of what?
- Why is it not relevant if it lives on the boundary? I always learned to be cautious with boundary (effects)
- What would change to the duality, if the $U(1)$ was taken into account?