Studying Disappearance of moduli for condensate of open strings and Negative open string norms after BRST cohomology? gave me a huge huge shock!
Suppose we have N completely wrapped Dp-branes over a $T^d$ compactification with $p\leqslant d$. There is a U(1) gauge symmetry localized on each brane. Supersymmetric open strings are oriented (assuming it's not type I). So, they each have a $\sigma=0$ end, and a $\sigma=\pi$ end. The charge of the brane U(1) is the sum over all open strings of the total number of $\sigma=0$ open string ends on it minus the total number of $\sigma=\pi$ ends. That's not always zero because the ends of an open string can lie on different branes.
Here's the clincher. The spatial manifold of the completely wrapped Dp-brane is compact. Because of this, the total U(1) charge on each brane has to be zero. This means the total number of $\sigma=0$ open string ends on it has to equal the total number of $\sigma=\pi$ ends. This is not a constraint which appears at the level of perturbative open string worldsheets! Where does this additional constraint come from?
This is not a nonperturbative incompleteness of open string worldsheets. This is a perturbative incompleteness of open string worldsheets. This constraint survives in the limit as the string coupling $g\to 0^+$, and the constraint doesn't get any "smaller" physically either in this limit.
How do you explain this at the level of open string worldsheets? Yes, I know it's easy to explain using string field theory (and I know how to do that!), but if you can't explain this without resorting to string field theory, then you are admitting worldsheet theory is the wrong theory, and that the "proper" theory is string field theory.
Suppose at first, all the N completely wrapped Dp lie at spatially distinct locations. Now, move them so they are stacked up right on top of each other. Then, in addition to the previous constraints, new constraints appear. They are best described in terms of Chan-Paton factors and U(N) representations. An open string with both ends on this stack transforms as an adjoint. An open string with only one end on this stack transforms as a fundamental, or antifundamental depending upon which end it is. The new constraint is that the total $U(N)$ charge coming from all the open strings has to transform trivially. Where did this new constraint come from? Even with arbitrarily small but nonzero relative displacements between the Dp-branes, this new constraint isn't present. You might say from Higgsing (and I know how to do that using string field theory), but please explain this at the level of open string worldsheets.
Clarification: At the level of perturbative field theory, it's not valid to use the Fock basis for the zero momentum modes. I know that. Instead, the correct variables to use are the Wilson lines, and the total charge Q is set to zero as a perturbative field constraint. But starting from string worldsheets, aren't you forced to still use the nonexistent Fock basis for the zero momentum modes anyway?