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?


1 Answer 1

This is not a constraint which appears at the level of perturbative open string worldsheets! Where does this additional constraint come from?

Your problem has absolutely nothing to do with any special features of string theory; the very fact that you introduced string theory into this discussion only has one effect, namely to completely mask the actual essence of your problem. Your problem is completely equivalent to the following situation and assertion:

You have a quantum field theory on a compact space $M^3\times R$ with a charged field, e.g. an electron field. Now, the total charge must vanish on a compact space but this is not a constraint which appears at the level of perturbative quantum field theory!

Now, in both situations, the statements are right as long as one is sloppy but they are wrong if one is looking at the states carefully. If one calculates the electric field from a charged particle on a compact spatial manifold, he finds out that there is simply no solution.

The previous comment refers to some "mixed perturbative/semiclassical treatment" and one could think that the constraint doesn't appear in the usual systematic perturbative expansions. Except that it does. Loop corrections to Green's functions "know" about the electric field around the electron. They implicitly calculate it for themselves, so when the electric field doesn't exist, the problem manifests itself as an anomaly. There's simply no way to preserve the gauge invariance. The QFT on a compact space differs from the QFT on a noncompact space because the loop momenta have to obey the quantization laws coming from the compactness. It makes a difference and all the loop corrections are simply different and one finds out that there is no regularization that could make sense out of the loop corrections to certain diagrams if the overall charge on the compact space is nonzero. These states are forgiven simply because they are not gauge invariant – although the failure of gauge invariance of these states isn't visible at the free, non-interacting level.

A refined version of your statement would be that the free quantum field theory (or string theory) allows you to create these states but these states actually don't exist in the full interacting theory. That's a valid statement but it is not an inconsistency of any sort. It just means that if you try to verify whether the states with the overall nonzero charge exist – are associated with actual states in the Hilbert space of the whole quantum field theory (including the gauge field: that's the QED case) or with marginal operators on the world sheet, to all orders (to deal with your case) – you will see that the test fails at the one-loop level so these states don't exist even though the classical, tree-level theory could lead one to believe that they do exist.

The case of $U(N)$ is analogous except that one has to deal with the usual options about the confinement or symmetry breaking or other possible fates of the non-Abelian gauge symmetry.

The OP's focus on the "world sheet description" is a straw man. The calculations involving world sheet clearly have the same spacetime physics at long distances as the calculations in QFT. After all, the QFT Feynman diagrams are just limits of the string diagrams. That's why it's totally obvious that the anomaly – or more generally, any obstacle one could encounter in the QFT description – clearly has a counterpart in the world sheet description, too. So the OP should first try to understand how the constraint $Q=0$ emerges in QFT because this is the real confusion he or she is having.

Let me finally phrase the resolution in one more way. The existence of electron excitations or open strings is "locally allowed" but the condition for the total charge to vanish on a compact manifold is a "global constraint", one that is visible only if one is able to check the observables (values of the field) in the whole compact spacetime. The free spectrum clearly can't be affected by global constraints. However, the global constraints do have an impact on the loop corrections, and it's the careful analysis with all the quantum corrections that reveals that these states don't actually exist in the full theory.


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