# Do open strings with both ends attached to the same D-brane really exist?

Perturbatively, open strings with both ends attached to the same D-brane definitely exist. But D-branes have nonzero p-form charges. These charges give rise to a nonzero flux with nonzero stress-energy tensor. This backreaction leads to curved spacetime. Being BPS, the metric becomes that of an extremal black brane with the event horizon at an infinite distance (logarithmic divergence) away.

I wish to apologize for asking bad questions. Please put up with a newbie like me.

The energy of a string due to its tension is proportional to the length of the string and also the time factor aka the gravitational redshift factor. A string is a worldsheet after all. An open string extending radially from the event horizon has an infinite length, but the gravitational redshift factor cancels that, and the total energy of the open string remains finite.

I have no question after all.

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your questions are so well stated it seems you already know the answer [: But, no, kidding aside, great question(s) +1 – user346 Apr 7 '11 at 14:32
WHAT?! A downvote on this?! Really?! – centralcharge Aug 6 '13 at 15:51

## 1 Answer

Dear Shankar, a good question. Andy Strominger would be telling us stories how, around 1995, Andy would speak about (black) $p$-branes during lunches in Santa Barbara. Then he would stop and Joe Polchinski would speak about D-branes during the rest of the lunch. They wouldn't notice that they were speaking about the same object - up to some moment when they did realize that. ;-)

So $p$-branes are physically the same objects as D-branes. However, you can't use both of these descriptions at the same moment. If you describe the object as a D-brane, you're treating it as a light perturbation of the background and you assume that the backreaction is small. The backreaction is effectively proportional to $g_{\rm string} N$ where $N$ is the number of D-branes in the stack. So at weak coupling, for a single (or a few) D-branes, there is no backreaction.

If the backreaction is large, then $g_{\rm string} N\gg 1$, or a similar inequality holds. In that case. the perturbative expansion based on open strings attached to the D-brane(s) and their world sheet loops wouldn't converge and it is better to switch to the dual description which sees the curved geometry of the extremal black $p$-brane.

However, you can never use both of these descriptions at the same moment! The picture of D-branes as sheets of paper inserted into a pre-existing space is valid or useful if $g_{\rm string} N\ll 1$ while the curved geometry with the event horizon is valid, useful, and nicely convergent in the opposite limit $g_{\rm string} N\gg 1$.

This "duality of regimes" also becomes decisive in the AdS/CFT correspondence. If 't Hooft's coupling $g_{\rm Yang-Mills}^2 N \ll 1$, the Yang-Mills perturbative expansion - inherited from the open string perturbative physics attached to D-branes - is convergent. When $g_{\rm Yang-Mills}^2 N \gg 1$, the new fifth holographic direction becomes visible and you should switch to the AdS higher-dimensional gravitational description. In fact, the AdS space is just the near-horizon limit of the backreacted geometry of the stack of D-branes, and in a similar way, Yang-Mills theory is the corresponding low-energy limit of the open "string field theory" living on the D-branes.

It is unacceptable to calculate D-brane loops and backreaction of the geometry at the same moment. It is as invalid a double counting as if you take the toroidal world sheet with the complex structure $\tau$ and you would double count the thin and fat tori with $\tau$ and $\tau'=-1/\tau$. Only the effects from one of the descriptions may be counted as independent ones, otherwise you're double-counting.

So if you consider the energy of open strings at the background of D-branes, you always calculate their energy from perturbative string theory, assuming the background without backreactions. If you also include higher-loop corrections, all the effects of the backreactions will be correctly accounted for and they will be relatively low. The physical open strings with Chan-Paton factors never terminate on the event horizon etc.

There is the holographic duality between the two descriptions - one that is also visible in AdS/CFT (which is just the near-horizon/low-energy limit of what you're considering). In the CFT description, there is an $SU(N)$ gauge symmetry - in the most canonical example of the AdS/CFT correspondence - and no gravity and no diffeomorphism symmetry. In the dual AdS gravitational description, the natural gauge group includes the bulk diffeomorphism of general relativity but there is no $SU(N)$ Yang-Mills symmetry. The two pictures have the same physical Hilbert spaces - only the physical ones, after you eliminate the gauge-variant states etc.

So it is also untrue that open strings attached to D-branes always want to become closed strings because they're shorter. In fact, many open string states attached to D-branes with both ends may be BPS - which proves that they can't decay to anything lighter, because of energy conservation. When counting their energy at weak 't Hooft coupling, you shouldn't include any event horizons or infinite enhancement of the energy: you must use the pre-existing metric. Alternatively, you may use the "closed string" description with the backreacted metric. But if you do so, you don't see open strings going through the complicated space. These open-string degrees of freedom get reinterpreted as closed-string degrees of freedom. You should never double-count them.

For example, the oscillations of the shape of a D3-brane in the transverse 6 dimensions are represented by scalar open string modes at weak coupling. At the strong coupling, the stack of D3-branes becomes a black $p$-brane and the variations of the shape of the brane within the 10-dimensional spacetime are represented by changes of the black-hole-like solution, which is clearly carried by closed-string modes such as the graviton. Double counting would be a mistake: it would be literally equivalent to taking the $N=4$ $SU(N)$ gauge theory in 3+1 dimensions and adding the five-dimensional curved bulk metric as well. That's not how the physics works: the dynamics is fully determined by the gauge theory, or fully determined by the gravitating (closed) string theory in the bulk - and these two descriptions are equivalent and you don't have to (and you shouldn't) combine them.

The Chan-Paton factors become literally invisible if you switch to the closed-string description (with nonzero backreaction). They're some hidden degrees of freedom, much like the bulk diffeomorphisms are hidden in the boundary CFT. There's no contradiction because the symmetry arising from the Chan-Paton factors is a gauge symmetry, so all physical states have to be singlets under this symmetry, anyway. The same holds in the opposite way, for the gauge symmetry in the bulk - e.g. the diffeomorphisms in $AdS_5$.

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Are you implying we have complementarity a la Niels Bohr between D-branes and p-branes? – Shankar Apr 8 '11 at 16:15
Dear @Shankar, the D-branes and $p$-branes are physically the same objects, not "two complementary objects". However, D-branes and $p$-branes are two descriptions of them. So maybe, to make your last hint relevant, you could say that D-branes are like a wave function in the position representation and $p$-branes in the momentum representation. They still describe the same wave function, but in two different ways. – Luboš Motl Apr 9 '11 at 7:07
Your "derivation" of the non-BPSness is "strange". The BPSness of a configuration may only be judged if one looks at the whole object "simultaneously". But if you still wanted to separate the open string to pieces, the bulk of the open string is just a "string" and a fundamental string locally preserves 1/2 of the supercharges. The end points may also preserve 1/2 of SUSY, and in fact, the same one. – Luboš Motl Apr 9 '11 at 7:10
To see that your conclusion is wrong, just take e.g. type I string theory on a circle, with Wilson lines. There are open string modes that change the values of the $SO(32)$ Wilson lines. Still, if you add condensates of these strings, you don't change anything about the unbroken supersymmetry, so these open string modes clearly preserve the same supercharges as the original type I background. It is not true that the SUSY preserved by the two endpoints has to be incompatible. The intersection is often nonzero, especially for unoriented theories like type I where the reason is obvious. – Luboš Motl Apr 9 '11 at 7:12