According to the book ''String Theory in a Nutshell'' by Elias Kiritsis (page 204, linked here), the tension of an $Op$-plane is

$$T_{Op} = -2^{p-4}T_{Dp}$$

With this picture, one gets -- for type IIA -- the fact that $T_{O8} = -2^{4} T_{D8} = -16 T_{Dp}$. So for two orientifold planes, one should expect 32 D8-branes.

But in a paper by Clifford Johnson (arXiv:hep-th/9805047) on page 9, it is stated that there are 16 D8-branes and 2 O8-planes.

I know that the orientifold projection introduces two fixed points and there are 16 D8 branes sitting at each point. But if one counts two O8-planes, shouldn't one also count 32 D8-planes?

  • $\begingroup$ I can't say enough to write an answer, but I think the question is a matter of compactification. $\endgroup$ – Lawrence B. Crowell May 31 '16 at 22:00

In the presence of orientifold planes, every object (away from the orientifold fixed points) including a D-brane has "two copies". So it's a matter of convention whether this pair of copies is counted as "one D-brane" or "two D-branes".

Because the whole half-space is a "copy" of the other one, it makes sense to only consider one-half of the space as "the space", and count this whole pair of D-branes as one.

However, this becomes doubly confusing if the D-branes are located at the orientifold plane itself. In that case, both copies mentioned above can move separately – and act as "half-D-branes", which may still be called "D-branes" in the other convention.

Now, the orientifold planes are non-dynamical and one can just count the number of the mutually disconnected fixed points of the orientifold action in the spacetime. For type II on $T^k$ with an orientifold, there are $2^k$ orientifold planes – they are O$(9-k)$-planes.

The precise number of the half-D-branes that cancel those orientifold planes' anomalies is always 32, like in the $SO(32)$ group of type I string theory (=type IIB string theory with spacetime-filling orientifold 9-planes). So the tension of one orientifold plane is equal to that of $32/2^k$ "half-D-branes". Or, writing $p=9-k$, the O$p$-plane has the tension of $32/2^{9-p}=2^p/16$ half-D$p$-branes.

Note that a group like $SO(32)$ may be broken to $U(16)$ – that breaking may be visualized as 32 half-D-branes moving away from the orientifold plane and becoming 16 full D-branes (which have 16 images on the other side). Well, for O9-planes, the D-branes can't moved away from the O-planes because the latter fill the whole spacetime. But D8-branes can move away from O8-planes and break e.g. $O(16)\times O(16)$ to $U(8)\times U(8)$ etc.

The only difference between the authors is that Kiritsis uses the term "D-brane" for what I called the "half-D-brane" above, while Clifford Johnson and most others use the term "D-brane" for the "pair of half-D-branes" or, equivalently, one D-brane away from the orientifold plane (plus its copy included in the package).

  • $\begingroup$ Thanks for the detailed answer @Luboš! That clarifies a lot of things. $\endgroup$ – leastaction Jun 1 '16 at 5:50

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.