How many orientifold planes does one need?

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?

• I can't say enough to write an answer, but I think the question is a matter of compactification. – 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).

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