Orientifold projection of open-string states (with $SO(32)$ Chan-Paton charges) of IIB superstring theory is said to result in an orientifold 9-plane with $\pm 16$ D9 brane charge. My question is why 16? Is it related to the rank of $so(32)$ algebra?
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It comes from R-R tadpole cancellation. If you do the calculation (I recommend [1, Chapter 9.4]) you see that the tadpoles cancel if for each O9-plane we add 32 D9-branes. Therefore $q_{O9} = -32\, q_{D9}$ in the double cover ("upstairs geometry").
After the orientifold projection we are left with half as many D-branes with twice the charge, hence $$ q_{O9} = -16\, q_{D9} $$ in the orientifold.
More remarks (see [1, Chapter 10.6]):
- For general $Op$-planes, the result is $$ q_{Op} = -2^{p-4} q_{Dp} $$ (in the upstairs geometry).
- If we consider more general orientifold projections $\Omega (-1)^F$ (where $F$ is the fermion number), we can indeed have $$ q_{Op} = \pm 2^{p-4} q_{Dp} $$ like you wrote.
[1] Blumenhagen, R., D. Lüst, and S. Theisen: Basic Concepts of String Theory.
