Anti-Quill Part 5: D-Branes and O-planes 1

 In Anti-Quill Part 4, we discussed Calabi-Yau manifolds and the essential idea of compactifications on them. Before going forward, it is important to note that a lot of care has to be taken in ensuring SUSY and structures are preserved. As an example, take a type IIB theory on $T^{6}=T^{2}\times T^{2}\times T^{2}$. We have two fluxes, the NS-NS fluxes $H_{3}$ and the RR flux $F_{3}$. If you define an involution operator $\pi : (z)_{3}\to (-z)_{3}$, the fixed points of $\pi$ would correspond to the O3-planes along with the parity conjugation operator $\mathcal{P}$. This has the effect of taking $p$-forms and turning them into -$p$-forms under $\pi$, after which we act with $\mathcal{P}$. In order for $H_{3}$ and $F_{3}$ to be of the same parity after the action of $\mathcal{P}\cdot \pi$, they must have odd parity under $\pi$, so that they have consistent parity with the O3-planes. Having said this, it is a nice simple exercise to convince yourself why this is important in view of tadpole cancellation. Another interesting exercise from van Riet is to take $T^{6}$ with angles $\theta _{i}$ and $\pi _{O3} : \theta _{i} \to -\theta _{I}$, with 64 O3-planes. Taking a simple $F_{3}$ flux $F_{3}=d\theta _{1} \wedge d\theta _{2}\wedge d\theta _{3}$, for dilaton stabilisation, what would be the corresponding $H_{3}$ flux?