dex $j$ is summed. In the IIA theory, these conditions are mapped to the conditions for the $5_3^{2,1}$-branes associated to the components $
E_{4\,x^iy^ix^jy^jx^k,x^ix^jx^k,x^k}$ of the potential $E_{9,3,1}$ (see Table \[Ebranestable\]). This can be shown by evaluating for these components the constraints coming from the generalised Chern-Simons term $$\frac{1}{2} \int E_{ 9,abc,a} \times ( {\rm flux} \cdot {\rm flux} )^{abc,a}_1 \quad ,$$ where the $( {\rm flux} \cdot {\rm flux} )$ term is given by $$\begin{aligned}
( {\rm flux} \cdot {\rm flux} )^{abc,a}_d & =-2P^{a,a[b|e|}f^{c]}_{de}+f^a_{cd}P^{c,abc}+f^a_{bd}P^{b,abc}+P^{a,a}Q^{bc}_{d}+Q^{ae}_{d}P_e^{abc}\nonumber \\ &-2P^{a[b|e}_{d}Q^{a|c]}_e+\tfrac{1}{2}P^{abcef}_{d} f^a_{ef}+P^a_{d}R^{abc}+\tfrac{1}{2}P^{a,abcef}H_{def} \quad . \label{quadratic931}\end{aligned}$$ As we have discussed in the previous subsection, since the $5_3^{2,1}$-branes correspond to the components of the potentials such that the indices $abc$ have to be inside the first $9$ indices, this implies that for these components the index $d$ in eq. differs from $a,b,c$. In particular, the components $
E_{4\,x^iy^ix^jy^jx^k,x^ix^jx^k,x^k}$ couple to the terms $( {\rm flux} \cdot {\rm flux} )^{x^i x^j x^k, x^k}_{y^k}$, which lead precisely to the tadpole conditions equivalent to eq. .

In section 3 we have seen that the additional $5_3^{2,1}$-branes that can be included in the IIA/O6 theory are mapped in the IIB/O3 theory to the $3_3^4$, $6_3^{1,1}$ and $5_3^{2,2}$-branes associated to the potentials $E_{8,4}$, $E_{9,2,1}$ and $E_{10,4,2}$ respectively. The tadpole conditions for all these branes can be easily determined using our rules. We write schematically the flux contributions to the tadpole conditions for all these branes as $$\begin{aligned}
 &P_1^2 \cdot H_3 \longleftrightarrow E_{8} \nonumber \\
 &P_1^2 \cdot  Q \longleftrightarrow E_{8,4},E_{9,2,1}\label{tadpolesforallPbranesIIBO3}\\  
 &P^{1,4} \cdot  Q \longleftrightarrow E_{10,4,2} \nonumber \quad .\end{aligned}$$ Similarly, one 