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Suppose I have two M5 branes on $X^9 \times \Sigma^2$, which share a few directions on $X$ and besides that each occupy some curves on $\Sigma$ that may intersect.

When is such a configuration BPS?

What are $X^9$ and $\Sigma^2$? I guess that $X^9$ is a 9-manifold and $\Sigma^2$ is a 2-manifold but what are they more precisely (for example are they flat $\mathbb{R}^9$ and flat $\mathbb{R}^2$ or something more complicated? where is the time direction?...)

I want to be relatively agnostic about the choice of $X^9$. $\Sigma^2$ should be allowed to be any Riemann surface, though. The time direction is along $X^9$.

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