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What does it mean that a M5-branes wraps a holomorphic curve in M-theory? In specific a lot of Vafa's paper involve various branes (not only M5) wrapping some cycles.

What does this really mean intuitively and physically?

Also, when a brane intersects a compact divisor in a (non-compact) CY3-fold what does it physically mean?

Almost identical: http://physicsoverflow.org/27416/what-exactly-mean-wrap-brane-brane-riemann-surface-sigma

Hard to tell if the question is after the basic concept or some subtleties of it. Let's first check if the basic concept is clear: a brane configuration of shape some manifold \(\Sigma \) inside a target spacetime \(X\) is a suitably well behaved map \(\Sigma \to X\). One says that such a configuration wraps cycles in \(X\) if it represents the corresponding element in the homology group of \(X\). For instance if \(\Sigma = T^2\)and \(X = Y \times T^2\) then the brane wraps that torus surface if the embedding map is the identity onto that torus over some point of \(Y\).

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