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  1. What is the difference between a divisor and a homology cycle?

  2. What is the difference between a D-brane wrapped around a divisor and a D-brane wrapped around a cycle?

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http://en.wikipedia.org/wiki/Divisor_(algebraic_geometry)

is a particular submanifold of complex codimension one that may be described by a holomorphic equation in the complex coordinates parametrizing the original manifold. It's usually homologically nontrivial as a cycle. But general cycles are much more general; not every cycle (not even a cycle of the right dimension), is a divisor.

For example, Calabi-Yau three-folds are 6-real dimensional. Divisors are holomorphic cycles with 4 real dimensions, so to say. D4-branes wrapped on such cycles will give you supersymmetric states in simplest cases, so they're stable which shows that their stability has to be protected by a nontrivial homology of the cycle. It's likely that all supersymmetric 4-cycles are divisors; you have to ask a better geometer to be sure. However, there are clearly many 4-cycles that are not supersymmetric and that are consequently not divisors.

And there are also 3-cycles and 2-cycles in Calabi-Yau three-folds which are surely not divisors because they even have wrong dimensions.

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