# What is an $NS2$ brane?

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This question concerns topological string theory.

The existence of a new brane called "an NS-2 brane" is predicted in (the second paragraph in the page 14 of) the paper [N=2 strings and the twistorial Calabi-Yau](https://arxiv.org/abs/hep-th/0402128) and confirmed to exist in [S-duality and Topological Strings](https://arxiv.org/abs/hep-th/0403167).

The argument that confirms the existence of such objects (last paragraph in page eight in [S-duality and Topological Strings](https://arxiv.org/abs/hep-th/0403167)) is based on the the fact that the A and B models are S-dual to each other over the same Calabi-Yau space. It is argued that the S-dual picture of a F1-string ending on a lagrangian submanifold in the A picture (S-)dualize to a D1-brane ending on the aforementioned NS-2 brane in the B model.

**My problem:** Although I understand that the NS-2 brane must exists in the B-model as the S-dual of a lagrangian submanifold in the A model, I can't understand the physical and mathematical significance of such objects.

**Question 1 (Physical significance):** My naive intuition says that **because the NS-2 brane has a real three dimensional worldvolume, then it should descend from the M-theory membrane** (by [embedding the topological string into M-theory](https://arxiv.org/abs/hep-th/9809187)). **Is this true?** And if the answer is positive, how can I check that that? (I'm asking for a chain of dualities that explicitly transform the M2-brane into the NS-2 brane).

I'm unsure about the M2 - NS2 identification probably because I don't understand the physical origin of a lagrangian submanifold in the A-model. Strings can end on lagrangian subspaces but as far I understand, lagrangian submanifolds are also three dimensional submanifolds but not M2 branes, aren't they?

**Question 2 (Mathematical significance):** The next cite can be read in the first paragraph in the page nine of [S-duality and Topological Strings](https://arxiv.org/abs/hep-th/0403167)

> "Their geometric meaning (referring to the NS-2 brane) is that they
> correspond to a source for lack of integrability of the complex
> structure of the Calabi-Yau in the B-model."

**Does that mean that the NS-2 brane is "charged" under the [Nijenhuis tensor](http://www.math.uit.no/ansatte/boris/Images/1/12DEF-NJ.pdf) of the target space?** A little bit more precisely, an NS-2 brane can be defined as any three dimensional geometry at which the integral of the (pullback) of the Nijenhuis tensor is non-zero?

Any comment or reference is welcome.

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