This question is about topological string theory and it was also posted in Physics Stack Exchange.
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 and confirmed to exist in S-duality and Topological Strings.
The argument that confirms the existence of such objects (last paragraph in page eight in S-duality and Topological Strings) 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). 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
"Their geometric meaning (talking about 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 of the target space? A little bit more precisely, an NS-2 brane can be defined as any three dimensional geometric place at which the integral of the (pullback) of the Nijenhuis tensor is non-zero?
Any advice/comment/reference is very welcome.