I've noticed that several references take for a fact that by studying Kähler gravity on a Calabi-Yau threefold one can demostrate that any lagrangian submanifold embedded in the threefold posees three dimensional $SL(2,\mathbb{C})$ Chern Simons theory as its worldvolume theory.

For an example of a paper were this is explicitly stated, see the first paragraph on page 28 of Quantum Foam and Topological Strings, this paper remits to its reference [22] for a proof; the problem is that this reference was never published, as far I know.

Question one: Does anyone know of any reference with a proof of this fact?

I've thought that maybe nobody needs a proof because the statement is obvious. Unfortunately, that's not my case. I can figure out that some relationship exist because some supersymmetric extensions of three dimensional gravity compute the Casson invariant of its three dimensional target space (see the last section of this paper), but the Casson invariant is related to tree level topological string B-model and not with Kähler gravity which is the tree level A-model theory. So I'm confused.

Question two: Does anyone know an intuitive picture of why the statement I'm hoping to understand is true?

Any reference, parallel reading suggestion or comment is helpful.



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