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The Strominge-Yau-Zaslow conjecture is roughly the following. Any Calabi-Yau $m$-manifold $X$ admits a special Lagrangian $T^m$ fibration (maybe at around a special point in its complex moduli space) and a mirror partner $Y$ is obtained by dualizing the tori $T^m$ with "instanton corrections" coming from singular fibers.

If my memory serves one of heuristics of SYZ conjecture comes from Kontsevich's homological mirror symmetry conjecture; the moduli space of sky-scraper sheaves $\mathcal{O}_y$ (the easiest B-branes) on $Y$ is $Y$ itself, and there should be a corresponding moduli space of A-branes on $X$. A-brane is a pair $(L,c)$ of Lagrangian submnaifold $L\subset X$ and a flat $U(1)$ connection $c$ on $L$. By computing cohomology groups $HF^*(L,L)\cong Ext(\mathcal{O}_y,\mathcal{O}_y)=H^*(T^m,T^m)$, we expect that our $L$ is a Lagrangian $T^m$. My first question is

How do you expect that $L$ is a spcecial Lagrangian manifold?

If we assume that $L$ is special Lagrangian, then its deformation space is known and is of dimension $3$, and we expect that $L$ sweeps over $X$. My second question is

Does the flat $U(1)$ connection $c$ on $L$ play any role in this story?

All we "deduced" from HMS conjecture is that $X$ admits a special Lagrangian fibration. Can we say anything more? Since HMS conjecture doesn't say anything about construction of mirror manifolds, I am afraid that we cannot say anything about dualizing these $L=T^n$ etc. My third question is

What is the A-brane object on $X$ that corresponds to the obvious $6$-brane $Y$?

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1 Answer 1

For your first question, you can see this answer

What is geometric intuition of special Lagrangian manifolds?

for some idea of how the special condition maybe natural from the point of view of homological mirror symmetry.

For your second question, if we grant this, then the local system is a natural addition, because we can naively expect that $HF((L,c),(L,c))$ is also $H^*(T^3)$. Meanwhile as you have said we expect that there is only a three dimensional family of special Lagrangians, so we need more objects to correspond to the six dimensional family of skyscraper sheaves. It is therefore natural to allow the $U(1)$ flat connections on $L$ and expect that they also correspond to points in our mirror.

For your third question, the answer is that line bundles on the mirror should correspond to sections of the Lagrangian torus fibration. Normally, we fix a section to begin with and just declare that that goes to the structure sheaf $O_Y$. One motivation for this idea is a similar naive reasoning with Exts as the one you mention in the question. Namely $Ext(O_Y, O_y)$ should be isomorphic to $\mathbb{C}$, so we expect that our Lagrangian hits each fiber once. A nice case to consider is that of the elliptic curve. If you examine the functor constructed in Polishchuk and Zaslow's paper on the elliptic curve, you will see that this is in fact how mirror symmetry works in this case, namely points will correspond to local systems over (0,1) curves and line bundles will correspond to (1,n) curves.

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