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Let $X$ be a Calabi-Yau threefold. The Strominger-Yau-Zaslow conjecture suggests that $X$ should have a special Lagrangian $T^3$-fibration (when $X$ lies near a large complex structure limit) and a mirror manifold $Y$ is obtained by "dualizing the smooth fibers $T^3$". I want to know what "dualizing the smooth fibers $T^3$" rigorously means.

To formulate my question,

What information is required to construct the mirror manifold?

To dualize smooth fibers, we definitely need a section of the fibration to specify an origins of each smooth fiber. Does the section need to be special Lagrangian as well?

Also, what do we need to know about the singular fibers in order to incorporate instanton corrections (and modify the gluing of the complex structure)?

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I'll fill in a few details here; more can be found in the references that Daniel gave.

Suppose first that $f:X\rightarrow B$ is a special Lagrangian $T^n$ fibration with only smooth fibres. If we just want to describe the dual as a complex manifold, we do the following. Hitchin showed that $f$ induces two affine structures on $B$. The important one for us is the one coming from the fact that $f$ is a Lagrangian fibration. Locally choose submanifolds $\gamma_1,\ldots,\gamma_n$ of $X$ smooth over $B$ whose restriction to each fibre provides an integral basis for $H_1$ of that fibre. We can use these to define one-forms $\lambda_i$ on $B$ via $f_*(\omega|_{\gamma_i})$ (where $\omega$ is the Kaehler form). Here $f_*$ denotes fibrewise integration. These forms are closed and hence locally there exists functions $y_i$ such that $dy_i=\lambda_i$. One checks that $y_1,\ldots,y_n$ form a coordinate system locally on $B$. Once the cycles $\gamma_i$ are chosen the $y_i$ are only well-defined up to constants, and we can always change basis, and as a result the coordinates are well-defined up to affine linear changes of coordinates, where the linear part of the affine linear transformation must be integral.

Now here is the dual: consider the local system $\Lambda$ of lattices contained in the tangent bundle $T_B$ of $B$, given locally by integral linear combinations of the tangent vectors $\partial/\partial_{y_1},\ldots,\partial/\partial_{y_n}$. This gives a lattice in each fibre of $T_B$. Define the dual torus bundle to be $T_B/\Lambda$, i.e., divide each tangent space out by the lattice generated by the above tangent vectors. The projection to $B$ gives the dual torus bundle.

Note that we did not need to have a section of the orginal fibration, although the dual does have a section. It is possible to twist the dual so that it doesn't have a section; this is discussed in my paper

Now this dual, which I'll write as $X(B)$, does carry the ``semi-flat'' complex structure. This is described via complex coordinates $q_i=\exp(2\pi \sqrt{-1}(x_i+\sqrt{-1}y_i))$, where $x_i$ is the function on the tangent bundle given by $dy_i$.

Now we allow singular fibres, i.e., consider $f:X\rightarrow B$ whose fibres are only smooth tori over an open subset $B_0\subseteq B$. So we get the dual $X(B_0)\rightarrow B_0$ and the question is how we compactify this. In particular, if we want to obtain a complex manifold, we need to deform the above semi-flat complex structure via instanton corrections. We might assume that $f$ is only a Lagrangian fibration, as that is the only part of the structure we used to get the dual as a complex manifold, and it is easier to construct Lagrangian fibrations. In this case there has been a certain amount of success describing instanton corrections explicitly in terms of counting holomorphic disks on $X$. This explicit program was initiated by Denis Auroux in the reference Daniel gave, but so far can only be done for very simple singular fibres and very few singular fibres.

The point of view taken by myself and Siebert, as well as by Kontsevich and Soibelman in the paper Daniel referenced, is that one discards the explicit description of the dual as a complex manifold, as this introduces too many analytic difficulties in general. Instead, Siebert and I replace this with a degenerating family of schemes over the spectrum of a formal power series ring. Most convergence issues disappear, and when the singularities are of a relatively simple form, a purely algorithmic approach can be taken to constructing this family. A posteriori one expects that the data produced by this algorithmic approach are in fact the instant corrections, i.e., are produced by counts of holomorphic disks on $X$. This most precise realization of this expectation is in my paper with Hacking and Keel, There we explicitly construct the mirror of a surface using the Gromov-Witten invariants of the surface, with no restriction on what the expected singular fibres of a Lagrangian fibration on the surface might be.

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I find Leung and his group follow the similar idea and get an explicit construction of mirror complex manifold for toric CY by instant corrections of open Gromov-Witten invariants: Do you have any comments on it? – Jay Jan 26 '13 at 21:16

This is more of a comment than an answer, because I think this is a complicated story and it may just be better to give the relevant references. In short, this has been solved partially by (Kontsevich and Soibelman) in

and then completely by (Gross and Siebert) when the singular locus of the fibration has real codimension 2. These works are mostly concerned with the problem of how we can recover the mirror given a singular affine structure with mild singularities. Gross has a big book on his website which surveys this body of work. He also recently released a shorter 65 page survey

which also touches on this topic extensively, though I haven't had a chance to look at it closely yet. It seems important to note that many Lagrangian fibrations in nature have codimension 1 singularities. Of course, nobody knows how to write down (special) Lagrangian fibrations with the desired properties on Calabi-Yau threefolds. Gross and Siebert seem to work around these hard analytical issues using toric degeneration and staying in the realm of algebraic geometry.

One could also consult work by Auroux e.g. for concrete examples where one works with actual Lagrangian fibrations and shows how wall-crossing is natural from the point of view of symplectic geometry. Note that Auroux is concerned with a slightly different setup than the OP (that of mirror symmetry in the complement of an anticanonical divisor). I think consulting those sources would be more informative than any short answer.

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