Do you understand SYZ conjecture
Hi-
Just saw this thread. Maybe I should comment. The conjecture can be viewed from the perspective of various categories: geometric, symplectic, topological. Since the argument is physical, it was written in the most structured (geometric) context -- but it has realizations in the other categories too.
Geometric: this is the most difficult and vague, mathematically, since the geometric counterpart of even a conformal field theory is approximate in nature. For example, a SUSY sigma model with target a compact complex manifold X is believed to lie in the universality class of a conformal field theory when X is CY, but the CY metric does not give a conformal field theory on the nose -- only to one loop. Likewise, the arguments about creating a boundary conformal field theory using minimal (CFT) + Lagrangian (SUSY) are only valid to one loop, as well. To understand how the corrections are organized, we should compare to (closed) GW theory, where "corrections" to the classical cohomology ring come from worldsheet instantons -- holomorphic maps contributing to the computation by a weighting equal to the exponentiated action (symplectic area). The "count" of such maps is equivalent by supersymmetry to an algebraic problem. No known quantity (either spacetime metric or Kahler potential or aspect of the complex structure) is so protected in the open case, with boundary. That's why the precise form of the instanton corrections is unknown, and why traction in the geometric lines has been made in cases "without corrections" (see the work of Leung, e.g.). Nevertheless, the corrections should take the form of some instanton sum, with known weights. The sums seem to correspond to flow trees of Kontsevich-Soibelman/ Moore-Nietzke-Gaiotto/Gross-Siebert, but I'm already running out of time.
Topological: Mark Gross has proven that the dual torus fibration compactifies to produce the mirror manifold.
Symplectic: Wei Dong Ruan has several preprints which address dual Lagrangian torus fibrations, which come to the same conclusion as Gross (above). I don't know much more than that.
Also-
Auroux's treatment discusses the dual Lagrangian torus fibration (even dual slag, properly understood) for toric Fano manifolds, and produces the mirror Landau-Ginzburg theory (with superpotential) from this.
With Fang-Liu-Treumann, we have used T-dual fibrations for the same fibration to map holomorphic sheaves to Lagrangian submanifolds, proving an equivariant version of homological mirror symmetry for toric varieties. (There are many other papers with similar results by Seidel, Abouzaid, Ueda, Yamazaki, Bondal, Auroux, Katzarkov, Orlov -- sorry for the biased view!)
Reversing the roles of A- and B-models, Chan-Leung relate quantum cohomology of a toric Fano to the Jacobian ring of the mirror superpotential via T-duality.
Help or hindrance?
Here are some papers on SYZ worth reading:
Hitchin's "The moduli space of special Lagrangian submanifolds" arXiv:dg-ga/9711002
M. Gross's survey
Hitchin's paper was written shorly after Mirror Symmetry is T-duality and it is a matematical explanation of the paper. Essentially Maclean proved that the moduli space of sL submanifolds is unobstructed and its tangent space is the space of harmonic 1-forms on the sL submanifold. A natural metric which you can put on the moduli space is the $L^2$ metric on harmonic forms. When the sL submanifold is a torus, the moduli space also has an "affine structure". This was already known from integrable systems, they are called action coordinates. They are affine because they are defined up to affine transformations (with linear part having integral coefficients). Hitchin shows that with respect to these coordinates the metric can be expressed as the Hessian of a function. Hitchin also shows that the moduli space has two affine strutures (this is because of the "special" condition). The two affine structures are related by Legendre transform using the Hessian (i.e.the metric). So one could say that mirror symmetry is "Legendre transform".
Things have developed a lot since Hitchin's paper, and M. Gross surveys these developements. How to do "quantum corrections" to the metric is a major open problem, there are many approaches. They seem all quite difficult.... Auroux in the paper mentioned above deals with it. I heard a talk of Fukaya where he wants do do it with Floer homology for families, but I do not know much about it. Then there is the program of Kontsevich and Soibelmann, using rigid analytic geometry and the Gross-Siebert program. It seems that quantum corrections could be understood in terms of "tropical geometry" in the moduli space of SL tori (an "affine manifold with sigularities"). In a recent paper of M. Gross ("Mirror symmetry for $\mathbb{P}^{2}$ and tropical geometry"), he explains how "period calculations" can be understood in terms of tropical geometry (at least for $\mathbb{P}^{2}$). Here you can find a link to a book of M. Gross where he explains the connection between tropical geometry and mirror symmetry.
You might also want to check out
Riemannian Holonomy Groups and Calibrated Geometry by Dominic Joyce.
In Chapter 9, he gives a nice introduction to SYZ that is very accessible. He also points out reasons why SYZ as originally formulated (even though it does not have a precise formulation) could not be true. He proposes modified versions of SYZ that he believes are likely to be closer to any eventual true statement. (I have not looked at Auroux's recent work, so I can't comment on that.)