The aim of this question is to understand SYZ conjecture ("Mirror symmetry is T-Duality").
I don't expect a full and quick answer but to find a better picture from answers and comments.


The whole idea is to construct the Mirror C.Y $Y$ from $X$ intrinsically as follows.
One considers the moduli of special Lagrangian tori with a flat $U(1)$ bundle on it in $X$.


Then we put a metric on this moduli (plus corrections coming from J-holomorphic disks) and expect that this moduli and the metric given on it is the mirror C.Y we were looking for.


Here are the things I can not understand:


---What is the metric given in the paper "Mirror symmetry is T-Duality"?
Where does it come from? (I can not understand the formulation of metric there).


and more importantly
--How do we deform the metric using J-holomorphic disks(instantons)?



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?


